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 UNIVERSITY OF CALIFORNIA 
 
 DEPARTMENT OF EDUCATION 
 
 GIFT OF THE PUBLISHER 
 
 No. 
 
 s 
 
 Education Department 
 
NEW 
 
 GRAMMAR SCHOOL 
 ARITHMETIC 
 
 BY 
 
 JOHN H. WALSH 
 
 ASSOCIATE SUPERINTENDENT OF SCHOOLS, THE CITY 
 OF NEW YORK 
 
 BOSTON, U.S.A. 
 D. C. HEATH & CO., PUBLISHERS 
 1905 
 

 Copyright, 1895 and 1903, 
 By JOHN H. WALSH. 
 
INTRODUCTION. 
 
 The New Grammar School Arithmetic forms with the 
 New Primary Arithmetic a complete course in elementary 
 school mathematics. 
 
 Each of the first four chapters of the New Grammar 
 School Arithmetic provides for a half year, beginning with 
 advanced matter, which is followed by a review and an 
 extension of the topics of the preceding grades. Each of 
 the next two chapters (V and VI) contains arithmetic work 
 for a year, which should be supplemented by portions of 
 the algebraic and geometrical material of Chapters VII and 
 VIII. It is recommended that at least a portion of the 
 work in equations of Chapter VII should precede the study 
 of Chapter V. 
 
 Among the special features of the New Grammar School 
 Arithmetic are the number and the variety of the problems ; 
 the systematic reviews, which cover oral and written drill 
 work even in the fundamental operations ; the attention 
 paid to short, direct, business methods of computation ; and 
 the spiral handling of the various topics. 
 
 iii 
 
 548N48 
 
CONTENTS. 
 
 CHAPTER I. 
 
 PAGES 
 
 Mixed Numbers 1 to 16 
 
 Addition, Subtraction, Multiplication, Division (Common 
 
 Denominators determined by inspection). 
 Review of Simple Numbers 17 to 24 
 
 Notation and Numeration, Special Drills, Fundamental 
 
 Processes. 
 Decimals (three places) 25 to 32 
 
 Addition and Subtraction, Multiplication and Division 
 
 by an Integer. 
 United States Money . . . . . . . . 32 to 43 
 
 Fractional Parts of a Dollar, Division of United States 
 
 Money. 
 Denominate Numbers 43 to 45 
 
 Time, Dry and Liquid Measures, Avoirdupois Weight, 
 
 Miscellaneous Examples. 
 Measurements 46 to 49 
 
 The Area of Rectangles. 
 
 Bills 50 
 
 Miscellaneous 51 to 58 
 
 Approximate Answers, Review Problems. 
 
 CHAPTER II. 
 
 Fractions 59 to 84 
 
 Greatest Common Divisor, Least Common Multiple, 
 Addition and Subtraction of Fractions, Cancellation, 
 Ratio, Multiplication and Division of Fractions. 
 
 Decimals 84 to 91 
 
 Multiplication of Decimals, Division of Decimals. 
 
 v 
 
vi Contents. 
 
 PAGES 
 
 United States Monet , 92 to 93 
 
 Fractional Parts of a Dollar 
 
 Denominate Numbers 94 to 99 
 
 Reduction, Addition, Subtraction, Multiplication, 
 and Division. 
 
 Measurements 99 to 101 
 
 Areas and Surfaces. 
 
 Bills 101 to 102 
 
 Review of Simple Numbers 102 to 118 
 
 Short Methods, Sight Exercises, Sight Approxima- 
 tions, Review Problems. 
 
 CHAPTER III. 
 
 Decimals 119 to 132 
 
 Notation and Numeration, Reduction, Addition, Sub- 
 traction, Multiplication, Division. 
 United States Money . . . . . . . 132 to 133 
 
 Denominate Numbers 133 to 139 
 
 Reduction, Addition, Subtraction, Multiplication, 
 
 Division (two denominations). 
 Measurements 139 to 144 
 
 Areas of Rectangles, Areas of Right-angled Triangles. 
 
 Bills 144 to 145 
 
 Percentage 145 to 147 
 
 Interest 148 to 152 
 
 Review of Simple Numbers and Fractions . . 152 to 162 
 
 Sight Approximations, Special Drills, Cancellation, 
 
 Ratio, Short Methods, Review Fractions. 
 Miscellaneous Problems 162 to 172 
 
 Oral and Written. 
 
 CHAPTER IV. 
 
 Denominate Numbers 173 to 189 
 
 Reduction, Descending and Ascending, Compound 
 Addition, Subtraction, Multiplication, and Division, 
 Avoirdupois Weight, Time between Dates. 
 
Contents. t \s:\: »/, vhV i 
 
 Percentage . • 189 to 194 
 
 Applications and Simple Interest. 
 Measurements ......... 195 to 209 
 
 Area of Rectangles, Square Measure, Solid Contents, 
 
 Cubic Measure, Surfaces of Rectangular Solids, 
 
 Angles, Triangles, Quadrilaterals. 
 Review op Simple Numbers and Fractions . . 209 to 218 
 . Special Drills, Sight Approximations, Fundamental 
 
 Processes, Cancellation, Review Fractions, Review 
 
 Decimals. 
 Review Problems 218 to 228 
 
 Miscellaneous, Oral, Written. 
 
 CHAPTER V. 
 
 Percentage 229 to 276 
 
 Finding Percentage, Base, Rate ; Commission, Insur- 
 ance, Duties, Taxes, Profit and Loss, Commercial 
 Discount, Interest, Partial Payments, Bank Discount, 
 Interest by Aliquot Parts. 
 
 Denominate Numbers 277 to 291 
 
 Reduction Descending and Ascending, Addition, Sub- 
 traction, Multiplication, Division, Review. 
 
 Review of Simple Numbers, Fractions, and Decimals 291 to 309 
 
 CHAPTER VI. 
 
 Ratio and Proportion , 310 to 328 
 
 Ratio, Proportion, Partitive Proportion, Partnership, 
 Compound Proportion. 
 
 Involution and Evolution 328 to 338 
 
 Square Root, Applications of Square Root, Cube Root. 
 
 Mensuration 339 to 357 
 
 The Circle, Areas of Circles, Areas of Triangles, Areas 
 of Quadrilaterals, Surfaces of Prisms and Cylinders, 
 Surfaces of Pyramids and Cones, Volumes of Prisms 
 and Pyramids, Volumes of Cylinders and Cones, Sur- 
 face of Sphere, Volume of Sphere, Circular Measure. 
 
viii . Contents. 
 
 ! ',' < f '< r < ' ', ' , . ' t ' PAGS8 
 
 Longitude and Solar Time . . . . . 358 to 363 
 
 Standard Time, Solar Time. 
 Review Problems ........ 363 to 366 
 
 Miscellaneous, Oral, Written. 
 
 Stocks and Bonds 367 to 372 
 
 Domestic Exchange ' . . . 373 to 377 
 
 Sight Drafts, Time Drafts, Bills of Exchange. 
 Interest . . . 378 to 380 
 
 Compound Interest, Annual Interest. 
 
 Metric System 380 to 384 
 
 Review Problems 384 to 414 
 
 Special Drills, Review of Fractions, Review of De- 
 nominate Numbers, Review of Commercial Discount, 
 Review of Interest, Review of Bank Discount, Exact 
 Interest, Miscellaneous — Oral and Written. 
 
 CHAPTER VII. 
 
 Algebraic Equations of One Unknown Quantity . 415 to 439 
 Coefficients, Clearing of Fractions, Positive and Nega- 
 tive Quantities, Addition, Subtraction, Removing 
 Parentheses. 
 
 Two Unknown Quantities 440 to 445 
 
 Three Unknown Quantities 445 to 449 
 
 Multiplication and Division 449 to 469 
 
 Exponents, and Terms. 
 
 Factoring 460 to 467 
 
 Fractions 468 to 470 
 
 Quadratics 471 to 479 
 
 CHAPTER VIII. 
 
 Geometry 480 to 503 
 
 Lines, Angles, Triangles, Quadrilaterals, Circles, 
 Problems in Construction, Calculating Heights and 
 Distances. 
 
SUGGESTIONS TO TEACHERS. 
 
 Additions and Omissions. — The teacher should freely supplement 
 the work of the text-book when it is found necessary to do so ; and 
 the pupils should not be required to continue the work under any 
 topic after they fully understand it, even though they may not have 
 solved all the problems given in connection therewith. 
 
 Oral and Written Work. —The heading "Written Problems" is 
 merely a general direction, and it should be disregarded by the teacher 
 when the pupils are able to do the work " mentally." The use of the 
 pencil should be required only so far as it may be necessary. It is a 
 pedagogical mistake to insist that the brighter pupils of a class should 
 set down a number of figures that they do not need. As an occasional 
 exercise, the pupils may be directed to give all the work required to 
 solve a problem, and to make a written explanation of each step in 
 the solution ; but it should be the teacher's aim to have the majority 
 of the examples done with as great rapidity as is consistent with abso- 
 lute correctness. It will be found that, as a rule, the quickest workers 
 are the most accurate. 
 
 Conduct of the Recitation. —It is often advisable, for some pur- 
 poses, to divide an arithmetic class into two sections, even where the 
 pupils are nearly equal in attainments. The members of one sec- 
 tion may work examples from their books while the others write the 
 answers to oral problems given by the teacher, etc. 
 
 Where a class is thus taught in two divisions, the members of each 
 should sit in alternate rows, extending from the front of the room to 
 the rear. Seated in this way each pupil is doing a different kind of 
 work from those on the right and the left, and he does not have the 
 temptation of a neighbor's work to lead him to compare answers. 
 
 To save time, explanations of new subjects may be given to the 
 whole class ; but much of the arithmetic work should be done in " sec- 
 tions," one of which is under the immediate direction of the teacher, 
 while the other is employed in "seat" work. The "seat" work of 
 pupils of the more advanced classes should consist largely of problems 
 solved without assistance. Especial pains have been taken to grade the 
 
Suggestions to Teachers. 
 
 .'PfdbjeHja sdas-to have' none beyond the capacity of the average pupil. 
 It is not necessary that all the members of a division should work the 
 same problems at a given time, or the same number of problems, or 
 that a new topic should be postponed until all of the previous problems 
 have been solved. 
 
 Whenever it is possible, each of the members of the division work- 
 ing under the teacher's immediate direction should take part in all the 
 work done. In mental arithmetic, for instance, while only a few may 
 be called upon for explanations, all of the pupils should write the 
 answers to each question. The same is true of much of the sight 
 work, the approximations, some of the special drills, etc. 
 
 Drills and Sight Work. — To secure reasonable rapidity, it is 
 necessary to have regular and systematic drills. These should be 
 employed frequently, but should not last longer than five or ten 
 minutes. A page of special sight drills is given in each chapter. 
 These may also be used in oral problems. 
 
 It often happens that as pupils go forward in school they lose much 
 of the readiness in oral and written work that they possessed in the 
 lower grades, owing to the neglect of their teachers to continue to 
 require quick, accurate review work in the operations previously 
 taught. In this book these special drills follow the plan of the 
 combinations of the earlier book, but gradually grow more difficult. 
 They should first be used as sight exercises, either from the books or 
 from the blackboard. 
 
 To secure valuable results from drill exercises, the utmost prompt- 
 ness in answers should be required. 
 
 Language. — While the use of correct language should be insisted 
 upon in all lessons, children should not be required in arithmetic to 
 give all answers in "complete sentences." Especially in the drills, 
 it is important that the results be expressed in the fewest possible 
 words. The teacher should be careful always to employ exact arith- 
 metical language and to require it from the pupils. 
 
 Objective Illustrations.— The chief reason for the use of objects 
 in the study of arithmetic is to enable pupils to work without them. 
 While counters, weights and measures, diagrams, or the like are neces- 
 sary at the beginning of some topics, it is important to discontinue 
 their use as soon as the pupil is able to proceed without their aid. 
 
 Approximate Answers. — An important drill is furnished in the 
 "approximations" (see Arts. 104, 180, 238, etc.). Pupils should be 
 required in much of their written work to estimate the result before 
 beginning to solve a problem with the pencil. Besides preventing an 
 
Suggestions to Teachers, , xT 
 
 absurd answer, this practice will also have the effect of causing 'a 
 pupil to see what processes are necessary. In too many instances, 
 work upon a problem is commenced before the conditions are grasped ; 
 this will be less likely to occur in the case of one who has carefully 
 "estimated" the answer. The pupil will frequently find, also, that 
 he can obtain the correct result without using his pencil. 
 
 Indicating Operations. — It is a good practice to require pupils to 
 indicate by signs all of the processes necessary to the solution of a 
 problem, before performing any of the operations. This frequently 
 enables a pupil to shorten his work by cancellation, etc. In the case 
 of problems whose solution requires tedious processes, some teachers 
 do not require their pupils to do more than to indicate the operations. 
 It is to be feared that much of the lack of facility in adding, multiply- 
 ing, etc., found in the pupils of the higher classes is due to this desire 
 to make work pleasant. 
 
 Sight Exercises. — Many pupils who find it difficult to solve prob- 
 lems read to them readily make the necessary calculations without a 
 pencil when they have the numbers before them on the blackboard, or 
 in their books. It may be found advisable to have a class first solve 
 the whole of a given set of oral problems from their books, and at a 
 later lesson write the answer to each question after it has been read 
 by the teacher. In the case of sight exercises too difficult to be solved 
 mentally, the set might be taken up one at a time by individual pupils, 
 after which the pupils might be required to write answers " at sight " 
 at a signal from the teacher. If the exercises are on the blackboard, 
 the teacher might use a pointer to indicate the example whose answer 
 was desired, not following the order in which they appeared on the 
 blackboard. A similar method might be employed in sight work done 
 from the books. 
 
NEW 
 GEAMMAK SCHOOL ARITHMETIC. 
 
 ~tto* 
 
 CHAPTER L 
 
 PAGK8 
 
 Mixed Numbers 1 to 16 
 
 Addition, Subtraction, Multiplication, Division (Common 
 
 Denominators determined by inspection). 
 Review of Simple Numbers . . . . . . 17 to 24 
 
 Notation and Numeration, Special Drills, Fundamental 
 
 Processes. 
 Decimals (three places) 25 to 32 
 
 Addition and Subtraction, Multiplication and Division 
 
 by an Integer. 
 United States Monet 32 to 43 
 
 Fractional Parts of a Dollar, Division of United States 
 
 Money. 
 Denominate Numbers 43 to 45 
 
 Time, Dry and Liquid Measures, Avoirdupois Weight, 
 
 Miscellaneous Examples. 
 Measurements 46 to 49 
 
 The Area of Rectangles. 
 
 Bills 50 to 51 
 
 Miscellaneous 51 to 58 
 
 Approximate Answers, Review Problems. 
 
 MIXED NUMBERS. 
 1. Preliminary Exercises. 
 
 How many halves in 1 ? How many fourths in 1 ? Six 
 halves = ? 12 fourths = ? 6 thirds = ? 12 sixths == ? 
 
 f = ? | = ? | = ? | = ? ll = ? Y = ? 
 
 1 
 
d ' ' ' " ... Chapter One. 
 
 r *«» 'A mixed* number is a whole number and a fraction 
 written together. 
 
 3. A proper fraction is a fraction whose numerator is less 
 than its denominator. 
 
 An improper fraction is a fraction whose numerator is 
 equal to or greater than its denominator. 
 
 4. Change each of the following improper fractions to a 
 whole number or to a mixed number : 
 
 ¥ 
 
 ¥ 
 
 V 
 
 ¥ ¥ i 
 
 5. Oral Exercises. 
 
 How many quarts in a gallon? 
 
 What part of a gallon is a quart ? 
 
 \ gallon = how many quarts ? \ = how many fourths ? 
 
 How many quarts in a peck ? What part of a peck is one 
 quart? One-half peck is how many quarts? One-half 
 = how many eighths ? 
 
 \ peck is how many quarts ? \ = how many eighths ? 
 
 \ =how many eighths ? 
 
 how many eighths? 
 
 il 
 
 I 1 
 
 si 
 
Mixed Numbers. 3 
 
 6. Draw a line one foot long. Draw a second line of the 
 same length ; divide it into halves. Divide a third line of 
 the same length into three equal parts. Divide three other 
 lines, one into fourths, one into sixths, and one into twelfths. 
 
 How many inches in a foot ? What part of a foot is one 
 inch ? J foot = how many inches ? \ — how many twelfths ? 
 
 1 = how many twelfths ? f = how many twelfths ? Change 
 \ to twelfths. Change j, f to twelfths. How many twelfths 
 
 _ 1 ? 2? 3 ? 4? 5? 6? 
 — $"• 3"' 6* ^' 6* 6 ' 
 
 2 _ ? 3 _ ? 4 _ ? _ 2 
 
 T2"— 6 TJ — f TJ — ^ — ^ 
 
 6 _ ? _ ? _ ? 8_?_? 9_? 
 
 T2— "6"— T — "2 T2"— ~5 — T T2~ — 4~ 
 
 10—1 44 = 1=1 = 1= i 
 
 How many inches in f ft + J ft. + J ft. + 1 ft. + ^ ft. ? 
 How many feet and inches ? 
 
 How many 12ths in i+i + i+J + yV? Change to a 
 mixed number. Change the fractional part to a different 
 fraction having the same value. 
 
 What fraction of a dime is 1 cent ? £ dime = how many 
 cents? £ = T V 
 
 £ dime = how many cents ? £ = T V Change f to tenths. 
 
 !• t !• 
 
 Add -i- dime, £ dime, and ^ dime. How many cents? 
 How many tenths = 1 + i + T V ? Can y° u change the 
 answer to a different fraction having the same value ? 
 
 7. Oral Problems. 
 
 1. I spent 1 of a dollar for a ball and T ^ of a dollar for 
 a bat. What part of a dollar did I spend for both ? 
 
 2. What is the cost of a pen-knife at f of a dollar, and 
 a book at i of a dollar ? 
 
 3. I need -J of a yard of ribbon for one hat and £ of a 
 yard for another. How much ribbon should I buy ? 
 
4 Chapter One. 
 
 4. Sold f of a pound of tea to one customer and £ to 
 another. How much was sold to both ? 
 
 5. What quantity of oats should I buy to give J of a 
 peck to one horse and | to another ? 
 
 6. If I sell i- of a dozen of oranges to one person and J 
 of a dozen to another person, what part of a dozen do I sell ? 
 
 7. f of an hour is how many minutes ? 
 
 8. I spent J of an hour reading and ^j- of an hour writ- 
 ing. What part of an hour did I spend at both ? 
 
 9. A boy is carrying 6 J pounds of flour, and 6 J pounds 
 of ham. What is the weight of his load ? 
 
 10. 18 hours are what part of a day ? 
 
 ADDITION OF MIXED NUMBERS. 
 
 8. In fractions the numbers above the line are called numera- 
 tors ; the numbers below the line are called denominators. 
 
 The numerator and the denominator are called the terms of a 
 fraction. 
 
 To add fractions they must have a common denominator. 
 
 A common denominator is a number that will exactly contain 
 all the denominators. 
 
 The least common denominator is the least number that will 
 exactly contain all the denominators. 
 
 9. Add 12}, 6fc 8J, 15f , |. 
 
 24 
 
 12} 
 
 12 
 
 61 
 
 16 
 
 H 
 
 6 
 
 15* 
 
 20 
 
 | 
 
 9 
 
 43| 
 
 ff 
 
 2f. Arts. 43f. 
 
Mixed Numbers. 5 
 
 An inspection of the denominators, 2, 3, 4, 6, 8, shows that 24 is 
 the smallest number that will contain each without remainder. This 
 is the least common denominator. 
 
 Instead of writing the least common denominator 24, with each 
 fraction, we may place it above, and write only the new numerators. 
 } = if, | = ££, i = fo etc. Write 12, 16, 6, 20, 9. The sum of these 
 numerators, 63, is written over the denominator 24, making the sum of 
 the fraction ff. This improper fraction is reduced to 2|$, and the 
 fractional part is reduced to f. f is placed under the fractions to be 
 added, and 2 is carried to the whole numbers, making 43. 
 
 Add the fractions and unite their sum with the sum of the 
 
 The fractional parts of answers should be reduced to lowest 
 terms. 
 
 10. Written Exercises. 
 
 
 
 
 Add: 
 
 
 
 
 
 1. 23* 
 
 2. 73* 
 
 3. 93| 
 
 4. 11$ 
 
 5. 18* 
 
 63* 
 
 H 
 
 2* 
 
 3* 
 
 7* 
 
 n 
 
 39& 
 
 7±A 
 
 20A 
 
 9A 
 
 3A 
 
 16* 
 
 6* 
 
 5+ 
 
 i 
 
 6. 12* 
 
 7. 19* 
 
 8. 73} 
 
 9. 5** 
 
 io. loo* 
 
 3A 
 
 n 
 
 98* 
 
 38* 
 
 75* 
 
 27f 
 
 34£ 
 
 i 
 
 23* 
 
 9* 
 
 Ji- 
 
 _± 
 
 33* 
 
 17* 
 
 49* 
 
 ll. 33| 
 
 12. 6^ 
 
 13. 103* 
 
 14. 218f 
 
 15. 444* 
 
 m 
 
 18* 
 
 84* 
 
 301* 
 
 518f 
 
 2*A 
 
 32* 
 
 25* 
 
 18* 
 
 37* 
 
 69A 
 
 94* 
 
 H 
 
 24| 
 
 95| 
 
6 Chapter One. 
 
 11. Written Problems. 
 
 1. A merchant sold 17f yards of muslin, 14J- yards of 
 silk, and as many yards of calico as of the other two 
 together. How many yards did he sell in all? 
 
 2. A boy has to walk from his home to a house If miles 
 east of his home, and from there to a place 2£ miles west 
 of his home. How far has he to walk ? 
 
 3. From a piece of cloth 17-J- yards, 5f yards, and 4f 
 yards were sold. How many yards were sold ? 
 
 4. A man walked 12^ miles Tuesday, 16f miles Wed- 
 nesday, 22 ^ miles Thursday. How far did he walk in 3 
 days? 
 
 5. A farmer owned 3 fields containing, the first 21| 
 acres, the second 27f acres, and the third 28^ acres. How 
 many acres were there in all ? 
 
 6. A man bought 3 loads of wood containing respec- 
 tively 1J, cords, If cords, and If cords. How many cords 
 of wood did he buy ? 
 
 7. A man has 10^ acres of wheat, 6f acres of corn, 20f 
 acres of barley, 16 § acres of rye. How many acres of grain 
 has he? 
 
 8. William lives 24£ rods from school, James 6^ rods 
 farther than William, and Charles 10^f rods farther than 
 James. How far does Charles live from school ? 
 
 9. Henry weighs 58 T 3 b r pounds, Peter 65} pounds, and 
 John 67f pounds, and their father as much as all three of 
 them. How much does their father weigh ? 
 
 10. A dealer mixed 2-J- pounds of black tea costing 32 
 cents per pound with 1£ pounds of green tea costing 40 
 cents per pound. How much per pound does the mixed tea 
 cost him ? 
 
Mixed Numbers. 7 
 
 SUBTRACTION OF MIXED NUMBERS. 
 
 12. Preliminary Exercises. 
 
 l-i=? li-i=? 10-*=? 10J-i=? 10i-li=? 
 
 In subtraction of mixed numbers, as in addition, the 
 fractions must have a common denominator. 
 
 Subtract : 
 
 
 
 
 
 1. left 
 
 2. 49$$ 3. 
 
 38$f 4. 
 
 i»A 
 
 5. 27A 
 
 13A 
 
 37$$ 
 
 29$$ 
 
 "ft 
 
 16ft 
 
 6. 28^ 
 
 7. 47$ 8. 
 
 36i$ 9. 
 
 25$$ 
 
 10. 32$$ 
 
 13* 
 
 29$ 
 
 1»A 
 
 19$$ 
 
 ISA 
 
 13. From 197$ take 68$. 
 
 
 
 
 15 
 
 Reduce the fractions to the least 
 
 common 
 
 denominator 
 
 197| 
 
 9 
 
 15, as in addition of fractions. {% being 
 
 greater than 
 
 68$ 
 
 10 
 
 ■fs, we change 197-& to 196 + 1£, or 196f 
 = i|. 196-68 = 128. Ans. 128}|. 
 
 *• tt-tt 
 
 128$* 
 
 14 
 T5 
 
 
 Reduce the fractions to the least common denominator, 
 and subtract the fractions and the integers separately. 
 
 14. "Written Exercises. 
 
 1. 36| 2. 63L 3. 27f 4. 05f 5. 105^ 
 -*i -9 T V -17i -25f -8J 
 
 6. 120$ 
 
 7. 39$ 
 
 8. 13$ 
 
 9. 99$ 10. 
 
 67$ 
 
 -84$ 
 
 -38$ 
 
 -7ft 
 
 -21$ 
 
 • 
 
 -59$ 
 
 11. lOOJy 
 
 12. 25A 
 
 13. 93A 
 
 14. 101$$ 15. 
 
 12$ 
 
 76$ 
 
 5$ 
 
 24$ 
 
 98$ 
 
 4$ 
 
16. 
 
 21. 
 
 26. 
 
 
 
 Chapter One. 
 
 
 
 23£ 17. 
 
 9A 
 
 ft 
 
 18. 133} 
 27* 
 
 19. 
 
 16^ 
 
 3* 
 
 20. 37J 
 29J 
 
 52| 22. 
 
 64£ 
 
 18£ 
 
 23. 125-ft 
 lOOf 
 
 24. 
 
 47i 
 8* 
 
 25. 72 T V 
 50£ 
 
 31f 27. 
 
 27A 
 
 63^ 
 44j. 
 
 28. 3^ 
 
 29. 
 
 25f 
 
 Hi 
 
 30. 102/j 
 86J 
 
 15. Oral Problems. 
 
 1. A man had $6^, and he spent $ 3 J. How much 
 money had he left ? 
 
 2. Take $ 8 \ from $ 12f. How many quarters of a 
 dollar are there in the remainder? 
 
 3. One-half of our books are in the case ; we have in all 
 184 books ; one-half of the remainder are on the table. 
 How many are on the table ? 
 
 4. If 6 apples cost 14 cents, what will 3 cost ? 
 
 5. How many hours from 10 a.m. to 10 p.m. ? 
 
 6. A man had 1000 acres of land and sold 996 J acres. 
 How many acres had he left ? 
 
 7. If a man earns $ 14£ in a week, and spends $ 8f , 
 how much does he save ? 
 
 8. Bought sugar for 5J cents a pound, and sold it for 6J 
 cents a pound. What was the gain on 200 pounds ? 
 
 9. What will 12f pounds of beef cost at 16 cents a 
 pound? 
 
 10. If a girl studies 5\ hours in school, and 1\ hours at 
 home each day, how many hours does she study in a week 
 of five days ? 
 
Mixed Numbers. 9 
 
 16. Written Problems. 
 
 1. The weight of a tub of butter, including the weight 
 of the tub, is 48J pounds. The tub weighs 9£ lb. What is 
 the butter worth at 24 cents per pound ? 
 
 2. A farmer had 7 bushels of potatoes. He used 2 
 bushels and 3 pecks for seed. What would the remainder 
 be worth at 20 cents per peck ? 
 
 3. How much heavier is a cheese weighing 40J pounds 
 than one which weighs 26f pounds ? 
 
 4. A farmer having 217 bushels of corn sold 95 }{ 
 bushels j how many bushels had he left? 
 
 5. A milliner gained 1-J dollars by selling a hat for 6£ 
 dollars ; what did it cost her ? 
 
 6. From a cask of oil containing 43f gallons, 17| gallons 
 were drawn ; how many gallons remained ? 
 
 7. A man having 25J dollars paid 6J dollars for coal, 
 2\ dollars for dry goods, and £ of a dollar for a pound of 
 tea ; how much had he left ? 
 
 8. A butcher buys an ox weighing alive 1200 pounds, 
 at 6 cents per pound. When killed and dressed, its weight 
 is f of the live weight. What is the butcher's profit, if he 
 sells the meat at an average of 15 cents per pound ? 
 
 9. A farmer sold 36^ dozen eggs to one storekeeper, 5J 
 dozen to another, 17f dozen to a third, 8f dozen to a fourth, 
 and 11 T 7 2- dozen to a fifth. How much did he receive for 
 them at 12 cents per dozen ? 
 
 10. A teacher's salary per month is 135^ dollars, and 
 his expenses average 51 J dollars : how much does he save 
 per month ? 
 
 11. A man gave \ of his money to his wife and i of it to 
 his daughter. He divided the remainder equally among his 
 three sons, each of whom received $1000. How much 
 money had he? 
 
io Chapter One. 
 
 MULTIPLICATION OF MIXED NUMBERS. 
 
 17. Preliminary Exercises. 
 
 i + i + } = ? 3timesJ = ? ixS = ? 
 
 6 times \ = ? 3 times £ = ? |x3 = ? 
 
 | X 9 = ? |xl5 = ? |xl7 = ? 
 
 | X 7 = ? f x 20 = ? £ x 12 = ? 
 
 £ X 5 = ? fxl3 = ? |xlO = ? 
 
 18. Multiplication of a mixed number by an integer. 
 
 Find the product of 235f by 39. 
 
 235f 
 
 Multiply 3 by 39; divide the result by 4: the 39 
 
 quotient, 29£, is 39 times f . Write the next partial 4 )117 
 product, 235 x 9 ; then the product of 135 by 3 tens. 29J 
 
 The sum of the three partial products gives the 2115 
 
 result, 9 194 J. 
 
 
 705 
 
 19. Oral Exercises. 
 
 
 9194J Ans 
 
 1. 1| X 9 = ? 3. 
 
 3|x5 = ? 
 
 5. 5fxl2 = ? 
 
 2. 2|x7=? 4. 
 
 4| x 8 = ? 
 
 6. 6^x10 = ? 
 
 20. Written Exercises. 
 
 
 
 1. 215§xl7 = ? 3. 
 
 417^x20 = ? 
 
 5. 619^x19 = ? 
 
 2. 316fxl5 = ? 4. 
 
 518£xl3=? 
 
 6. 720 T \x23 = ? 
 
 7. 163| 9. 
 X75 
 
 509£ 
 X213 
 
 11. 6089f 
 X1004 
 
 8. 103| 10. 
 Xl7 
 
 308| 
 X156 
 
 12. 1607f 
 X2340 
 
Mixed Numbers. n 
 
 21. Multiplication of an integer by a mixed number. 
 Multiply 276 by 280|. 
 
 276 
 
 280# 
 Multiply 276 by the numerator, 3; divide the Q\eo8 
 product by the denominator, 8 ; the quotient, 103£, is ^.n 4 ^ 
 the product of 276 by f . Multiply 276 by 8 tens and 
 by 2 hundreds, etc. 
 
 2208 
 552 
 77383^ Ans. 
 
 To multiply a whole number by a fraction, place the product 
 of the numerator by the whole number over the denominator, 
 and reduce, if possible. 
 
 22. Written Exercises. 
 
 1. 13x7| =? 4. 17x10^ = ? 7. 102x22f = ? 
 
 2. 19x8^ = ? 5. 21xll| =? 8. 204x34f = ? 
 6. 27 x 12f m ? 9. 468 x 56f = ? 
 
 12. 4060 14. 3579 T V 
 X 2050| x 4300 
 
 3. 
 
 23 x 9^- = 
 
 9 
 
 10. 
 
 387 
 x400f 
 
 
 11. 
 
 698 
 Xl35f 
 
 
 13. 3050 15. 4987^- 
 X 2060f x 2469 
 
 23. Oral Problems. 
 
 1. How many ounces in 6% pounds ? 
 
 2. I sold 3^ yards of silk and 2f yards of velvet. How 
 many yards in all did I sell ? 
 
 3. From 60 take 24. Find \ of the remainder. 
 
 4. | of 100 rods = ? 6. | of 81 yards = ? 
 
 5. (fof60)-9 = ? 7. f of 56 pounds = ? 
 
12 Chapter One. 
 
 8. | of a yard and 12 inches are how many inches ? 
 
 9. If one-half a pound of soap costs 10 cents, what will 
 three pounds cost ? 
 
 10. John is going a journey of 100 miles ; if he travels 
 | of the distance in the cars and the rest in a coach, how 
 many miles will he travel in the coach ? 
 
 11. How many times must I fill my glass, which holds 
 £ a pint, to fill my pitcher, which holds a gallon ? 
 
 12. If a boy is in school h\ hours a day, how many 
 hours is he in school in 200 days ? 
 
 24. Written Problems. 
 
 1. What is meant by f of any number or thing ? Make 
 a drawing to show what you mean. 
 
 2. What is the cost of 15^- acres of land at $45 an acre ? 
 
 3. Reduce £, f, §, and ^ to fractions having a common 
 denominator. 
 
 4. What is the cost of a side of beef containing 252 
 pounds at 9J cents a pound ? 
 
 5. A hotel uses 18f pounds of beef in a day. What will 
 be the weekly bill at 22 cents a pound ? 
 
 6. A man walks 3^ miles in one hour. How far can he 
 walk in 9 hours ? 
 
 7. From a piece of muslin containing 37£ yards, three 
 pieces each measuring 7-J- yards were sold. How much 
 remained in the piece ? 
 
 8. At $ 7.86 a barrel, what will 18| barrels of flour cost ? 
 
 9. Bought 6 bushels of apples at 62 \ cents a bushel, 
 and sold them at 12£ cents a half-peck. What was the 
 gain? 
 
 10. In a school containing 945 pupils ^ of the number 
 were boys ; how many boys in the school ? 
 
 11. What is the cost of 15 acres of land at $ 45£ an acre ? 
 
Mixed Numbers. 13 
 
 12. If a quart of cream is worth 22 cents, what are two 
 gallons worth ? 
 
 13. At 9 cents a quart, what is the cost of 2\ gallons 
 of vinegar ? 
 
 14. What is the total quantity of molasses in 4 casks 
 containing, respectively, 40 J, 25f 27-^-, and 55| gallons ? 
 
 15. The Post-office Department bought 6670 pounds of 
 twine at 19 J cents a pound ; 372 pounds of sponge at 65£ 
 cents a pound, and 40£ dozen of ink at $2 a dozen. 
 What was the total cost of the purchase ? 
 
 DIVISION OF MIXED NUMBERS. 
 
 25. Preliminary Exercises. t , 
 
 How many times is \ of a dollar contained in $ 1 ? How 
 many times is 1 of a pint contained in 1 pint ? \ of a gal- 
 lon in 1 gallon ? 
 
 How many times is ^ of a dollar contained in $ 2 ? In $ 3 ? 
 In $5? 
 
 How many times is -J- of a dollar contained in $1.50? 
 In $2.50? In $3.50? In $4.50? 
 
 How many times is 1 half contained in 3 halves ? In 
 5 halves ? In 7 halves ? In 9 halves ? 
 
 3-1-1=? 5^_\—9 l-±-l = ? 5.-^1 = ? 
 
 2-Y* 2 ' 2 ' 2 ' 2 * T * ^ ' 
 
 How many times is f contained in f ? In f ? In -^ ? 
 
 In^f? 
 Divide 1| by If 4£ by If 7| by If 10J by If 
 Divide 3 by If 6 by If 9 by If 12 by If 15 by If 
 Divide 5 by If 6Jbylf 10 by If lljbylf 15 by If 
 Divide i by f. £by|. 1} by f . 1$ by f. 2J by $ 
 
 3 by J. 3fby{. 
 
14 Chapter One. 
 
 26. Written Exercises. 
 
 1. Divide 250 by 12$. 
 
 250 = 500 halves. 12$ = 25 halves. 
 600 halves -=- 25 halves = 500 ^ 25 = 20, Ans. 
 Proof : 20 x 12$ = 250. 
 
 2. Divide 62\ by 25. 
 
 62$ = 125 halves. 25 = 50 halves. 
 
 125 halves -r- 50 halves = 125 -r- 50 = 2£f = 2$, Ans. 
 
 Proof : 25 x 2$ = 62$. 
 
 3. Divide 1387$ by 18}. Ans. 74 
 
 18| = 75 fourths. 75)5550 
 
 Change 1387 1 to fourths by multiplying by 4. ^25 
 
 1387$ x 4 = 5550 ; that is, 1387$ = 5550 fourths. 300 
 
 75 fourths is contained in 5550 fourths 74 times. 300 
 
 Reduce the dividend and the divisor to improper fractions of 
 the same denominator, and divide the numerator of the divi- 
 dend by the numerator of the divisor. Prove the correctness 
 of the answer by multiplying the quotient by the divisor. 
 
 27. Written Exercises. 
 
 Div 
 1. 
 
 ide: 
 60+ \ 
 
 11. 
 
 75 + 12$- 
 
 21. 
 
 62| + 12$. 
 
 2. 
 
 60 + 1* 
 
 12. 
 
 150 + 12$- 
 
 22. 
 
 187$ + 12^ 
 
 3. 
 
 60+ i 
 
 13. 
 
 75+ 6J 
 
 23. 
 
 81 J + 6J 
 
 4. 
 
 60 + 1* 
 
 14. 
 
 150+ 6J 
 
 24. 
 
 193}+ 6J 
 
 5. 
 
 60+ \ 
 
 15. 
 
 62 + 15$- 
 
 25. 
 
 77$- + 15$ 
 
 6. 
 
 60 + If 
 
 16. 
 
 105 + 17$. 
 
 26. 
 
 192^ + 17$ 
 
 7. 
 
 60+ * 
 
 17. 
 
 69+ 5} 
 
 27. 
 
 97}+ 5} 
 
 8. 
 
 60 + 2* 
 
 18. 
 
 93+ 7} 
 
 28. 
 
 193}+ 7} 
 
 9. 
 
 60+ } 
 
 19. 
 
 100 + 33$ 
 
 29. 
 
 166$ + 33$ 
 
 10. 
 
 60 + 3J 
 
 20. 
 
 150 + 16f 
 
 30. 
 
 133$ + 16 $ 
 
Mixed Numbers. 15 
 
 31. 601-^2 34. 87i-f-6J 37. 60 -- 3f 
 
 32. 60 
 
 33. 15f 
 
 71 35. 621 --6J 38. 241 
 If 36. 60f-=-3 39. 87£ 
 
 If 
 
 28. Oral Problems. 
 
 1. I paid 18 cents for 11 pounds of lard. What is the 
 
 price per pound ? 
 
 36 cents for 3 pounds. 
 
 2. At f dollar per yard, how many yards of silk can be 
 bought for $ 9 ? 
 
 36 quarter dollars -=- 3 quarter dollars. 
 
 3. If one fish cost 25 cents, how much would 2\ fish 
 cost? 
 
 4. A man bought 30 apples at the rate of 3 for 5 cents. 
 How much did he give for them ? 
 
 5. If I pay 6 cents for a dozen apples, how much does 
 each apple cost ? 
 
 6. How many times is 41 contained in 27 ? 
 
 7. If 21 bushels of oats will keep a horse one week, how 
 long will 18 bushels keep him ? 
 
 8. If $ 97 is \ of a sum of money, what is that sum ? 
 
 9. What is the cost of 12 doz. eggs at the rate of 2 eggs 
 for 3 cents ? 
 
 10. If 3 boys can cut a cord of wood in 8 hours, how 
 long will it take 4 boys to cut a cord ? 
 
 11. If i of a melon costs 15 cents, what will two melons 
 cost at the same rate ? 
 
 12. It takes 2\ yards of cloth for a pair of trousers. 
 How many pairs can be made from 30 yards of cloth ? 
 
 13. Paid $ 12.90 for 3 pieces of lace. How much did 
 each cost ? 
 
 14. If 3 straw hats cost 63 cents, what will be the cost 
 of 5? 
 
1 6 Chapter One. 
 
 29. Written Problems, 
 
 1. A farmer distributed 15 bushels of corn among several 
 persons, giving them If bushels apiece; among how many- 
 persons did he divide it ? 
 
 2. A man bequeathed to his son $ 3500, which was -f- of 
 what he left his wife. How much did he leave his wife ? 
 
 Suggestion. — } of wife's share = $ 3500. Multiplying by 7 : 
 5 times wife's share = $24,500. 
 
 3. If f of a farm is valued at $ 1728, what is the value 
 of the whole ? 
 
 4. A man walks 4f miles in one hour, how far can he 
 walk in 9 hours ? 
 
 5. At | of a cent a foot, how many feet of wire can be 
 bought for $1.26? 
 
 6. The sum of 69f dollars was divided equally among 
 5 men ; what was each one's share ? 
 
 7. At | dollars per yard, how many yards of cloth can 
 be purchased for $98? 
 
 8. In how many days can a horse eat 66 bushels of oats 
 if he eats f of a bushel a day ? 
 
 9. A man bought chairs at 4f dollars apiece for 114 
 dollars, and then sold them at 6 J dollars apiece ; how much 
 did he gain? 
 
 10. A man sold 9f bushels of seed for $61.60; find the 
 price per bushel. 
 
 11. What part of 24 is 3 ? What part of 24J is 8} ? 
 
 12. What would be the cost of 24^ pounds of beans at 
 the rate of 11 cents for 3£ pounds ? 
 
Review of Notation and Numeration. 17 
 
 30. Notation and Numeration. 
 
 The largest number that can be written with six figures 
 is 999,999. 
 
 1,000,000, is called one million. 
 
 Write in figures two million. Three million. Four 
 million. Six million. Eight million. Ten million. 
 
 31. Bead the following : 
 
 1. 1,234,567 6. 11,034,065 11. 30,100,021 
 
 2. 3,000,560 7. 14,602,500 12. 35,000,600 
 
 3. 5,009,008 8. 17,386,925 13. 401,023,160 
 
 4. 7,090,070 9. 20,007,316 14. 760,030,020 
 
 5. 9,843,000 10. 25,000,005 15. 980,750,000 
 
 32. Write in figures : 
 
 1. Seventy-eight million, one hundred eight thousand, 
 ninety-six. 
 
 2. Three million, eight. 
 
 3. Fourteen million, seven thousand, five. 
 
 4. Nine hundred eighty-seven thousand, six hundred 
 fifty-four. 
 
 5. Twenty million, thirty thousand, forty. 
 
 6. Three hundred seven million, nine hundred four thou- 
 sand, six. 
 
 7. Nine hundred ninety-nine million, nine hundred 
 ninety-nine thousand, nine hundred ninety-nine. 
 
 8. Four hundred seventy-six million, three hundred 
 thousand. 
 
 9. Thirty-four thousand, eighteen. 
 
 10. Sixty-four million, thirty-two thousand, sixteen. 
 
 11. Add the foregoing. 
 
1 8 Chapter One. 
 
 REVIEW OF FUNDAMENTAL OPERATIONS. 
 
 Practice in the fundamental operations should not be neglected. 
 Business men complain that elementary and high school graduates 
 cannot add. 
 
 Read the following numbers. Add each column. 
 
 1. 
 
 27,083,549 2. 508,900,007 
 
 3. 2 
 
 43,576,908 
 
 
 3,006,005 4,629,880 
 
 
 5,987,600 
 
 
 20,080,070 25,936,097 
 
 
 380,070 
 
 
 1,647,893 134,870,603 
 
 
 68,000 
 
 
 206,045 59,009,300 
 
 
 593,056 
 
 
 73,000 7,000,004 
 
 
 2,384,672 
 
 
 180,059 686,909 
 
 
 59,876,004 
 
 4. 
 
 9,256,874 5. 348 
 
 6. 
 
 7,293 
 
 
 863,052 2,967 
 
 
 82,538 
 
 
 24,635,998 36,847 
 
 
 786,324 
 
 
 7,007,007 243,837 
 
 
 94,649 
 
 
 6,875,634 183,634 
 
 
 256,834 
 
 
 3,987,456 986,246 
 
 
 3,983,387 
 
 
 35,068 8,216 
 
 
 54,619 
 
 
 705 586,237 
 
 
 760,888 
 
 
 33. Oral Exercises. 
 
 
 
 Give answers : 
 
 
 
 1. 
 
 1200x6 6. 1300x9 11. 2100x4 
 
 16. 
 
 1400x8 
 
 2. 
 
 1800x4 7. 2300x3 12. 1400x6 
 
 17. 
 
 2400 x 4 
 
 3. 
 
 2500x3 8. 3200x2 13. 4100x2 
 
 18. 
 
 1300 x 7 
 
 4. 
 
 1700x5 9. 1500x4 14. 1600x5 
 
 19. 
 
 1200 x 9 
 
 5. 1400x7 10. 1200x8 15. 2200x3 20. 6300x2 
 
Review of Fundamental Operations. 19 
 
 34. Written Exercises. 
 Multiply: 
 
 1. 9,207x3014 
 
 2. 5,482 x 798f 
 
 3. 5,290 x 6075 
 
 4. 9,204 x 678J 
 
 5. 75,074 x 395 
 
 6. 68,431 x 924| 
 
 7. 95 x 95 x 95 
 
 8. 185 x 19 x 78 
 
 9. 87£x 23 x 36 
 
 10. 706 x 304x509 
 
 11. 48Jx 32 x 74 
 
 12. 538 x 247 x 125 
 
 35. Oral Exercise 
 
 «. 
 
 
 
 
 Divide : 
 
 
 
 
 
 1. 960-240 
 
 6. 
 
 8400-^2100 
 
 11. 
 
 10800-1200 
 
 2. 780 h- 260 
 
 7. 
 
 8600-4300 
 
 12. 
 
 10400 -j- 1300 
 
 3. 960 -5- 480 
 
 8. 
 
 8800-2200 
 
 13. 
 
 6000 -r- 1500 
 
 4. 720-180 
 
 9. 
 
 9600-3200 
 
 14. 
 
 5700-1900 
 
 5. 2170-310 
 
 10. 
 
 9900 -;- 3300 
 
 15. 
 
 12000 -h 2400 
 
 The foregoing exercises are given as a preparation for the long 
 division drill that follows. Each of the above set has an exact quo- 
 tient, easily determined at sight. 
 
 The object of the following set is to drill pupils to obtain rapidly 
 the correct quotient figure in a long division example. A pupil giving 
 4 as the answer to No. 1 should be asked to give the product of 241 by 4. 
 
 36. Long division drill. (Omit remainders.) 
 
 1. 960-241 
 
 2. 779^-260 
 
 3. 959-480 
 
 4. 720-181 
 
 5. 1160 -r- 130 
 
 6. 8,400-2110 
 
 7. 8,500-4300 
 
 8. 8,800-2199 
 
 9. 9,599-3199 
 10. 10,000 -j- 3330 
 
 11. 10,800-1205 
 
 12. 10,300-1300 
 
 13. 6,100-^1550 
 
 14. 5,700 - 1899 
 
 15. 12,020-5-2410 
 
<zo Chapter One. 
 
 37. Divide: 
 
 1. 34,463-5-370 7. 703,705 -j- 12,345 
 
 2. 823,150-1298 8. 420,135- 6,789 
 
 3. 639,712-624 9. 370,088 -j- 5,986 
 
 4. 345,738^-7210 10. 510,940-=- 4,900 
 
 5. 861,704-351 11. 639,215- 9,783 
 
 6. 857,384-3004 12. 345,678 -j- 7,095 
 
 38. Sight Exercises. 
 
 Note. — First, combine the quantities within the parentheses, ( ); 
 next, complete the combinations within the brackets, [ ], if any ; then 
 perform the remaining operations. 
 
 28 + (40 -j- 2) = 28 + 20 
 [30 -- (6 -4- 2)] x 5 = [30 -- 3] x 5 = 10 x 5 
 
 Perform indicated operations at sight : 
 
 1. 18 + (30x4) 7. } of (240 + 60) 
 
 2. 7 + (2x8)-4 8. (7 + 2)x(8-4) 
 
 3. [(7 + 2)x8]-4 9. 7 + [2x(8-4)] 
 
 4. l-(i + i) 10. 1-i + i 
 
 5. (6xJ) + i ii. 6x(i+i) 
 
 6. \ of \ of 600 12. jxl2x| 
 
 SPECIAL DRILLS. 
 
 Note. — In adding, subtracting, and multiplying without using the 
 pencil, it is inadvisable to begin with the units : 68 and 34, for in- 
 stance, are more readily combined mentally, by adding 68 and 30 (88) 
 and 4. In the recitation, the pupil should say 88, 92 ; or 92, merely. 
 
 630 + 280 = 630 + 200 + 80. 
 
 39. Give sums : 
 
 56 + 25 32 + 48 750 + 190 225 + 54 
 
 47+47 29 + 28 390 + 120 315 + 21 
 
 22 + 68 65 + 26 480 + 150 437 + 60 
 
Review of Fundamental Operations. 21 
 
 40. Give remainders : 
 
 92 - 58 = 92 - 50 - 8. Say 42, 34. 
 840 - 280 = 840 - 200 - 80. Say 640, 560. 
 
 81-56 750-190 750-560 279-54 
 
 94-47 510-120 510-390 386-63 
 
 60-28 630-150 630-480 457-37 
 
 72-39 820-160 820-660 568-25 
 
 41. Give products : 
 
 87 x 2 = (80 x 2) + (7 x 2). Say, 160, 174. 
 
 410x6 83x7 43 x 5 12x70 
 
 310x9 99x2 26 x 7 18x30 
 
 420 x 4 65 x 3 24 x 8 16 x 40 
 
 630x3 49x4 22 x 9 13x50 
 
 740x2 37x5 18x11 11x60 
 
 42. Give quotients : 
 
 168-3 168-56 1470-7 1470-210 
 
 196-4 196-49 2790-^-9 2790-^310 
 
 190-5 190-^38 1680-5-4 1680 -*- 420 
 
 192-6 192-32 1890 -r- 3 1890-630 
 
 196 -r- 7 196-*- 28 1480 -h 2 1480 -s- 740 
 
 43. Give answers : 
 
 2^ + lJ If-- { fof 66 12J+| 
 
 2J + 1J 2}-H 84 x f Si^-f 
 
 2£ + l£ 8|-2| foflOO 51 + f 
 
 21 + li H-H 186x i 3| + f 
 
 2 J + H 6J-4J |ofl20 4£ + | 
 
11 Chapter One. 
 
 44, Oral Problems. 
 
 1. Paid 59^ for muslin and 25^ for trimming. How 
 much, was paid for both ? 
 
 2. A boy had 75^. How much had he after spending 
 25^ for a knife and 15^ for a ball ? 
 
 3. If 8 pounds of raisins cost $ 1.04, what is the price 
 per pound ? 
 
 4. At $ 1.89 per yard of silk, what will be the cost of 
 J of a yard ? 
 
 5. If 32 pounds of flour cost 96 cents, how many pounds 
 can be bought for 60 cents ? 
 
 6. One girl has 16 cents, another has 24 cents, a third 
 has 8 cents. How many dolls at 16 cents each can be 
 bought with their money ? 
 
 7. What will be the weight of 3 bushels of corn, weighing 
 56 pounds per bushel ? 
 
 8. How many ounces in 9 pounds avoirdupois ? 
 
 9. How many pounds in 8 packages, each weighing 10 
 ounces ? 
 
 10. Find the cost of 3 pounds and 2 ounces of butter at 
 32 cents per pound. 
 
 11. Bought 4 pounds of sugar at 6 cents a pound, and a 
 pound of butter at 36 cents. How much change from $ 1 ? 
 
 12. Four boys have 144 marbles among them. If the 
 marbles were equally divided, how many would each have ? 
 
 13. A man earns $100 per month, and spends $76. 
 How much does he save ? 
 
 14. If a man saves $ 32 per month, how many months 
 will it take him to save $ 960 ? 
 
 15. Paid $27.90 for 9 jackets. What did they cost 
 apiece ? 
 
Review of Fundamental Operations. 23 
 
 16. Mr. B's farm contains 520 acres. How many acres 
 will he have left after selling 180 acres ? 
 
 17. William's kite string is 435 yards long, John's is 62 
 yards longer. What is the length of John's string ? 
 
 18. A farmer raised 168 bushels of grain. He sold £ of 
 it. How many bushels did he sell ? 
 
 19. A piece of ribbon measuring 6 \ yards is cut into pieces 
 a quarter of a yard long. How many pieces are there ? 
 
 20. If it takes 18f yards of cloth to make 3 suits, how 
 many yards does it take for 1 suit ? 
 
 21. James has 150 marbles, Thomas has -f as many. 
 How many marbles have both ? 
 
 22. A newsdealer received $6.36 for papers sold at 3 
 cents each. How many papers did he sell ? 
 
 23. If it takes 4|- days for one man to do a piece of work, 
 how long will it take 2 men to do the same work ? 
 
 24. A farm is divided into 4 fields, each containing 49 
 acres. How many acres are there in the farm ? 
 
 25. From a piece of cloth containing 10 \ yards, 5f yards 
 are sold. How many yards are left? 
 
 26. Find the cost of 28 pounds coffee at $ \ per pound. 
 
 27. How much does a farmer receive for 28 cows which 
 he sells at $30 each? 
 
 28. Find the number of hours in a week. 
 
 29. How many pieces, each three-quarters of a yard long, 
 can be cut from six yards of wire ? 
 
 30. 3600 seconds are equal to how many minutes ? 
 
 31. If 25 yards of material are needed for a dress, how 
 many yards will be required for 30 dresses ? 
 
 32. At 7 for a cent, what will 98 marbles cost ? 
 
24 Chapter One. 
 
 45. "Written Problems. 
 
 1. The sum of three numbers is 150. Two of the num- 
 bers are 68 and 43. What is the third ? 
 
 68 + 43 -f ? = 150 
 
 2. The divisor is 24; the dividend is 264. Find the 
 quotient. 
 
 3. The product is 228; the multiplicand is 19. What 
 is the multiplier ? 19 x ? = 228 
 
 4. The minuend is 175; the subtrahend is 87. What is 
 the remainder? 
 
 5. The remainder is 92; the subtrahend is 89. Find 
 the minuend. ? — 89 = 92 
 
 6. The minuend is 176, and the remainder is 99. What 
 is the subtrahend ? 
 
 7. The multiplier is 15; the multiplicand is 46. What 
 is the product? 
 
 8. The multiplier is 16; the product is 272. What is 
 the multiplicand? 
 
 9. The dividend is 300; the divisor is 17. Find the 
 remainder. 
 
 10. The quotient is 15 ; the remainder is 3 ; the divisor 
 is 8. What is the dividend? 
 
 8) P 
 15| 
 
 11. The dividend is 273 ; the quotient is 21. What is 
 the divisor ? 
 
 12. The dividend is 267; the quotient is 13; the remain- 
 der is 7. What is the divisor? 
 
 ? )267 
 13| 
 
 13. How many acres of land could you buy for $ 76,225, 
 if one acre cost $37? 
 
Decimals. 25 
 
 NOTATION OF DECIMALS. 
 
 46. A decimal fraction is one in which the unit is divided 
 into tenths, hundredths, thousandths, etc. 
 
 47. Preliminary Exercises. 
 
 In the number 25 ; what does the 2 stand for ? 
 
 In the number 467, what does the 4 represent ? The 6 ? 
 The 7? 
 
 In the number 33,333, give the value of the first 3 (com- 
 mencing at the left). Of the second. Of the third. Of 
 the fourth. Of the fifth. 
 
 The last 3 is what part of the number represented by the 
 fourth 3 ? The third 3 is what part of the second ? Each 
 3 is what part of the 3 to its left ? Upon what does the 
 value of each 3 in this number depend ? 
 
 In the number XXXIII, what is the value of the first X ? 
 Of the second ? Of the third ? 
 
 When we write $784,365, the 7 stands for seven times 
 how many dollars ? The 8 for eight times how many dol- 
 lars? The 4 for four times how many dollars? The 3 
 stands for three times what part of a dollar ? The 6 stands 
 for six times what part of a dollar ? The 5 stands for five 
 times what part of a dollar ? 
 
 Hundreds. Tens. Units. Decimal Point. Tenths. Hundredths. Thousandths. 
 
 7 8 4 .36 
 
 784.365 is read 784 and 365 thousandths. 
 37.5 is read 37 and 5 tenths. 
 6.492 is read 6 and 492 thousandths. 
 400.75 is read 400 units and 75 hundredths. 
 
 Note. — In reading a number containing an integer and a decimal, 
 the word and may be placed between the two, as is shown above. 
 To avoid mistakes, the word units should be used after the integer 
 in reading such numbers as 200.005. Say : Two hundred units and 
 five thousandths. 
 
16 Chapter One. 
 
 48. Read the following : 
 
 1. .7 5. 3.275 9. 100.025 
 
 2. 34.9 6. 32.4 10. .125 
 
 3. .36 7. 1.025 11. .005 
 
 4. .95 8. .35 12. 1.348 
 
 49. Express in decimals : 
 
 1. 7 tenths. 
 
 2. 36 and 47 thousandths. 
 
 3. One hundred twenty -five thousandths. 
 
 4. One hundred units and twenty-five thousandths. 
 
 5. 47 hundredths. 
 
 6. Four hundred units and six tenths. 
 
 7. Four hundred six thousandths. 
 
 8. 3 and 56 hundredths. 
 
 9. 65 hundredths. 
 10. 6 and 5 tenths. 
 
 Note. — Since ffo equals ^, .50 = .5. The cipher at the right of 
 .60 has, therefore, no value, •fflfo = ^ ; .700 is, therefore, the same 
 as .7. In giving answers, reject ciphers at the right of the decimal. 
 
 ADDITION OF DECIMALS. 
 
 50. Add: 
 
 
 
 
 1. .7 
 
 2. 3.84 
 
 3. 28.978 
 
 4. 5.6 
 
 4.18 
 
 68.075 
 
 .28 
 
 .387 
 
 .005 
 
 .5 
 
 5.375 
 
 26.93 
 
 5.67 
 
 24.698 
 
 18.758 
 
 8.754 
 
 10.555 97.113 
 
 Write the numbers so that the decimal points stand in a 
 column. Add as in whole numbers, and place the point in 
 the sum directly under the points in the addends. 
 
Decimals. Y] 
 
 51. Written Exercises. 
 
 1. .027 + 1.39 + 48.6 + 72.978 ' 
 
 2. 234.96 + .675 + 50.4 + 6.02 + 1.001 
 
 3. 3.047 + 54.79 + .097 + .76 + .862 
 
 4. .8 + .38 + .479 + 27.87 + 375 
 
 5. .445 + 34.75 + 306.973 + .004 + 48.56 
 
 6. .81 + 12.654 + 234.79 + 8.6 + .603 + 42.96 
 
 7. 45.78 + .237 + 6.987 + 18 + 372.003 + 37.5 
 
 8. 4.745 + 36.58 + 725.894 + 9.87 + 75.357 + 86.74 
 
 9. 59.3 + 83 + 9.64 + 48.565 + 6.98 + 8.795 + 963.826 
 
 10. 13.387 + 72.563 + .7 + .603 + 7.245 + .483 + 9.25 
 
 11. 8.3 + 2.576 + 3.42 + 1.5 + 6.279 + .003 + 1.417 
 
 SUBTRACTION OF DECIMALS. 
 
 52. From 37 take 3.7. 
 
 37 may be written 37.0 
 subtract 3.7 
 
 33.3 Ans. 
 In practice, the pupil should not waste time in writing the unneces- 
 sary ciphers at the right of the decimals in the minuend. 
 
 182.01 1. 28.6 
 
 -4.624 -.009 -1.003 
 
 177.386 .991 27.597 
 
 Write the numbers so that the decimal point in the subtra- 
 hend stands directly under the decimal point in the minuend. 
 Subtract as in whole numbers, and place the point in the remain- 
 der under the points above. 
 
 53. Written Exercises. 
 
 
 Find answers : 
 
 
 1. 1-.057 3. 6-3.324 
 
 5. 3-1.568 
 
 2. 1-.245 4. 4-2.491 
 
 6. 7-4.736 
 
28 Chapter One. 
 
 7. 3.587-1.34 14. 681.38-94.572 
 
 8. 91.352-72.456 15. 1000-465.874 
 
 9. 42.007-17.658 16. 30.053-18.7 
 
 10. 68.217-39.4 17. 2568.91-1925 
 
 11. 9.34-5.672 18. 1.234 - .825 
 
 12. 45.268-23.068 19. 473.5-298.572 
 
 13. 219.843-187.95 20. 57.083-44.95 
 
 MULTIPLICATION OF A DECIMAL BY AN INTEGER. 
 54. Three times 3 tenths equals how many tenths ? 
 .3x3 = what? .3x4 = ? .3x12 = ? 
 
 1. Multiply 2.7 by 8. 
 
 8 times 7 tenths = 56 tenths = 5.6. Write .6. 8 times \ 
 
 2 = 16;carry5 - Mb. 21S 
 
 2. Multiply .275 by 12. 275 
 The product of 275 thousandths by 12 is 3300 thousandths, X 12 
 
 which equals 3 and 300 thousandths, or 3 and 3 tenths. 3.300 
 
 Ans. 3.3 
 
 Multiply as in whole numbers, and point off in the product 
 decimal places equal to the number in the multiplicand, reject- 
 ing unnecessary ciphers at the right of the decimal. 
 
 55. "Written Exercises. 
 
 
 
 Multiply : 
 
 
 
 1. .36 x 3 
 
 6. 
 
 .048 x 375 
 
 2. 57.2x7 
 
 7. 
 
 12.67 x 300 
 
 3. 6.4 x 122 
 
 8. 
 
 6.57 x 9 
 
 4. .67 x 4 
 
 9. 
 
 8.76 x 43 
 
 5. 38.4x25 
 
 10. 
 
 005 x 360 
 
Decimals. 29 
 
 56. Oral Exercises. 
 Give products : 
 
 1. 6.84x10 6. .961x100 
 
 2. 68.4x10 7. .57x1000 
 
 3. 3.28x10 8. .09x1000 
 
 4. 5.71x100 9. .026x100 
 
 5. 5.71x1000 * 10. 5.17x10 
 
 Note. — The pupil should deduce the rule for multiplying a decimal 
 by 10, 100, 1000. 
 
 57. To multiply an integer by a decimal. 
 Multiply 35 by 6.4. 
 
 35 6.4 
 
 "• 4 Since the product of 35 by 6.4 is equal to the ^ 5 
 
 14.0 product of 6.4 by 35, there will be one decimal 32.0 
 
 210 place in the product. 192 
 
 224.0 Ana. 224. 224.0 
 
 In multiplying an integer by a decimal, or a decimal by 
 an integer, point off in the product as many decimal places 
 as there are decimal places in the multiplier or the mul- 
 tiplicand. 
 
 58. Multiply: 
 
 1. 122 by 6.4 6. 5430 by .8 
 
 2. 512 by .003 7. 748 by .97 
 
 3. .056 by 987 8. 964 by .347 
 
 4. 97 by .005 9. 570 by .11 
 
 5. 275 by 1.2 10. 570 by 1.1 
 
30 Chapter One. 
 
 DIVISION OF A DECIMAL BY AN INTEGER. 
 59. Preliminary Exercises. 
 
 1. 8.64 
 
 4-2 
 
 
 6. .6664-6 
 
 2. 48.24 
 
 -=-4 
 
 
 7. .048-8 
 
 3. .465 
 
 -4 
 
 
 8. .814-9 
 
 4. 8.40 
 
 4-5 
 
 
 9. .12 4-5 
 
 . 5. 8.4 
 
 4-5 
 
 
 10. .34 4-4 
 
 60. Where it is necessary, ciphers may be annexed to the right of 
 the decimal in the dividend. 
 
 1. 8) .12 
 .015 
 
 2. 
 
 15).06 
 .004 
 
 1.875 
 5. 64)120. 
 64 
 
 .012 
 
 
 .413 
 
 56.0 
 
 3. 125)1.50 
 1.25 
 
 4. 
 
 21)8.673 
 8.4 
 
 51.2 
 4.80 
 
 .250 
 
 
 .27 
 
 4.48 
 
 .250 
 
 
 .21 
 
 .320 
 
 
 
 .063 
 
 .320 
 
 .063 
 In dividing a decimal by an integer, point off in the quotient 
 as many decimal places as there are decimal places in the 
 dividend (including the ciphers annexed). 
 
 Note. — In practice, however, the decimal point may be placed in 
 the quotient under (or over) the decimal point in the dividend. 
 
 61. "Written Exercises. 
 Divide : 
 
 1. 25 )1.00 6. 11 )70.07 
 
 2. 4 )21.80 7. 24)36J5 
 
 3. 8]^ 8. 18p76 
 
 4. 1 3)3.913 9. 25)TET 
 
 5. 1 2)48.12 10. 32)62.000 
 
Decimals. 
 
 
 
 62. Perform the indicated divisions : 
 
 
 A = ' + 25 25)1.00 
 
 
 1. i = 
 
 6. 
 
 TTS" — 
 
 2. i = 
 
 7. 
 
 w- 
 
 3. | = 
 
 8. 
 
 w- 
 
 4- A = 
 
 9. 
 
 H^= 
 
 5. | = 
 
 10. 
 
 A = 
 
 63. Give quotients at sight : 
 
 
 
 1. 932-100 
 
 6. 
 
 684-100 
 
 2. 86-1000 
 
 7. 
 
 57.6-*- 10 
 
 3. 328-10 
 
 8. 
 
 24.3-100 
 
 4. 9^1000 
 
 9. 
 
 8.75-*- 10 
 
 5. 48-1000 
 
 10. 
 
 932.5-^100 
 
 31 
 
 Note. — The pupil should deduce the rule for dividing by 10, 100, 
 1000. 
 
 64. Written Problems. 
 
 1. A man had 10.5 yards of cloth, and used 4.125 yards 
 to make a coat. How many yards did he have left ? 
 
 2. Find the cost of 2.578 acres of land, at $ 37 an acre. 
 
 3. Find the amount of .87 and 8.7. Find the difference 
 between .906 and 90.6. 
 
 4. Write in figures : Seventy-six thousand four hundred 
 nine, and eighty-two thousandths. Nine hundred thousand 
 nine hundred units, and thirty-one hundredths. 
 
 5. A franc is 19.3 cents. Find the cost in United States 
 money of goods amounting to 1250 francs. 
 
 6. A merchant bought 1800 meters of silk. How many 
 yards did he buy, a meter being 39.37 inches ? 
 
3 2 Chapter One. 
 
 7. A kilogram is 2.2046 pounds. What is the difference 
 in weight between the English ton of 2240 pounds and a 
 French ton of 1000 kilograms ? 
 
 8. A cubic foot of water weighs 1000 ounces. How many- 
 pounds does a cubic foot of gold weigh, gold being 19.4 
 times as heavy as water ? 
 
 9. There are 128 cubic feet in a cord. How many tons 
 of 2000 pounds are there in a cord of pine wood, the latter 
 being .66 times as heavy as water? 
 
 10. A man buys three plots of ground containing 35.27, 
 17.8, and 40.375 acres, respectively. Find the total cost at 
 $36 per acre. 
 
 11. How many pints are there in 2.375 gallons ? 
 
 12. What decimal of a peck is a quart ? 
 
 13. What will be the cost of carrying 468 tons of coal at 
 $0,125 per ton? 
 
 14. A farmer sold one-eighth of his farm of 224.2 acres 
 at $ 62.50 per acre. How much did he receive for it ? 
 
 UNITED STATES MONEY. 
 65. Learn the following table : 
 
 10 mills = 1 cent. 
 100 cents = 1 dollar. 
 
 1 dime = 10 cents. 
 1 eagle = 10 dollars. 
 
 ADDITION AND SUBTRACTION OF UNITED STATES MONEY. 
 
 66. Add the following without placing the amounts in 
 columns : 
 
 1. $8.34, $40.39, $638.27, $594.38, $1.97. 
 
 2. $0.03, $8.05, $600.00, $38.72, $198.52, $0.63. 
 
United States Money. 23 
 
 3. $432.84, $96.25, $3.64, $782.46, $800.06, $6.50. 
 
 4. $3.60, $40.05, $91.86, $350.48, $84.00, $287.63. 
 
 5. $98.27, $0.60, $600.39, $8.09, $37.38, $503.07. 
 
 6. $202.97, $42.23, $453.60, $7.18, $63.54, $0.37. 
 
 7. $8.43, $0.54, $2.57, $85.13, $425.31, $8.27. 
 
 8. $486.54, $84.62, $1.96, $8.13, $35.84, $236.49. 
 
 9. $83.61, $523.00, $23.04, $0.86, $35.82, $584.60. 
 10. $34.80, $93.54, $200.41, $324.86, $50.14, $8.75. 
 
 The foregoing examples may be added directly from this book 
 or from the blackboard, the pupils writing on their slates or papers 
 nothing but the answers. 
 
 67. Subtract the following without rearranging them. 
 Find the sum of the minuends, the sum of the subtrahends, 
 and the sum of the remainders. 
 
 1. $1,000.00- $876.49 = 
 
 2. $549.37- $99.89 = 
 
 3. $345.93- $76.04 = 
 
 4. $1786.08- $1097.19 = 
 
 5. $345.00- $187.23 = 
 
 6. ? — T = 
 
 7. $3545.37- $966.38 = 
 
 8. $82.46- $7.59 = 
 
 9. $5074.02- $4987.63 = 
 
 10. $77.84- $9.88 = 
 
 11. $4680.35- $4679.46 = 
 
 12. ? - 
 
34 Chapter One. 
 
 MULTIPLICATION OF UNITED STATES MONEY. 
 
 68. Find the cost of: 
 
 1. 197 barrels of flour, at $ 5.66 per barrel. 
 
 2. 486 bushels of wheat, at $ 1.04 per bushel 
 
 3. 237 tons of plaster, at $ 6.72 per ton. 
 
 4. 809 tons of hay, at $ 11.45 per ton. 
 
 5. 74 carloads of bran, at $ 20.62^ per load. 
 
 6. 208 sheep, at $ 4.65 per head. 
 
 7. 673 barrels of mackerel, at $ 10.60 per barrel. 
 
 8. 984 bushels of onions, at $ 1.09 per bushel. 
 
 9. 99 pounds of butter, at 24 cents per pound. 
 
 10. 208 pounds of coffee, at 28 cents per pound. 
 
 69. What will be the cost of 157 pounds of sugar, at 5p 
 per pound. 
 
 At 5^ per pound 157 pounds will cost 157 times 5^. In 157 
 
 practice, however, we multiply 157 by the smaller number 5. 5 
 
 Ans. $7.85. jgg 
 
 11. 1376 yards of muslin, at 6f £ 
 
 12. 2084 bushels of corn, at 47^. 
 
 13. 1864 pounds of beef, at 5|£ 
 
 14. 988 pounds of turkeys, at 13|£ 
 
 15. 296 bushels of potatoes, at 47|£ 
 
 16. 1272 pounds of dried apples, at 2f# 
 
 17. 488 pounds of lard, at 10J f. 
 
 18. 2240 pounds of sugar, at 4|^. 
 
 19. 5176 pounds of wool, at 30|y. 
 
 20. 4892 bushels of wheat, at 99f £ 
 
United States Money. 35 
 
 DIVISION OF UNITED STATES MONEY. 
 
 70. Oral Exercises. 
 
 How often is 1 quart contained in 1 gallon ? 1 pint in 
 1 quart ? 2 quarts in 1 gallon ? 1 inch in 1 foot ? 2 inches 
 in 1 foot ? 3 inches in 1 foot ? 4 inches in 1 foot ? 6 inches 
 in 1 foot ? 6 inches in 2 feet ? 8 inches in 2 feet ? 1 ounce 
 in 1 pound ? 1 ounce in 2 pounds ? 4 ounces in 2 pounds ? 
 1 fourth in 1 half ? 1 third in 1 ? 
 
 How often is 1 cent contained in $ 1 ? 2 cents in a dollar ? 
 4 cents in 2 dollars ? 25 cents in 25 dollars ? 
 
 $25 = 2500j* ; 2500/* -4- 25^ = 100, Ans. 
 
 Note. — When the divisor is a concrete number, i.e. a number asso- 
 ciated with objects, the dividend must be a like concrete number ; in 
 which case the quotient will be an abstract number, i.e. a mere number. 
 
 3 dollars, 4 coats, 7 apples, are concrete numbers ; 3, 4, 7, are ab- 
 stract numbers. 
 
 When the divisor is abstract and the dividend concrete, the quotient 
 is concrete. 
 
 71. Give answers at sight: 
 
 1. $4-10^ 11. $1-*-^ 
 
 8. 95-f- 5l 12. $3-=-$ J 
 
 3. $12-*- 4^ 13. $84-5-50^ 
 
 4. $86-+- 6^ 14. $l-=-16^ 
 
 5. $63 -j- 3? 15. $16 -=-16^ 
 
 6. $7-f-25^ 16. $16-*-16# 
 
 7. $20-33^ 17. $16 -=-33^ 
 
 8. $36-=- 3^ 18. $16-=-25^ 
 
 9. $40-=-50j* 19. $16-50^ 
 10. $9-f-10^ 20. $12-f-20^ 
 
36 Chapter One. 
 
 72. At 36 cents each, how many spellers can be bought 
 
 for $27? 
 
 75 
 
 $ 27 = 2700 cents. Since 1 speller costs 36 cents, the Qayrmf) 
 
 number of spellers that can be bought for 2700 cents will be oro 
 
 2700 - 36 = 75. Ana. 75 spellers. ^-r 
 
 180 
 
 -IQf) 
 
 73. Written Problems. — 
 
 1. At $ 2.75 per day, how long will it take a man to earn 
 $ 110? (11,000-*- 275.) 
 
 2. How many yards of muslin, at 12 cents per yard, can 
 be bought for $ 126? 
 
 .3. A farmer spent $ 140 for sheep at $ 5.60 each. How 
 many did he buy ? 
 
 4. A grocer pays $ 74.50 for tea at % of a dollar per 
 pound. What is the weight of the tea ? 
 
 5. When rye is worth 87 cents per bushel, how many 
 bushels can be purchased for $ 261 ? 
 
 6. At 12} cents per pound, how many pounds of meat 
 will cost $ 175.25 ? 
 
 7. If 75 spellers cost $ 27, what is the price of 1 speller? 
 
 If 75 spellers cost $ 27, 1 speller will cost ^ of $ 27. 
 
 75 )$ 27.00 
 
 The divisor 75 is an abstract number. The dividend being a con- 
 crete number, the quotient will be concrete, viz. $ .36. 
 
 8. A woman paid $ 24 for 36 yards of dress goods. What 
 did she pay per yard ? 
 
 9. At 6 for a dollar, how many rabbits can be bought for 
 
 $87? 
 
 10. The cost of 13 houses was $36,887.50. What was 
 the price of each ? 
 
United States Money. 37 
 
 FRACTIONAL PARTS OF A DOLLAR. 
 
 SHORT METHODS. 
 
 74. What will be the cost of 16 base-balls at 25 cents 
 each? 
 
 At $1 each, 16 base-balls cost 16 quarter-dollars, or $4. 
 
 75. Oral Exercises. 
 
 At 25 cents per pound, yard, dozen, etc., what will be paid 
 for: 
 
 1. 32 base-balls ? 7. 37 dozen lemons ? 
 
 2. 52 pounds coffee ? 8. 25 bushels tomatoes ? 
 
 3. 48 straw hats ? 9. 41 panes of glass ? 
 
 4. 84 yards ribbon ? 10. 33 packages of candy ? 
 
 5. 36 second readers ? 11. 49 Koman candles ? 
 
 6. 56 vases ? 12. 60 bars of soap ? 
 
 76. At 50 cents, give the cost of: 
 
 1. 46 pounds tea. 7. 76 grammars. 
 
 2. 28 pairs of scissors. 8. 57 boxes of pens. 
 3.. 38 penknives. 9. 49 picture books. 
 
 4. 84 third readers. 10. 83 dolls. 
 
 5. 44 pounds candy. 11. 27 games. 
 
 6. 32 caps. 12. 75 feather dusters. 
 
 77. How many cents in one-eighth of a dollar ? 
 
 At one-eighth of a dollar each, what will be the cost of 
 24 bars soap ? 
 
 At $1 each, 24 bars cost $ 2 /, or $3. 
 
38 Chapter One. 
 
 Give the cost of the following at 12£ cents per pound, etc. 
 
 (ft): 
 
 1. 16 pounds meat. 5. 80 jars of jelly. 
 
 2. 48 dozen eggs. 6. 96 cans of condensed milk. 
 
 3. 72 straw hats. 7. 104 yards sheeting. 
 
 4. 64 gallons oil. 8. 88 pounds currants. 
 
 78. How many cents in one-third of a dollar ? 
 
 At one-third of a dollar each, what will be the cost of 12 
 bottles of cologne ? 
 
 At $ l each, 12 bottles cost $ >/, r $4. 
 
 Give the cost, at 33^ cents per yard, pound, etc., of: 
 1. 36 yards of ribbon. 4. 27 bushels of oats. 
 
 £. 63 pairs of cuffs. 5. 54 pecks of walnuts. 
 
 3. 48 pounds of butter. 6. 72 dozen oranges. 
 
 79. How many cents in three-fourths of a dollar ? 
 If sleds cost $ | each, what is paid for 16 sleds ? 
 
 At $£ each, 16 sleds would cost $^, or $4 ; at $f each, the cost 
 is 3 times $4, or $12. 
 
 Give the cost of the following at 75 cents per yard, etc. : 
 
 1. 48 yards silk. 4. 28 gallons syrup. 
 
 2. 24 bushels peaches. 5. 36 base-balls. 
 
 3. 84 pounds tea. 6. 32 concert tickets. 
 
 80. Find the cost of 13 pairs of gloves at 75 cents per 
 pair. 
 
 Since 13 is not exactly divisible by 4, this problem should be handled 
 as follows : 
 
 13 pairs at $£ per pair cost $-^, or $0}, or $9.75. 
 
 Give the cost of the following at 75 cents per bushel, etc. i 
 
 1. 11 bushels rye. 4. 7 mats. 
 
 2. 15 gallons ice-cream. 5. 21 bushels potatoes. 
 
 3. 9 cloth caps. 6. 18 pairs of skates. 
 
United States Money. 
 
 39 
 
 81. Parts of a Dollar. 
 
 6£ cents = & of $ 1 
 
 8£ cents = ^ of $ 1 
 12^ cents = } of $1 
 16| cents = i of $1 
 25 cents = \ of $1 
 33$ cents = i of $1 
 
 82. Oral Exercises. 
 
 Give the cost of 72 articles at 
 
 1. 12 J cents each. 
 
 2. 33^ cents each. 
 
 3. 16| cents each. 
 
 37£ cents = | of $ 1 
 50 cents -\ of $1 
 62£ cents = £ of $1 
 66f cents = $ of $1 
 75 cents = £ of $ 1 
 87£ cents = | of $1 
 
 4. 25 cents each. 
 
 5. 50 cents each. 
 
 6. 37 J cents each. 
 
 37£ cents = $ f . At $ \ each, the cost of 72 articles would be $ 9 ; 
 at$|, $27. 
 
 7. 62-J- cents each. 
 
 8. 87 J cents each. 
 
 83. Multiply: 
 
 1. 6J cents x 16 
 
 2. 8| cents x 24 
 
 3. 12i cents x 88 
 
 4. 16| cents x 54 
 
 5. 25 cents x 240 
 
 6. 33J cents x 66 
 
 7. 50 cents x 186 
 
 8. 37J cents x 48 
 
 9. 62^ cents x 32 
 
 9. 66| cents each. 
 10. 75 cents each. 
 
 10. 66| cents x 33 
 
 11. 75 cents x 128 
 
 12. 87£ cents x 88 
 
 13. $1.33£x 24 
 
 14. 9*112} X 16 
 
 15. $2.25 x 12 
 
 16. $3.75 x 12 
 
 17. $4.37£x 8 
 
 18. 95.16} x 6 
 
40 Chapter One. 
 
 84. Find the cost of: 
 
 1. 86 neckties, at 50 cents each. 
 
 2. Six dozen handkerchiefs, at 25 cents apiece. 
 
 3. 32 yards of silk, at $1.12J per yard ($4). 
 
 4. 64 arithmetics, at 75 cents each. 
 
 5. 84 geographies, at $1.25 each (f 1J). 
 
 6. 96 pounds of tea, at 75 cents a pound. 
 
 7. 84 pairs of gloves, at $ 1.50 per pair. 
 
 8. 72 yards of cloth, at $ 2.12^ per yard. 
 
 85. Written Exercises. 
 
 Note. — Pupils should be taught to perform operations without 
 placing the numbers under each other. In working examples 1 to 8, 
 one figure is written at a time, beginning at the right. The answers 
 to examples 9 to 12 are found by division, one figure being written at 
 a time. In examples 13 to 20, the cents should be changed to fractions 
 of a dollar. 
 
 Write answers. 
 
 1. 687 pounds, at 4^. 
 
 2. 976 yards, at 6^. 
 
 3. 938 coats, at $ 7. 
 
 4. 695 pounds, at 20£ 
 
 5. 12 bushels, at $ 1.43. 
 
 6. 11 sheep, at $ 7.47. 
 
 7. 9 tons, at $22.75. 
 
 8. 13 sacks of salt, at $ 1.11. 
 
 9. 352 yards, at 12^. 
 10. 1728 hats, at 50£ 
 
 11. 933 yards, at 33^. 
 
 12. 2504 dolls, at 25^ 
 
 13. 248 pounds, at 75^. 
 
 14. 186 pounds, at 66%#. 
 
 15. 8 barrels, at $ 16.37£. 
 
 16. 16 gallons, at $3.62£. 
 
 17. 124 bushels, at $ 1.50. 
 
 18. 96 pounds, at $1.25. 
 
 19. 120 gallons at $ 2.33f 
 
 20. 64 sacks, at $1.12 J. 
 
United States Money. 41 
 
 86. Oral Exercises. 
 
 At 50 cents each, how many penknives can be bought for 
 fl? For $2? For $3? For $10? For $ 20 ? 
 
 At 25 cents each, how many readers can be bought for 
 $1? For $2? For $3? For $10? For $20? 
 
 At 12£ cents per yard, how many yards can be bought for 
 $1? For $2? For $3? For $10? For $20? 
 
 At 33J cents per pound, how many pounds can be bought 
 £or$l? For $2? For $3? For $10? For $20? 
 
 87. At 25 cents each (four for $ 1) : 
 
 1. How many base-balls can be bought for $9 ? 
 
 2. Straw hats, for $12? 
 
 3. Koman candles, for $18? 
 
 4. Readers, for $ 15 ? 
 
 5. Vases, for $21? 
 
 6. Bars of soap, for $3J? 
 
 7. Packages of candy, for $4£? 
 
 8. Yards of ribbon, for $5.75 ? 
 
 9. Bushels of tomatoes, for $10.50? 
 
 10. Pounds of coffee, for $12.75 ? 
 
 88. At 50 cents (two for $1) : 
 
 11. Pounds of tea, for $ 43 ? 
 
 12. Penknives, for $20.50 ? 
 
 13. Pounds of candy, for $94 ? 
 
 14. Third readers, for $ 17.50 ? 
 
 15. Caps, for $ 21 ? 
 
 16. Grammars, for $37? 
 
42 Chapter One. 
 
 17. Boxes of pens, for $72? 
 
 18. Dolls, for $64? 
 
 19. Pairs of scissors, for $ 19 ? 
 
 20. Feather dusters, for $26.50? 
 
 89. At 121 cents (eight for $ 1) : 
 
 21. Gallons of oil, for $8? 
 
 22. Dozen of eggs, for $ 11 ? 
 
 23. Pounds of meat, for $21 ? 
 
 24. Quarts of plums, for $ 1 J- ? 
 
 25. Jars of jelly, for $f ? 
 
 26. Yards of sheeting, for $1J? 
 
 27. Cans of milk, for $2J? 
 
 28. Pounds of currants, for $3.12£? 
 
 29. Whisk brooms, for $4,371? 
 
 30. Collars, for $5.62J? 
 
 90. At 331 cents (three for $1) : 
 
 31. Yards of ribbon, for $ 6 ? 
 
 32. Pairs of cuffs, for $ 12 ? 
 
 33. Pounds of butter, for $ 18 ? 
 84. Bushels of oats, for $32? 
 
 35. Pecks of walnuts, for $1J ? 
 
 36. Dozen of oranges, for $ 1$ ? 
 
 37. Straw hats, for $2.33£? 
 
 38. Dolls, for $3.66f? 
 
 39. Penknives, for $4.33 J? 
 
 40. Pounds of candy, for $5.66}? 
 
Denominate Numbers. 
 
 43 
 
 91. At 16f cents (six for $ 1) : 
 
 41. Collars, for $4? 
 
 42. Pounds, for $ 21 ? 
 
 43. Yards, for $ J? 
 
 44. Ounces, for f± ? 
 
 45. Packages, for 66 1 cents ? 
 
 92. Oral Exercises. 
 Divide at sight : 
 
 51. $ 24.50 by 50 cents. 
 
 52. 9 12.25 by 25 cents. 
 
 53. $ 26 by 33J cents. 
 
 54. 9 14.50 by 121 cents. 
 
 55. $ 17 by 16| cents. 
 
 46. Quarts, for f 1.16}? 
 
 47. Gallons, for $1.50? 
 
 48. Pecks, for f 2§T 
 
 49. Feet, for $ 3.33^ ? 
 
 50. Yards, for $ 4.66| ? 
 
 56. $ 18.75 by 25 cents. 
 
 57. $ 11.87$- by 12£ cents. 
 
 58. $ 13.33J- by 33J cents. 
 
 59. $ 37.50 by 50 cents. 
 
 60. $ 13.33 1 by 16f cents. 
 
 DENOMINATE NUMBERS. 
 93. Learn the following tables : 
 
 TIME. 
 
 60 seconds (sec.) = 1 minute (min.) 
 
 60 minutes = 1 hour (hr.) 
 
 24 hours = 1 day (da.) 
 
 7 days = 1 week (wk.) 
 
 AVOIRDUPOIS WEIGHT. 
 
 16 ounces (oz.) = 1 pound (lb.) 
 
 2000 pounds = 1 ton (T.) 
 
 The hundredweight (100 pounds) is written cwt. 
 
 DRY MEASURE. 
 
 2 pints (pt.) 
 8 quarts 
 4 pecks 
 
 = 1 quart (qt.) 
 = 1 peck (pk.) 
 = 1 bushel (bu.) 
 
44 Chapter One. 
 
 LIQUID MEASURE. 
 
 2 pints (pt.) = 1 quart (qt.) 
 
 4 quarts = 1 gallon (gal.) 
 
 A gill (gi.) is equal to one- fourth of a pint. It is very rarely used. 
 
 LINEAR MEASURE. 
 
 12 inches (in.) = 1 foot (ft.) 
 
 3 feet = 1 yard (yd.) 
 5 £ yards = 1 rod (rd.) 
 
 320 rods = 1 mile (mi.) 
 
 1 mi. = 320 rd. = 1760 yd. = 5280 ft. = 63,360 in. 
 
 A furlong is equal to 40 rods, } mile. 
 
 A hand, used in measuring the height of horses, = 4 in. A knot, 
 used in measuring distances at sea, = 1.15 mi. A fathom, used in 
 measuring the depth of the sea, = 6 ft. 
 
 SQUARE MEASURE. 
 
 144 square inches (sq. in.) = 1 square foot (sq. ft.) 
 
 9 square feet = 1 square yard (sq. yd.) 
 
 30 \ square yards = 1 square rod (sq. rd.) 
 
 160 square rods = 1 acre (A.) 
 
 640 acres = 1 square mile (sq. mi.) 
 
 1 A. = 160 sq. rd. = 4840 sq. yd. = 43,560 sq. ft. 
 
 A section of land is a mile square. 
 
 Roofing, flooring, and slating are often estimated by the square, 
 which contains 100 square feet. 
 
 94. Written Exercises. 
 
 1. How many hours in 7 J- days ? 
 
 2. How many hours in 7 days 12 hours ? 
 
 3. How many minutes in 2 hours? How many seconds ? 
 
 4. A man buys 12 bushels and 3 pecks apples at $ 1 per 
 bushel. What is the cost ? 
 
Denominate Numbers. 45 
 
 5. What will be the cost of 3 pecks 7 quarts chestnuts 
 at 8 cents per quart ? 
 
 6. How many pints are there in 5 gallons of ice-cream ? 
 
 7. How many half-pints are there in 10 gallons of ice- 
 cream ? 
 
 8. How many 4-ounce packages can be made from 5 
 pounds of pepper? 
 
 9. A boy pays $ 1.50 for 1 gallon and 2 quarts of ice- 
 cream. What is the price per quart ? 
 
 10. How many gallons of lemonade will be needed to 
 give 96 people £ pint each? 
 
 11. How many seconds in 5 hours? 
 
 12. How many hours in 1 we^k ? 
 
 13. Change 13 hours and 20 minutes to minutes. 
 
 14. Change 15 bushels 4 pecks to pecks. 
 
 15. How many ounces in 47 pounds 5 ounces ? 
 
 16. How many pounds and ounces in 237 ounces ? 
 
 17. Change 1494 minutes to hours and minutes. 
 
 18. Find the number of hours in 6 weeks. 
 
 19. Change 60 pounds to the decimal of a hundredweight. 
 
 20. How many inches are there in 12 feet 2 inches ? 
 
 21. How many pounds in 14i tons ? 
 
 22. How many pounds in f of a ton ? 
 
 23. What will 400 pounds of coal cost at $5 per ton? 
 
 24. What decimal of a ton is 1500 pounds ? 
 
 25. How many days and hours in £ of a week ? 
 
 26. Find the number of yards in 3 pieces of cloth, each 
 containing 16 yards 2 feet. 
 
 27. When coal is $7.50 per ton, what will be the cost of 
 3000 pounds ? 
 
46 Chapter One. 
 
 MEASUREMENTS. 
 95. Preliminary Exercises. 
 
 One square inch. 
 
 Draw a square each side of which 
 is one inch. This is called a square 
 inch. Cut out of paper several one-inch 
 squares. 
 
 Draw a rectangle two inches long, one 
 inch wide. How many paper one-inch 
 squares will exactly cover it ? 
 
 Draw a rectangle three inches long, two inches wide. 
 Divide it into one-inch squares. How many one-inch 
 squares are there in the lower row? How many rows? 
 How many square inches in the rectangle ? 
 
 How many square inches in a rectangle 6 inches long, 3 
 inches wide ? 
 
 How many square inches in a rectangle 4 inches long, 
 4 inches wide ? 
 
 How many square inches are there in a rectangle 12 
 inches long, 3 inches wide ? In a rectangle 1 foot long, 
 3 inches wide ? In a rectangle 1 foot 1 inch long, 4 inches 
 wide? 
 
 Note. — The foregoing exercises should be accompanied by accu- 
 rate drawings on paper or on the blackboard, which should lead the 
 pupils to see that the unit in the given examples is the square inch. 
 They should be made aware that the number of squares in the lower 
 row corresponds to the length of the rectangle in inches; and that 
 the number of rows corresponds to the width of the rectangle. From 
 this they should deduce the rule : 
 
 The number of square inches in the surface of a rectangle is 
 equal to the number of inches in its length taken as many times 
 as there are inches in its width. 
 
 This product is called the area of the rectangle. 
 
Measurements. 47 
 
 96. The area of a surface is the number of times that it 
 contains another surface, taken as the unit of measurement. 
 Thus, the statement that the area of a surface is 8 square 
 inches means that a square inch is contained in the surface 
 8 times. 
 
 97. The sum of all the sides of a figure is called its 
 perimeter. 
 
 98. Written Exercises. 
 
 Find the area of each of the following rectangles in square 
 inches. Find the perimeter of each in feet and inches. 
 
 1. 13 in. by 14 in. 7. 13 in. by 42 in. 
 
 2. 17 in. by 9 in. 8. 27 in. by 31 in. 
 
 3. 18 in. by 7 in. 9. 18 in. by 22 in. 
 
 4. 23 in. by 15 in. 10. 64 in. by 29 in. 
 
 5. 21 in. by 19 in. 11. 1 ft. by 7 in. 
 
 6. 37 in. by 14 in. 12. 1 ft. 1 in. by 11 in. 
 Note. — Change each dimension to inches before multiplying. 
 
 13. 1 ft. 3 in. by 12 in. 17. 2 ft. 6 in. by 1 ft. 3 in. 
 
 14. 1 ft. by 1 ft. 18. 3 ft. 7 in. by 2 ft. 9 in. 
 
 15. 1 ft. 4 in. by 1 ft. 19. 4 ft. 11 in. by 1 ft. 8 in. 
 
 16. 2 ft. 6 in. by 1 ft. 20. 5 ft. 3 in. by 2 ft. 11 in. 
 
 99. -Oral Exercises. 
 
 How many square feet in a rectangle 2 feet long, 1 foot 
 wide? 
 
 How many square feet in a rectangle 6 feet long by 5 feet 
 wide? 
 
 How many square feet in a rectangle 9 feet long by 7 feet 
 wide ? 
 
 Note. — The unit in the following examples is the square foot. 
 
48 Chapter One. 
 
 100. "Written Exercises. 
 
 Find the area in square feet of each of the following rec- 
 tangles. Find the perimeter of each in feet. 
 
 1. 12 ft. by 14 ft. 6. 29 ft. by 12 ft. 
 
 2. 15 ft. by 17 ft. 7. 15J. ft. by 12 ft. 
 
 3. 19 ft. by 11 ft. 8. 15 ft. 6 in. by 12 ft. 
 
 4. 23 ft. by 15 ft. 9. 18| ft. by 16 ft. 
 
 5. 18 ft. by 16 ft. 10. 18 ft. 9 in. by 16 ft. 
 Note. — Change the inches to fractions of a foot. 
 
 11. 23J ft. by 18 ft. 16. 36 ft. by 23 ft. 5 in. 
 
 12. 24 ft. 8 in. by 18 ft. 17. 13 ft. by 24J ft. 
 
 13. 19 ft. 3 in. by 16 ft. 18. 13 ft. 4 in. by 24 ft. 
 
 14. 24 ft. by 17 ft. 9 in. 19. 26 ft. 8 in. by 15 ft. 
 
 15. 24 ft. by 16 ft. 1 in. 20. \2\ ft. by 12 ft. 
 
 101. Suggestive Examples. 
 
 1. Measure the top of the desk, disregarding fractions 
 of an inch, and calculate the surface in square inches. 
 
 2. Measure the blackboard, and find how many square 
 feet in its surface. (Do not include fractions of a foot.) 
 
 3. Calculate the number of square inches in a pane of 
 glass in the schoolroom window. 
 
 4. Find the number of square feet in the floor of the 
 classroom. 
 
 5. Find the number of square feet in the classroom 
 ceiling. 
 
 6. Estimate the height of the classroom, and calculate 
 the number of square feet in the front wall. 7. In the rear 
 wall. 8. In the right-hand wall. 9. In the left-hand wall. 
 
Measurements. 49 
 
 102. Written Problems. 
 
 Suggestion. — When the surface is required in square inches, 
 change each dimension to inches ; when required in square feet, ex- 
 press each dimension in feet, or in feet and the fraction of a foot ; 
 when required in square yards, etc. , express each dimension in yards, 
 etc. 
 
 1. How many square feet are there in the surface of a 
 field 125 feet long, 87.5 feet wide ? 
 
 (1 square foot x 125 x 87.5.) 
 
 2. A rug is 2 yards long, If yards wide. How many 
 square yards does it contain ? 
 
 (1 square yard x 2 x If.) 
 
 3. How many square yards are there in a strip of carpet 
 6 yards long, 27 inches (f yard) wide ? 
 
 4. Find the number of square meters in a room 12 meters 
 long, 9.75 meters wide. 
 
 5. At 50 cents per square yard, what will be the cost of 
 the oil-cloth needed to cover a floor 18 feet (6 yards) long, 
 15 feet (5 yards) wide ? 
 
 6. What will be the cost, at $ 1.50 per square yard, of 
 carpeting a room 6^- yards long, 15 feet wide ? 
 
 7. At 3 cents a square foot, how much must be paid for 
 10 boards, each 16 feet long, ^ foot wide ? 
 
 8. A field is 30 rods long and 24 rods wide. How many 
 square rods will it contain after a strip 24 rods long and 2 
 rods wide is taken from it for a road ? 
 
 9. How many square yards of plastering will be re- 
 quired for a ceiling 18 feet long, 15 feet wide ? 
 
 10. If a roll of wall paper is 24 feet long and 18 inches 
 wide, how many square yards does it contain ? 
 
So 
 
 103. 
 
 Mrs. M. O'Donnell. 
 
 Chapter One. 
 
 BILLS. 
 
 Chicago, July 31, 1904. 
 
 Bought of Seaver Brothers. 
 
 l\ yd. Plaid 
 16 yd. Cambric 
 12 pr. Socks 
 
 1 Wrapper 
 
 h yd. Silk 
 
 1 pr. Gloves 
 
 2 spools Silk 
 
 $1.00 
 .05 
 .25 
 
 .65 
 
 .08 
 
 08 
 
 1. Copy the above, filling in the cost of each item and the 
 total. 
 
 In these examples, the total cost of each item should be written 
 in its place without any side calculation. Pupils should be drilled in 
 short, direct methods of computation, being required to omit unneces- 
 sary figures. 
 
 In No. 2, for instance, 64 is multiplied by 5|, as follows: 
 | of 64 is 8 ; carry this to the product of 5 and 4, making 28 ; write 
 8. 6 times 6 are 30, add 2, making 32. Total, 328. 
 
 2. Otto Haas buys of Murphy & Cooper 64 pounds of 
 sugar @ 5J^ ; 28 pounds of lard @ 9\0; 24 pounds of coffee 
 @25^; 1 barrel flour @$ 5.75; and 12 gallons of molasses 
 @ 25^. Make out the bill. 
 
 3. Make out a bill for 10 pairs of men's shoes, at $ 4.75 ; 
 4 pairs of boys' shoes, at $ 1.47^ ; 6 pairs slippers, at $ .87^ ; 
 9 pairs of girls' shoes, at $2.43; 8 pairs of women's shoes, 
 at $3.37f s 
 
R 
 
 eview. 
 
 51 
 
 4. Make out a bill for 8-J- pounds of ham, at 14^ per pound ; 
 
 3 J pounds of beefsteak, at 20^ ; 9 pounds of corned beef, at 
 12^ ; 10 \ pounds of chicken, at 24^ ; 12 pounds of roast beef, 
 at 18£ 
 
 5. Make out a bill for 14 dozen collars, at $ 1.50 per 
 dozen; 6 dozen pairs of culfs, at $2.75 per dozen pairs; 
 
 4 dozen shirts, at $ 9 per dozen ; 3 dozen ties, at $ 2.25 per 
 dozen ; 17 dozen pairs of socks, at $ 2.10 per dozen pairs. 
 
 104. Eeview Exercises. Approximate Answers. 
 
 Note. — These drills are intended to lead a pupil to such an ex- 
 amination of his answers to other problems as will prevent him from 
 being satisfied with one that is very far astray. 
 
 It is not expected that every pupil will give exactly the same 
 answer. In No. 5, for instance, the cost of 99 yards is asked at $ 1.95 
 per yard. One pupil may consider 100 yards at $2, or $200 ; a sec- 
 ond may keep the rate at $1.95, and say $195 ; a third might come 
 still closer ; each of such answers, however, should be accepted as an 
 approximation. 
 
 1. What will be the cost of 39f pounds butter at 20^ 
 per pound ? 
 
 Nearly 40 pounds at 20f. The cost is nearly what ? Solve. 
 
 2. A man has 4200 pounds of flour which he wishes to 
 put into barrels containing 196 pounds each. About how 
 many barrels will he need ? 
 
 Each barrel contains nearly how many pounds ? Solve. 
 
 3. A merchant bought a hogshead of molasses, contain- 
 ing 47| gallons, at 50 cents per gallon. About how much 
 did it cost ? 
 
 4. How many lots at f 1975 each can be bought for 
 $12,000? 
 
 5. Sold 3 pieces of cloth, 33 yards to the piece, at $1.95 
 per yard. Give the approximate amount of the bill. 
 
§2 Chapter One. 
 
 6. 28|f + 371$ = nearly what ? 
 
 7. 175£-j- 24 T % = nearly what ? 
 • 8. 18| X 9J = nearly what ? 
 
 9. 87 T V - 49 J| = nearly what ? 
 10. 4£ x 4f x 4^ = nearly what ? 
 
 105. Oral Eeview Problems. 
 
 1. What will be the cost of 8 pounds of meat at 15 cents 
 per pound ? 
 
 2. Gave $ 1 in payment for a 25-cent ball, and four 10- 
 cent bats. How much change did I receive ? 
 
 3. At the rate of 3 oranges for 5 cents, what will be the 
 cost of a dozen oranges ? 
 
 4. A gross is 12 dozen. How many pens in -J- gross ? 
 
 5. How many inches in 4 yards ? 
 
 6. At 5 cents per pint, how much would be paid for a 
 bushel of chestnuts ? 
 
 7. A person used 2 gallons and 3 quarts of milk one 
 week, and 3 gallons and 1 quart the next week. How many 
 gallons are used in the two weeks ? 
 
 8. Multiply 15 by 5. Take 18 from the product. 
 
 9. How many 9's in 3 times 21 ? 
 
 10. 12 times 6 are how many times 8 ? 
 
 11. To 9 times 7 add 10. Take 15 from the sum. 
 
 12. One can has in it 4 gallons of milk, and another has 
 in it 6 quarts. How many pints are in both ? 
 
 13. 27 + 15 + 18 + 25 + 9 = ? 
 
 14. James had half a dollar to spend ; he bought 14 cents* 
 worth of candy, and spent the rest of his money for oranges 
 at 4 cents each. How many oranges did he buy ? 
 
Review. 53 
 
 15. A woman bought 7 pounds of rice at 12? a pound, 
 and paid for it with a dollar bill. How much money did 
 she receive in change? 
 
 16. A man paid one dollar for a bag of peanuts containing 
 3 pecks. He sold them at $ 0.10 a quart. How much did he 
 gain? 
 
 17. Book, 75?; pencil, 8^; slate, 15? = ? 
 
 18. 20 boxes of berries at 15? = ? 
 
 19. At 6 cents each, how many bananas for $1? How 
 many cents over ? 
 
 20. Bought 3 pounds of raisins worth 12 cents a pound; 
 2 dozen bananas at 25 cents a dozen. I gave the man a 
 dollar bill. How much did he give back? 
 
 21. How many hours are there in a week? 
 
 22. If John earned 16? Monday, 9? Tuesday, 20? 
 Wednesday, 15? Thursday, 8? Friday, and 12? Saturday, 
 how much did he earn in the whole week ? 
 
 23. What will 3 bushels of sand cost, at 4? a peck ? 
 
 24. Mrs. Hall divided 84 oranges equally among 14 girls. 
 How many oranges did each girl receive ? 
 
 25. If you give 24 cents for one thing, and 19 cents for 
 another, what will both things cost ? 
 
 26. If a quart of milk is worth 7?, what is the value of 
 two gallons ? 
 
 27. Find the cost of 60 oranges at 20 cents per dozen. 
 
 106. Written Eeview Problems. 
 
 1. A man walks 14£ miles in 4f hours. How many 
 miles an hour is that ? 
 
 2. If a milk can holds 23 quarts and 1 pint, how many 
 half-pints does it hold ? 
 
54 Chapter One. 
 
 3. Bought 87 pounds of tea at 45 cents a pound ; sold it 
 at 63 cents a pound. How much was gained ? 
 
 4. In a school there were 356 girls and 259 boys ; if 25 
 girls and 32 boys leave, how many pupils remain in the 
 school ? 
 
 5. Which are worth more, 63 cows at $ 38 apiece, or 56 
 horses at $ 75 apiece ? How much more ? 
 
 6. Suppose your mother gave you a 5-dollar bill to buy 
 articles for the Sunday dinner, and you bought 6 lb. of roast 
 beef at 25 cents a lb., 1 pk. spinach at 45 cents, 2 qt. of 
 onions at 12J cents, 1 doz. oranges at 12 cents, 2 qt. of milk 
 at 7 cents. How much change would you bring home to 
 your mother ? 
 
 7. If a railway mail clerk earns $ 800 in a year, how 
 much will he have left after paying his board at the rate of 
 1 16 a month ? 
 
 8. How many pieces of second-class matter (newspapers) 
 are there in 644 pounds, each piece weighing 8 ounces ? 
 
 9. The postmaster at Norwich made requisition for the 
 following postage stamps : 27 sheets of 1-cent, 97 sheets of 
 2-cent, 35 sheets of 5-cent, and 17 sheets of 10-cent stamps. 
 What was the money value of these stamps, there being 100 
 stamps in each sheet ? 
 
 10. The whole number of pieces of mail matter handled 
 at 212 post-offices was 2,164,517,880. What was the average 
 number of pieces for each office ? 
 
 11. A merchant pays $30 for 65 vases. He sells 17 of 
 them at 50 cents each, and receives 48 cents each for the 
 others. What is his profit? 
 
 12. One boy had 15 marbles, another had 19, a third had 
 17, a fourth had 13. What was the average number of 
 marbles for each boy ? 
 
Review. 55 
 
 13. A teacher divided 200 foreign postage stamps among 
 the eight boys of his class. He gave one-fourth of them to 
 the first boy, one-fifth of the remainder to the second boy, 
 and then divided the rest equally among the other six boys. 
 How many did each of the latter receive ? 
 
 14. If 23 buggies cost $4025, what are 80 buggies 
 worth ? 
 
 15. How many gills in 7 quarts and 1 pint ? 
 
 16. How many bushels in 384 quarts? 
 
 17. Change 864 pints to gallons. 
 
 18. A farmer exchanged 16 cows, worth $ 40 each, for a 
 span of horses. What are the horses worth apiece ? 
 
 19. A boy bought a bicycle for $35. He rented it to 
 another boy for 3 months at $ 2 a month, and then sold it 
 for $33.50. Did he gain or lose, and how much ? 
 
 20. John had 16 marbles, Henry half as many, and 
 Frank as many as both the other boys. How many more 
 marbles had Frank than John ? 
 
 21. How many quarts in 12 bushels ? 
 
 22. How many feet of string will be required to go 
 around a room 30 feet long and 25 feet wide ? 
 
 23. If I buy a bushel of walnuts for $3, and sell them at 
 5 cents a pint, how much do I make ? 
 
 24. Write 83, 47, 69, and 56 in Eoman numbers. 
 
 25. A man works 9 months, 26 days per month, and 
 receives $ 702. What are his daily wages ? 
 
 26. A merchant buys 136 vases for $272. If 36 are 
 broken, how much must he charge apiece for the others to 
 gain $28 on all? 
 
56 Chapter One. 
 
 27. On Monday, the receipts of a store are $180; on 
 Tuesday, $30 less ; on Wednesday, $ 110 less than the total 
 of Monday and Tuesday. What are the receipts for the 
 three days ? 
 
 28. The yearly rent of a house is $480. What is the 
 rent for 2 years 4 months ? 
 
 29. A mechanic works 300 days per year, at $ 4 per day. 
 If his daily expenses for 365 days average $ 3, how much 
 money does he save each year ? 
 
 30. A woman pays $5.20 for 3 pounds of tea and 56 
 pounds of sugar. What is the price per pound of the sugar, 
 if the tea costs 80^ per pound ? 
 
 31. A man had $7500. He paid -J of it for a house, 
 $ 575.60 for repairs, and $ 387.75 for furniture. How much 
 money had he left ? 
 
 32. How much hay will be required to feed 18 horses 
 a month of 30 days, if each horse receives 15 pounds. a day? 
 
 33. A person pays a debt of $576, giving 40 ten-dollar 
 bills, 30 two-dollar bills, 6 one-dollar bills, and the re- 
 mainder in five-dollar bills. How many of the last did 
 he give ? 
 
 34. A drover buys 64 sheep for $400. He sells -$- of 
 them at $7 each, and the remainder at $8 each. What 
 is his profit? 
 
 35. A merchant sells 56 yards of cloth for $ 84, gaining 
 $ 14. What did it cost him per yard ? 
 
 36. A package of coffee, costing 60 cents, was sold for 75 
 cents, the profit on each pound being 5 cents. What was 
 the selling price per pound ? 
 
 37. How many yards of cloth, at $1.75 per yard, can be 
 bought for $105? 
 
Review. 57 
 
 38. A tailor buys a piece of cloth for $50. From it 
 he makes 4 pairs of trousers, which he sells at $ 7 per pair, 
 and 4 coats, for each of which he receives $15. Thread, 
 buttons, lining, etc., cost him $ 16. How much does he get 
 for his labor ? 
 
 39. A man sold a certain number of papers for 50 cents. 
 If he had sold nine more, he would have received 95 cents. 
 How many papers did he sell ? 
 
 40. How long is a post which is 5|- feet above water, 
 one-half of its length in the water, and one-fourth of its 
 length in the mud ? (Make a diagram.) 
 
 41. Eight pounds of black tea costing 35^ per pound are 
 mixed with twelve pounds of green tea costing 50^ per 
 pound. What is the cost of 20 pounds of the mixed tea ? 
 
 42. How many bushels and pecks are there in 1442 
 pounds of corn weighing 56 pounds per bushel? 
 
 43. How is division proved ? 
 
 44. Multiply by 208 the quotient of (169,668 -j- 36). 
 
 45. Add seventy-two dollars, eleven cents; fifteen. dollars, 
 nine cents; eighty-seven cents; three hundred fifty dollars; 
 and one dollar, four cents. 
 
 46. Which is greater and how much ? 
 
 486 x 29 or 26,845 - 19,976. 
 
 47. Write in Eoman numerals 1905, 1775, and 560. 
 
 48. If a railway mail clerk spends ten cents a day for 
 street-car fare, how much will he spend in six months of 30 
 days each? 
 
 49. Add nine thousand eleven, seventy thousand forty- 
 four, five hundred thousand four hundred ten, fifty-four 
 thousand twenty-one. 
 
 50. Multiply $40.25 by 96. 
 
58 Chapter One. 
 
 51. From $300,000 take $7050.75. 
 
 52. How many days will 36 bushels of oats last 12 
 horses, if each horse eats 12 quarts a day ? 
 
 53. If a barrel of flour is worth $4.50, how many barrels 
 can be bought for $441? How much will all the flour 
 weigh if each barrel holds 196 pounds? 
 
 54. Suppose your slate is 12 inches long, 9 inches wide, 
 and 15 inches across diagonally. How long a string is 
 needed to go around the outside and along the diagonal ? 
 Make a diagram to explain your work. 
 
 55. The total cost of the Union Pacific railroad, which 
 is 1819 miles long, was $157,092,478. What was the aver- 
 age cost per mile ? 
 
 56. An officer who was paid $3.50 a day stayed in the 
 service until he had earned $ 143.50. How many days had 
 he worked? 
 
 57. A cargo of potatoes was discharged in tubs contain- 
 ing 250 pounds each, which were filled 1785 times. A 
 bushel of potatoes weighs 60 pounds. How many bushels 
 were landed ? 
 
 58. How long will it take 50 clerks to count $1,500,000 
 in silver coin, one-half of which is in half-dollars and the 
 other half in quarter-dollars, each clerk counting at the rate 
 of fifty pieces a minute ? Express the answer in hours. 
 
 59. Write in figures one million one thousand one dollars 
 and one cent. 
 
 60. Multiply 657,934 by 3209. 
 
 61. The War Department expended $ 1765.25 for muci- 
 lage at $5.75 a dozen quarts. How many quarts were pur- 
 chased ? 
 
CHAPTEE II. 
 
 PAGES 
 
 Fractions 69 to 84 
 
 Greatest Common Divisor, Least Common Multiple, 
 Addition and Subtraction of Fractions, Cancellation, 
 Ratio, Multiplication and Division of Fractions. 
 
 Decimals 84 to 91 
 
 Multiplication of Decimals, Division of Decimals. 
 
 United States Money 92 to 93 
 
 Fractional Farts of a Dollar. 
 
 Denominate Numbers 94 to 99 
 
 Reduction, Addition, Subtraction, Multiplication, and 
 Division. 
 
 Measurements 99 to 101 
 
 Areas and Surfaces. 
 
 Bills . . . 101 to 102 
 
 Review of Simple Numbers 102 to 118 
 
 Short Methods, Sight Exercises, Sight Approximations, 
 Review Problems. 
 
 FACTORS AND MULTIPLES. 
 
 107. The factors of a number are the integers whose 
 product makes the number. 
 
 Note. — An integer is any whole number. 
 
 2 and 3 are factors of 6. 
 2, 3, and 5 are factors of 30. 
 
 108. A number that contains another number an exact 
 number of times is a multiple of that number. 
 
 24 is a multiple of 12; 36, 48, etc., are also multiples of 12. 
 30 is a multiple of 2, 3, 5, 6, 10, 15. 
 
 69 
 
60 Chapter Two. 
 
 109. Preliminary Exercises. 
 
 1. 95 is a multiple of what two numbers ? 
 
 2. Give the two factors of 51. 
 
 3. What number is a multiple of both 8 and 6 ? 
 
 4. Mention another number that is a multiple of both 8 
 and 6. 
 
 5. Find the smallest number that can be exactly divided 
 by 8 and 12. 
 
 6. Give the two factors of 91. 
 
 7. 57 is a multiple of what two numbers ? 
 
 8. What is the smallest number that can be exactly 
 divided by 4, 6, and 8 ? 
 
 PRIME NUMBERS. 
 
 110. A number that has no factors is a prime number* 
 
 Note. — 1 is not considered a factor. 
 
 1. 2, 3, 5, 7, etc., are prime numbers. 
 
 111. 1. Name the prime numbers between 10 and 20. 
 
 2. Between 20 and 30. 4. Between 50 and 70. 
 
 3. Between 30 and 50. 5. Between 70 and 100. 
 
 112. Oral Exercises. 
 
 Give the prime factors, commencing with the smallest. 
 
 1. 15 6. 40 11. 64 16. 80 
 
 2. 16 7. 48 12. 72 17. 81 
 
 3. 24 8. 54 13. 74 18. 82 
 
 4. 32 9. 56 14. 76 19. 84 
 
 5. 36 10. 60 15. 77 20. 85 
 
by 
 
 ne 
 
 2 
 3 
 3 
 
 90 
 
 45 
 
 15 
 
 5 
 
 17. 
 
 576 
 
 18. 
 
 840 
 
 19. 
 
 1152 
 
 20. 
 
 1728 
 
 21. 
 
 2016 
 
 Fractions. 6i 
 
 113. Written Exercises. 
 
 1. Find the prime factors of 180. 
 
 Divide 180 by its smallest prime factor, 2. Divide the 2 |180 
 quotient 90 by its smallest prime factor, 2. Divide 45 by- 
 its smallest prime factor, 3. Divide 15 by its smallest prime 
 factor, 3. The quotient 5 is a prime number. 
 
 The prime factors of 180 are 2, 2, 3, 3, 5, Ans. 
 
 2. 86 7. 92 12. 100 
 
 3. 87 8. 93 13. 120 
 
 4. 88 9. 94 14. 210 
 
 5. 90 10. 95 15. 240 
 
 6. 91 11. 96 16. 360 
 
 GREATEST COMMON DIVISOR. 
 
 114. A common factor of two or more numbers is a num- 
 ber that wijl divide each of them without remainder. 
 
 The largest number that is a factor of two or more numbers 
 is called the greatest common divisor. 
 
 115. Oral Exercises. 
 
 Find the greatest common divisor of: 
 
 1. 27 and 48 6. 34 and 51 
 
 2. 25 and 35 7. 32 and 48 
 
 3. 36 and 54 8. 45 and 75 
 
 4. 26 and 39 9. 40 and 65 
 
 5. 42 and 63 10. 54 and 69 
 
62 Chapter Two. 
 
 LOWEST TERMS. 
 
 116. How can you tell that a number is divisible by 2 ? 
 By 5? 
 
 A number is divisible by 3 when the sum of its digits 
 (figures) is divisible by 3 ; it is divisible by 9 when the sum 
 of its digits is divisible by 9. 
 
 A number is divisible by 4 when the number expressed 
 by its last two figures is divisible by 4. 
 
 When is a number divisible by 25 ? 
 
 A fraction is reduced to lowest terms by dividing the 
 numerator and the denominator by their greatest common 
 divisor. 
 
 117. "Written Exercises. 
 
 1. Reduce £f§ to its lowest terms. 
 
 A look at both terms shows that 3 is a common factor. This 
 reduces the fraction to fife. 41 is a prime number, and is not a factor 
 of 100, so that T ^y cannot be reduced to lower terms. 
 
 2. Reduce Jj-f- to its lowest terms. 
 
 4 + 3 + 2 = 9; 6 + 2 + 1=9. 
 Since the sum of the digits of each term is divisible by 9, this num- 
 ber is a common factor, and reduces the fraction to £$, etc. 
 
 3. Reduce £§f to its lowest terms. 
 5 is clearly a common factor, etc. 
 
 Reduce to lowest terms : 
 
 *• m *■ m "• ftt 
 
 «• m »• m is- m 
 
 6- H( io. ft 14. |H 
 
 7. w ii. u i5- m 
 
 118. Reduce to its lowest terms T Vfr 
 
 In this example, it is not easy to ascertain by inspection any num- 
 ber that will divide both terms. In such cases, the greatest common 
 
Fractions. 63 
 
 divisor is found by dividing the denominator by the numerator. The 
 remainder is divided into the numerator, and each subsequent remainder 
 is divided into the corresponding divisor until there is no longer a 
 remainder. This last divisor is the greatest common divisor of the two 
 numbers. 
 
 The numerator, 169, is contained in the 5 
 
 denominator, 1001, 5 times with 156 re- 169)1001 
 mainder. This remainder is contained in 845 1 
 
 the numerator, 169, one time with 13 re- 156)169 
 
 mainder. This remainder is contained in 156 12 
 
 the previous divisor, 156, 12 times with no 13)156 
 
 remainder. 13 
 
 13 is the greatest common divisor. ~Kq 
 
 169 + 13 13 , 26 
 
 = == lowest terms. — 
 
 1001 -=- 13 77 
 
 Note. — In reducing fractions to lowest terms, the method of 
 finding the greatest common divisor given above should not be resorted 
 to if it is possible to get along without it. 
 
 119. Written Exercises. 
 
 
 
 
 
 Keduce to lowest terms : 
 
 
 
 
 i- » 
 
 5. 
 
 m 
 
 9. 
 
 u 
 
 2- Hi 
 
 6. 
 
 22T 
 
 10. 
 
 TFT 
 
 3. « 
 
 7. 
 
 frV 
 
 11. 
 
 ttf 
 
 4- II 
 
 8. 
 
 t¥* 
 
 12. 
 
 m 
 
 Suggestion. — Do not waste time in finding the greatest common 
 divisor. 
 
 13. ^ 17. {fa 21. Jy& 
 
 14. #V 18. £k 22. fitffo 
 
 15. t^ 19. A% 23. T^fo 
 16- TWir 20- libs 24. T U 
 
64 Chapter Two. 
 
 LEAST COMMON MULTIPLE. 
 
 120. The smallest number that is a multiple of two or more 
 numbers is called the least common multiple of such numbers. 
 
 121. Oral Exercises. 
 
 Give the least common multiple of: 
 
 1. 
 
 16 and 24 
 
 6. 
 
 2, 3, 5, 9, 10 
 
 2. 
 
 12 and 15 
 
 7. 
 
 2, 3, 5, 6, 9, 10 
 
 3. 
 
 3, 9, 11 
 
 8. 
 
 3, 6, 9, 12, 4, 18 
 
 4. 
 
 4, 12, 16 
 
 9. 
 
 2, 7, 14, 3, 9 
 
 5. 
 
 2, 3, 4, 5, 6 
 
 10. 
 
 5, 10, 20, 25, 50 
 
 2 3 9 J U 14 2 12 
 
 3 
 
 9 7 6 
 
 122. Written Exercises. 
 
 Find the least common multiple of 3, 9, 7, 14, 6, 14, 2, 12. 
 
 3 is stricken out since it is a 
 factor of 6, which is one of the 
 numbers. 7 is a factor of 14, 3 7 2 
 
 one 14 is stricken out. 6 is a 
 
 factor of 12. 2 is a factor of 12. The least common multiple of the 
 remaining numbers, 9, 14, and 12, is to be found. 
 
 Divide these numbers by a prime number that is exactly contained 
 in any two of them, bringing down the numbers that are not multiples 
 of the divisor. 
 
 Taking 2 as a divisor, bring down 9, and write quotients 7 and 6. 
 
 3 being a factor of two of the three numbers, 9, 7, 6, is taken as the 
 next divisor. 3 is written as a quotient, 7 is brought down, 2 is a 
 quotient. 
 
 As there is no factor common to any two of the numbers, 3, 7, 2, 
 we find the least common multiple by multiplying together the two 
 divisors and these three numbers. 
 
 2 x 3 x 3 x 7 x 2 = 252 L. C. M. 
 
 123. Find the L. CM. of: 
 
 1. 4, 6, 3, 5, 8, 20 2. 9, 15, 15, 4, 4, 12, 25 
 
 Strike out 4, 3, 5. Strike out one 16 and two 4's. 
 
Fractions. 
 
 65 
 
 3. 2, 3, 5, 7, 5, 14, 10, 12, 24 
 
 4. 2, 3, 5, 6, 8, 10, 15, 16, 80 
 
 5. 20, 30, 40, 50 
 
 6. 2, 3, 4, 6, 8, 12, 16, 24 
 
 7. 24, 12, 5, 3, 10, 18 9. 18, 5, 9, 40, 16 
 
 8. 11, 3, 7, 77, 33 10. 10, 12, 15, 21 
 
 ADDITION AND SUBTRACTION OF FRACTIONS. 
 
 124. In adding or subtracting fractions, they must be reduced 
 to a common denominator. 
 
 The least common denominator is the least common multiple 
 of the denominators. 
 
 125. Add the fractions, }, ft, f, ft, f f, &. 
 £ 20 30 45 12 
 
 10 
 
 U 45 6 
 
 45 £ 
 L. C. M. = 2 x 2 x 45 = 180 
 
 Strike out 4 and 6. 
 
 Strike out 15, a factor of 45. 
 
 Strike out 5 and 3, factors of 45. 
 
 
 180 
 
 f 
 
 135 
 
 H 
 
 99 
 
 1 
 
 150 
 
 « 
 
 102 
 
 H 
 
 92 
 
 & 
 
 105 
 
 Ans. 8Hf 
 
 H* = 
 
 3H§- 
 
 To add, reduce the fractions to a common denominator, add 
 the numerators, and place the sum over the common denom- 
 inator. Reduce if possible. 
 
 To subtract, reduce the fractions to a common denominator? 
 subtract the numerators, and place the difference over the 
 common denominator. Reduce if possible. 
 
66 Chapter Two. 
 
 Note In the following examples, determine the least common 
 
 denominator by inspection, if possible. 
 
 126. Add: 
 
 1. 8f, 5}, 3J 6. |, f, A, A, H 
 
 2. 45f, 20|, 8f, 9£ 7. 63^, 3|, 24, 5f, 7^ 
 
 3. 32}, 19f i 6J, 8£| 8. 5&, 18^, 34, 7|, 8} 
 
 4. 2£, 20, 3f , ^, 54. 9. 4^,7^,84,6^,^ 
 
 5. 84, 454 , 2J4, 4J, |4 10. 17^ rffo 6&, ^ ^ 
 11. Work No. 9 as an example in decimals. 
 
 12- Work No. 10 as an example in decimals. 
 
 127. Subtract: 
 
 13. 65H- 57^ 18. 251^- - 27^ 
 
 14. 18tf- 9f| 19. 755£f -283ff 
 
 15. 100||- 15|f 20. 123£} - 80£f 
 
 16. 102^- 27« 21. 100^- 89^ 
 
 17. 208^-1284$ 22. 67^ - 58^ 
 
 23. Work No. 21 as an example in decimals. 
 
 24. Work No. 22 as an example in decimals. 
 
 128. Perform the operations indicated : 
 
 OK 21 + 5 21 
 
 9A 
 
 25 + 5 25 
 21 21-5 
 
 
 
 <0O. 
 
 25 25-5 
 
 
 
 27. 
 
 (37|-llf)- 
 
 -(28rV 
 
 -19A> 
 
 28. 
 
 14f-8i-3f + 4J 
 
 
 29. 
 
 (8ft + 6|)- 
 
 •(»A- 
 
 <® 
 
Fractions. 67 
 
 30. 4f x 16 x 8£ 
 
 31. (M**k}Mm-H+m 
 
 32. (8i + 4J)-(2i + li) 
 
 33. (3^x36)x8f 
 
 34. 4| + 3i-6f + 17i-9f 
 
 129. Oral Problems. 
 
 1. A person travelling from New York to Albany (140 
 miles apart) has gone 102J miles. How much farther has 
 he to go ? 
 
 2. There are 196 pounds of flour in a barrel. How 
 many pounds in J barrel ? 
 
 3. How many fourths in 24 J ? 
 
 4. Eeduce f$ to lowest terms. 
 
 5. Change ±$Q- to a mixed number. 
 
 6. Add i, ^, and J. 
 
 7. From a chest of tea containing 45 J pounds, 14| 
 pounds are sold. How many pounds remain ? 
 
 8. From ^ of a dollar take 10| cents. 
 
 9. How many cents in ^ + ^ -f- y% of a dollar? 
 
 10. A farmer has 60| acres of pasture and 20| acres of 
 woodland. How many acres in both ? 
 
 11. Considering the circumference of a circle as 3^ times 
 its diameter, find the circumference of a circle whose diam- 
 eter is 8 feet. 
 
 12. Mary is 12^ years old; Jane is 3fJ- years older* 
 How old is Jane in years and months ? 
 
 13. In a year and a half William will be 7 years 2 months 
 old. How old is he now ? 
 
68 Chapter Two. 
 
 14. What number multiplied by 3 equals 231 ? 
 
 15. What number between 7 and 12 is a prime number ? 
 
 16. A boy received 9 marks in arithmetic, 8 in penman- 
 ship, and 7 in reading. What was his average mark ? 
 
 17. -f of a class consists of boys. How many girls in the 
 class, if it contains 49 pupils ? 
 
 18. When July 1 falls upon Tuesday, what will be the 
 date of the third Tuesday of July ? 
 
 19. If July 1 falls upon Thursday, upon what day will 
 the first of August fall ? 
 
 20. A man bought 20|- pounds of sugar; he sold 10| 
 pounds at one time and 6J pounds at another. How much 
 had he left ? 
 
 21. If 3 quarts 1 pint of oil cost 7 cents, what will 1 gal- 
 lon 1 quart cost ? 
 
 22. How much will have to be paid for 7 cows at $50 
 each, and 4 horses at $ 150 each ? 
 
 23. f = how many hundredths? 
 
 24. What are the two factors of 87 ? 
 
 25. Find the G. C. D. of 36 and 54. 
 
 26. If eggs are sold at the rate of 21 for 25 cents, how 
 much will be paid for 5 \ dozen ? 
 
 Suggestion. — Every member of the class should be required to 
 solve one of the foregoing examples as a sight problem, first reading it 
 from the book, and then giving the answer. No time should be wasted 
 in "analyzing" the problems, unless some pupil desires the explana- 
 tion of one that he does not understand. 
 
 At another time, the teacher should read, say, five or ten problems, 
 requiring the answer to each to be written, at a given signal, and the 
 pencil laid down before the next is read. No alteration of an answer 
 should be permitted. 
 
Fractions. 69 
 
 130. Written Problems. 
 
 1. A horse travelled 48^- miles in one day, 56| the 
 next, 40i| the third, and 45f-J the fourth. How far did he 
 travel in all ? 
 
 2. To the sum of 6£ and 19| add their difference. 
 
 3. From a bin containing 25f bushels of grain there 
 were taken out 5f bushels at one time and 6J at another. 
 How much remained ? 
 
 4. A merchant sold 4 pieces of cloth containing 21\ 
 yards, 26f yards, 29f yards, and 28±- yards, respectively. 
 How much did he receive for the cloth at 96 cents per yard ? 
 
 5. Keduce ||-| to lowest terms. 
 
 6. A man has 8^- bushels of peanuts. He puts them 
 into bags holding -^ bushel. How many bags does he fill ? 
 
 7. A 160-acre farm consists of five fields ; the first 
 contains 17-f acres, the second 29£ acres, the third 35^j- 
 acres, the fourth 22-fe acres. How many acres are there in 
 the fifth field ? 
 
 8. Prom a piece of silk that contained 28£ yards, there 
 were sold 2\ yards, 6£ yards, and 13| yards. Find the value 
 of the remainder at $1.20 per yard. 
 
 9. Three pieces of cloth bought at $2 per yard cost 
 $150. The first piece measures 23£ yards, the second 
 measures 30f yards. How many yards in the third piece ? 
 
 10. What part of a person's income remains after he 
 spends J, -^, and \ of it? 
 
 11. A boy loses \ of his marbles, and he gives away \ of 
 them. If he has 17 marbles left, how many had he at first ? 
 
 12. A dealer sells If gross, 3J gross, and 8| gross of lead 
 
 pencils at 36 cents per dozen. How much does he receive 
 
 for all ? 
 
 1 gross = 12 dozen. 
 
70 Chapter Two. 
 
 13. There are four towns, A, B, C, and D, on a certain 
 railroad running east and west. A is 41^- miles west of C ; 
 D is 39J miles east of B ; B is 22£ miles west of C. How 
 many miles from A to D ? Make a diagram. 
 
 CANCELLATION. 
 
 131. Preliminary Exercises. 
 
 1. Divide 64 by 16. The quotient is 4. 
 
 2. Divide \ of 64 by £ of 16 ; i.e. 32 -*- 8. 
 
 3. Divide \ of 64 by J of 16 ; i.e. 16 -f- 4. 
 
 4. Divide \ of 64 by £ of 16 ; i.e. 8 -h 2. 
 In each case the quotient is 4. 
 
 In example 2 we took out of the dividend 64 the factor 2, making 
 the new dividend 32 ; and we took out of 16 the same factor, making 
 the new divisor 8. 
 
 In example 3 we took what factor out of the divisor and the divi- 
 dend ? What common factor was taken out in example 4? 
 
 Rejecting the same factor from the divisor and the dividend 
 does not change the quotient. 
 
 In reducing -fj to J what factor common to the numerator 
 and the denominator of the first fraction is rejected ? Is 
 the value of the first fraction altered by this rejection? 
 
 Cancellation is the striking out of common factors from the 
 divisor and the dividend. 
 
 132. Oral Exercises. 
 
 36 x!4 42 x23 67 x 36 83 x 36 
 
 9 21 18 ' ' 12 
 
 2 -f x16 6 -2^ x46 10 -i x82 14 -if x48 
 
 3 ' 12x i 7 - 32x I "' 4X i 15 ' 15X | 
 25x18 33x12 89 x 13 44x17 
 
 36   ' 99 26 34 
 
Fractions. 71 
 
 RATIO. 
 
 133. Preliminary Exercises. 
 
 1. If oranges are worth 28 cents a dozen, what will be 
 the cost of 3 oranges ? 
 
 2. What part of a dozen is 3 ? 
 
 3. What is the ratio of 3 to 12 ? 
 
 Ratio is the relation between two like numbers. It is found 
 by dividing the first by the second. 
 
 4. What is the ratio of 12 to 16 ? 
 
 5. If 16 apples cost a certain sum, what part of this sum 
 should be paid for a dozen apples ? 
 
 134. Written Exercise. 
 
 1. If 17 horses cost $4000, what will be the cost of 51 
 horses at the same price for each ? 
 
 Since the ratio between 51 and 17 is f^ or 3, 51 horses will cost 
 3 times §4000, or $12,000. 
 
 2. If 15 eggs cost 25 cents, what will 10 dozen cost? 
 
 10 x 12 
 The ratio of 10 dozen eggs to 15 eggs is — - — • 
 
 15 
 
 Multiply 25 cents by - 10 x 12 . 
 15 
 
 In this case, 15 is not contained in any number 
 
 above the line. We divide 15 and 10 by 5, cancel- 2 4 
 
 ing them and writing quotients 3 and 2 alongside. 25 X %$ X \% 
 
 3 is contained in 12 4 times ; so we cancel 3 and 12. J.fi 
 
 Our answer now is 25 cents x 2 x 4 = 200 cents, p 
 
 or $2. 
 
 3. Eighteen men can do a piece of work in 26 days. 
 How long will it take 13 men to do the same work? 
 
 Thirteen men will do the work in \\ of the time required by 18 
 men. 
 
72 Chapter Two. 
 
 4. Seventeen barrels of flour, 196 pounds each, were put 
 into bags holding 49 pounds each. How many bags of 
 flour were put up? 
 
 5. At the rate of 23 cents for 7 pounds, how much would 
 be paid for 91 pounds of flour? 
 
 6. A bank pays $ 4 interest a year on every $ 100. How 
 much interest will be paid for 3 years on $650? 
 
 7. At $ 7.50 per thousand for bricks, what must I pay for 
 63,200 bricks? 
 
 8. If flour is $ 6 per barrel (196 lb.), what must be paid 
 for a 49-pound bag? 
 
 9. A grocer buys 84 dozen eggs, and sells them at the 
 rate of 21 eggs for 25 cents. What does he receive for 
 them? 
 
 10. A miller buys 9840 pounds of wheat at 90 cents per 
 bushel of 60 pounds. How much does he pay for it ? 
 
 11. What will be the cost of 64 sheep, if 18 cost $ 198 ? 
 
 12. If 18 men can do a piece of work in 42 days, how 
 long will it take 21 men to do the same work? 
 
 13. What will be the cost of 66 dozen pens at 42 cents 
 per gross of 12 dozen? 
 
 14. A certain quantity of hay feeds 15 horses 56 days. 
 How long will it feed 14 horses ? 
 
 15. A merchant bought 9 pieces of cloth, each containing 
 24 yards, for $ 189. What was the price per yard ? 
 
 MULTIPLICATION OF FRACTIONS 
 
 135. Preliminary Exercises. 
 
 What is i of 2 fifths ? Of 4 sevenths ? Of 6 elevenths ? 
 
 What is \ of i? Of£? Of J? Of J? Show by diagram 
 
 What is \ of |? Off? Off? Off? 
 
 What is \ of f? fof£? fof£? 
 
Fractions. 73 
 
 What is -§- of I? I off? foff? 
 
 What is -I of i? ioff? fofj? foff? 
 
 What is the half of 1£? 0f2i? 0f3|? Of 4^? 
 
 What is one-third of 1£ ? J of 1\ ? £ of 2\ ? f of 2J ? 
 
 136. "Written Exercises. 
 
 1. Multiply £ by f. 
 This means to find # of f . 
 
 Since * of * = A» i of * = A. and } of f = &, or | x f = &. 
 
 One fraction is multiplied by another by placing the product 
 of the numerators over the product of the denominators in the 
 form of a fraction. 
 
 Note. — Cancel when possible. 
 
 2. Multiply I by A- 
 
 i°f A = A £<>f A = 2timesA = f 
 
 3 
 
 Cancel 2 and 10, writing 5 under 10. Cancel % ^ ft _ 3 
 
 3 and 9, writing 3 above 9. £ 20 5 
 
 Show by a diagram that 2 times ^ is \. 
 
 5 
 
 3. Multiply 12i by 3^. 
 Reduce the mixed numbers to improper fractions. 
 
 17 7 
 
 g2 40 = 119 == 
 
 3 
 
 4. Multiply 117 by 3|. 
 
 The multiplication of an integer by a mixed 
 number, or of a mixed number by an integer, can 13 
 be considered as multiplication of fractions, the 2ML v _ — 377 
 integer being written as an improper fraction with 1 ^ 
 1 for the denominator. 
 
74 Chapter Two. 
 
 137. Multiply: 
 
 1. | by 96 16. &x$% 
 
 2. 128 by | 17. 3fbyl2£ 
 
 3. * by | 18. $ x 4« 
 
 ibyf 19. fbyfbyH 
 
 5. * by* 20. fj of |f of 4. 
 
 6. 3^ by 72 21. |} x f X * 
 
 7. 24fbyl8 22. ft of ft of ff 
 
 8. 69fby32 23. J of 65f 
 
 9 
 10 
 
 11. 2£by3f 26. 4^x5^ 
 
 12 
 13 
 
 111^ by 28 24. fof55f 
 
 67by^ 25. 6Jx7| 
 
 14. 
 15. 
 
 AX2J- 27. § of 4^x3* 
 
 17iby6f 28. |of3|x4 I V 
 
 6*xf 29. Hx2Jx3J 
 
 4Jby8f 30. 2£x2Jx2J 
 
 138. Perform the indicated operations : 
 Note. — \ of 3 J is the same as \ x 3 J, or 3 J x $. 
 
 1. 
 
 J of (3H-6J) 
 
 
 6. 
 
 (8} x 21) -(* of 15© 
 
 2. 
 
 <3*-2»X| 
 
 
 7. 
 
 5i + 6J + 7J 
 
 3. 
 
 iof(5i-3f) 
 
 
 8. 
 
 18f-8|-7J 
 
 4. 
 
 (24J + 16i)H-8 
 
 
 9. 
 
 f off of (3* + If) 
 
 6. 
 
 (3i+2*)x(3i- 
 
 •2fi 
 
 10. 
 
 (l 8i -6f) +11 
 
 139. Oral Exercises. 
 
 1. Sold a house lot for $30, which was J of what it 
 cost me. What was the cost of the lot ? 
 
 2. A man can mow 6 J acres of grass in a day. How 
 much can he mow in 6 days ? 
 
Fractions. 75 
 
 3. A man bought 15 bushels of corn for 1\ dollars. 
 How much did a bushel cost ? 
 
 4. A boy is 18 years old and his age is f of the age of 
 his father. How old is his father ? 
 
 5. Cloth is worth fo of a dollar a yard. What is f of 
 a yard worth ? 
 
 6. At the rate of 5 cents for \ of a pie, for how many 
 pies will a man receive $ 1.60 ? 
 
 7. What would |- of a yard of carpet cost at f of a 
 dollar a yard ? 
 
 8. I had ^2 °f a pound of candy and gave away f of it. 
 What part of a pound did I give away ? 
 
 9. What will 15 yards of ribbon cost at 6§ cents a yard ? 
 
 10. What will 2f gallons of ice-cream cost at If dollars 
 a gallon ? 
 
 140. "Written Exercises. 
 
 1. A man worked 6 days at 2} dollars per day, his son 
 5 days at If dollars, his daughter 4 days at £ of a dollar. 
 What were their total earnings ? 
 
 2. A merchant bought a piece of cloth for 28| dollars 
 and was obliged to sell it for -| of what it cost him. How 
 much did he lose ? 
 
 3. A hotel in one month used 31 pounds of coffee and 
 7f times as much sugar. How much sugar was used ? 
 
 4. A man gave 124 T 5 g- acres of land to his two sons, 
 giving f of it to the elder and ■§■ to the younger. How 
 many acres did each receive? 
 
 5. If it requires 21| days for a man to dig a ditch, 
 what part can he dig in 15 days ? 
 
7 6 Chapter Two. 
 
 6. If a bird can fly lOf miles in f of an hour, how far 
 can it fly in 2£ hours ? 
 
 7. What would be the cost of a side of veal containing 
 52 pounds at 9 \ cents a pound ? 
 
 8. What will 16 pairs of shoes cost at % 3} a pair ? 
 
 9. A man who owed $ 7825 failed and could pay only 
 | of his debts. How much could he pay ? 
 
 10. I bought a house and lot and made a payment of 
 % 4500, which was f of the cost. What was the cost of the 
 property ? 
 
 DIVISION OF FRACTIONS. 
 141. Preliminary Exercises. 
 
 1. If 3 yards of calico cost 18 cents, what is the price 
 per yard ? 
 
 18 j* -T- 3, or \ of 18/?. The latter may be written 18 j* x \. 
 
 2. If \\ yards of dress goods cost 18^, what is the price 
 
 Per yard? i 8 ^ H , or 18^|. 
 
 To divide 18 by f , we can change 18 to halves and proceed as fol- 
 lows : *£■ + f = 36 + 3. 
 
 The following are the steps : 18 is multiplied by 2, and the product 
 
 18 v 2 
 is divided by 3, or ° * , which is the same as 18 x f . 
 
 o 
 
 That is, 18 + | = 18 X f. 
 
 3. If 3 yards of dress goods are required to make a waist, 
 how many waists can be made out of 18 yards ? 
 
 The number of waists = 18 -r- 3 = \ of 18 = 18 x \. 
 That is, 18 -?- 1 = 18 x \. 
 
 4. If an apron requires \\ yards of material, how many 
 aprons can be made out of 18 yards ? 
 
 The number of aprons = 18 -*• \\ = 18 -*- \ = 18 x f. 
 
Fractions. 77 
 
 5. If it takes three-quarters of a pound of flour to make 
 a loaf of bread, how many loaves can be made with 18 
 pounds of flour? 
 
 The number of loaves = 18 -j- f = 18 x $ . 
 
 6. At three-quarters of a dollar each, how many dolls can 
 be bought for a dollar and a half ? 
 
 To divide by $ (examples 1 and 3), we multiply by $. 
 To divide by f (examples 2 and 4), we multiply by f. 
 To divide by £ (examples 5 and 6), we multiply by f. 
 
 To divide by a fraction, multiply by the divisor inverted. 
 
 7. Divide 8 by |. 
 
 8-=-§ = 8xf = 10, Ans. 
 
 8. Divide f by 10. 
 
 9. Divide 6f by 9. 
 
 142. Divide: 
 
 1. |+4 4. ^-A 7. A-f-H 9. t+3| 
 
 2. f + 10 5. H-t\ 8. 3f + * 10. i-| 
 
 3. lf+5 6. A+A 
 
 143. "Written Exercises. 
 
 1. Divide A by & 
 
 3 5 
 
 16 * 20 J0 2 4 * 
 4 
 
 2. Divide 15^ by 13. 
 
 Changing the mixed number to an improper fraction, we have, 
 
 13 
 
7« 
 
 8 
 
 Chapter Two. 
 
 
 
 Divide : 
 
 
 
 
 3- A+A 
 
 8. A^ 
 
 13. 
 
 8A + 3J 
 
 4. 5-f-lf 
 
 9- tt + *A 
 
 14. 
 
 9* + 3* 
 
 5. 8$ -5- 11 
 
 io. f+* 
 
 15. 
 
 18J + 1H 
 
 6. 4^-17 
 
 ii. A^-l 
 
 16. 
 
 2H+6# 
 
 7. 24J-20 
 
 12. A+A 
 
 
 
 Note. — The pupil should prove his answers to each of the foregoing 
 examples by multiplying the quotient by the divisor. If his answer is 
 correct, this product will equal the dividend. 
 
 144. Perform operations indicated : 
 
 17. (3fx4i)-10J 24. (tfx«) + (4f x6» 
 
 18. (13f-7f)xf 25. 34f-17f + 20« 
 
 19. (20 x })-*-! 26. 18| + 24^-36iJ 
 
 20. (20 + f)x| 5|x9x7| 
 
 21. 20 + (|xf) W ' 4f xf 
 
 22. (20-*-$)-s-f 51 x 73 x 31, x 6 ^ 
 
 23. (14 j x 7) - (9 x 10 1) 2fx 4^x31 
 
 145. Oral Problems. 
 
 Give analysis of each : 
 
 1. If base-balls are worth f of a dollar each, what will 
 be the cost of 16 base-balls ? 
 
 Note. — The pupil is frequently at a loss to determine whether a 
 given problem in fractions involves multiplication or division. In 
 such a case, he should substitute for the fraction a whole number to 
 ascertain the proper operation. While in example 1 a pupil would 
 analyze without hesitation : " If base-balls are worth $ | each, 16 
 balls would cost 16 times $£," he might stumble at No. 2. By read- 
 ing the problem, "Paid a certain sum for base-balls at $3 each," he 
 would see that the number of balls is ascertained by division. His 
 analysis would then be, "If base-balls are $f each, I can buy as 
 
Fractions. 79 
 
 many balls as there are $£ in $ 12." The work would be 12 -s-f = 
 12 x f . He could complete the solution by finding $ of 12, taking $ 
 of 12 as 4, etc. Another method of solving this problem mentally, 
 is to change the price to a whole number and to make a corresponding 
 change in the cost. "Paid 4 times $12 for base-balls at 4 times $ J 
 each ; i.e. % 48 for balls at $ 3 each." 
 
 2. Paid $ 12 for base-balls, at J of a dollar each. How 
 many were bought ? 
 
 3. What is the cost of 2 feet of ribbon at 30 cents per 
 yard? 
 
 4. Find how much a yard of ribbon is worth, if -| yard 
 costs 20 cents. 
 
 5. If it takes f yard of material to make a child's waist, 
 how many can be made from a piece containing 24 yards ? 
 
 6. If 18 jackets require 24 yards of cloth, how much is 
 needed for 1 jacket ? 
 
 7. A man had 60 acres of land. How many acres had 
 he left after selling J- of his land ? 
 
 8. After spending £ of his money, a person had $26 
 remaining. How much money had he at first ? 
 
 9. When tea is $ .50 per pound, how much can be 
 bought for $ .75 ? 
 
 10. If tea is worth f of a dollar per pound, how much 
 can be bought for i of a dollar ? 
 
 11. When silk is selling at $ .75 per yard, how much can 
 be bought for one-fourth of a dollar ? 
 
 12. Find the cost of a gallon of milk at the rate of 9 cents 
 for 3 pints. 
 
 13. | of a gallon of milk costs 9 cents. What is the 
 price per gallon ? 
 
 14. f of what number is 12 ? 
 
 15. 1 yard and 1 foot of wire cost 16 cents. How mucb 
 must be paid for a yard ? 
 
8o Chapter Two. 
 
 146. Written Problems. 
 
 1. How much does a man earn in a day if he earns 45J 
 dollars in a month of 26 working days ? 
 
 2. When flour is 5 \ dollars per barrel, how many barrels 
 can be bought for 294 dollars ? 
 
 3. If coffee is 37£ cents per pound, how many pounds 
 can be bought for 60 dollars ? 
 
 4. A man divided 16 dollars among some boys, giving 
 to each If dollars. How many boys received a share ? 
 
 5. Paid 38^ dollars for 6 J cords of wood. What was 
 the price per cord ? 
 
 6. How many steps will it take to walk 2640 feet, each 
 step being 2 J feet in length ? 
 
 7. A man put 40^ bushels of barley into bags holding 
 1-| bushels. How many bags were required ? 
 
 8. In 2 J acres of land, how many building lots of f of 
 an acre ? 
 
 9. If | of a farm is worth $ 8000, what is { of it worth? 
 
 10. The product of two factors is 9^ ; one factor is 3£. 
 What is the other ? 
 
 SPECIAL DRILLS — REVIEW. 
 
 147. Give sums at sight: 
 
 1. 59 + 75 = 59 + 70 + 5 = 
 
 2. 48 + 63 • 5. 88 + 22 8. 66 + 56 
 
 3. 69 + 47 6. 94 + 38 9. 29 + 94 
 
 4. 67 + 83 7. 61+39 10. 65 + 86 
 
 11. 560 + 390 = 560 + 300 + 90 = 
 
 12. 270 + 280 14. 430+480 16. 420 + 280 
 
 13. 640 + 260 15. 250 + 390 17. 780 + 260 
 
Review. 81 
 
 18. 225 + 154 = 225 + 150 + 4 = 
 
 19. 315 + 421 21. 540 + 355 23. 172 + 304 
 
 20. 437 + 260 22. 248 + 131 24. 517 + 329 
 
 148. Give remainders at sight : 
 
 1. 134 — 75 = 134-70-5 = 
 
 2. 150-83 5. 124-89 8. 100-61 
 
 3. 132-94 6. 112-56 9. 124-35 
 
 4. 122-56 7. 180-89 10. 132-38 
 
 11. 750 — 290 = 750-200-90 = 
 
 12. 510-220 14. 820-560 16. 910-550 
 
 13. 630-380 15. 730-440 17. 380-290 
 
 18. 279 — 154 = 279-150-4 = 
 
 19. 386-263 21. 668-325 23. 386-123 
 
 20. 457-237 22. 279-125 24. 721-468 
 
 149. Give products at sight: 
 
 1 . 49 X 4 = 4 forties + 4 nines. 
 
 2. 47 x 3 3. 48 x 4 4. 43 x 5 5. 46 x 6 6. 38 X 7 
 
 7. 123 X 3 = 3 times one twenty three = three sixty nine. 
 
 8. 431 x 2 10. 332 x 3 12. 232 x 3 
 
 9. 122 x 4 11. 242 x 2 13. 31 x 24 
 
 14. 47 X 25 = k of 47 hundred = llf hundred = 1175. 
 
 15. 25 X 38 = 38 X 25 = 1 of 38 hundred = 9f hundred. 
 
 16. 32 x 25 18. 44 x 25 20. 49 x 25 
 
 17. 25 x 33 19. 25 x 45 21. 63 x 25 
 
 150. Give quotients at sight : 
 
 1 . 925 -f- 25 = 9J hundred -r- J hundred = 9J -*• 1 = 91 X 4. 
 
 2. 875-4-25 4. 725-^-25 6. 575-4-25 
 
 3. 625 -s- 25 5. 450-5-25 7. 350-5-25 
 
22 Chapter Two. 
 
 151. Oral Problems. 
 
 1. Find the cost of a pound of tea at 75 cents, and a 
 piece of ham at 56 cents. 
 
 2. A farmer sold 58 sheep from his flock of 121 sheep. 
 How many remained ? 
 
 3. What will be paid for 8 pounds of coffee at 35^ per 
 pound ? 
 
 4. A laborer received $ 4.88 for four days' work. How 
 much did he earn per day ? 
 
 5. At $40 each, how many cows can be purchased for 
 $2000? 
 
 6. Bought 20 pounds of sugar at 5^ per pound, and 2\ 
 pounds of butter at 30^. What was the amount of my bill ? 
 
 7. A piece of cloth measuring 31^ yards was divided 
 into 2 equal parts. Find the length of each. 
 
 8. How many weeks in a year of 366 days ? 
 
 9. If I pay 25 cents for 3 pounds of cherries, how many 
 pounds can I buy for $1.25 ? 
 
 10. Find the cost of a bushel and a peck of peanuts at 
 the rate of 5 cents per quart. 
 
 11. A farmer had 164 acres of land. How much had he 
 left after selling 87 acres ? 
 
 12. Find the total number of pounds in 3 tubs of butter 
 weighing respectively 25 pounds, 34 pounds, and 36 pounds. 
 
 13. At 60^ per pound, how much tea can be bought for 
 $5.85? 
 
 14. A drover paid $ 219 for oxen, at an average price of 
 $ 73. How many did he buy ? 
 
 15. What will be the cost of 20 bushels of wheat at 
 $1.04£ per bushel? 
 
 16. At 24^ per pound, how many ounces of butter can be 
 bought for 18^? 
 
Review. 83 
 
 17. A woman pays $ 540 per year for a house. What is 
 the rent per month ? 
 
 18. How many weeks in 294 days ? 
 
 19. At 72^ per yard, what will be the cost of 2 ft. 11 in. 
 of lace ? 
 
 20. How much does a grocer receive for a barrel of flour, 
 196 pounds, retailed at 3 cents per pound ? 
 
 21. If 47 men can do a piece of work in 4 days, how long 
 will it take 1 man to do the same work ? 
 
 22. Find the cost of 36 acres of land at $25 per acre. 
 
 23. If it takes 3^ yards of cloth to make a coat, how 
 many coats can be made from 24J yards ? 
 
 24. How much will be paid for 84 yards of silk at $lf 
 per yard ? 
 
 25. If a certain quantity of provisions will last one man 
 215 days, how long will it last 43 men ? 
 
 26. How many square yards are there in a rectangular 
 field 36 yards long and 25 yards wide ? 
 
 152. Written Exercises. 
 
 1. What is the sum of 94,625; 215; 5172; 819,365; 
 121? 
 
 2. Bought 172 acres of land for $860. What was that 
 an acre ? 
 
 3. In a classroom there are 54 pupils ; each pupil spent 
 $ 2.75 for books this year. How much money was spent 
 for books by the whole class ? 
 
 4. By the census of 1890, Massachusetts had a popula- 
 tion of 2,238,943 ; in 1900, it had a population of 2,805,346, 
 What was the gain ? 
 
 5. How many boxes of strawberries at $.15 a box can 
 I get for $ 1.20 ? 
 
84 Chapter Two. 
 
 6. What is a proper fraction? An improper fraction? 
 Define numerator, denominator, a mixed number. 
 
 7. Add J, |, |, and ± 
 
 8. If 7 pairs of shoes cost $12J, how much will one 
 pair cost ? 
 
 10. What is the product of -fa, ^, if, and f$ ? 
 
 11. 8ix7| = ? 
 
 12. Paid | of a dollar for potatoes, % of a dollar for 
 apples, and ^ of a dollar for sugar. How much did I pay 
 for all ? 
 
 13. Divide 2\ by If 
 
 14. Find the difference between 4§ and 3f. 
 
 MULTIPLICATION OF DECIMALS. 
 153. Oral Problems. 
 
 1. When the French franc is worth 19.3 cents, what is 
 the value of the 20-f ranc piece in United States money ? 
 
 2. What is the equivalent of 10 German marks, the mark 
 being quoted at 23^- cents ? 
 
 3. A man paid 100 pounds sterling for a piano. Find 
 the cost in U. S. money at $ 4.8665 per pound sterling. 
 
 Note. — $ 4.8665 may be read 4 dollars 86 cents 6 mills and 5 tenths 
 of a mill, a mill being one-tenth of a cent. 
 
 4. A meter contains 39.37 inches. How many inches in 
 100 meters ? 
 
 5. One kilogram = 2.2046 pounds. What is the equiva- 
 lent of 1000 kilograms, in pounds ? 
 
 Note. —.2046 is read 2046 ten-thousandths. 
 
 6. How many square yards are there in a piece of ground 
 40 yards long and 12.5 yards wide ? 
 
Decimals. 85 
 
 * 
 
 7. How many ounces in 2.5 pounds ? 
 
 8. Change .75 hour to minutes. 
 
 9. Find the perimeter of a square, each side of which 
 measures 10.25 feet. 
 
 154. Written Exercises. 
 
 1. Multiply 38.4 by 6.37. 
 
 Place the units' figure (6) of the multiplier under 38.4 
 the last figure (4) of the multiplicand. 6 times 6.37 
 
 4 tenths = 24 tenths = 2.4 ; write .4 under, the multi- 230.4 
 
 plier 6, and carry 2 ; etc. Next multiply by .3, 11.52 
 or fV A x t% = T<fr> or - 12 - Write 2 in the hun - 2.688 
 
 dredths' place, and carry 1 tenth; etc. Multiply 244 f 08 
 finally by .07, or T ^. & x T % = T $fo, or .028. 
 Write 8 in the thousandths' place, etc. Ans - * 44 - w °- 
 
 Note. — By writing the units' figure of the multiplier under the last 
 figure of the multiplicand, and by taking care to place the right-hand 
 figure of each partial product under the corresponding figure of the 
 multiplier, the decimal points in the partial products and the total will 
 naturally fall under the decimal point in the multiplicand. 
 
 2. Multiply 12.34 by 56.7. 
 
 12.34 While pupils should occasionally begin to ^2 34 
 
 gg j multiply by the left-hand figure (5) of the gg j 
 
 n-ij r\ multiplier, some may prefer to begin with the ft^RQft" 
 
 74 04. right-hand figure (7). It will be noted that „. \. 
 
 8 £oo the number of decimal places in the product n^r, \ 
 
 ,,„„'' equals the sum of those in the multiplier and „ *„ r 
 
 699.678 2 u . .. . * 699.678 
 
 the multiplicand. 
 
 Multiply as in ivhole numbers, and from the right of the 
 product point off as many decimal places as there are decimal 
 places in both factors. 
 
 155. Multiply: 
 
 1. 32x2.5 3. 6.4x4.5 
 
 2. 3.2x25 4. 7.2x3.75 
 
16. 
 
 18.4 x 20.25 
 
 17. 
 
 11.16 X 42.40 
 
 18. 
 
 66.6 x 3.3£ 
 
 19. 
 
 6.24 x 1.75 
 
 20. 
 
 400.04 x 39.25 
 
 86 Chapter Two. 
 
 5. 12.8 x 5.7 8. 5.625 x 8.4 
 
 6. 9.6 x 1.125 9. 1.875 x 12.8 
 
 7. 34.9x2.34 10. 42.36x2.95 
 
 Note. — The pupil should correct any error he may make in placing 
 the decimal point by estimating the approximate answer. The answer 
 to example 3, for instance, is more than 2 times 32 and less than 3 times 
 32. In example 3, it is more than 4 sixes and less than 6 sevens. 
 
 11. 1.75 x 64 
 
 12. 8.375 x 40 
 
 13. 24.5 x 18.2 
 
 14. 9.6 x 12£ 
 
 15. 7.43 x 3.6 
 
 156. Oral Problems. 
 
 1. I owned 40 acres of land and sold .25 of it. How 
 many acres did I sell ? 
 
 2. A boy bought 15 hens, which was .6 of what he al- 
 ready had. How many had he at first ? 
 
 3. A lawyer charged me .11 of the money for collecting 
 $ 100. How many dollars did he charge ? 
 
 4. If I earn $ 8 in a week, how much can I earn in 7.5 
 weeks ? 
 
 5. .75 of a class of 44 were promoted. How many were 
 not promoted ? 
 
 6. What is the surface of a table 4 feet wide and 6.25 
 feet long ? 
 
 7. .5 of a yard is how many feet ? How many inches ? 
 
 8. A man bought 3.5 yards of cloth at $ 5 a yard. What 
 was the price ? 
 
 9. 25 miles is .5 of the distance between two cities. 
 What is the distance? 
 
 10. In a box were 100 oranges; .08 of them became 
 spoiled. How many sound ones were left ? 
 
Decimals. 87 
 
 157. Written Problems. 
 
 1. How many yards are there in 25 pieces of carpeting 
 if each piece contains 32.75 yards ? 
 
 2. A mill uses 95.6 tons of coal per day. How many 
 tons will it use in 42.25 days ? 
 
 3. A cubic foot of water weighs 62.5 pounds ; ice is .92 
 as heavy as water. What is the weight of a cubic foot 
 of ice ? 
 
 4. I bought 3 loads of wood, the first containing 1.04 
 cords, the second 1.05 cords, and the third .946 cord. 
 What did it cost me at $ 4.50 a cord ? 
 
 5. A gallon of water weighs 8.33 pounds. What is the 
 weight of a gallon of milk which is 1.03 times as heavy as 
 water ? 
 
 6. A wheel in making one revolution travels 15.03 feet. 
 How far will it travel in 25 revolutions ? 
 
 7. A ship sails 18.54 miles in an hour. How far will she 
 sail in 15.5 hours ? 
 
 ^. Find the cost of concreting a cellar 24.5 feet long by 
 14.25 feet wide, at 30 cents per square foot. 
 
 9. A quantity of provisions will last 25 men 12.75 days. 
 How long will it last one man ? 
 
 10. Two men start from the same place and travel in op- 
 posite directions, one at the rate of 3.85 miles per hour, and 
 the other at the rate of 4.12 miles per hour. How far apart 
 will they be at the end of 13 hours ? Make a diagram. 
 
 DIVISION OF DECIMALS. 
 
 158. Divide 42 by 2.1. 
 
 Changing the decimal fraction in the divisor to a common fraction, 
 we have 42 -2^ = ^-H= ^ x tf = *&> 
 
 42-2.1 = 420-7-21. 
 
 Note. — When we change the divisor 2.1 to 21, we have multiplied 
 it by 10, and the same change must be made in the dividend. 
 
88 Chapter Two. 
 
 Make the divisor a wJiole number, and make a correspond- 
 ing change in the number of decimal places in the dividend. 
 This reduces the numbers to the same denomination. If neces- 
 sary to complete the operation, ciphers may be annexed to the 
 dividend. The number of decimal places in the quotient is 
 equal to the number in the dividend as changed. 
 
 9. 50 -r- .25 
 
 10. 72 - .5 
 
 11. 960 - .03 
 
 12. '.847 -.007 
 
 13. 27 -=- .002 
 
 14. 10 -r- .8 
 
 15. 1.263 -f- .03 
 
 16. 19.63 -5- .013 
 
 17. Diyide 196.3 by .013. 
 
 Remove the decimal point in the divisor three 
 places to the right, and make a corresponding 
 change in the dividend, adding two ciphers. 
 
 To show where the decimal point originally be- 
 longed, draw a cancellation mark through it, 
 instead of erasing it. 
 
 When the divisor is thus made a whole number, the decimal point 
 In the quotient will be placed under (or over) the new decimal point 
 In the dividend. 
 
 1.736 - 16 17.36 -j- .16 .01736 -i- 1.6 
 
 .1085 Ans. 108.5 Ans. .01085 Ans. 
 
 159. Written Exercises. 
 
 Divide : 
 
 
 1. 
 
 80-2.5 
 
 2. 
 
 8+2.5 
 
 3. 
 
 840 -f- 1.2 
 
 4. 
 
 se^A 
 
 5. 
 
 36 -.9 
 
 6. 
 
 12.6 -f- 6.3 
 
 7. 
 
 48 + 15 
 
 8. 
 
 18.36 + .6 
 
 L6)1.7360 /16.)17/36.0 l/6.)/0.17360 
 

 
 Decimals. 
 
 
 18. 
 
 .504 --.024 
 
 26. 
 
 392 -- 3.2 
 
 19. 
 
 5.04 -- .24 
 
 27. 
 
 48 -5- 3000 
 
 20. 
 
 50.4 -- 2.4 
 
 28. 
 
 92 -*- .23 
 
 21. 
 
 504 -5- 24 
 
 29. 
 
 .875 - 125 
 
 22. 
 
 168 -s-,7 
 
 30. 
 
 381.17 -f- 8.11 
 
 23. 
 
 36 -5- .12 
 
 31. 
 
 .624 -r- 9.75 
 
 24. 
 
 .875 -5- .25 
 
 32. 
 
 48.195 -5- 3.57 
 
 25. 
 
 123.6 -v- .01 
 
 33. 
 
 829.31 -5- .019 
 
 89 
 
 160. Divide 381.6 by 95.032. 
 
 Note. — The sign (+) after the last 
 figure of the quotient indicates that there 
 is a remainder. 
 
 4.015 + 
 
 95/032.)381/ 600.000 
 380128 
 147200 
 95032 
 
 521680 
 
 161. Divide, carrying out the quotient to 3 places of 
 
 decimals 
 
 34. 31 -5- 13 
 
 35. 4.5 -5- 17 
 
 36. 920.07-5-46 
 
 162. Write answers at sight : 
 
 37. 7.049-5-1.6 
 
 38. 81.22-5-3.275 
 
 39. 246.3 -5- 93.473 
 
 Note. — To multiply .042 by 100, the decimal point is moved two 
 places to the right ; i.e. .042 x 100 = 4.2 ; .042 x 200 = 4.2x2 = 8.4. 
 
 1. .042 x 200 
 
 2. .13 x 300 
 
 3. .014x50 
 
 4. 8.1 x 60 
 
 5. 40 x .7 
 
 6. 25 x .8 
 
 7. 234 x. 2 
 
 8. .73 x 30 
 
 9. .121x4000 
 
 10. .061x500 
 
 11. .03x1000 
 
 12. .012 x 700 
 
 Note. — Remember that 369 -=- 1000 = T 8 ^ = 369 thousandths =.369, 
 To divide 369 by 3000, therefore, divide .369 by 3. 
 
90 Chapter Two. 
 
 13. 369-^-3000 17. 2460 + 3000- 21. 4.68 + 20 
 
 14. 219 + 300 18. 196 + 4000 22. 30.5 + 500 
 
 15. 48.6 + 60 19. 6 + 500 23. 18.8 + 200 
 
 16. 1.89 + 90 20. 27.9 + 300 24. 248 + 4000 
 
 163. Written Exercises. 
 
 1. 1728 + 1200 2. 172.8 + 1200 3. 1.728 + 1200 
 
 1200 )17.28/ 1200 )1.72/8 1200 ).Ol/728 
 
 1.44 Ans. .144 Ans. .00144 Ans. 
 
 Cancel the ciphers in the divisor, and remove the decimal 
 point in the dividend a corresponding number of places to the 
 left, prefixing ciphers if necessary. 
 
 164. Divide: 
 
 1. 2436 + 3000 7. 45 + 800 
 
 2. 136.5 + 1300 . 8. 25.2 + 240 
 
 3. 84.8 + 80 9. 345.6 + 1200 
 
 4. 100.1 + 700 10. 4004 + 110 
 
 5. 1 + 40 11. 5.28 + 60 
 
 6. 2.2 + 50 12. 907.5 + 1500 
 
 165. Oral Problems. 
 
 1. I cut 8.72 yards of cloth into 8 equal pieces. How 
 long was each piece ? 
 
 2. I divided .75 of a pound of candy equally among 3 
 girls. What part of a pound did each receive ? 
 
 3. I divided .5 of a pound of cherries among 4 children. 
 What part of a pound did each receive ? 
 
 4. 49 rods is .7 of the distance round a field. How 
 many rods of fence will enclose the field? 
 
 5. 24 yards of matting cover .8 of my floor. How 
 many yards more must I buy? 
 
Decimals. 91 
 
 6. 40 pounds are .4 of my weight. What do I weigh ? 
 
 7. I spent 2.5 dollars, which was .5 of what I had. How 
 much had I ? 
 
 8. 36 square inches are .25 of a square foot. How many- 
 square inches in a square foot ? 
 
 9. A collector receives .05 of all the money he collects. 
 How much did he collect to earn 9 15 ? 
 
 10. At 75 cents each, how many chairs can be bought 
 for $12? 
 
 166. Written Problems. 
 
 1. If 35.84 cubic feet of water weigh a ton, what will be 
 the weight of 2458.6 cubic feet ? * 
 
 2. How many francs are there in $150? (A franc 
 equals 19.3^.) 
 
 3. If a barrel of flour costs $5.75, how many barrels can 
 be bought for $ 258.75 ? 
 
 4. If $ 640.05 are paid for 75.3 tons of coal, what is the 
 price per ton ? 
 
 5. There are 31.5 gallons in a barrel. How many bar- 
 rels are there in 2787.75 gallons ? 
 
 6. I have 96 cubic feet of wood ; this is .75 of a cord. 
 How many cubic feet in 1 cord ? 
 
 7. A man earns $162 in 13.5 weeks. What are his 
 wages per week? 
 
 8. I bought a farm of 71.5 acres for $6220.50. What 
 did it cost me per acre ? 
 
 9. There are 2150.4 cubic inches in a bushel. How 
 many bushels are there in 9676.8 cubic inches ? 
 
 10. The wheel of a bicycle is 7.25 feet around. How 
 many times will it turn in going a mile, or 5280 feet ? 
 
gi Chapter Two. 
 
 UNITED STATES MONEY. 
 FRACTIONAL PARTS OF A DOLLAR. 
 
 167. Oral Problems. 
 
 1. How many 50-cent base-balls can be bought for f 15 ? 
 
 (15 -r- 1, i.e. 15 x 2) 
 
 2. How many 75-cent base-balls can be bought for $ 15 ? 
 
 (15 h- f, i.e. 15 x |) 
 
 3. At 75^ per pound, how much tea can be bought 
 for $ 1 ? 
 
 4. How many hats, at $1.25 each, can be bought 
 for $15? (15 ^-1|) 
 
 5. Paid $16 for coffee at 25^ per pound. How many 
 pounds were purchased ? 
 
 6. At 33^ per pound, how many pounds of butter can 
 be bought for $ 32 ? 
 
 7. Find the number of yards of ribbon, at 12-J-^ per yard, 
 that will cost $45. 
 
 8. At 6J£ per bar, how many bars of soap will cost $ 11 ? 
 
 9. If 4 pieces of violet soap are sold for 25^, how many 
 can be bought for $ 9 ? 
 
 10. $ 24 is paid for corn at 75^ per bushel. How many 
 bushels are bought ? 
 
 11. I spent $30 for lace at 66^ per yard. How many 
 yards did I buy ? 
 
 12. For $ 36 how many pairs of rubber shoes can be 
 bought at 37^ per pair? 
 
 13. Oats are 62^ per bushel. How many bushels will 
 $40 buy? 
 
 14. A farmer pays 87^ per bushel for seed rye. If his 
 bill amounted to $ 21, how many bushels did he purchase ? 
 
United States Money. 93 
 
 15. A storekeeper sold $ 33 worth of collars, at 16f ^ 
 each. How many did he sell ? 
 
 16. At the rate of 3 for 50^, how many collars can be 
 bought for $25? 
 
 17. Corn is worth 20^ per can. How many cans will 
 cost $ 32 ? 
 
 18. Find the cost of 35 yards of cloth, at $ 1.25 per yard. 
 
 19. At $ 1.25 per yard, how many yards of cloth can be 
 bought for $ 35 ? 
 
 20. How many pairs of gloves, at $ 1.75 per pair, will 
 cost $ 28 ? 
 
 21. When coal is $ 5.25 per ton, how many tons can be 
 bought for $ 42 ? 
 
 22. Cost of 16 pairs of shoes at $ 2.75 ? 
 
 23. 33 jackets at $ 3.33J ? 24. 18 yards cloth at $ 2.16| ? 
 
 25. Paid $ 26 for cloth at $ 2.16| per yard. How many 
 yards did I buy ? 
 
 26. Find the cost of 16 pairs of skates at $ 1.87^- per pair. 
 
 27. If sheep cost $3. 12^ each, how many can I get 
 for $ 75 ? 
 
 28. How many 25-cent balls can be bought for $ 8.75 ? 
 
 29. Divide 775 by 25. 30. Divide $8.25 by 75^. 
 
 31. How many square feet are there in a lot 96 feet long, 
 100 feet wide ? In a lot 96 feet long, 25 feet wide ? 
 
 32. Find the total cost of 32 head of cattle at $ 75 per 
 head. 
 
 33. How much must be paid for 32 cows at $37.50 each ? 
 
 34. If sheep are worth $3.75 each, how much will a 
 farmer receive for 32 sheep ? 
 
 35. If a train goes at the rate of 25 miles per hour, how 
 many hours will it take to go 675 miles ? 
 
16 oz. 
 Xl7 
 
 112 
 16 
 
 272 oz. 
 Add 4 oz. 
 
 276 oz. 
 4qt. 
 
 37 gal. 3 qt. 
 
 151 qt. 
 
 94 , Chapter Two. 
 
 DENOMINATE NUMBERS. 
 
 Note. — For the tables of Denominate numbers used in these les- 
 sons, see section 93, pages 43-44. 
 
 168. Written Exercises. 
 
 1. Change 17 lb. 4 oz. to ounces. 
 
 Since there are 16 ounces in 1 pound, in 
 17 pounds there are 272 ounces, etc. 
 
 Add 4 oz. 
 
 Arts. 
 
 2. Change 37 gal. 3 qt. to quarts. 
 In this example, we are to multiply 4 
 
 quarts (the number in a gallon), by 37, and 
 
 to add 3 quarts to the product. In practice, \§± q^ Ans. 
 
 however, 4 is taken as the multiplier, and 
 
 the three quarts are added in. We say 4 sevens are 28, and 3 are 31, 
 
 writing the 1 ; 4 threes are 12, and 3 are 15. 
 
 3. Change 45 bushels to quarts. 
 
 Write as here shown, placing 4 pk. 8 qt. 
 
 above pecks the number of pecks 45 k u . Opk. qt. 
 
 in a bushel, and above quarts the jca pk # 1440 qt. 
 
 number of quarts in a peck. Multi- . .. 1At i , 
 
 ply 4 pecks by 45, and write the 
 product, 180 pecks, in the proper column ; multiply 8 quarts by 180, etc. 
 
 Change : 
 
 4. 63 qt. 1 pt. to pints. 
 
 5. 27 bu. 3 pk. to pecks. 
 
 6. 48 pk. 7 qt. to quarts. 
 
 7. 84 pk. to pints. 
 
 8. 7 mi. 60 rd. to rods. 
 
 9. 13 hr. 20 rnin. to minutes 
 &0. 18 wk. 3 da. to days,, 
 
Denominate Numbers. 95 
 
 11. Change 151 quarts to gallons and quarts. 
 Write above 151 quarts the number of quarts 4 Q ^ 
 
 in a gallon. Divide 151 by 4 to obtain the num.- 15T~ot~ 
 
 ber of gallons, 37. Write the remainder, 3, in ^Z ;~ 7; — 7~ A 
 
 f . . 6 .' ' ' 37 gal. 3 qt. An& 
 
 the column of quarts. 
 
 12. Change 228 inches to yards and feet. 
 
 Divide the number of inches, 228, by 12, to 3 f^ ^2 in 
 
 obtain the number of feet, 19. Write this to -in ft 228 in 
 
 the left of 228 inches. Reduce to yards by divid- a v a 1 f 4- An* 
 ingby3. 
 
 13. 87 pints to quarts and pints. 
 
 14. 250 feet to yards and feet. 
 
 15. 1650 rods to miles and rods. 
 
 16. 864 hours to weeks. 
 
 17. 296 quarts to bushels and pecks. 
 
 18. 315 ounces to pounds and ounces. 
 
 19. 743 months to years and months. 
 
 20. 15,000 minutes to days and hours. 
 
 21. Add 3 ft. 6 in., 9 ft. 5 in., 12 ft. 3 in. 
 
 Write the feet in one column and the inches 3 ft. 6 in. 
 in another. The sum of the column of inches is 9 ft. 5 in. 
 
 14 inches, or 1 foot 2 inches. Write 2 inches, 12 ft. 3 in. 
 
 and carry 1 foot to the next column. 25 ft. 2 in. Ans. 
 
 22. 30 min. 15 sec. + 30 min. 18 sec. + 45 min. 24 sec. 
 
 23. 9 yr. 3 mo. -f 18 yr. 7 mo. + 22 yr. 2 mo. 
 
 24. 19 wk. 4 da. + 7 wk. 5 da. + 8 wk. 
 
 25. 9 mi. 169 rd. + 84 rd. -f- 3 mi. 67 rd. 
 
 26. 7 yd. 1 ft. + 33 yd. -f- 19 yd. 2 ft 
 
 27. 18 gal. 1 qt. + 16 gal. 2 qt. + 15 gal. 3 qt 
 
 28. 5 pk. 3 qt. + 6 qt. + 7 pk. 1 qt. 
 
$6 Chapter Two. 
 
 29. 24 bu. 3 pk. + 24 bu. 3 pk. + 24 bu. 3 pk. 
 
 30. 12 qt. 1 pt. + 12 qt. 1 pt. + 12 qt. 1 pt. + 12 qt. 1 pt 
 
 31. Multiply 12 qt. 1 pt. by 7. 
 
 7 times 1 pint = 7 pints =3 quarts 1 pint. Write 12 at 1 pt 
 1 pint in the proper column, and carry 3 quarts. „ n 
 
 7 times 12 quarts = 84 quarts. Carrying 3, we <*» 7 -j , a 
 get 87 quarts. 
 
 32. 12 qt. 1 pt. x 4. 37. 15 wk. 3 da. x 5. 
 
 33. 24 bu. 3 pk. x 3. 38. 7 yr. 3 ino. x 10, 
 
 34. 5 pk. 3 qt. x 9. 39. 40 min. 35 sec. x 2. 
 
 35. 18 gal. 1 qt. x 8. 40. 9 ft. 5 in. x 12. 
 
 36. 33 yd. 1 ft. x 6. 
 
 41. From 25 ft. 3 in. take 18 ft. 7 in. 
 
 Take 7 inches from 1 foot 3 inches, or 25 ft. 3 in. 
 15 inches. Carry 1 foot to 18 feet, making — 18 ft. 7 in. 
 19 feet, etc. 6 ft. 8 in. Ans. 
 
 42. 50 min. 13 sec. — 27 min. 30 sec. 
 
 43. 12 yr. 1 mo. — 5 yr. 11 mo. 
 
 44. 50 wk. 4 da. - 18 wk. 6 da. 
 
 45. 25 ft. -18 ft. 7 in. 
 
 46. 33 yd. 1 ft. - 18 yd. 2 ft. 
 
 47. 240 gal. 1 qt. - 94 gal. 2 qt. 
 
 48. 83 pk. 3 qt. - 59 pk. 1 qt. 
 
 49. 170 bu. 1 pk. - 85 bu. 2 pk. 
 
 50. 135 qt. 1 pt. - 67 qt. 1 pt. 
 
 51. Divide 87 gal. 2 qt. by 5. 
 
 Dividing 87 gallons by 5, we get 17 gallons, 
 and 2 gallons remainder. Change 2 gallons 5 )87 gal. 2 qt. 
 to 8 quarts, add in 2 quarts, making 10 quarts. 17 gal. 2 qt. Ans. 
 Dividing 10 quarts by 5, we get 2 quarts. 
 
Denominate Numbers. 97 
 
 52. 50 min. 35 sec. -7- 5 57. 387 gal. -*- 6 
 
 53. 156 yr. 9 mo. ^ 9 58. 222 bu. 3 pk. -*- 9 
 
 54. 73 wk. 2 da. ^ 3 59. 150 qt. -*- 4 
 
 55. 50 mi. 135 rd. -f- 7 60. 75 bu. -f-8 
 
 56. 253 yd. 1 ft. -- 10 
 
 61. Divide 87 qt. by 43 qt. 1 pt. 
 
 Change the divisor to 87 pints. 43 ^ ^ p^ \ gj ^ 
 Change the dividend to the same de- g<r ^ )\J4. D t 
 
 nomination. 87 pints is contained 2 a Ans. 
 
 times in 174 pints. 
 
 62. 50 min. 35 sec. -r- 10 min. 7 sec. 
 
 63. 78 bu. -j- 9 bu. 3 pk. 
 
 64. 5 lb. 1 oz. -f- 9 oz. 
 
 65. 14 ft. 2 in. -s- 1 ft. 5 in. 
 
 169. Oral Exercises. 
 
 1. 3 pints is what part of a gallon? 
 
 (3 pints is what part of 8 pints ? ) 
 
 2. What part of a gallon is 1 qt. 1 pt. ? 
 
 3. Find the ratio of £ to |. 
 
 (Divide 10 fifteenths by 9 fifteenths.) 
 
 4. Find the ratio of f to f . 
 
 5. How many square feet in a rectangle 12 feet long, 13 
 feet wide? 
 
 6. J of a day is how many hours and minutes ? 
 
 7. 14 ounces is what part of 2 pounds ? 
 
 8. I foot is what part of a yard ? 
 
 9. A strip of tape 3 yards long is cut into four equal 
 pieces. How many feet and inches are there in each piece? 
 
 10. At $ 30 per month, how much rent will be paid in 
 i year, 8 months ? 
 
98 Chapter Two. 
 
 11. 2 \ months is what part of a year ? 
 
 12. At -f of a dollar per pound, how much tea can I get 
 for $1? 
 
 13. How many square yards in a room 15 feet long, 18 
 feet wide ? 
 
 14. A lot is 25 feet by 100 feet. How many feet of 
 fence will it take to enclose it ? 
 
 15. 1 pk. 1 qt. is what part of a bushel ? 
 
 16. 15 is what part of 4 dozen ? 
 
 17. Reduce ff to lowest terms. 
 
 170. Written Problems. 
 
 1. Add 4 da. 6 hr., 9 da. 11 hr., 3 da. 7 hr. 
 
 2. What part of a week is 1 da. 18 hr. ? 
 
 3. If a man receives $ 60 interest per year, how much 
 will he receive in 3 yr. 1\ mo. ? 
 
 4. Reduce 3 da. 18 hr. to minutes. 
 
 5. How many days and hours are there in 8100 
 minutes ? 
 
 6. -ff^ of a day is how many hours ? 
 
 7. How many hours and minutes in .4 day ? 
 
 8. A man receives $1456 per year of 52 weeks. What 
 is his salary per week? 
 
 9. Find the cost of 1 bu. 1 pk. 1 qt. of potatoes at 8 
 cents per half-peck. 
 
 10. A piece of meat weighing 27 lb. 12 oz. is divided 
 among 6 persons. How many pounds and ounces does 
 each receive? 
 
 11. How many bushels and pecks are there in 5 bags, 
 each containing 1 bu. 1 pk ? 
 
 12. How many gallons, quarts, and pints of ice-cream 
 will be needed to give a half-pint to each one of 67 persons? 
 
Measurements. 99 
 
 13. Find the cost of 7 lb. 10 oz. of tea at 40 cents per 
 pound. 
 
 14. From a bin containing 20 bushels of wheat there 
 were sold 10 bu. 3 pk. How much remained ? 
 
 15. How many yards in a mile ? How many feet ? How 
 many inches ? 
 
 16. A field is 16 rods long, 12 rods wide. How many 
 square yards does it contain? What is the perimeter in 
 rods? In feet? 
 
 17. How many rails each 30 feet long will be needed for 
 a single track road (two tracks) 40 miles long ? 
 
 18. A boy steps 33 inches. How many steps will he take 
 in going 2 miles ? 
 
 19. December 20 the sun rises at Boston at 7.26 a.m. and 
 sets at 4.30 p.m. How long is it between sunrise and sun- 
 set? How much longer is the day at Charleston, S. C, 
 where the sun rises at 6.58 a.m. and sets at 4.57 p.m. ? 
 
 20. On June 21 the sun rises at Boston at 4.23 a.m. and 
 sets at 7.40 p.m. On the same day it rises at Charleston at 
 4.53 a.m. and sets at 7.11 p.m. What is the length of the 
 day at each place ? 
 
 MEASUREMENTS. 
 
 171. How many square yards in a floor 6 yards long, 5 
 yards wide ? 
 
 How many square yards in a ceiling 18 feet long, 15 feet 
 wide? 
 
 172. Written Exercises. 
 
 1. How many square yards are there in a piece of 
 ground 60 feet long and 30 yards wide ? 
 
 60 feet = 20 yards. In a plot 20 yards by 30 yards the area = 1 
 square yard x 20 x 30 = 600 square yards, Ans. 
 
ioo Chapter Two. 
 
 Calculate the number of square yards in the following. 
 First reduce each side to yards. 
 
 2. 18 yd. by 21 yd. 7. 33 ft. by 36 yd. 
 
 3. 54 ft. by 63 ft. 8. 27 ft. by 96 ft. 
 
 4. 72 in. by 108 in. 9. 54 ft. by 72 in. 
 
 5. 19 yd. by 47 yd. 10. 48 ft. by 45 ft. 
 
 6. 67 yd. by 89 yd. 11. 54 in. by 72 ft. 
 First indicate the operations ; then cancel. 
 
 12. Find the number of square yards in a room 18 ft 
 4 in. long, 22 ft. 6 in. wide. 
 
 18 ft. 4 in. = 18 1 ft. = Mi yd. = ^ yd. 
 3 3 9 
 
 22 ft. 6 in. = 22} ft. = ^ yd. = ^ yd. 
 
 O D 
 
 5 
 
 Area = 1 sq. yd. x 5§ x ** . Canceling, §L*i2 = ?75 = 45| 
 
 96 JfxG 6 * 
 
 Ans. 45 1 sq. yd. 
 
 13. How many square yards in a room 13 ft. 1 in. long, 
 27 ft. wide ? 
 
 13 ft. 1 in. = 157 in. = ^ yd. 27 ft. = 9 yd. 
 
 . Ar ea =1 s, y d.xfx0. 80 ^f-«* 
 
 4 Ans. 39£ sq. yd. 
 
 14. How many square inches in 12 panes of glass, each 5 
 inches long, 7 inches wide ? 
 
 15. A piece of cloth is 48 yards long, 24 inches wide. 
 How many square yards does it contain ? 
 
 16. A merchant imports 8 pieces of cloth, 36 yards to the 
 piece. How many square yards of cloth are there, if it is 
 32 inches wide ? 
 
 17. A tight board fence 6 feet high surrounds a lot 25 feet 
 front by 100 feet deep. How many square feet of boards 
 in the front fence ? In the back fence ? In each side 
 fence ? In the whole ? (Make diagrams.) 
 
Bills. 
 
 101 
 
 18. A room is 18 feet long, 15 feet wide, l/> feet- high. 
 How many square feet in the floor ? *.'.•?•>•? »* , ,• 
 
 Draw a rectangle to represent the ceiling. Write the dimensions in 
 their proper places, and write in the centre the number of square feet 
 in its surface. Draw diagrams of the four walls ; give dimensions 
 and surface of each. 
 
 19. How many faces has a cube ? If one edge of a cube 
 measures 4 inches, how many square inches are there in 
 the entire surface ? 
 
 Suppose you wish to make a cube out of a single piece of paste- 
 board. Make a drawing to show the shape of the piece needed, 
 without allowing anything for overlapping parts. 
 
 20. The United States government charges a duty of 4^ 
 per square yard on imported cotton cloth. What duty must 
 the importer pay on a piece containing 24 yards, f yard 
 wide? 
 
 21. What will be the cost at $ 1 per square yard for 
 making a sidewalk 12 feet wide and 30 feet long? 
 
 BILLS. 
 173. New York, Oct. I, 1904. 
 
 Mrs. William Johnson, 
 
 Bought of Furey & Company. 
 
 
 1904 
 
 
 
 
 
 
 
 Aug. 
 
 18 
 15 
 
 19 
 
 1,4 yd. Carpet 
 8 Oak Chairs 
 1 Rocker 
 
 18 yd. Oil-cloth 
 
 f .90 
 
 1.75 
 
 .50 
 
 12 
 
 — 
 
 
 
 
 27 
 
 1 Parlor Suit 
 
 
 75 
 
 — 
 
 
 
 Sept. 
 
 19 
 
 6 Kitchen Chairs 
 
 .75 
 
 
 
 
 
 
 
 1 Table 
 
 
 4 
 
 50 
 
 
 
 
 26 
 
 86 yd. Matting 
 
 M 
 
 
 
 9 
 
 
 
 
 
 
io2 Chapter Two. 
 
 ,1.- Copy 'jthe . bill on the preceding page. Supply the 
 missing amounts. 
 
 2. Charles W. Wise has bought the following goods of 
 Thos. F. Farley & Co. : 
 
 Jan. 3, 1904, 50 pounds of sugar, at h\$ ; 4 pouDds of tea, 
 at 62-£/. Jan. 4, 10 pounds of coffee, at 32|^ ; 2 barrels of 
 flour, at % 5.75. Jan. 9, 24 bars of soap, at 16|^ ; 42 pounds 
 of starch, at 8^. 
 
 Make out a bill dated Feb. 1, 1904. 
 
 3. Make out a bill for the following articles bought dur- 
 ing March and April. Supply the names of buyer and 
 seller, also the dates : 
 
 23£ yards of silk, at 80^; If yards of lace, at $2.40; 
 64 yards of muslin, at 6 J^ ; 8 spools of sewing silk, at 7^ ; 
 4 pairs of stockings, at 65$; 6 yards of linen, at 87|^; 
 -J- dozen collars, at $ 2.10. 
 
 4. Make out a bill for the following goods, bought 
 June 15: 
 
 3 cases of torpedoes, at $ 2.20 ; 12 boxes of firecrackers, 
 at $1.62-J-; 3 gross pinwheels, at $1.35; 5 gross sky- 
 rockets, at $ 3.25 ; 2 dozen balloons, at $ 2.25 ; 45 lanterns, 
 at 9£ 
 
 Note. — The date is written only at the top of the bill when all the 
 articles are bought at one time. 
 
 SHORT METHODS — REVIEW. 
 
 If the school is to train for life, it must accustom pupils to use 
 modes of calculation followed in the business world. 
 
 In their previous work, pupils have employed $^ instead of 25?, 
 -$£ instead of 12^?, etc. They have, for instance, found the cost of 
 32 pounds at 25? per pound, by multiplying $£ by 32. While the 
 result in example 4 is the same, 25 pounds at 32?, the analysis is dif- 
 ferent. The following is suggested : 
 
 100 pounds at 32? would cost $ 32, \ of 100 pounds would cost \ of 
 432, or #8. 
 
 The rule generally given for the multiplication by 26 is to annex 
 
Review. 103 
 
 two ciphers to the multiplicand and to divide by 4. In practice, the 
 ciphers need not be annexed actually or mentally. To multiply 19 by 
 25, the pupil divides 19 by 4, getting 4 for quotient ; to this he adds 75 
 for the 3 remainder, getting 475 for the result. 
 
 174. Oral Exercises. 
 
 1. Multiply by 25 : 
 
 16, 19, 21, 23, 25, 29, 33, 36, 42, 48. 
 
 2. How many square feet in a lot 84 feet long, 25 feet 
 wide? 
 
 3. What is the weight of 25 barrels of flour, each 
 weighing 196 pounds? 
 
 4. Find the cost of 25 pounds of coffee at 32^ per 
 pound. 
 
 5. What will a woman have to pay for 25 yards of silk 
 at $1.60 per yard? 
 
 6. A man sold 25 cows at $44 each. How much did 
 he receive for them? 
 
 7. Multiply 64 by 12i 
 
 8. Find the cost of 12^- bushels of wheat at 96^ per 
 bushel. 
 
 9. At $12.50 per barrel, how much should I pay for 
 56 barrels of pork? 
 
 10. How many pens in 12^- gross? (144 to gross.) 
 
 11. Find the cost of 121 pounds of tea at 56 jt per pound. 
 
 12. How many square yards in a field 96 yards long, 75 
 yards wide? 
 
 175. Sight Exercises, 
 
 To multiply 427 by 25 the pupil considers 4 as the divisor. He 
 writes on his paper 1, then 0, then 6, annexing 75 for the 3 remaining. 
 Ans. 10,675. 
 
 Example 5 : 25 x 686 is the same as 686 x 25. 
 
 Example 9 : To multiply by 250, consider three ciphers annexed to 
 the multiplicand. 
 
104 Chapter Two. 
 
 Example 11 : Divide by 8, annexing two ciphers to the quotient 
 when there is no remainder. Annex 12£ when the remainder is 1 ; 25, 
 when the remainder is 2 ; etc. 
 
 Example 19: Consider three ciphers annexed and divide by 8. 
 
 Write only the answers : 
 
 1. 
 
 837 x 25 
 
 8. 
 
 25 x 2174 
 
 15. 
 
 12 J x 1084 
 
 2. 
 
 763 x 25 
 
 9. 
 
 837 x 250 
 
 16. 
 
 12£ x 2196 
 
 3. 
 
 934 x 25 
 
 10. 
 
 763 x 250 
 
 17. 
 
 12£ x 3670 
 
 4. 
 
 508 x 25 
 
 11. 
 
 864 x 12£ 
 
 18. 
 
 \2\ x 6281 
 
 5. 
 
 25 x 686 
 
 12. 
 
 776 x 121- 
 
 19. 
 
 864 x 125 
 
 6. 
 
 25 x 301 
 
 13. 
 
 236 x 12£ 
 
 20. 
 
 776 x 125 
 
 7. 
 
 25 x 1039 
 
 14. 
 
 404 x 12\ 
 
 21. 
 
 125 X1020 
 
 176. Sight Exercises. 
 
 Pupils do much unnecessary work in rearranging numbers and in 
 writing fractions over again with a common denominator. A few of 
 these examples should be written on the blackboard from time to time, 
 and the teacher should require the pupil to write nothing but the 
 answers. 
 
 Add: 
 
 
 
 
 1. 3£ + 5£ 
 
 4. 11J + 4J. 
 
 7. 
 
 8i + 6J 
 
 2. 4£ + 8f 
 
 5. 7f + 9^ 
 
 8. 
 
 15| + 8J 
 
 3. 9| + 7f 
 
 6. 5f + 2| 
 
 9. 
 
 9f + 5£ 
 
 177. Subtract at 
 
 sight : 
 
 
 
 10. 23^~19| 
 
 14. 9£-2* 
 
 18. 
 
 35J -Si 
 
 11. 16f - 9f 
 
 15. lOJ-5^ 
 
 19. 
 
 11* -6* 
 
 12. 18f - S\ 
 
 16. 14J-8A 
 
 20. 
 
 43A-8J 
 
 13. 15f - Si 17. 27£-7| 21. 50J- - 4f 
 
Review. 105 
 
 178. Multiply at sight, 18| x 4. 
 
 f x 4 = 3. 4 eights are 32, and 3 are 35 (put down 5). 4 ones 
 are 4 and 3 are 7 (put down 7). Ans. 75. 
 
 The pupil should write only the answers. 
 
 22. 
 
 27} x 10 
 
 27. 
 
 15} x 3 
 
 32. 
 
 37} x 3 
 
 23. 
 
 33} x 12 
 
 28. 
 
 13} x 4 
 
 33. 
 
 45|x5 
 
 24. 
 
 16f x8 
 
 29. 
 
 20} x 11 
 
 34. 
 
 23} x 4 
 
 25. 
 
 17|x8 
 
 30. 
 
 40f x5 
 
 35. 
 
 17} x 6 
 
 26. 
 
 19|x6 
 
 31. 
 
 16|x7 
 
 
 
 179. Divide at sight. 
 
 "When the divisor is an integer less than 12, the pupil should not 
 reduce the mixed number in the dividend to an improper fraction, 
 by 3, the pupil first gets the whole number of the 
 
 Ans. 89}. 
 
 In dividing 248£ by 4, the pupil obtains 62 as the quotient of 248 
 by 4 ; he then finds \ of \, which is \. Ans. 62} 
 
 In dividing 202£ by 5, the quotient is 40, and the remainder is 2} 
 which is reduced to f . One-fifth of this is } Ans. 40} 
 
 In dividing 183| by 6, the quotient is 30 with a remainder of 3£, 
 
 Ans. 30 J|. 
 
 46. 7 )97^ 
 
 47. 10 )87} 
 
 48. 4 )66} 
 
 49. 3 )94} 
 
 50. 5 )83} 
 
 SIGHT APPROXIMATIONS. 
 
 180. Give approximate answers in whole numbers. Solve 
 for the exact answers. 
 
 1. 17 & X 3fJ; or, about 17 x about 4. 
 
 2. 25^ -7- 1|; or, about 25 -r- J nearly. 
 
 or^. *of¥ = lf 
 
 
 
 36. 3)45f 
 
 41. 
 
 8)37} 
 
 37. 4)56} 
 
 42. 
 
 9)48} 
 
 38. 12)36} 
 
 43. 
 
 6)25} 
 
 39. 5)72} 
 
 44. 
 
 7)10} 
 
 40. 11)834- 
 
 45. 
 
 6)751 
 
io6 
 
 
 Chapter Two. 
 
 
 3. 
 
 61 x H 7. 
 
 799f|-=-99H 
 
 4. 
 
 300^ + llff 8. 
 
 7t%x7A 
 
 5. 
 
 86|xH *• 
 
 TixllA 
 
 6. 
 
 35J + 3H 10. 
 
 M* x f 
 
 181. Give answers in whole numbers : 
 
 1. 5.75 x 9.999; or, 5.75 x 10 nearly. 
 
 2. 24.002 + .4999 ; or, 24 -?- nearly f 
 
 3. 25.125x11.834 7. 799.9 x .103 
 
 4. 36.843 -j- 6.105 8. 7.999x7.999 
 
 5. 86.4 -.983 9. 7.001x12.003 
 
 6. 32.04x5.001 10. 64.001 -f- .249 
 
 182. Give the cost, approximately, of : 
 
 1. 49 horses at $ 199 each. ($200 x 49.) 
 
 2. 199 yd. 2 ft. 11 in. of cloth at $2.50 per yard. 
 
 3. 3 lb. 15 oz. of butter at 25? per lb. 
 
 4. 398 coats at $ 12 each. 
 
 5. 7 bu. 3 pk. 7 qt. potatoes at $2 per bushel. 
 
 6. 798 base-balls at 25 cents each. 
 
 7. 19 gal. 3 qt. 1 pt. alcohol at $2.49 per gallon. 
 
 8. 995 lb. tea at 59f cents per pound. 
 
 9. 7 houses at $4995 each. 
 
 10. 507 pounds of hay at 99 cents per 100 pounds. 
 
 183. Oral Eeview Exercises. 
 
 1. What is | of 60 ? fof35? 
 
 2. A man sold a boat for $ 8, which was \ of what it 
 cost him. What did it cost him ? 
 
 3. A man having $35, gave away -J- of it. How much 
 had he left ? 
 
Review. 107 
 
 4. How many inches are there in f of a yard ? f of a 
 yard ? -J of a yard ? 
 
 5. If 6 eggs cost 12 cents, what will 5 dozen cost ? 
 
 6. How much is f less \ ? £ less i ? 
 
 7. Change to improper fractions : 7$, 9J, 6f, 2^, 6£. 
 
 8. How many apples at 2^ apiece are worth as much as 
 
 4 peaches at 5fi apiece ? 
 
 9. Which is the greater and how much : f of $ 24, or | 
 of I 25 ? 
 
 10. Change to mixed numbers : &£-, A^, -fj, - 4 /, -^. 
 
 11. There are 45 pupils in school and -J of them are girls. 
 How many are boys ? 
 
 12. Add 8J and 7$. 5f and 1\. 
 
 13. If it takes 5 men 15 days to do a piece of work, how 
 long will it take 10 men to do it ? 
 
 14. What will 2 bushels of corn cost, if £ peck costs 15 
 cents ? 
 
 15. If it costs 25 cents to set one shoe, what will it cost 
 to shoe a span of horses all around ? 
 
 16. Bought 5 yards of ribbon at 16^, and 3 yards of linen 
 at 75^, and gave a two-dollar bill. What was my change ? 
 
 17. If 7 yards cost 84^, how many yards can be pur- 
 chased for $1? 
 
 18. If 6 oranges cost 15^, how much will 8 cost ? 
 
 19. 1-J- pecks of peanuts cost $ 0.48 ; what will one quart 
 cost? 
 
 20. Two boys walked in opposite directions ; one walked 
 
 5 miles an hour, the other 4 miles an hour. How far apart 
 were they in six hours ? 
 
 21. If I of a yard of cloth cost 6^, how much cloth can 
 be bought for 40? ? 
 
108 Chapter Two. 
 
 22. At £ a dollar per day for board, how many days' 
 board can you get for $ 7.50 ? 
 
 23. Charles picked £ peck of berries, William £ peck, 
 and Alfred £ peck. How much did they all pick ? 
 
 24. How much more is } of 80^ than f of 75^ ? 
 
 25. A boy bought 3^ pounds of butter for his mother. 
 How many ounces did he buy ? 
 
 26. If a man is 50 years old now, how old was he 22 
 years ago ? 
 
 27. Mary works 4 hours and 40 minutes, and Nellie 
 works 2 hours and 20 minutes. How many hours do they 
 both work ? 
 
 28. If you should receive 15 cents at one time, 26 cents 
 at another time, and 14 cents at another time, how much 
 would you receive in all ? 
 
 29. If you had f of a dollar, and should buy a pound of 
 soda for 8^ and a pound of tea for 45^, how much would 
 you have left ? 
 
 30. If you give a boy $ 10, how many mills do you give 
 him? 
 
 31. 50-12-9-19 = 
 
 32. 72—7 times 9 = what number? 
 
 33. 45 is how much less than 5 times 12? 
 
 34. (f of 80) + 25 = 
 
 35. (35 + 15) -(14 -9) = 
 
 36. $ = how many sixths ? 
 
 37. 2\ — how many fourths? 
 
 38. Give the exact divisors of 20. 40. From \ take J. 
 
 39. Give the three factors of 30. 41. 2\ + \ — \ = 
 42. At 12 cents a dozen, what will a gross of buttons cost? 
 
Review. 109 
 
 43. How many inch cubes will exactly cover a square 
 foot of surface? 
 
 44. What does f of anything mean ? 
 
 45. 1 gallon 2 quarts and 1 gallon 1 quart are how many 
 quarts ? 
 
 46. If 4 yards of muslin cost 48 cents, how much will 
 one-third of a yard cost ? 
 
 47. Paid $4.86 for 6 bushels of rye. What was the price 
 per bushel ? 
 
 48. Bought 3 dolls at 49 cents each. Total cost ? 
 
 49. If 12 hats cost $ 7, what will be paid for 36 hats ? 
 
 50. If 2 pounds and 5 ounces butter cost 74 cents, what 
 will be the cost of 3 pounds and 2 ounces ? 
 
 51. How many bottles holding 1\ pints will be needed to 
 contain 2\ gallons ? 
 
 52. A bag of flour contains \ of a barrel of 196 pounds. 
 How many pounds does the bag contain ? 
 
 53. What will be the cost of a dozen heads of cauliflower 
 at the rate of 2 for 25 cents ? 
 
 54. Twenty examples are given out. A pupil that cor- 
 rectly answers all receives 100 per cent. What per cent 
 will a pupil receive that solves 16 examples ? 
 
 55. A woman receives $40 interest a year. How much 
 does she receive in 3 years and 6 months ? 
 
 56. A man bought some cows at $ 35 each, and the same 
 number at $ 45 each. What was the average price ? 
 
 57. A girl received 100 credits in each of three studies, 
 and 80 credits in the fourth. What was the total number 
 of credits in the four studies ? What was her average ? 
 
 58. A square floor contains 144 square feet. How many 
 feet long and wide is it ? 
 
no Chapter Two. 
 
 59. -J yard cloth costs $ f . What is the price per yard ? 
 
 Note. — In dividing one fraction by another mentally, reduce both 
 to their common denominator. 
 
 | price of a yard = $ £. ■& price of a yard = $ fo Multiplying by 
 12, 8 times price of a yard = $ 9. 
 
 60. A man owning f of a vessel sells f of his share. What 
 part of the vessel does he then own ? 
 
 61. A barrel contains 196 pounds of flour; the barrel 
 weighs 24 pounds. What is the weight of both? 
 
 62. A family uses 3J pounds of sugar per day. How long 
 will 24^- pounds last ? 
 
 63. How much will be the cost of 3 pounds of 25-cent coffee 
 and 1 pound butter at 36^ ? 
 
 64. If | of a pound of candy costs 30^, what will be the 
 cost of J of a pound ? 
 
 Note. — 6 eighths cost 30^, what will 7 eighths cost ? 
 
 65. A tailor has a piece of cloth containing 2\ yards; he 
 sells If yards. What part of the piece does he sell ? 
 
 66. How many quarts in 1 bushel 1 peck and 1 quart? 
 
 67. Keduce ff to lowest terms. 
 
 68. 24 J- yards of cloth are used for 7 coats. How many 
 yards in each coat ? 
 
 69. If cloves are worth 20^ per \ pound, how much will 
 be paid for 7 ounces? 
 
 70. At 3 oranges for 5^, what will be the cost of 1^ 
 dozen oranges? 
 
 71. My purchase amounts to $1.29. I give the store- 
 keeper a $2 bill. How much change do I receive? 
 
 72. A bushel of nuts was sold for 5^ per quart. How 
 much money did it bring ? 
 
Review. ill 
 
 73. How many days in the summer months, June, July^, 
 and August ? 
 
 74. John had 40 cents. After earning 24 more, he spent 
 his money for marbles at 4 cents each. How many did he 
 buy? 
 
 75. George was sent to the store with 50^. He bought 
 6 pounds of rhubarb at 2^ a pound, and two bunches of 
 radishes at 5^ a bunch. How much money had he left ? 
 
 76. At $ 10 a ton what will be the cost of 1000 pounds ? 
 
 77. There are 16 rooms in a building with 50 desks in a 
 room. How many desks in all ? 
 
 78. Edgar earned $2.75 one week, and $2.50 the next 
 week. How much did he earn in both weeks ? 
 
 79. $ 6 is f of how many dollars ? 
 
 80. Charles began work at 2.45 p.m. and stopped at 
 5.15 p.m. How long did he work ? 
 
 81. 29 + 18 + 30 + 9 + 8 + 7=? 
 
 82. ^ of 22 is how many times 4 ? 
 
 83. Bought a horse for $45 and a saddle for $35, and 
 then sold them, gaining $20. For how much were they 
 sold? 
 
 84. Add these numbers : 12, 15, 9, 13, 11, 7, and 24. 
 
 85. If you buy 6 yards of tape at 7 cents a yard, and 
 4 yards of silk at 7 dollars a yard, what will you give for 
 both tape and silk ? 
 
 86. Bought 8 firkins of butter for $72, and gave 2 of 
 them for 9 yards of cloth. What was a yard of the cloth 
 worth ? 
 
 87. Mr. Brown mixed 3 pounds of black tea worth 40 
 cents a pound with 1 pound of 60-cent green tea. What is 
 the mixed tea worth a pound ? 
 
112 Chapter Two. 
 
 184. Written Eeview Exercises. 
 
 1. In 6987 days how many minutes? 
 
 2. Find the cost of 1,588,000 pounds of coal at $5.98 
 a ton. 
 
 3. How many cords of wood, at $ 7.85 a cord, can be 
 purchased for $ 59,730.65 ? 
 
 4. Divide $ 3,245,530 by 468. 
 
 5. Bought 8 bushels 3 quarts valuable seed at seven 
 dollars and eight cents a quart. How much did the seed 
 cost? 
 
 6. What is the cost of 19 gallons 2 quarts of cologne at 
 90^ a quart? 
 
 7. Divide f of $ 60,800 equally among 75 persons. 
 
 8. Bought a house for $23,650, and land for $73,640. 
 For how much must I sell them to gain $ 4500 ? 
 
 9. Find the greatest common divisor of 45 and 135. 
 
 10. A grocer bought 7200 gallons of oil, one-third of it 
 leaked out, and he sold the remainder at 25 cents a gallon. 
 How much did he receive for it ? 
 
 11. From two and four-tenths yards take .445 of a yard. 
 
 12. Add the numbers from 490 to 505 (inclusive). 
 
 13. If 56 pounds of sugar cost $ 3.08, what will 24 pounds 
 cost? 
 
 14. If 42 gallons 3 quarts 1 pint of cream cost $ 27.44, 
 what will 32 pints cost ? 
 
 15. A man's bill at a provision store was $ 6.66. He 
 had bought two pecks of peas for $0.54 and some beans for 
 $ 0.36. The rest of the bill was for sirloin steak at $ 0.32 
 per pound. How many pounds of meat had he bought? 
 
 16. From 1,890,070 take 990,979. 
 
 17. If a train travels 45 miles per hour, how far will it go 
 from half-past 9 in the morning to a quarter of 7 in the 
 evening ? 
 
Review. 113 
 
 18. A mechanic saved $ 35 per month for 11 months, and 
 $ 20 the twelfth month. His expenses averaged $ 3 each 
 day of the year. What were his daily wages for the 300 
 days he worked ? 
 
 19. A 160-acre farm consists of 5 fields. The first con- 
 tains 17.38 acres, the second 29.4 acres, the third 35.073 
 acres, the fourth 25.875 acres. How many acres are there 
 in the fifth field ? 
 
 20. How many seconds in 7 hours 15 minutes ? 
 
 21. Find the total cost of 2 dozen rockets at $7.50 per 
 gross, 3 dozen Roman candles at $9.60 per gross, and 24 
 dozen pin wheels at $ 1.35 per gross. 
 
 (1 gross = 12 dozen.) 
 
 22. Three lots of tea were sold for $330. The second 
 contained twice as much as the first, and the third three 
 times as much as the first. The third lot contained 330 
 pounds. Find the selling price of the tea per pound. 
 
 23. A barrel of molasses contained 40 gallons. One- 
 fourth of it leaked out. If the molasses cost 45 cents per 
 gallon, what price must be charged for the remainder so 
 that there will be no loss? 
 
 24. If 12£ dozen rockets cost $5.75, what will 15 dozen 
 cost? 
 
 25. Show by drawings that -J = -j^, and that f = |. 
 
 26. Write the first five prime numbers that are greater 
 than 7. 
 
 27. Find the greatest common divisor of 1220 and 2013. 
 
 28. Find the least common multiple of 12, 15, 14, 6, 21, 
 21, and 24. 
 
 29. Find the prime factors of 1140. 
 
 30. Add 8fc -£, f and } of 7*. 
 
 31. From 14^ pounds of butter, 5f pounds were sold to 
 one person and 3£ to another. How much remained ? 
 
H4 Chapter Two. 
 
 32. A man bought 4 bushels of wheat for 3f dollars. 
 What fraction of a dollar did one bushel cost? 
 
 33. If f of a bushel of oats will last a horse one day, 
 how long will 4J- bushels last? 
 
 34. In two months Ann will be 15 years old. How old 
 was she nine months ago ? 
 
 35. A boy has to walk from his home to a house If miles 
 east of his home, from there to a place 2\ miles west of his 
 home, and then home. How far has he to walk ? 
 
 36. I lost f of my money, then found f of what I had 
 lost, and then had 64 cents. How much had I at first ? 
 
 37. Quotient 24J, divisor 3^. What is the dividend? 
 The product is 2|, and one factor is -§. What is the other 
 factor ? 
 
 38. Bought Z\ yards of muslin at 7 cents a yard, 5^- 
 yards of ribbon at 3J cents a yard, and 2\ yards of cloth at 
 $ 1.75 per yard, and gave a ten-dollar note in payment. 
 How much change did I receive? 
 
 39. Write seven million nine thousand nineteen. 
 
 40. A milliner sells 3 pieces of ribbon at 18 cents per 
 yard. They measure 4| yards, 3f yards, and 5^- yards 
 respectively. What does she receive for the ribbon ? 
 
 41. How many feet and inches in T 5 ^- of a yard ? 
 
 42. To make powder, a man mixes 1\ pounds of saltpetre, 
 IfV pounds of sulphur, and as much charcoal as sulphur. 
 How many pounds of powder will there be ? 
 
 43. Four men form a partnership ; the first furnishes -£■ 
 of the capital, the second -f, and the third -j^-. What frac- 
 tion of the capital is furnished by the fourth ? 
 
 44. I pay 15 cents more for a half-pound of tea than I 
 pay for a quarter-pound of the same tea. What is its price 
 per pound ? 
 
Review. 115 
 
 45. After doing £ of a piece of work, a man requires 3 
 days more to finish it. How many hours does he take to do 
 the whole work if he works 8 hours per day ? 
 
 46. If 1 pound 7 ounces of coffee cost 46 cents, what will 
 3 pounds 9 ounces cost ? 
 
 47. Add 6 hours 50 minutes and 17 hours 10 minutes. 
 
 48. 15 men do a piece of work in lOf days. How long 
 would it take 5 men to do the same work? 
 
 49. To make a cloak, 3 yards of cloth 1^ yards wide are 
 required. How much cloth £ yard wide would be required? 
 
 50. In 3 years 4 months a gas company manufactures 
 4,200,000 cubic feet of gas. How many cubic feet are 
 manufactured per year ? 
 
 51. If 2| dozen hats cost $80, what will be the cost of 3 
 hats? 
 
 52. A boy hires a boat at 20 cents per hour. How 
 much should he pay if he uses it from 20 minutes before 9 
 a.m. until 10 minutes past 1 p.m. ? 
 
 53. A and B kill an ox. A takes f and B the remainder. 
 If B's share weighs 361^ pounds, what is the weight of the 
 ox? 
 
 54. A grocer buys 30 dozen eggs at 18 cents per dozen. 
 He sells them at the rate of 15 eggs for 25 cents. What is 
 his profit ? 
 
 55. How many cents in T 5 g- of a dollar ? 
 
 56. What fraction of 18f is 6f ? ' 
 
 Suggestion. — What fraction of 18 is 6 ? Which is the divisor ? 
 
 57. A farmer buys a horse for $140, and sells it at an 
 advance of -fa of the cost. What is the selling price ? 
 
 58. In 1903, A was 36 years old and B was If times 
 as old. In 1894, B was how many times as old as A ? 
 
n6 Chapter Two. 
 
 59. From the sum of 18^ and 25£ take their difference. 
 
 60. If 2f acres of land cost $ 220, what will be the cost 
 of 17 $ acres? 
 
 Note. — Indicate the operations, and cancel. 
 
 61. A can do a piece of work in 6 days, B can do it in 6 
 days, C can do it in 6 days. How long will it take all 
 three working together ? 
 
 62. Find the value of 1 f t£*M . 
 
 63. A man sold a horse for } of its cost, losing $40. 
 What did the horse cost him ? 
 
 64. I have an oblong piece of land which is 96 feet long 
 and 72 feet wide. There are three gateways; one is two 
 feet wide, one is three feet wide, and the other is four feet 
 wide. How many feet of fence will it take to go around 
 the field? 
 
 65. Add: $83.34; $67.58; $50.37; $62.50; $35.75; 
 $62.50; $35.75; $63.81; $67.59; $86.37; $37.50; 
 $15.09; $57.32; $49.63. 
 
 66. A boy bought a suit of clothes for $21, boots for 
 $3.50, overcoat for $15, and gloves for 50 £ Paid for 
 these things in work at $1.25 per day. How many days 
 did he work? 
 
 67. If $36.53 will buy 6£ yards of cloth, how much will 
 ■J- yard cost ? 
 
 68. If two quarts of peaches cost 25^, what will half a 
 bushel cost? 
 
 69. How many geographies at 90^ apiece can be bought 
 for $54? 
 
 70. Find the least common multiple of 6, 24, 32, 48, 96. 
 
 71. Add: 87.5; 7004.3; 500.004; 21,090; 5040.29. 
 
Review. 117 
 
 72. Spent $290 for horses, $286.75 for carriages, 
 $150.80 for harness, and $12.75 for blankets. Gave 4 
 fifty-dollar bills and 2 one-hundred-dollar bills. What did 
 I still owe ? 
 
 73. How many bushels of oats will a span of horses eat 
 in 4 weeks, if they eat 24 quarts a day? 
 
 74. How many bottles, each holding \ pint, will it take 
 to hold 725 gallons and 2 quarts of oil ? 
 
 75. How many pounds of rice at 12^ a pound, will pay 
 for 4 bushels 2 pecks of nuts at 8^ a pint ? 
 
 76. A man had $ 600. He bought a horse for $ 225, a 
 carriage for $ 190.12, and a harness for $ 40.76. He then 
 gave away £ of what he had left. What did he still have? 
 
 77. Find the greatest common divisor of 18, 24, 36. 
 
 78. The least common multiple of 12, 20, and 30. 
 
 79. Find the cost of 18,756 feet, of lumber at $30 per 
 1000 ft. 
 
 80. A field is 14.25 rods long by 7.4 rods wide. What is- 
 its area in square rods ? 
 
 81. A rod is 16.5 feet; how many feet are there in 24 
 rods ? How many rods are there in 231 feet ? 
 
 82. How many marks are there in $100? (A mark is- 
 equal to 23.8 cents.) 
 
 83. Add 3 and 4 tenths, 96 thousandths, 100 and 5 thou- 
 sandths, 27 hundredths. 
 
 84. From 2700 take 27 hundredths. 
 
 85. Multiply 8 and 4 tenths by 9 and 25 hundredths. 
 
 86. Divide 96 and 75 hundredths by 322 and 5 tenths. 
 
 87. A load of hay, at 75 cents per 100 pounds, cost 
 $13.98. What was the weight of the hay ? 
 
xi8 
 
 Chapter Two. 
 
 88. The circumference of a circle is 3.1416 times the 
 diameter. How many inches in the circumference of a 
 circle whose diameter is 20 inches ? 
 
 89. Show by a diagram the number of pieces of wire # 
 yard long that can be made from 4 yards of wire. 
 
 90. Show by a diagram that three-fourths of 1 is equal 
 to one-fourth of 3. 
 
 91. If two-thirds of a yard of material will make an 
 apron, how many aprons can be made from two yards ? Show 
 by a diagram. 
 
 92. A boy paid 6 cents for three-eighths of a pie. What 
 would be the cost of the whole pie at the same rate? 
 Make a drawing. 
 
 93. Seven-eighths of an 
 acre of land is sold for 
 $140. What is the price 
 of an acre? 
 
CHAPTER III. 
 
 PAGES 
 
 Decimals 119 to 132 
 
 Notation and Numeration, Reduction, Addition, Sub- 
 traction, Multiplication, Division. 
 
 United States Monet 132 to 133 
 
 Denominate Numbers 133 to 139 
 
 Reduction, Addition, Subtraction, Multiplication, 
 Division (two denominations). 
 
 Measurements 139 to 144 
 
 Areas of Rectangles, Areas of Right-angled Triangles. 
 
 Bills 144 to 145 
 
 Percentage 145 to 147 
 
 Interest 148 to 152 
 
 Review of Simple Numbers and Fractions . . 152 to 162 
 
 Sight Approximations, Special Drills, Cancellation, 
 Ratio, Short Methods, Review Fractions. 
 
 Miscellaneous Problems 162 to 172 
 
 Oral and Written. 
 
 DECIMALS. 
 
 185. Preliminary Exercises, 
 
 1. Write seven tenths as a common fraction. As a 
 decimal. 
 
 2. Write three hundredths as a common fraction. As a 
 decimal. 
 
 3. Write thirty-one thousandths as a common fraction. 
 As a decimal. 
 
 119 
 
ISO Chapter Three. 
 
 4. Kead the following : 
 
 .3 .09 .043 
 
 .17 .007 .241 
 
 5. Write each of the foregoing decimals as a common 
 fraction. 
 
 186. Notation and Numeration of Decimals. 
 
 1. 7 tenths, or T 7 7 , is written .7. 
 
 2. 3 hundredths, or -j-J^, is written .03. 
 
 3. 53 hundredths, or ^j-, is written .53. 
 
 4. 9 thousandths, or Tir Vu"> * s written .009. 
 
 6. 19 thousandths, or j^fo, is written .019. 
 
 6. 419 thousandths, or ^j^, is written .419. 
 
 7. 67 ten-thousandths, or l0 6 7 00 , is written .0067. 
 
 8. 1031 hundred-thousandths, or y^nHBiF' * s written .01031. 
 
 Note. — In the foregoing examples, it will be observed that the num- 
 ber of places to the right of the decimal point is equal to the number 
 of ciphers in the denominator of the corresponding common fraction. 
 
 187. Write the following as decimals : 
 
 1. 314 ten-thousandths. 
 
 Since y^^ has a denominator containing four ciphers, the decimal 
 roust have four places ; a decimal cipher must be written after the 
 decimal point. Ans. .0314. 
 
 To write a decimal, write the numerator, and from the right, 
 point off as many decimal places as there are ciphers in the 
 denominator, prefixing decimal ciphers, if necessary. 
 
 Note. — Ciphers between the decimal point and the first significant 
 figure of the numerator are called decimal ciphers. 
 
Decimals. 121 
 
 2. 217 hundred-thousandths. 
 
 3. 83 hundredths. 
 
 4. 7 millionths. 
 
 5. 345 thousandths. 
 
 6. 27 ten-thousandths. 
 
 7. 325 and 7 thousandths. 
 
 Ans. 325.007. This is called a mixed decimal, which consists of an 
 integer and a decimal. 
 
 188. The word and is used in reading mixed numbers or 
 mixed decimals to separate the integer from the common 
 fraction or the decimal. 
 
 8. 42 and 56 hundred-thousandths. 
 
 9. 150 and 62 millionths. 
 10. 489 and 3 hundredths. 
 
 189. Eead the following: 
 
 1. .0346. 
 
 Since there are four decimal places, the denominator is 1 with four 
 ciphers, 10000. Ans. 346 ten-thousandths. 
 
 2. 654.15 6. 25.006347 
 
 3. .000209 7. 3.259 
 
 4. 60.0207 8. .002468 
 
 5. 684.007 9. 200.0035 
 
 200.0035 read as 200 and 35 ten-thousandths might be mistaken for 
 235 ten-thousandths. It should be read 200 units and 35 ten-thou- 
 sandths, or 200 whole number and 35 ten-thousandths. 
 
 10. 1000.0006 12. 2300.00021 
 
 11. 300.075 13. 400.000007 
 
122 
 
 Chapter Three. 
 
 190. Changing Common Fractions to Decimals. 
 Eeduce -£% to a decimal. 
 
 ■fc means 3 -4- 32. Performing the indicated division, 
 we obtain the quotient .09375. -fa = .09375, Ans. 
 
 Divide the numerator, ivith the necessary ciphers 
 annexed, by the denominator. The number of 
 decimal places in the quotient will be equal to the 
 number in the dividend. 
 
 .09375 
 
 Eeduce to decimals 
 
 1. 
 
 "T017 
 
 2. 
 
 A 
 
 3. 
 
 A 
 
 4. 
 
 H 
 
 5. 
 
 H 
 
 6. 
 
 A 
 
 7. 
 
 sort 
 
 8- * 
 
 y. 
 
 4000 
 
 10. 
 
 2 00 
 
 11. 
 
 H 
 
 12. 
 
 TT5" 
 
 13. 
 
 ¥ 
 
 14. 
 
 « 
 
 32)3.00000 
 288 
 
 
 120 
 96 
 
 
 240 
 224 
 
 
 160 
 160 
 
 15. 
 
 250" 
 
 16. 
 
 F2T 
 
 17. 
 
 W 
 
 18. 
 
 T% 
 
 19. 
 
 T5~8" 
 
 20. 
 
 "ST"? 
 
 21- rrfcr 
 
 191. Changing Decimals to Common Fractions. 
 What is the denominator of a decimal fraction ? 
 
 What prime numbers are contained in 10 ? What are the 
 only factors of 10 ? The prime factors of 100 ? Of 1000 ? 
 
 Can yoVg he reduced to lower terms ? Why ? Can jfo 
 be reduced to lower terms ? Why ? Can y^jj^rr be reduced 
 to lower terms ? How can we tell by merely looking at a 
 decimal whether or not it can be reduced to a common 
 fraction of lower terms ? 
 
 192. Written Exercises. 
 
 . Reduce the following to common fractions — lowest terms. 
 Do not find the greatest common divisor. 
 
 1. Reduce .0064 to a common fraction — lowest terms. 
 
 •0064 = T &V* = rib = vfc, -4ns. 
 
Decimals. 123 
 
 2* Reduce .039 to a common fraction. 
 
 •039 = xfjyi Ans. 
 
 This cannot be reduced to lower terms, since 39 is not divisible by 2 
 or 5. 
 
 3. Reduce .900 to a common fraction — lowest terms. 
 
 tW(5 = tVff = tV Ans. 
 
 Omit the decimal point. Write in the form of a common 
 fraction, and reduce to lowest terms. 
 
 Ciphers at the right of a decimal cancel ciphers in the denominator ; 
 they do not, therefore, affect the value of the decimal, and they should 
 be omitted. 
 
 193. Reduce to common fractions : 
 
 1. 
 
 .0075 
 
 8. 
 
 .37500 
 
 15. 
 
 .0009 
 
 2. 
 
 .36 
 
 9. 
 
 .144 
 
 16. 
 
 .816 
 
 3. 
 
 .0275 
 
 10. 
 
 .0006 
 
 17. 
 
 .15625 
 
 4. 
 
 .44 
 
 11. 
 
 .27 
 
 18. 
 
 .0375 
 
 5. 
 
 .03125 
 
 12. 
 
 .027 
 
 19. 
 
 .00625 
 
 6. 
 
 .486 
 
 13. 
 
 .00365 
 
 20. 
 
 .096 
 
 7. 
 
 .3750 
 
 14. 
 
 .96 
 
 21. 
 
 .326 
 
 ADDITION OF DECIMALS. 
 
 194. Add the following, reducing the common fractions 
 to decimals. 
 
 1. 18£ + 9.084 + 25^ + 163 + 2.09 + .0975 
 
 18.75 
 9.084 
 Write the decimals so that tenths, hundredths, etc., 25.05 
 stand in the same column, etc„ 163. 
 
 2.09 
 .0975 
 
124 Chapter Three. 
 
 Write the numbers so that decimal points stand in a column. 
 Add as in integers, and place the point in the sum directly 
 under the points above. 
 
 2. 275^ + 58.64 4 8.6796 4 30J- 4 8f 4 99 
 
 3. 84 1 ^ + 93 T ^4-3 I i^ + TlHir + 684.1 + i 
 
 4. 250 + 1875.93 4- Iff + A + 608.94 + .0005 
 
 5. 8.6796 + 96.8 + 18f + 34^ + 1876 
 
 6. 40^ + 7.2832 4 86.3 4 128.46 4 2^ 
 
 7. 540 41.32 4-576 4 1^ + 68^ 4 395£ 
 
 8. 5.308 4 .25 4 567.8 + 8.4825 4 49.795 + 8^ 
 
 9. 7.08 + 23.04 4 8^ 4 .348 + 3^ 4. 7.00019 
 10- 8^^4-8^4 507 + 28^4 6.8819 
 
 SUBTRACTION OF DECIMALS. 
 
 195. Give answers in decimals : 
 
 1. 275.3 -87 A 
 
 275.3 
 
 Arrange the decimals as in addition, tenths under ^~ ^ . 
 
 tenths, etc. ^ nn \„ . 
 
 188.26 Ans. 
 
 Write the numbers so that the decimal point of the subtra- 
 hend is directly under the decimal point of the minuend; sub- 
 tract as in integers, and place the point in the remainder 
 directly under the points above. 
 
 2. 
 
 387f - 99.0127 
 
 7. 
 
 2345-345^- 
 
 3. 
 
 woo -T^n 
 
 8. 
 
 168^-54.8759 
 
 4. 
 
 62.365-48| 
 
 9. 
 
 618.42 -;576J 
 
 5. 
 
 198}- 13.6431 
 
 10. 
 
 1847H-344rfo 
 
 6. 
 
 24A-9rtt* 
 
 11. 
 
 622.5-6.243 
 
Decimals, 125 
 
 196. Oral Problems. 
 
 1. Reduce -fa to a decimal. 
 
 2. Express the decimal .3 J as a simple fraction. 
 
 3. What decimal of a ton is 125 pounds ? 
 
 4. One hundred fifty marbles are divided among a certain 
 number of boys. Each receives 12 and there are 6 remaining. 
 How many boys are there ? 
 
 5. Express the decimal .62^ as a simple fraction. 
 
 6. What decimal of a peck is 7 quarts ? 
 
 7. If 8 men can do a piece of work in 6 days, in how 
 many days can 4 men do it ? 
 
 8. If Maria spends $ .75 a day, in how many days will 
 she spend $9? 
 
 9. If you had 3J oranges to divide among your friends, 
 giving each \ of an orange, to how many friends would you 
 give? 
 
 10. -J- of 14 is -J- of what number ? 
 
 11. Change .75 yards to feet and inches. 
 
 12. At 16|^ a yard, what will 12 yards of ribbon cost ? 
 
 13. At 80^ a pound, what do 4 ounces of tea cost ? 
 
 14. If I have 12 yards of ribbon, to how many girls can I 
 give f of a yard each ? 
 
 15. A boy lives 10^- rods from his school. How far does 
 he walk in a day to attend two sessions of school ? 
 
 197. Written Problems. 
 
 1. In the written number 54,372, the value expressed by 
 the 5 is how many times the value expressed by the 2 ? 
 
 2. Find the sum of two and twenty-five thousandths, 
 five and twenty-seven ten-thousandths, forty-seven and one 
 hundred twenty-six millionths, one hundred fifty and seven 
 ten-millionths. 
 
126 Chapter Three. 
 
 3. In a mass of alloy weighing 291.42685 pounds, there 
 were found 40.0921 pounds of silver, 160.09090 pounds of 
 copper, 22.002 pounds of iron, and .426900 pounds of zinc. 
 The remainder was lead. What was the weight of the lead ? 
 
 4. How many bushels of oats at f of a dollar a bushel 
 will pay for f of a barrel of flour at $ 5.40 a barrel ? 
 
 5. Add 3.684; 19.5; .00875; 15,863.625; 8.7; and 
 100.4875. 
 
 6. Change to a common fraction in its lowest terms 
 .009375. Change -fe to a decimal. 
 
 7. If f pound of tea costs $■$■, how many pounds can be 
 bought for $7.50? 
 
 8. Change to common fractions .0075 and .625. 
 
 9. Change to decimals ^-, • 2 - 9 -, and 5 J, and add the results. 
 
 10. Reduce to common fractions, and then find the sum 
 of the common fractions: .12^-, .3^-, .16f. 
 
 1 1 . Add three hundred seventy-six ten-thousandths, forty- 
 five hundred-thousandths, five hundred sixty-eight thou- 
 sandths, fourteen and fifteen hundredths. 
 
 12. At 24 cents per gallon, what will be the cost of 16 
 gal. 3 qt. of milk ? 
 
 MULTIPLICATION OF DECIMALS. 
 
 198. Give answers in decimals : 
 
 1. Multiply .000486 by 29.5. 
 
 Place the units' figure (9) of the multiplier under the last figure (6) 
 of the multiplicand. 486 millionths multiplied by 2 tens gives a prod- 
 uct of 972 hundred-thousandths, or .00972 ; place the 
 right-hand figure (2) of this product under the 2 of the 
 
 multiplier, etc. — 
 
 00Q79 
 The result, .0143370, contains seven decimal places, -w^i* 
 
 which is equal to the six in the multiplicand plus the one ^374 
 
 in the multiplier. Rejecting the unnecessary cipher at 2430 
 
 the right, the product is .014337, Am. .0143370 
 
Decimals. 
 
 2. Multiply 29.5 by .000486. 
 
 The units' figure of the multiplier may be 
 considered as zero. 
 
 Ans. .014337. 
 
 127 
 
 29.5 
 
 0.000486 
 .01180 
 2360 
 1770 
 
 .0143370 
 
 Multiply as in whole numbers, and from the right of the 
 product point off as many decimal places as there are decimal 
 places in both factors. 
 
 Multiply : 
 
 
 
 1. 24.75 x 3.02 
 
 6. 
 
 1.876 x 34 
 
 
 
 2. 98| x .00046 
 
 7. 
 
 3.48 x 4.8665 
 
 3. 148^x12.5 
 
 8. 
 
 .43J x 1A 
 
 4. 380^- x .012 
 
 9. 
 
 192.38 x .238 
 
 5. .09375x1.48 
 
 10. 
 
 26.4 x .016 
 
 DIVISION OF DECIMALS. 
 
 199. 1. Divide 7.345 by .29. 
 
 Make the divisor a whole number by mov- 
 ing the decimal point two places to the right, 
 which multiplies the divisor by 100 ; and make 
 a corresponding change in the dividend. Di- 
 viding 734.5 by 29 gives a quotient of 25.3275+. 
 Since the quotient is to be limited to three 
 decimal places, 8 followed by a minus sign is 
 substituted for the 7, to indicate that the 
 fourth decimal figure is at least 5. 
 
 25.327 
 
 /29)7/34.500 
 
 58_ 
 
 154 
 
 145 
 
 95 
 
 5L 
 80 
 58_ 
 220 
 203 
 17 
 
 Ans. 25.328- 
 
i*8 
 
 Chapter Three. 
 
 2. Divide 753 by 4.18. 
 
 Kemoving the decimal point in the divisor 
 two places to the right multiplies the divisor 
 by 100. Annex two ciphers to the dividend. 
 
 As the fourth decimal figure in the quo- 
 tient is greater than 5, the 3 is changed to a 
 4, followed by a minus sign. 
 
 Arts. 180.144 - 
 
 3. Divide .8756 by 4326. 
 
 The decimal point in the quotient is placed 
 over the new decimal point in the dividend, 
 the necessary decimal ciphers being sup- 
 plied. A + sign is placed after the last 
 quotient figure to show that the next quo- 
 tient figure is less than 5. 
 
 180.143 
 
 4/18.)753/00.000 
 418 
 
 3350 
 3344 
 
 60.0 
 41.8 
 18.20 
 16 72 
 1480 
 1254 
 226 
 Arts. .0002024 + 
 
 4326).8756000 
 8652 
 10400 
 8652 
 
 17480 
 
 17304 
 
 176 
 
 Make the divisor a whole number by removing the decimal 
 point, and make a corresponding change in the dividend. 
 The number of decimal places in the quotient will be equal to 
 the number in the dividend as changed. 
 
 200. Written Exercises. 
 
 
 
 Divide : 
 
 
 
 1. 4.054 -j- 18.25 
 
 10. 
 
 62.478 + 4279 
 
 2. 123.5 -j- 384 
 
 11. 
 
 346.25 -f- 64.8 
 
 3. 471 -*- 5.325 
 
 12. 
 
 9.1342 -4- 208.3 
 
 4. .3126 -h .0134 
 
 13. 
 
 1784 -f- 29.57 
 
 6. 12.345 -r- .0047 
 
 14. 
 
 343.71 -f- 1.127 
 
 6. .8756 -*- 4.322 
 
 15. 
 
 83.087 -5- 5.37 
 
 7. 8 -f- 122 
 
 16. 
 
 137.84 -5- 7.91 
 
 8. .3678 -*- .9125 
 
 17. 
 
 38.9008 -*- .523 
 
 9. 48.45 -*- .089 
 
 18. 
 
 .81074 h- .0091 
 
Decimalso 
 
 129 
 
 201. Solve by short division: 
 
 1. Divide 18.756 by 3000. 
 
 Cancel the ciphers in the divisor, thereby 
 dividing it by 1000. Move the decimal point 
 in the dividend three places to the left, which 
 divides it by 1000. Place the decimal point 
 in the quotient under the new decimal point 
 in the dividend. 
 
 3000 ).O18/756 
 
 .006252 Ans. 
 
 2. 
 
 48.36-5-4000 
 
 11. 
 
 48.64-5-200 
 
 3. 
 
 .4824-5-12000 
 
 12. 
 
 .00531-5-90000 
 
 4. 
 
 11.011-5-700 
 
 13. 
 
 96.51-5-60 
 
 5. 
 
 3.6504-90 
 
 14. 
 
 87.5^-500 
 
 6. 
 
 45.63-5-1500 
 
 15. 
 
 183.275-5-10000 
 
 7. 
 
 130.13-5-1100 
 
 16. 
 
 1.7632-5-1600 
 
 8. 
 
 .8712-5-60 
 
 17. 
 
 1.5639-5-130 
 
 9. 
 
 3.075-5-5000 
 
 18. 
 
 614.4-5-120 
 
 10. 
 
 .07056-5-140 
 
 19. 
 
 .8008-7000 
 
 202. Perform indicated operations. 
 
 Change the divisor to a whole number, making corresponding 
 change in the dividend. Cancel. 
 
 2. 
 
 7 
 34.2 x/OT 
 
 tm 
 
 .249 x 3.92 
 .098 
 
 .083 x .72 
 288 
 
 .6876 x .27 
 .081 
 
 234 .001 
 
 239.4 5. ffi/ff x -#3--.234 
 
 I&0 
 
 6. 
 
 7. 
 
 3.1416 x 2.3 
 
 .7854 
 
 7.72 x 65 
 19.3 
 
 450 x 23.8 
 1.19 
 
130 Chapter Three. 
 
 34.3 x 8.1 2.75 x .801 
 
 * .49x100 ' 1.1x6 
 
 .576 x 6.3 .306 x 8.75 
 
 ' 14.4x25 ' .9x68 
 
 203. Eeduce to common fractions — lowest terms. 
 
 1. Eeduce .3^ to a common fraction — lowest terms. 
 
 .3£ is a complex decimal ; that is, a decimal and a common fraction 
 written together. It may be written as the complex fraction ^3, 
 which means S\ -h 10. 
 
 Note. — A complex fraction is one which has a fraction in the 
 numerator or in the denominator or in both. 
 
 2. Eeduce .006 \ to a common fraction — lowest terms. 
 
 .006* = . 00625 = T ^b; 
 dividing both terms by 25, we get jffo ; 
 dividing both terms by 25, we get T ^, Ans. 
 
 3. .33J 6. .01£ 9. .04£ 
 
 4. .16| 7. .06| 10. .76ff 
 
 5. .142^ 8. .833| 11. .037£ 
 
 204. Oral Exercises. 
 
 1. Divide 6 by .03. 
 
 2. f is what part of 2 ? 
 
 3. What is the product of one hundred by one-hundredth? 
 
 4. Subtract 25 thousandths from 5. 
 
 5. What will 150 pounds of coffee cost at the rate of 3 
 pounds for 50 cents ? 
 
 6. What will be the cost of 3 pecks of cherries at 2 
 cents a pint? 
 
Decimals. 131 
 
 7. Divide -§ by f . 
 
 8. At 3 oranges for 5 cents, what will be the cost of 4 
 dozen oranges ? 
 
 9. If a man walks ^ of a mile in 10 minutes, bow far 
 can be walk in an hour and a half ? 
 
 10. A woman bought 12 yards of cloth at 70^ a yard ; 
 she paid $ 5 in cash, and the rest in butter at 20^ a pound, 
 How many pounds of butter did she give ? 
 
 205. Written Exercises. 
 
 1. Divide the sum of .736 and 1.2854 by their difference= 
 
 2. Divide .1 by .2, and .35 by 35, and find the product 
 of the quotients. 
 
 3. Eeduce -^fa to a decimal, and divide it by .3125. 
 
 4. Divide .12096 by .032. 
 
 5. Multiply .00273 by 3000.456, and divide the .product 
 
 by .08. 
 
 6. Divide 12.3125 by .000625. 
 
 7. Divide 51.5 by 412, and 412 by 51.5. 
 
 8. Multiply 31.5 by 27.9, and divide the product by 
 9.765. 
 
 4 9t 
 
 9. Eeduce ^=^-. 
 
 3-H 
 
 10. Find the value of ' 0Q21 * 3004 . 
 
 .024 
 
 11. What will be the duty on 175 kilograms of wool at 
 33 cents per pound ? (1 kilogram = 2.2046 pounds.) 
 
 12. How much is the fraction f increased or diminished 
 when 2 is added to each of its terms (numerator and denom- 
 inator) ? 
 
132 Chapter Three. 
 
 13. Find the cost of 360 meters of cloth at $1.10 per 
 yard (1 meter = 39.37 inches). 
 
 14. Find the cost in United States money of 386 hats at 
 24 francs each (1 franc = 19.3 cents). 
 
 15. Find the cost in United States money of 480 meters 
 of cloth at 1.10 marks per meter (1 mark = 23.8 cents). 
 
 16. A merchant bought 30 pieces of cloth, each contain- 
 ing 41.6 yards, for $3,875 per yard, and 25 pieces of 36.8 
 yards each, for $ 4.125 per yard. He sold the entire lot for 
 $ 3.96 per yard. How much did he gain or lose ? 
 
 17. An importer received a box of chemicals weighing 
 122 grams, each gram containing 15.432 English grains, on 
 which he paid a duty of $.05 per grain. What was the 
 amount of duty ? 
 
 18. A dealer exported 374.319 bushels of corn, receiving 
 in exchange coal at the rate of 1 ton of coal for 15.124 
 bushels of corn. How much coal did he receive? 
 
 19. .75 is what part of 3.25 ? 
 
 20. Keduce .005025 to a common fraction. 
 
 UNITED STATES MONEY. 
 206. Written Exercises. 
 
 1. Find the cost of 24,400 bricks @ $ 6.25 per M. 
 
 M means 1000. 24,400 = 24.4 M. Since the cost per thousand is 
 $6.25, 24.4 thousand will cost 24.4 times $6.25. 
 
 2. 760 pounds of hay @ 95 cents per cwt. (100 pounds) 
 
 ($.95x7.6) 
 
 3. 48,600 laths @ $ 2.80 per M. 
 
 4. 39,250 stamped envelopes @ $21.30 per thousand. 
 6. 1875 pounds of straw @ 68 cents per cwt. 
 
Denominate Numbers. 133 
 
 6. 108,745 Philadelphia bricks @ $22 per M. 
 
 7. 14,860 oranges @ 75^ per hundred. 
 
 8. 2376 eggs @ 13J^ per dozen. 
 
 9. 4500 cigars® $35 per M. 
 
 10. 28 dozen wax candles @ $13.50 per gross (144). 
 
 Solve by cancellation where possible : 
 
 1 1 . 38,648 pounds of wheat @ 90 f per bushel (60 pounds). ' 
 
 Since there are 60 pounds in a bushel, 38,648 pounds = 
 
 8.09 
 
 ^^ bushels. At 90 cents per bushel, the cost is $ '^ x38648 , etc. 
 60 60 
 
 Note. — In cancelling, be careful not to strike out a cipher in 60 
 and one in .90, without inserting a decimal cipher. 
 
 12. 18,964 pounds of coal @ $5 per ton (2000 pounds). 
 
 13. 48,576 pounds of oats @ 36/ per bushel (32 pounds). 
 
 14. 69,104 pounds of rye @ 91-J-^ per bushel (56 pounds). 
 
 15. 74,816 pounds of corn @ 48^ per bushel (56 pounds). 
 
 DENOMINATE NUMBERS. 
 207. Written Exercises. 
 
 1. Change 12 pounds and 9 ounces to ounces. 
 Since there are 16 ounces in 1 pound, in 12 
 
 16 oz. 
 
 pounds there are 12 times 16 ounces, or 192 ounces. 
 
 i 
 In 12 pounds 9 ounces, there are 192 ounces + 9 
 
 ounces, or 201 ounces. \ f * D - jj oz " 
 
 The work may be arranged in this way. Above 201 oz. 
 
 the ounces, write the number of ounces in a pound, 
 
 viz. 16. Multiply 16 ounces by 12, adding in the ^ ns ' 201 oz. 
 
 9 ounces. 
 
134 Chapter Three. 
 
 Change : 
 
 1. 20 rods and 3 yards to yards. 
 
 2. 2 miles to yards. 
 
 3. 3 days and 17 hours to hours. 
 
 4. 24 minutes and 15 seconds to seconds, 
 
 5. 8 tons and 1675 pounds to pounds. 
 
 6. 43 gallons and 8 quarts to quarts. 
 
 7. 75 gallons to pints. 
 
 8. 19 bushels and 3 pecks to pecks. 
 
 9. .03125 ton to pounds and ounces. 
 10. -J yard to feet and inches. 
 
 208. "Written Exercises. 
 Change : 
 
 1. 975 ounces to pounds and ounces. 
 
 2. 396 inches to yards. 
 
 3. 517 hours to days and hours. 
 
 4. 1694 seconds to minutes and seconds. 
 
 5. 9314 pounds to tons and pounds. 
 
 6. 987 pints to gallons, quarts, and pints. 
 
 7. 1485 quarts to pecks and quarts. 
 
 8. 185 pecks to bushels and pecks. 
 
 9. 840 hours to weeks. 
 
 10. 12 hours to the fraction of a week. 
 
 11. 28 inches to the fraction of a yard. 
 
 12. 10 ounces to the decimal of a pound. 
 
 13. 3 quarts to the decimal of a bushel. 
 
Denominate Numbers. 
 
 135 
 
 209. Written Exercises. 
 Add: 
 
 1. 13 lb. 6 oz. 10 oz. +9 oz. + 6 oz. = 25 oz. = 1 lb. 9 oz. 
 
 5' lb. 9 oz. Write 9 ounces and carry 1 to column of 
 
 25 lb. 10 oz. pounds. Ans. 44 lb. 9 oz. 
 
 2. 19 yd. 1 ft. 
 
 2 ft. 
 3 yd. 1 ft. 
 
 3. 5 min. 30 sec. 
 11 min. 25 sec. 
 
 9 min. 18 sec. 
 
 4. 4 ft. 9 in. 
 
 2 ft. 6 in. 
 
 7 ft. 7 in. 
 
 6. 18 gal. 3 qt. 
 9 gal. 1 qt. 
 
 2 qt. 
 
 210. Subtract: 
 
 1. 81b. 
 4 lb. 7 oz. 
 
 2. 15 yd. 1 ft. 
 
 9 yd. 2 ft. 
 
 3. 17 hr. 
 
 9 hr. 50 min. 
 
 A. 1 yd. 1 ft. 1 in. 
 2 ft. 9 in. 
 
 5. 25 gal. 1 qt. 
 6 gal. 3 qt. 
 
 6. 
 
 7. 
 
 11 bu. 3 pk. 
 6 bu. 2 pk. 
 
 2pk. 
 
 Ipk. 
 Ipk. 
 
 6qt. 
 7qt. 
 5qt. 
 
 3 wk. 
 
 6 wk. 
 1 wk. 
 
 5 da. 
 
 6 da. 
 3 da. 
 
 11 T. 
 4T. 
 
 165 lb. 
 
 983 lb. 
 
 1756 lb. 
 
 Change 8 lb. to 7 lb. 16 oz. 
 
 16 oz. — 7 oz. = 9 oz. 
 7 lb. - 4 lb. = 3 lb. 
 
 Ans. 3 lb. 9 oz. 
 
 6. 89 bu. 2 pk. 
 67 bu. 3 pk. 
 
 7. 3 pk. 2 qt. 
 2 pk. 7 qt. 
 
 8. 11 wk. 1 da. 
 
 9 wk. 5 da. 
 
 9. 5T. 896 1b. 
 
 1984 lb. 
 
136 Chapter Three. 
 
 211. Multiply: 
 
 1. 12 lb. 7 oz. x 3 
 
 3 times 7 ounces are 21 ounces, or 1 pound 6 ounces. Write 6 
 ounces. 3 times 12 pounds are 36 pounds, and 1 pound to carry are 
 37 pounds. Ans. 37 lb. 5 oz. 
 
 2. 3 hr. 10 min. x 7 7. 7 min. 18 sec. x 10 
 
 3. 4 T. 985 lb. x 11 8. 9 gal. 3 qt. x 2 
 
 4. 7 bu. 3 pk. x 9 9. 2 ft. 9 in. x 8 
 
 5. 3 wk. 4 da. x 4 10. 1 yd. 1 ft. 6 in. x 6 
 
 6. 4 yd. 1 ft. x 5 11. 3 yr. 4 mo. x 7 
 
 212. Divide: 
 
 1. 9 1b. 2oz. -s-2 
 
 \ of 9 pounds is 4 pounds and 1 pound remainder, ^q •,, « 
 
 or 16 ounces. Add to this 2 ounces, giving 18 ounces ' . * _   
 
 4 lb. 9 oz. 
 for the dividend. \ of 18 ounces is 9 ounces. 
 
 Ans. 4 lb. 9 oz. 
 
 2. 31 gal. 2 qt. -f- 9 7. 19 ft. 2 in. -- 10 
 
 3. 19 hr. 21 min. --3 8. 34 T. 936 lb. -- 4 
 
 4. 26 bu. 1 pk. 4-5 9. 17 wk. 1 da. 4-6 
 
 5. 41 min. 44 sec. 4- 8 10. 52 yd. ft. 9 in. 4- 11 
 
 6. 18 yd. 2 ft. 4- 7 11. 23 yr. 4 mo. 4- 7 
 
 213. Divide: 
 
 1. 18 lb. 4 oz. by 4 lb. 9 oz. 
 
 18 lb. 4 oz. = 292 oz. 
 4 lb. 9 oz. = 73 oz. 
 292 oz. -r- 73 oz. = 4, Ans. 
 
 Note. — Change the divisor and the dividend to the same denomi- 
 nation. The answer is an abstract number. 
 
Denominate Numbers. 137 
 
 2. 16 yd. by 2 yd. 2 ft. 
 
 3. 51 hr. 36 min. by 6 hr. 27 min. 
 
 4. 47 min. 42 sec. by 5 min. 18 sea 
 
 5. 84 yr. 7 mo. by 12 yr. 1 mo. 
 
 6. 19 da. 3 hr. by 2 da. 3 hr. 
 
 7. 3 mi. 40 rd. by 125 rd. 
 
 8. 103 T. 808 lb. by 8 T. 1234 lb. 
 
 9. 52 gal. 2 qt. by 3 gal. 2 qt. 
 
 10. 68 bu. 1 pk. by 5 bu. 1 pk. 
 
 11. 30 ft. 8 in. by 1 ft. 11 in. 
 
 12. 52 yd. 9 in. by 4 yd. 2 ft. 3 in. 
 
 13. 51 wk. 3 da. by 2 wk. 6 da. 
 
 214. Oral Problems. 
 
 1. What will be the weight of 16 hams that average 10 
 lb. 5 oz. each ? 
 
 2. From a chest of tea containing 54 pounds there were 
 sold 27 lb. 7 oz. How many pounds remain ? 
 
 3. Seven bushels of potatoes are divided among 8 per- 
 sons. How many pecks and quarts does each receive ? 
 
 4. How many square inches in the surface of a sheet of 
 paper measuring 11 inches by 13 inches ? 
 
 5. How many feet and inches in £ yard ? 
 
 6. What decimal of a pound is 14 ounces ? 
 
 7. A man buys a bushel of hickory nuts. After he sells 
 2 pk. 4 qt., what fraction of the bushel has he left ? 
 
 8. A dealer puts 30 gallons of milk in cans holding 
 1 qt. 1 pt. each. How many cans does he fill ? 
 
 9. At $ 24 per month, how much rent will a man pay 
 in 1 year and 5 months ? 
 
138 Chapter Three. 
 
 10. 75 hundredths of a pound is how many ounces? 
 
 11. How many feet in 5 rods ? 
 
 12. 7 qt. 1 pt. of milk is divided among 5 people. How 
 many quarts and pints does each receive ? 
 
 13. What fraction of 2 lb. 3 oz. is 1 lb. 4 oz. ? 
 
 14. Three-eighths of a ton is how many pounds ? 
 
 15. Change 9 hr. 36 min. to the fraction of a day. 
 
 215. Written Problems. 
 
 1. 32 hams weigh 464 pounds. What is the average 
 weight ? 
 
 2. 595 gallons of oil are put into 14 barrels. How many 
 gallons and quarts does each contain ? 
 
 3. If there are 42 gallons and 2 quarts in a barrel of oil, 
 how much oil will there be in 15 barrels ? 
 
 4. A piece of cloth containing 57 yards is divided 
 equally among six persons. What is the length of each one's 
 share ? 
 
 5. How many minutes in a day? 
 
 6. July 1 is the last school day. How many days' 
 vacation will there be, if school begins September 6 ? 
 
 7. How many hours and minutes are there from half- 
 past 3 Saturday afternoon to a quarter before 9 Monday 
 morning ? 
 
 8. How many steps, 2 ft. 6 in. long, must a man take 
 in walking 1200 feet ? 
 
 9. A man owns a plot of ground 420 feet long, 240 feet 
 wide. How many rods of fence will be required to enclose 
 it? 
 
 10. A train goes from Jersey City to Washington, 228 
 miles, in 4 hr. 12 min. How many miles an hour does it 
 travel ? How long does it take the train to go one mile ? 
 
Measurements. 139 
 
 11. On Monday a boarding-house uses 3 gallons 2 quarts 
 of milk ; on Tuesday, 4 gallons ; on Wednesday, 3 gallons 
 1 quart; on Thursday, 4 gallons 2 quarts; on Friday, 6 
 gallons ; on Saturday, 5 gallons 2 quarts ; on Sunday, 3 gal- 
 lons. How much is used during the week, and what is the 
 average per day ? 
 
 12. June 21 the sun rises at New York at 4.23 a.m. and 
 sets at 7.40 p.m. How long is the night ? 
 
 13. From 3£ bushels take 3 pecks. 
 
 14. What is the number of rods in the perimeter of a 
 field 206 ft. 3 in. wide and twice as long ? 
 
 MEASUREMENTS. 
 
 216. Written Exercises. 
 
 How many square inches in each of the following rec- 
 tangles ? First change each dimension to inches. 
 
 1. 42 in. by 36 in. 6. 9 ft. by 11 ft. 
 
 2. 71 in. by 18 in. 7. 27 in. by 30 in. 
 
 3. 3 ft. 1 in. by 4 ft. 2 in. 8. 65 in. by 92 in. 
 Note. —37 in. by 50 in. 9. 7 ft. 3 i n . by 2 yd. 
 
 4. 5 ft. 3 in. by 6 ft. 4 in. 10> 3 yd by 6 ft 6 in 
 
 5. 12 ft. by 18 ft. (108 in. by 78 in.) 
 
 217. How many square feet in each of the following rec- 
 tangles ? First change each dimension to feet, or to feet and 
 a fraction. 
 
 11. 18 ft. by 24 ft. 15. 31 ft. by 4 ft. 
 
 12. 36 in. by 4 ft. 16 - 3 ft. by 1J yd. 
 
 (3 ft. by 4 ft.) 17. 42 in. by 4 ft. 
 
 13. 6 yd. by 8 yd. 18. 25 ft. by 17 ft. 6 in. 
 
 (18 ft. by 24 ft.) 19. 42 in. by 48 in. 
 
 14. 1 yd. by 48 in. 20, 13 yd. by 15 yd. 
 
140 Chapter Three. 
 
 218. How many square yards in each of the following 
 rectangles ? Change each dimension to yards, or to yards 
 and a fraction. 
 
 21. 18 yd. by 25 yd. 26. 36 yd. by 24 in. 
 
 22. 15 yd. by 1 yd. 1 ft. 27. 17 ft. 6 in. by 32 in. 
 
 23. 27 ft. by 36 ft. 28. 22 ft. 9 in. by 18 in. 
 
 24. 54 ft. by 2 ft. 6 in. 29. 108 in. by 90 in. 
 
 25. 24 yd. by 27 in. 30. 180 ft. by 54 in. 
 
 219. Oral Exercises. 
 
 1. If a table is 3 yards long and 2 yards wide, how many 
 square feet in it ? 
 
 2. If it takes 24 yards of carpet, a yard wide, to cover a 
 floor, how many yards f yard wide will be needed for the 
 same floor ? 
 
 3. How many square inches in \ of a square foot ? 
 
 4. A room is 21 feet long and 18 feet wide. What will it 
 cost, at 5 cents per yard, for a strip of moulding around the 
 walls ? 
 
 5. How many square yards of carpet would be needed 
 for the floor of the above room ? 
 
 6. A field is 40 rods long and 26 rods wide. What is the 
 distance around it ? 
 
 7. What will it cost to carpet a room 18 feet long, 15 feet 
 wide, at 75 cents per square yard ? 
 
 8. What is the cost of fencing a lot 24 rods long by 20 
 rods wide, at $ 1.12 J per rod? 
 
 9. My field is 100 rods long and 75 rods wide. How 
 much is it worth at $ 2 a square rod ? How much will it 
 cost to fence it at $ 1 a rod ? 
 
 10. How many yards of fence will be required to enclose 
 a rectangular field 98 yards long and 50 yards wide ? 
 
Measurements. 141 
 
 220. Written Problems. 
 
 Make a diagram in each case : 
 
 1. A lot 25 feet by 100 feet has on it a house 25 feet by 
 55 feet. How many square feet are there left for a yard ? 
 
 2. How many square feet are there in the floor of a 
 room 24 feet long, 18 feet wide ? 
 
 3. How many square yards are there in the ceiling of 
 the same room ? 
 
 4. Find the number of square yards of plastering needed 
 for the end wall of a room 18 feet wide, 9 feet high, after 
 deducting for two windows each 6 feet high, 4£ feet wide. 
 
 5. How many square yards of plastering will be needed 
 for the opposite wall of the same room, 18 feet wide, 9 feet 
 high, after deducting for a door 7J feet high, 6 feet wide ? 
 
 6. Calculate the number of square yards of plastering 
 needed for two side walls of a room 24 feet long, 9 feet 
 high, after deducting for a fireplace 6 feet square on one 
 side. 
 
 7. A house 30 feet by 60 feet, with an addition 15 feet 
 square, is built upon a lot 100 feet square. How many 
 square feet of ground are covered by the building ? How 
 many square feet remain for a garden ? 
 
 8. Measure the top of a brick and calculate the number 
 of square inches in its surface. How many square inches 
 in the surface of the bottom of the brick? Measure one 
 side, and calculate its surface. How many square inches 
 are there in the surface of the opposite side ? How many 
 square inches in each end ? 
 
 9. Measure a crayon box, and calculate the number of 
 square inches in each face. 
 
142 
 
 Chapter Three. 
 
 10. Calculate the number of square feet in the floor of 
 the classroom. In the ceiling. In each side wall. In each 
 end wall. 
 
 11. What will it cost to put moulding around a room 
 shaped like the drawing, allowing 3 inches on every corner 
 for matching, the moulding being worth 5f ^ a foot? 
 
 
 10 ft. 
 
 
 
 
 
 • 
 
 £ 
 
 -«*< 
 
 
 
 
 
 6 ft. 
 
 
 
 
 
 € 
 
 
 
 
 05 
 
 22 ft. 
 
 
 
 
 12. The circumference of a circle is 3.1416 times the 
 diameter. What is the diameter of a circular track 1760 
 yards in circumference ? Find to two places of decimals. 
 
 13. Show the difference between 2 square inches and 2 
 inches square. 
 
 14. How many paving tiles 6 inches square are needed to 
 cover a floor 18 feet long, 10 feet wide ? 
 
 15. How many flagstones, each 4 feet long and 2 feet 
 wide, will be needed to lay a crossing 32 feet long and 6 
 feet wide ? What will be the cost of them at the rate of 
 $50 for 100 stones? 
 
Measurements. 143 
 
 AREAS OF RIGHT-ANGLED TRIANGLES. 
 
 221 . Preliminary Exercises. 
 
 The square shown in the diagram is divided into two parts 
 by a diagonal. One side of the square measures 10 feet. 
 
 1. Mark in each triangle its area. 
 
 Square. Eectangle. 
 
 2. Divide a rectangle 20 feet by 12 feet into two parts 
 by a diagonal. Mark in each triangle its area. 
 
 3. Draw a right-angled triangle 3 inches by 4 inches. 
 Calculate its area in square inches. 
 
 4. How many square yards in the sur- / jj 
 face of a right-angled triangle whose base 
 measures 30 feet, and whose perpendicular 
 measures 224 feet ? / n £ 
 
 222. Written Exercises. 
 
 Find the area in square feet of the following right-angled 
 triangles. (Change each dimension to feet.) 
 
 1 . Base 20 yards, perpendicular 30 feet. 
 
 Area = 1 square foot x £ (60 x 30) = 900 square feet, Ans. 
 
 Tlie number of square feet in the area of a right-angled 
 triangle is equal to one-half the product of the number of feet 
 in the base by the number of feet in the perpendicular, 
 
144 
 
 Chapter Three. 
 
 2. Base 16 inches, perpendicular 3 feet. 
 
 3. Base 30 inches, perpendicular 1 yard. 
 
 4. Base 3 feet 6 inches, perpendicular 5 feet. 
 
 5. Base 2 yards 1 foot, perpendicular 1 yard 9 inches. 
 
 6. Base 50 yards, perpendicular 36 yards. 
 
 7. Base 112J- feet, perpendicular 30 yards. 
 
 8. Base 90 inches, perpendicular 2 feet. 
 
 9. Base 12^- yards, perpendicular 13^ yards. 
 
 10. Base 1 rod, perpendicular 1\ feet. 
 
 11. Base 33 \ feet, perpendicular 18 feet 6 inches. 
 
 BILLS. 
 223. Philadelphia, Sept. 24, 1905. 
 
 Mr. William J. Hurley, 
 
 To John J. Petit & Son, Dr. 
 
 To 50 lb. Pipe 
 
 5\t 
 
 To 8 Faucets 
 
 75$ 
 
 To 1 Sink 
 
 
 To 3^ days' Labor 
 
 94.7S 
 
 75 
 
 $ 
 
 1. Copy and complete the above bill. 
 
 2. Albert Janson has done 3J days' work, @ $ 3.50 per 
 day, for Ephraim Whitlock. He charges for 850 feet lumber, 
 at $ 2 per hundred; 5 pounds of nails, at 9^ per pound; 
 3 locks, @ 50^ ; 2 bolts, at 10j*. Make out his bill. 
 
 3. A gardener furnishes 3 rose bushes, at 75^; 4 grape- 
 vines, at 50^ ; 11 fuchsias, at 30^ ; 25 pansies, at 10^. He 
 charges $3.25 per day for 2\ days' labor. Make out his bill. 
 
Percentage. 145 
 
 4„ An upholsterer charges $ 3 per day for repairing some 
 furniture. He supplies 6 pounds of hair, at 50^ per pound ; 
 17 yards of plush, at $ 1.75 per yard; 3 papers of tacks, at 
 10^; cord, gimp, etc., 47^. He works 4 days. Make out 
 his bill. 
 
 Note. — The foregoing bills are for work done and materials sup- 
 plied. Notice how the heading differs from those in Arts. 103 and 173. 
 
 5. Make out and receipt a bill for four articles bought 
 to-day by John Harrigan from Metz and Fagan, grocers 
 (Art. 103). 
 
 6. Make out a bill containing ten items bought by Mrs. 
 A. S. Jacobs, at different times during October, 1905, from 
 Frederick Loeser & Co., dealers in dry goods (Art. 173). 
 
 7. Make out a bill for labor done and materials furnished 
 by Joseph Minew, gardener. 
 
 PERCENTAGE. 
 
 224. Per cent means hundredths. 
 
 Six per cent means six hundredths, jfo, or .06. It is writ- 
 ten 6%. 
 
 225. Oral Exercises. 
 
 1. What is 6% of 200? 
 
 6% means jfo. To find 6% of 200, we multiply 200 by ^fo, or 
 200 x .06. Arts. 12. 
 
 2. What is yffr of 300 ? 6. 6% of 150 
 
 3. Find .06 of 400 7. 6% of 250 
 
 4. 6 per cent of 500 8. 6% of 125 
 
 5. 6% of 50 9. 6<f of 75 
 
146 Chapter Three. 
 
 10. 6% of 60 16. \% of 600 
 
 11. 6% of 160 17. \% of 600 
 
 12. 4% of 125 18. 2\% of 600 
 
 13. 7% of 500 19. 3J% of 400 
 
 14. 5% of 240 20. I % of 400 
 
 15. 1% of 600 21. 9% of 90 
 
 In solving examples in percentage, the work is frequently shortened 
 by changing the per cent to a common fraction in its lowest terms. 
 
 76% = * = *. Arts. 
 
 226. What fraction equals : 
 
 1. 25% 5. 20% 9. 6f% 
 
 2. 121% 6. 50% 10. 37^% 
 
 3. 33$% 7. 6J% 11. 62£% 
 
 4. 16|% 8. 3J% 12. 87£% 
 
 227. 1. Find 50% of 96. 
 
 60 % of 96 = I of 96 = 48, Ans. 
 
 2. 25% of 72 10. 150% of 140 
 
 3. 12^% of 120 11. 250% of 140 
 
 4. 6J% of 48 12. 125% of 140 
 
 5. 33£% of 36 13. 1% of 140 
 
 6. 16|% of 126 14. 1% of 350 
 
 7. $±% of 72 15. 2% of 350 
 
 8. 100% of 140 16. 3% of 350 
 
 9. 200% of 140 17. 4% of 350 
 
Percentage. 147 
 
 228. Written Problems. 
 
 Note. — The pupils should find but little difficulty in solving these 
 problems, which will serve to show a few applications of percentage. 
 There is no need of preliminary explanation of terms the meaning of 
 which can readily be determined from the context. 
 
 1. A merchant sells a lot of cotton for f 1872.50. He 
 receives 2% of this amount for selling it. How much does 
 he receive ? He receives $ 1872.50 x .02. 
 
 2. How much will it cost me to insure goods to the 
 amount of $ 18,760 at one per cent ? 
 
 3.. A dealer imports books worth $ 548.40, on which he 
 pays duty to the government at the rate of 25%. What is 
 the amount of the duty ? 
 
 4. Eighty per cent of a class of 55 pupils are promoted. 
 How many are not promoted ? 
 
 5. A man buys a house for $ 16,000 and sells it at an 
 advance of 3 per cent over the cost. How much does he 
 gain ? 
 
 6. A clerk spends for rent 18 per cent of his income of 
 $ 1850 per year. What rent does he pay ? 
 
 7. A girl spelled correctly 95 per cent of 60 words. 
 How many did she miss? 
 
 8. Tea costing 40 cents per pound is sold at a profit of 
 50 per cent. What is the selling price ? 
 
 9. I loan $ 600 at 6% interest per year. How much in- 
 terest should I receive from January 1, 1903, to January 1 ? 
 1905? 
 
 10. I loan a person $ 600 on July 1, 1903. He agrees to 
 pay me 5% of the amount loaned per year as interest. How 
 much interest should I receive July 1, 1904 ? 
 
 11. A house is valued at $6000. How much taxes must 
 the owner pay at the rate of $1.25 per $100 valuation ? 
 
148 Chapter Three, 
 
 INTEREST. 
 229. Oral Exercises. 
 
 Note. — A preliminary talk with the class should develop the fact 
 that a person hiring a horse is charged for its use, say so much an 
 hour ; that a person hiring a house is charged so much a month or a 
 year for its use. A person borrowing money is also charged for the 
 use of money. As the sum charged for the rent depends upon the 
 size and value of the house, so the sum charged for the use of money 
 depends upon the sum loaned. 
 
 A charge for use of money is called interest. The sum on 
 which the interest is paid is called the principal. The price 
 or rate is a certain per cent for a year. 
 
 1. What will be the interest on $ 100 for 1 year at 4% ? 
 
 $ 100 x .04 = $ 4, Ans. 
 
 2. On 9 200 for a year at 5% ? 
 
 3. On 1 300 for a year at 6% ? 
 
 4. On $ 400 for a year at 7% ? 
 
 5. On $ 250 for a year at 4% ? 
 
 At 4% per year, what will be the interest : 
 
 6. On $ 200 for 1 year? 
 
 7. On $ 300 for 2 years ? 
 
 8. On $ 100 for 3 years ? 
 
 9. On f 200 for 1\ years ? 
 
 10. On $200 for 1 year 6 months ? 
 
 11. What will be the interest on $ 200 for 3 years at 5% ? 
 The interest for 1 year will be $ 200 x .05, or $ 10 ; for three years 
 
 it will be 3 times $ 10, or $ 30, Ans. 
 
 12. On $ 300 for 2 years at 6% ? 
 
 13. On $ 400 for 6 years at 3% ? 
 
 14. On $ 100 for 5 years at 7% ? 
 
Interest. 149 
 
 15. On $ 250 for 2 years at 4% ? 
 
 16. On $ 100 for 1 year 6 months at 6% ? 
 
 17. On $ 200 for 3 months at 4% ? 
 
 At 4% per year, what will be the interest: 
 
 18. On $ 200 for 6 months ? 
 
 $200x.04xi. 
 
 19. On 1 300 for 4 months ? 
 
 20. On | 400 for 3 months ? 
 
 21. On $ 300 for 2 months ? 
 
 22. On 9 150 for 1 month ? 
 
 23. Find the interest on $ 24 for 1 year at 5%. 
 
 24. On $ 36 for 1 year at 4%. 
 
 25. On $ 67 for 1 year at 3%. 
 
 230. "Written Exercises. 
 Find the yearly interest on : 
 
 1. $286.50 at 4% $286.50 
 
 Multiply the principal by the rate, 4 %, written as _ 
 
 a decimal. $ 11.4600 
 
 Ans. $11.46. 
 
 2. $ 485 at 6% 12. $ 168 at 3| % 
 
 3. $375.40 at 5% 13. $244at5j% 
 
 4. $ 379 at 3% 14. $ 890 at 7 T \% 
 
 5. $ 486 at 4i% 15. $ 63.75 at 4% 
 
 6. $186.75 at 4% 16. $937.50 at 6% 
 
 7. $199.50 at 2% 17. $ 980.40 at 5% 
 
 8. $636 at 3J% 18. $159.60 at 2\% 
 
 9. $84.70 at 6% 19. $ 1357.37 at 7% 
 
 10. $ 93.25 at 8% 20. $ 2146.18 at H% 
 
 11. $1257 at 7% 21. $369.40 at 3|% 
 
150 Chapter Three. 
 
 Find the interest on : 
 
 $290 for 2 years at 4%. 
 
 The interest for 1 year is $ 290. 
 $11.60. Multiplying by 2, we .04 
 
 get the interest for 2 years, $ 11.60 interest for 1 year 
 
 $23.20. 2 
 
 Ans. $ 23.20 interest for 2 years 
 
 Multiply the principal by the rate expressed as hundredths, 
 and this product by the time expressed in years and fraction 
 of a year. 
 
 22. $ 1400 for 3 years at 4|%. 
 
 23. $ 2840 for 4 years at 5%. 
 
 24. $ 1250 at 6% for 3 years. 
 
 25. $ 5360 at 5-|% for 2 years. 
 
 26. $ 380 at 3% for 4J years. 
 
 27. $ 780 for 1 year 4 months at 6%. 
 Note. — 1 year 4 months = 1£ year. 
 
 28. $ 2560 for 2 years 6 months at 5%. 
 
 29. $ 1025 for 3 years 3 months at 4%. 
 
 30. $ 1296 for 7 months at 7%. 
 Note. — 7 months = ^ year. 
 
 31. $ 648 for 5 months at 5%. 
 
 32. $ 275 for 4 months at 3%. 
 
 33. $ 1000 for 11 months at 6%. 
 
 231. Oral Problems. 
 
 1. I bought a house for $4000, and sold it for 80% oi 
 the cost. For what did I sell it ? 
 
 2. A merchant whose income is $2000 a year spends 
 75% of it. How much does he save? 
 
Review. 151 
 
 3. John has $ 30 in the bank, Mary has 16f % as much. 
 How much has Mary ? 
 
 4. If I buy goods for $ 400 and sell them at a loss of 
 5%, how much do I lose ? 
 
 5. A farmer had 100 sheep and sold 20% of them. How 
 many did he sell ? 
 
 6. Cloth shrinks 5% of its length in sponging. What 
 is the shrinkage of a piece which contained 40 yards before 
 sponging? 
 
 7. In a school of 400, 60% are boys. How many girls 
 in the school ? 
 
 8. What is the interest on $ 100 for 2 years at 4% ? 
 
 9. What is the interest on $ 50 for one year at 6% ? 
 10. What is the interest on $ 200 for 2 years at 3J% ? 
 
 232. Written Problems. 
 
 1. A man receives a salary of $1800 a year; he pays 
 15% of it for board, 8J% for clothing, and 16% for other 
 expenses. What are his yearly expenses ? 
 
 2. My expenses during the month of April were $ 185.68 ; 
 my expenses in May were 12-|% less than in April. What 
 were my expenses in May ? 
 
 3. A lawyer collected 80% of a debt of $2360 and 
 charged 5% commission on the sum collected. How much 
 did the creditor receive ? 
 
 4. A house was insured for $3600 at 1|%. What was 
 the cost of the insurance ? 
 
 5. What is the interest on $ 550 for 2 years 6 months 
 at 4% ? 
 
 6. What is the interest on $ 1200 for 3 years at 5% ? 
 
 7. A merchant sold goods that cost $ 2180 at a gain of 
 33£%. How much did he receive for them ? 
 
152 Chapter Three. 
 
 8. What is the interest on $ 720 for 1 year 6 months 
 at7%? 
 
 9. I bought 1260 pounds of sugar at 4 J- cents a pound 
 and sold it at a gain of 10%. How much did I sell it for ? 
 
 10. What is the interest on $ 350 for 2 years at 3J% ? 
 
 APPROXIMATIONS. 
 
 These approximation examples should not be neglected. Pupils, 
 besides finding them useful in preventing gross errors in their calcula- 
 tions, will be enabled later to obtain exact results to similar examples 
 by an extension of the methods used in obtaining approximate results. 
 In a following chapter will be found suggestions as to the product by 
 99, 24, 99|, etc. 
 
 Some pupils can probably give the exact answer to No. 5 — 96 lb. 
 at 25f would be $ 24; at \f less per lb., the cost would be VL? (\f x 96) 
 less than $ 24. The exact answers to Nos. 2, 3, 8, 9, and 10 can be 
 obtained in a similar manner. 
 
 After the examples have been used for sight exercises in approxi- 
 mate answers, they should be solved for the exact answers. 
 
 Suggestions. — (1) 24 @ |f (2) 24 @ $125. (3) 64 @ $}. 
 (4)485@$1. (11) $27-4-$±. (12) $800 + 11 J. (13) $24-*-$$. 
 
 233. Give approximate answers, at sight: 
 
 1. 23| lb. of tea @ 50^. 
 
 2. 24 horses @ $ 124.95. 
 
 3. 64 yd. of carpet @ 87^0. 
 
 4. 485 bu. of wheat @ 99f £ 
 
 5. 96 lb. of coffee @ 24^. 
 
 6. 840 yd. of dress goods @ 33^. 
 
 7. 360 yd. of oil cloth @ 66f£ 
 
 8. 48 cwt. of straw @ 62f£ 
 
 9. 92 hats @ $1.49f. 
 
 10. 128 lb. of lard @ 12f£ 
 
Review. 1 53 
 
 234. Give approximate answers in whole numbers : 
 
 11. $27-5-24^ 21. 17.3x3.98765 
 
 12. $ 299.96 -T- $ 1.49 J 22. 256.008 x .249875 
 
 13. $ 24.05 -*- 37^ 23. 25.1234 x 15.93 
 
 14. $ 15.03 -r- 12|^ 24. 6.12 x 6.12 
 
 15. $ 60 -v- $ 2.49|f 25. 86.4 x. 996 
 
 16. $ 32 -4- 33-f^ 26. 33.333 x 5.004 
 
 17. $ 69.95 ^ 87^ 27. 799.387 x .125 
 
 18. 9 60 -T- 62^ 28. 7.999 x 7.99 
 
 19. $ 64 -*- 66^ 29. 7.33 x 11.0083 
 
 20. $ 27.95 -4- $ 1.75 30. 64.002 x .3750 
 
 SPECIAL DRILLS. 
 
 Note. — It is important for pupils to keep up their previous practice 
 in handling large numbers without a pencil, and to increase the size of 
 the numbers from year to year. 
 
 To add 135 and 89, the pupil first adds 80, then 9. 
 
 135 + 80(215)+9 = 224 
 
 235. Give sums : 
 
 256 + 56 576 + 76 437 + 73 832 + 99 
 
 394 + 77 646 + 85 768 + 48 543 + 78 
 
 690 + 450 = 690 + 400 + 50 
 
 350 + 680 440 + 590 570 + 640 750 + 250 
 
 770 + 260 '620 + 480 330 + 880 980 + 670 
 
154 Chapter Three. 
 
 236. Give differences : 
 
 To subtract 56 from 312 ; first deduct 50, then 6. 
 
 312 - 56 = 312 - 50 (262) - 6 = 256 
 
 224-89 652-76 500-73 931-99 
 
 471-77 731-85 816-48 621-78 
 
 1200 - 610 = 1200 - 600 - 10 
 
 1140 - 690 1130 - 870 1210 - 570 
 
 1030-350 1100-620 1650-980 
 
 237. Give products : 
 
 98 x 4 = 90 x 4 (360) + 8x4 (32) = 392 
 89 x 5 67 x 7 98 x 4 79 x 3 
 
 78 x 6 75 x 9 66 x 8 89 x 2 
 
 238. Oral Problems. 
 
 Note. — These problems should first be solved as sight exercises 
 from the book. Afterward, one should be read by the teacher and the 
 answer written by all the pupils at a given signal. These problems 
 require no analysis. They contain numbers similar to those of the 
 special drills on the previous page. 
 
 1. I sold 375 bushels of wheat to one miller and 87 to 
 another. How many bushels did I sell ? 
 
 2. Bought goods to the amount of $4.29. How much 
 change from a $ 5 bill ? 
 
 3. What will be the cost of 89 tons of coal at $ 5 per 
 ton? 
 
 4. If 49 hats cost $ 147, what is the cost of one hat ? 
 
 5. 567 marbles are divided among 9 boys. How many 
 does each receive ? 
 
 6. What will be the cost of a barrel of flour at $5.25 
 and 8 pounds of sugar at 6^ ? 
 
Review. 155 
 
 7. How much must be paid for 55 pounds of raisins, at 
 8^ per pound ? 
 
 8. Find the cost of 320 pounds of hay at 60^ per hun- 
 dred pounds. 
 
 9. A father earned $14.60, his son earned $7.80. 
 What were the earnings of both ? 
 
 10. There are 36 inches in a yard. How many yards 
 are there in 324 inches ? 
 
 11. The product is 925, the multiplier is 25. What is 
 the multiplicand ? 
 
 12. What price was paid for 20 sheep, at $8.75 per 
 head? 
 
 13. A man saved $320 per year for 5 years. How 
 much more would he require to make $ 2000 ? 
 
 14. Mr. Jones sold a lot for $675, thereby losing $85. 
 What did he pay for it ? 
 
 RATIO. 
 
 239. Written Problems. 
 
 Note. — Indicate operations, and cancel where possible. 
 
 1. If 56 men can pave a street in 24 days, how long will 
 it take 32 men to pave it ? 
 
 Analysis. — One man will take 56 times as long as 56 men ; and 32 
 men will do the work in ff of the time required by 56 men. 
 
 Problems of this kind, involving only multiplication and division, are 
 sometimes shortened by cancellation. Instead of multiplying 24 days 
 by 66, and dividing the product by 32, the pupil should indicate these 
 
 6 7 
 
 operations, then cancel : ?* days x ^ = 42 days, Arts. 
 
 i 
 
156 Chapter Three. 
 
 2. When a vessel sails 168 miles a day, she completes 
 her voyage in 14 days. In what time would she complete 
 it if she sailed 196 miles a day ? 
 
 At 168 miles per day, the voyage requires 14 days. 
 At 196 miles per day, it would require 14 days x |f§. 
 
 3. If a field would support 64 sheep for 21 days, how 
 long would it support 48 sheep ? 
 
 4. If 42 men could build a wall in 24 days, how many 
 men could build it in 18 days? 
 
 The pupil must first determine what is asked. In this problem, it 
 is the number of men. The given number of men, 42, must first be 
 written in the multiplicand. 
 
 To build a wall in 24 days requires 42 men. To build it in a shorter 
 time would require more men, hence the ratio is ff . 
 
 5 . If 21 horses are worth as much as 35 cows, how many 
 horses are worth as much as 55 cows ? 
 
 6. A girl that wrote 36 letters to a line, took 15 lines in 
 writing a piece of dictation. How many lines would a girl 
 that wrote 30 letters to a line require for the same dictation ? 
 
 7. If a boy that steps 27 inches at a time takes 1000 steps 
 in going home from school, how many steps will be taken 
 by a boy that steps 30 inches ? 
 
 8. If 1920 bricks will build a wall 15 yards long, how 
 many bricks will be required for a similar wall 24 yards 
 long? 
 
 9. A train going 44 miles an hour, went a certain distance 
 in 9 hours. How long would it take a train going 36 miles 
 an hour to make the same trip ? 
 
 10. Find the cost of one-fourth of a barrel of flour 
 at the rate of 22 cents for 7 pounds. 
 A barrel of flour weighs 196 pounds. 
 
Review. 1 57 
 
 11. Six men can do a certain piece of work in eighteen 
 days. How long would it take eighteen boys to do the 
 same work, if one man can do as much work as two boys ? 
 
 12. If a certain quantity of flour will last 48 persons 57 
 days, how .long will it last 38 persons ? 
 
 SHORT METHODS. 
 
 240. Sight Exercises. 
 
 1. 68x25 68 x 25 = \ of 6800 
 
 9* v 4.Q 25 x 49 = 49 x 25 = J of 49 hundred 
 
 Z. ZOX^y = 12i hundred = 1225, ^na. 
 
 3. 88 X 12 J i of 88 hundred 
 
 4. 24x75 13. 48x37£ 
 
 5. 82xl2J 14. 92x50 
 
 6. 72x25 15. 32x33J 
 
 7. 25x51 16. 88x25 
 
 8. 66x33£ 17. 25x97 
 
 9. 48 x 75 18. 16 x 87J 
 
 10. 24 x 62£ 19. 66 x 66% 
 
 11. 96x25 20. 16x66£ 
 
 12. 25x81 21. 18xl6| 
 
 241. Written Exercises. 
 
 1. 9347 x25 934700-4 
 
 2. 863x75 (86300 x 3) ~ 4 
 
 3. 8123 X 12£ 812300-8 
 
 Dividing 8123 hundred by 8 gives a quotient of 1015f hundred, the 
 fraction of which the pupil should write at once as 37^ units without 
 dividing out 300 units by 8. While he sets down the work in this way, 
 8 )812300, he should be able to write the remainder of the answer when 
 
 1015 
 he reaches the annexed ciphers. 
 
158 Chapter Three. 
 
 4. 6483x33 J | of 6483 hundred 
 
 5. 8123 x 125 i of 8123000 
 
 6. 9347x250 14. 33J x 3870 
 
 7. 9347 x2£* 15. 66|x3456 
 
 8. 9347x75 16. 16f x 1266 
 
 9. 6483 x66f 17. 8408 x62£ 
 
 10. 6488 x37£ 18. 3875 x 37 J 
 
 11. 4896 x87| 19. 1925 x 12£ 
 
 12. 1284 x 62J 20. 7314 x 250 
 
 13. 75x2468 21. 6480 x 125 
 
 242. Oral Problems. 
 
 1. What will be the cost of 49 pounds of coffee at 25^ 
 per pound ? 
 
 2. I paid $14.75 for eggs at 25^ a dozen. How many 
 dozen did I buy ? 
 
 3. What will be paid for 88 bushels of wheat at 87^ 
 per bushel ? 
 
 4. How many bushels of corn at 621^ per bushel can be 
 bought for $ 150 ? ($ i 50 + 1 $) 
 
 5. How much will be paid for 99 yards of dress goods 
 at 33 £ f per yard? 
 
 6. How many yards of carpet at 66%jt per yard can 
 be bought for $84? 
 
 7. Find the cost of 15 dozen collars at 12 J^ each. 
 
 8. Paid $24 for cuffs at 16|^ per pair. How many 
 dozen pairs were bought ? 
 
 9. What will be the cost of 128 pounds of tea at 75^ 
 per pound ? 
 
Review. 1 59 
 
 10. A bale of cotton at 6 \y per pound cost $ 25. What 
 was the weight of the cotton ? 
 
 11. A farmer sold hay at 75^ per hundredweight, receiv- 
 ing for it $ 39. How many hundredweights did he sell ? 
 
 12. How many barrels of mess pork at $12.50 per barrel 
 can be bought for $175? 
 
 13. What will be the cost of 84 yards of carpet at $ 1.25 
 per yard ? 
 
 14. When wheat sells at $ 1.12 \ per bushel, how many 
 bushels can be bought for $ 199 ? 
 
 15. At $3.50 each, what will be paid for 42 coats ? 
 
 16. Find the cost of 28 hats at $2.75 each. 
 
 17. A real estate agent sold 97 lots at $250 each. How 
 much did he receive for them ? 
 
 ($250 = \ of $1000) 
 
 18. What will be the cost of 248 horses at $125 each ? 
 
 19. At ^ cent each, how many penholders can I buy for 
 $4.32? 
 
 20. Paid $3075 for cows at $75 each. How many were 
 bought ? 
 
 REVIEW OF FRACTIONS. 
 
 Note. — Practice in the sight work such as is given in the following 
 examples will enable pupils to dispense with some of the aids they 
 found necessary to employ during the earlier stages of work in frac- 
 tions. These exercises should be answered one at a time from the 
 book or the blackboard, preferably the latter. At a later lesson, the 
 teacher should require the answers to five or ten examples selected pro- 
 miscuously, to be written from the book or the blackboard, the ex- 
 amples to be announced by the teacher by number. At the same, or 
 another lesson, the teacher should read a few, the answers to be 
 written one at a time. In these examples pupils should not take pen 
 or pencil until the signal is given to write the answer. No change 
 should be made in an answer after it is written. 
 
160 Chapter Three. 
 
 243. Write answers at sight : 
 
 1. Add 32£ and 15f . 
 
 Mentally changing the fractions to twelfths, the pupil proceeds as 
 follows : ^ + 15(47^) + & = 47ft = 48^, Ans. 
 
 2. 24£ + 15f. 4. 62{ + 23^. 
 
 3. 50f + 20f 5. 40f + 33f 
 
 6. From 78-J take 20£. 
 
 Suggestion. — 20| from 78 (and £) leaves 57 \ (and |), 57£ + |. 
 
 Ans. 67^. 
 
 7. 80£-40£. 9. 33£-16J. 
 
 8. 43|-12£. 10. 54|-30£. 
 
 11. Multiply 20| by 6. 
 
 Six times 20(120) + 6 times f (4) = 124, Ans. 
 
 12. 12fx8. 14. 12|x9. 
 
 13. 30f x 10. 15. 11J X 6. 
 
 16. Divide 24f by 2. 
 
 i of 24(12) +io£|a) = 12i, Jns. 
 
 17. 48^ + 6. 19. 80|-l-4. 
 
 18. 23^-^-3. 20. 55f£^5. 
 
 21. Divide 24£ by 4. 
 
 iof24(6) + Jof|(^) = 6A, ^ 
 
 22. 60^3. 24. 28f + 7. 
 
 23. 40^-^4. 25. 36£ + 9. 
 
 26. Divide 21J by 5. 
 
 21$ contains 5, 4 times, with a remainder of 1J, or 5 fourths. 
 6 fourths -=-5=1 fourth. Ans. 4 J. 
 
 27. 17^ + 4. 29. 26J-5-8. 
 
 28. 19£-i-6. 30. 19£-*-3. 
 
Review. 161 
 
 31. Divide 18f by 7. 
 
 18$ -*- 7 = 2, with 4f remainder. | of 4$ = \ of ^ = U- An8 - 2 H« 
 
 32. 25J-5-2. 34. 191-5-4. 
 
 33. 31^ -v- 3. 35. 22J-J-5. 
 
 244. Written Exercises. 
 Perform indicated operations : 
 
 i. (lj + D + cef + i) 5 . 52£ X (ij-W) 
 
 i + * ' 5 
 
 ' fof4^|oflf '15-8 
 
 4. 231-^(3^ + 11) 8. ^of(3f-2| + 9i) 
 
 245. Find answers : 
 
 9. Simplify l±^zM. 
 
 10. Find the sum of f, ^, £, & ^. 
 
 11. Keduce ~T + i~? to a simple fraction. 
 
 12. Divide (Hi + !)by(fx}ix W- 
 
 13. Simplify i±i of } of ] 
 
 71 — 31 
 
 14. iz 2s = 9 
 
 15. Find the value of 4 fr + (t of tV) . 
 
 (|ofl»-f 
 
 16. (2J + l|) + (2i + 3i) = ? 
 
 17. Find the value of 2\ times the quotient of (3 — 2J)- 
 
 18. 3f + 14-7f + 5— ft-? 
 
r6i Chapter Three. 
 
 246. Multiply. Do not reduce to improper fractions. 
 12f 
 
 X4i 
 
 
 51 
 
 4 times 12f = 51 ; £ of 12f = 4*. 
 
 41 
 
 
 m\ 
 
 
 i. 
 
 18fx6J 6. 16fx7£ 
 
 2. 
 
 25f x8J 7. 48fxl2£ 
 
 3. 
 
 16fx5£ 8. 37fxl0£ 
 
 4. 
 
 36£x9£ 9. 36fx9J 
 
 5. 
 
 221x6^ 10. 32|x8| 
 
 247. Oral Eeview Problems. 
 
 1. What per cent does a boy receive if he solves 16 
 examples of the 20 given out ? 
 
 2. What is the interest on $ 200 at 4% for 2 years ? 
 
 3. If 2| yards of calico cost 22 cents, how many yards 
 can be bought for 60 ^ ? 
 
 4. How old, Dec. 1, 1904, was a boy born Sept. 1, 1891 ? 
 
 5. What is the cost of 3500 bricks at $ 6 per M ? 
 
 6. How many sheep, at $5 each, should be given in 
 exchange for 12 horses, worth $ 200 each ? 
 
 7. 75 men can do a certain piece of work in 9 days. 
 How long will it take 45 men to do the same work ? 
 
 8. If 4 barrels of oil each containing 42 gallons are 
 emptied into a tank of 200 gallons' capacity, how many 
 more gallons will the tank hold ? 
 
 9. Change .375 yard to feet and inches. 
 
 10. How many half-pints in 2 gal. 1 qt.? 
 
 11. How many eggs in 15 dozen and 6 eggs ? 
 
Review. i6j 
 
 12. f = how many 98ths ? 
 
 13. Find the greatest common divisor of 12, 18, 27. 
 
 14. Find the least common multiple of 8, 9, 12. 
 
 15. How many yards in 5 pieces of cloth, each contain- 
 ing 12J yards ? 
 
 16. Divide 29| by 7. 
 
 17. "When silk is 75^ per yard, how many yards can be 
 bought for $9.75? 
 
 18. If 2| yards ribbon cost 42 cents, what will 3f yards 
 cost? 
 
 19. If eggs are sold at the rate of 18 for 25 cents, what 
 will be the cost of 6 dozen eggs ? 
 
 20. Three men require 22 days to do a certain piece of 
 work. How long would it take 11 men to do the same 
 work? 
 
 21. A farmer divides his farm of 425 acres into fields of 
 12 J acres each. How many fields has he ? 
 
 22. What will be the cost of 46 tons of hay, at $ 12 J per 
 ton? 
 
 23. What is the weight of 25 firkins of butter, each con- 
 taining 56 pounds ? 
 
 24. At $ 1.75 per yard, how many yards of cloth can be 
 bought for $49? 
 
 25. If the interest of $ 1 is 6^ a year, what is the interest 
 of three dollars for two years ? 
 
 26. If 4 boxes of raisins cost $7, what will 12 boxes cost? 
 
 27. A man having 75 dollars bought 7 sheep, and had $5 
 left. What did he pay for each sheep ? 
 
 28. A boy had 59 peaches and found 22 more ; he then 
 divided all of them equally among 9 boys. How many did 
 he give to each ? 
 
164 Chapter Three. 
 
 29. I bought 2 J pounds of sugar at one store and 3 J 
 pounds at another. How many pounds did I buy in all ? 
 
 30. If I of a load of hay is worth $ 14, what will two 
 loads be worth ? 
 
 31. 2J + 1J = ? 32. 2fxl£=? 
 
 33. f of my money equals 63^. What is % of it ? 
 
 34. Least common multiple of 8, 12, 15, 24? 
 
 35. If 5 men can do a piece of work in 12 days, in how 
 many days can 3 men do twice as much work ? 
 
 36. John lost \ of his money and has 96^ left. How 
 much had he at first ? 
 
 37. At 6^ a quart, what will 10 quarts 1 pint of milk cost ? 
 
 38. I bought a dozen oranges at the rate of 4 oranges for 
 3$, and sold them at the rate of 3 oranges for 0. How 
 much did I make ? 
 
 39. How long would it take 3 men to cut 12 cords of 
 wood, if 4 men can cut 8 cords in 2 days ? 
 
 40. John sold 24 tops at the rate of 3 tops for ten cents, 
 and with the money bought pictures at 8^ each. How many 
 pictures did he buy ? 
 
 41. How many pounds of cheese at -fa of a dollar per 
 pound can be bought for j of a dollar ? 
 
 42. 18 is f of -£ of what number ? 
 
 43. If one man can do a piece of work in llf days, in 
 what time can 12 men do it ? 
 
 44. How many times is £ contained in 2 J ? 
 
 45. If oranges are 37 J cents per dozen, what will be the 
 cost of a box containing 480 oranges ? 
 
Review. 165 
 
 248. Written Eeview Problems. 
 
 1. At 70 cents per 100 pounds, what will be the amount 
 of duty on an invoice of 3622 steel rails, each rail being 
 27 feet long and weighing 60 pounds to the yard ? 
 
 2. A man had property valued at $6500. What will 
 be his taxes at the rate of $10.80 per $1000 ? 
 
 3. Multiply seventy thousand fourteen hundred-thou- 
 sandths by one hundred nine millionths, and divide the 
 product by five hundred forty-five. 
 
 4. What number multiplied by 43f will produce 265f ? 
 
 5. What decimal of a bushel is 3 quarts ? 
 
 6. A man sells f of an acre of land for $ 93.75. What 
 would be the value of his farm of 150f acres at the same 
 rate? 
 
 7. A coal dealer buys 375 tons coal at $4.25 per ton of 
 2240 pounds. He sells it at $ 4.50 per ton of 2000 pounds. 
 What is his profit ? 
 
 8. Bought 60 yards of cloth at the rate of 2 yards for $ 5, 
 and 80 yards more at the rate of 4 yards for $ 9. I imme- 
 diately sold the whole of it at the rate of 5 yards for $ 12. 
 How much did I gain ? 
 
 9. A man purchased 40 bushels of apples at $1.50 per 
 bushel. Twenty-five hundredths of them were damaged, 
 and he sold them at 20 cents per peck. He sold the 
 remainder at 50 cents per peck. How much did he gain 
 or lose ? 
 
 10. If oranges are 37^- cents per dozen, how many boxes, 
 each containing 480, can be bought for $ 60 ? 
 
 11. A man can do a piece of work in 18f days. What 
 part of it can he do in 6| days ? 
 
 12. How old to-day is a boy that was born Oct. 29, 1896 ? 
 
1 66 Chapter Three. 
 
 13. At the rate of $ 5 per ton, what should be paid for 
 125 pounds of coal ? 
 
 14. From ten and five hundredths take the sum of six 
 ten-thousandths and 15 millionths, multiply the remainder 
 by one-tenth, and divide the product by 5000. 
 
 15. Keduce the following common fractions to decimals, 
 and perform the operations indicated : 
 
 ("oWC X   "2U") ~-~ 2 00000* 
 
 16. A man died in 1903, aged 94; his son died in 1887, 
 aged 47. How old was the man at the birth of his son ? 
 
 17. Multiply the sum of 6§ and 4| by their difference. 
 
 18. What will be the cost of 86,400 feet of gas at $1.25 
 per thousand feet ? 
 
 19. What time elapsed between the discovery of America, 
 Oct. 14, 1492, and Jan. 1, 1904 ? 
 
 20. How many hats can be bought for $237.25, at the 
 rate of $13 per dozen ? 
 
 21. A clerk receives a salary of $1500 per year, and his 
 expenses are $968. In what time can he save enough to buy 
 133 acres of land at $28 per acre ? 
 
 22. What will be the rent of a house for 1 yr. 10 mo. at 
 $45 per month? 
 
 23. The product is .00087, the multiplicand is 7.25. What 
 is the multiplier ? 
 
 24. A man sells cloth at $2.88 per yard, losing .04 of the 
 cost. How much did he pay per yard ? 
 
 25. A farm hand agreed to work for $300 per year and a 
 horse worth $60. If he leaves at the end of 9 months, how 
 much is due him if he has already received $100 and the 
 horse ? 
 
Review. 167 
 
 26. A train running 36 miles per hour leaves a station at 
 9 a.m. At 10.30 a.m. a second train leaves and runs at the 
 rate of 30 miles per hour. How many miles apart are the 
 trains at noon, if they run in the same direction ? 
 
 27. Multiply twenty thousand nine hundred eight by six- 
 teen. Divide the result by seven. 
 
 28. Divide two hundred sixteen by thirty-six thousandths. 
 Take seventy -five hundredths from the quotient. 
 
 29. If one acre yields 14 bu. 3 pk. cranberries, how much 
 will 40 acres yield ? 
 
 30. Find the difference between 3£ x 6f and 7^ -5- If. 
 
 31. An errand boy receives $2.75 per week. In how 
 many weeks will he earn enough to buy a pair of boots 
 worth $3.25, a coat worth $4.75, a hat worth $1.50, and 6 
 handkerchiefs worth 25 cents each? 
 
 32. How many cords of wood at $5J a cord must I give 
 for 78f bushels of wheat at $1.20 a bushel, and 84 bushels 
 of rye at $1 a bushel ? 
 
 33. Mr. Louis Scott bought from Thomas Green, at Phil- 
 adelphia, Jan. 10, 1904, the following : 67 pairs of boots at 
 $3.25 per pair; 75 pairs of gaiters at $1.12 per pair; 35 
 pairs of slippers at 70 cents per pair ; 50 pairs of rubbers at 
 62£ cents per pair. Make out and receipt the bill. 
 
 34. What will £ of a yard of cloth cost, if £ of a yard 
 costs $1.60 ? 
 
 35. Divisor 3£; quotient 400. Find dividend. 
 
 36. Dividend .014 ; quotient 2000. Find divisor. 
 
 37. Divide 118.35 by .04£, and add 3.0045 to the quotient. 
 
 38. If If yards of cloth are worth 11 \ dollars, what is a 
 yard worth ? 
 
1 68 Chapter Three. 
 
 39. If a roll of carpet, containing 75 yards, is worth 
 $ 132, what is f of a yard worth ? 
 
 40. How many quarts of berries at 11 cents a quart will 
 it take to buy 2| yards of cloth at 16J cents a yard ? 
 
 41. A man sold -J- and ^ of his farm and had 26 1 acres 
 left. How many acres had he at first ? 
 
 42. A boy sleeps | of his time, plays -J- of it, and goes 
 to school one-half the remainder. How many hours is he 
 in school each school day ? 
 
 43. Write in four other ways the quantity or value ex- 
 pressed by .16. 
 
 44. Bought 3 bu. 2 pk. of oats for $ 1.38 and retailed 
 them at $ .12| a peck. What was the gain ? 
 
 45. From a hogshead of molasses containing 54 gal. 2 qt. 
 there was sold 23 gal. 1 pt. What was the value of the 
 remainder at 8 cents a quart ? 
 
 46. What is the result, if the sum of 5 yd. 2 ft., 3 yd. 
 1 ft., and 14 yd. 1 ft. be taken from 42 yards ? 
 
 47. Eeduce £ of a day, ^j- of an hour, and T 4 7 of a minute 
 to common denominator, and add. 
 
 48. Bought a carriage for $180, and after paying 10% 
 for repairs, sold it at a profit of 25% of the total cost. 
 Find gain and selling price. 
 
 49. A man sold a horse for $ 125, and received in pay- 
 ment 12£ yards of cloth at $ 3.25 a yard, and the balance 
 in tea at $ .62 \. How many pounds of tea did he receive ? 
 
 50. Find equivalent per cents for the following : £, |, -J, 
 
 h &, f • 
 
 51. If 64 tons of iron cost $4816, how many tons can 
 be bought for $ 1730.75 ? 
 
 52. Change 28 gal. 3 qt. to quarts. 
 
Review. 1 69 
 
 53. A man carried to a store 75f bushels of potatoes, and 
 received for them 27-J^ a bushel. How many yards of cloth, 
 at 17f ^ a yard, would have paid for them ? 
 
 54. What will 75 men earn in 18f days, if each earns 2\ 
 dollars each day ? 
 
 55. What will 8 yd. 2 ft. 6 in. of silver wire cost at 8f t 
 an inch ? 
 
 56. A young man spent $195^ during his first term at 
 college, which was f of his year's allowance. What was his 
 year's allowance, and what had he left for the remainder of 
 the year ? 
 
 57. A man paid $ 18.60 for a load of hay weighing 
 2 1 tons. At the same rate, what should he pay for \ of 
 a ton? 
 
 58. Divide 4.5006 by .015. 
 
 59. One man owns -^j- of an estate ; another owns f|-f 
 of it ; and a third man owns -£^ of it. What part of the 
 whole do they own together ? 
 
 Note. — Reduce the fractions to lowest terms, by inspection. 
 
 60. If it takes 11 men 45§ days to do a piece of work, 
 how many days will it take one man to do the same work ? 
 
 61. I owned f of a house, and sold f of my share for 
 $1750. What was the value of the whole house at that 
 rate? 
 
 62. A grocer, after selling -J-, f , / ¥ , and \ of a quantity of 
 sugar, had 102 pounds left. How many pounds did he have 
 at first ? 
 
 63. A dealer in grain bought wheat at 94^ a bushel to 
 the amount of $59.22, and sold it for $70.56. What was 
 the selling price per bushel ? 
 
 64. If I of a cord of wood is worth $ 3.75, what will J of 
 a cord cost ? 
 
170 Chapter Three. 
 
 65. A man who had $ 50 J-, received $ S\ more, spent 
 $ 17}, lost $ 4y%, and collected $ 15£ of a debt How much 
 money had he then ? 
 
 66. 12f is what part of 29 ? 
 
 67. What must a carpenter pay for the following : 6500 
 shingles, at $ 4.75 per thousand ; 15,964 feet of boards, at 
 $ 39.25 per thousand ; 4849 feet of planks, at $ 45.32 per 
 thousand ? 
 
 68. A farmer sold -§ of his wheat for $ 796| and received 
 for it $ ly 1 ^ per bushel. How many bushels did he have at 
 first, and how many did he sell ? 
 
 69. If 123 tons of coal cost $ 848.70, what will be the 
 cost of 265 tons ? 
 
 70. A dealer sold -^ of his wheat to Mr. Adams, -J of it 
 to Mr. Baker, and y% of it to Mr. Charles ; then he had 630 
 bushels left. How much had he at first ? 
 
 71. Mr. Blank bottled 135 gallons of ink in bottles that 
 held I of a pint ; he sold it for \2\$ a bottle. How much 
 did he receive? 
 
 72. Three times a number, increased by -^ of the 
 number, equals 22. What is the number? 
 
 73. A grocer having a capital of $ 10,000, invested \ of 
 it in tea at -fa of a dollar per pound, ^ of the remainder in 
 coffee at \ of a dollar a pound, and -fe of the rest in sugar 
 at 5 cents per pound. What quantity of each did he buy, 
 and what money had he left ? 
 
 74. What will be the cost of 53,715 pounds of wheat at 
 90 cents per bushel of 60 pounds ? 
 
 75. A drover sold 15 cattle, weighing 1468 pounds each, 
 at $ 4.40 per hundred pounds. How much did he receive ? 
 
 76. After losing | of his money, a man had $ 75 left 
 How much had he at first? 
 
Review. 171 
 
 77. What will be the cost of 24 gallons 3 quarts of milk 
 at 4 cents per pint ? 
 
 78. A man bought a house for $6250 and sold it for 
 $6500. What fraction of the cost is the profit? What 
 decimal ? 
 
 79. At $30 per month, how much rent would a man pay 
 from July 1, 1904, to May 1, 1906? 
 
 80. How many sheep at $6.75 each should be given in 
 exchange for 54 horses worth $ 160 each ? 
 
 81. A man spent three-tenths of his money for clothes, 
 and one-fifth of it for rent, and had $ 75 left. How much 
 did his clothes cost? 
 
 82. What would be the cost of 48,500 stamped envelopes 
 at $21.30 per thousand ? 
 
 83. The width of a room is £ of its length. How many 
 square feet in the floor, if the width is 
 
 15 feet ? 
 
 84. If 2 lb. 6 oz. of tea cost 95 cents, '/*» 
 how many pounds and ounces can be -- 
 bought for $2.35? 
 
 85. John and James went out together, John had 38 
 cents. When one of the boys had spent 18 cents and the 
 other had spent 16 cents, they had 24 cents left between 
 them. Find the amount of money James had. 
 
 86. Find \ of the sum of f and f . 
 
 87. What is \ of the difference between $ and -|? 
 
 88. What fraction added to f gives f ? 
 
 89. Change 1^- hour to seconds. 
 
 90. £ of what number equals 180 ? 
 
 21 a 
 
 91. The half of a number added to its - 
 
 fourth part equals 21f. What is the number? 
 
172 Chapter Three. 
 
 92. A farm is sold for $5700, at a loss of -fa of the cost. 
 What was the cost ? 
 
 93. When it is noon at Philadelphia, it is 15 seconds and 
 10 minutes past 5 p.m. at Paris. What time is it at Phila- 
 delphia when it is noon at Paris ? 
 
 94. A, B, and C buy a house. 
 
 A furnished } of the cost, B fc 1 A 1 B I C 1 
 
 and C$1200. What did A and * * $ 1200 ° 
 
 B pay, respectively ? 
 
 95. After James has spent f of his money and \ of the 
 remainder, he has but $ 1.50 left. How much had he at 
 first? 
 
 96. A man buys oranges at $1.20 per 100. How many 
 would he have to sell, at 25# per dozen, to gain $3.18? 
 
 97. From a piece of cloth measuring 28 J yards, there 
 have been sold 2| yards, 6| yards, 13} yards. If the re- 
 mainder is worth $13.10, what was the value of the whole 
 piece ? 
 
 98. A man left for charitable purposes $3600, which was 
 I of his money. The remainder was divided equally among 
 8 relatives. How much did each relative receive ? 
 
 $3600. 
 
 k % i 
 
 Charitable purposes 
 
 tit 
 
 Eight relatives 
 
CHAPTER IV. 
 
 PAGES 
 
 Denominate Numbers 173 to 189 
 
 Reduction, Descending and Ascending, Compound 
 Addition, Subtraction, Multiplication, and Division, 
 Avoirdupois Weight, Time between Dates. 
 
 Percentage 189 to 194 
 
 Applications and Simple Interest. 
 
 Measurements 195 to 209 
 
 Area of Rectangles, Square Measure, Solid Contents, 
 Cubic Measure, Surfaces of Rectangular Solids, 
 Angles, Triangles, Quadrilaterals. 
 
 Review of Simple Numbers and Fractions . . 209 to 218 
 Special Drills, Sight Approximations, Fundamental 
 Processes, Cancellation, Review Fractions, Review 
 Decimals. 
 
 Review Problems . . 218 to 228 
 
 Miscellaneous, Oral, Written. 
 
 DENOMINATE NUMBERS. 
 
 249. Preliminary Exercises. 
 
 How many quarts in 5 gal. ? 
 How many quarts in 5 gal. 3 qt. ? 
 How many pints in 23 qt. ? 
 How many pints in 23 qt. 1 pt. ? 
 How many pints in 5 gal. 3 qt. 1 pt. ? 
 
 REDUCTION DESCENDING. 
 
 250. Keduce 5 gal. 3 qt. 1 pt. to pints. 
 
 In the first few examples, write 4 (the number of quarts in a gallon) 
 above the quarts, and 2 (the number of pints in a quart) above the 
 
 173 
 
174 Chapter Four. 
 
 pints. In 5 gallons there are 5 times 4 quarts, or 20 quarts ; adding 
 the 3 quarts, we have 23 quarts, as the 
 
 equivalent of 5 gallons 3 quarts, which is 4 qt. 2 pt. 
 
 written in the column of quarts. In 23 5 „ a ^ 3 qt. 1 pt. 
 
 quarts 'there are 23 times 2 pints ; adding 1 ~ ._ ~~ 
 
 pint, we have 47 pints as the equivalent of 5 ^ ' 
 
 gallons 3 quarts 1 pint. This is written in the Ans. 47 pt. 
 column of pints, the 23 quarts being cancelled. 
 
 Changing a denominate number to an equivalent denominate 
 number of a lower denomination is called reduction descending. 
 
 251. Written Exercises. 
 Reduce to pints : 
 
 1. 16 gal. 1 qt. 1 pt. 6. 31 J gal. 
 
 2. 27 gal. 2 qt. 7.9 gal. 2 J qt. 
 
 3. 16 gal. 8. 10 gal. 2 qt. 1 pt. 
 
 4. 16 gal. 1 pt. 9. 27 gal. 1 pt. 
 
 5. 34 gal. 3 qt. 1 pt. 10. 4 gal. 3 qt. 1£ pt. 
 
 REDUCTION ASCENDING. 
 
 252. Change 67 pt. to gallons, quarts, and pints. 
 
 Place 2 (the number of pints in a quart) above 67 pints. In 67 pints 
 
 there are 33 quarts and 1 pint. Write 
 
 33 quarts to the left of 67 pints, and 4 qt. 2 pt. 
 
 the 1 pint remainder in the column of 03 qt qj p^ 
 
 pints. Change the 33 quarts to 8 gal- ~ r - : ; : 7 A 
 % * * ,00 8 gal. 1 qt. 1 pt. Ans. 
 
 Ions 1 quart, and cancel 33 quarts. * * 
 
 Changing a denominate number to an equivalent denominate 
 number of a higher denomination is called reduction ascending. 
 
Denominate Numbers. 175 
 
 253. Written Exercises. 
 Change to gallons, etc. 
 
 1. 156 qt. 6. 177 pt. 
 
 2. 79 qt. 7. 139 pt. 
 
 3. 408 pt. 8. 171 qt. 
 
 4. 1302 pt. 9. 63 qt. 
 
 5. 63 pt. 10. 711 pt. 
 
 254. Review the tables of Long Measure, Dry Measure, Liquid 
 Measure, Avoirdupois Weight, and Time, Art. 93, pages 43-44. 
 
 Change : 
 
 1. 17 yd. 1 ft. 9 in. to inches. 
 
 2. 4 mi. 100 rd. 4 yd. to yards. 
 
 3. 74 bu. 2 pk. 7 qt. to quarts. 
 
 4. 156 lb. 11 oz. to ounces. 
 
 5. 63 yd. ft. 3 in. to inches. 
 
 6. 19 bu. pk. 3 qt. to quarts. 
 
 7. 11 rd. 3£ yd. to feet. 
 
 8. 63 gal. 3 qt. to pints. 
 
 9. 3 bu. 6 qt. to quarts. 
 
 10. 17 T. 369 lb. to pounds. 
 
 11. 15 hr. 16 min. to seconds. 
 
 12. 4 wk. 6 da. 11 hr. to hours. 
 
 Note. — Reduce a denominate fraction or a denominate decimal to 
 lower denominations by multiplying. 
 
 13. -f- of a week to hours. 
 
 14. ■£% of a mile to yards. 
 
 15. .00125 ton to ounces. 
 
 16. 1876 inches to yards, etc. 
 
 17. 475 ounces to pounds, etc. 
 
176 Chapter Four. 
 
 18. 729 quarts to bushels, etc. 
 
 19. 8675 minutes to days, etc. 
 
 20. 4972 pounds to tons, etc. 
 
 21. 972 rods to miles, etc. 
 
 22. 117 pints to gallons, etc. 
 
 23. 9483 seconds to hours, etc. 
 
 24. 877 quarts to bushels, etc. 
 
 25. 1495 ounces to pounds, etc. 
 
 26. 373 inches to yards, etc. 
 
 27. 216 quarts to gallons, etc. 
 
 28. 876 rods to miles, etc. 
 
 29. 319 pints to gallons, etc. 
 
 30. 3520 yards to miles. 
 
 255. Oral Exercises. 
 
 1. How many hours in § of a day ? 
 
 2. How many hours in -^ of a day ? 
 
 3. How many minutes in £ of an hour ? 
 
 4. How many hours and minutes in £ of a day ? 
 
 ■J day ss 4$ hours ; f hour = 48 minutes. \ day = 4 hours 48 
 ninutes. 
 
 5. How many quarts and pints in f of a gallon ? 
 
 6. How many hours and minutes in .2 day ? 
 
 .2 day = 4.8 hours ; .8 hour = 48 minutes. .2 day = 4 hours 48 
 minutes. 
 
 7. How many quarts and pints in .375 gallon ? 
 
 8. Change .3 day to hours and minutes. 
 
 9. Change .625 bushel to pecks and quarts. 
 
 10. What part of a gallon is 1 pint ? 
 
 11. What part of a gallon is 3 pints ? 
 
Denominate Numbers. 177 
 
 12. What part of a gallon is 1 qt. 1 pt. ? 
 
 13. What decimal of a gallon is 1 qt. 1 pt. ? 
 
 14. What decimal of a gallon is 2 qt. 1 pt. ? 
 
 15. What part of 2 gallons is 2 qt. 1 pt. ? 
 
 16. Change .375 bushel to pecks and quarts. 
 
 17. What decimal of a bushel is 4 quarts ? 
 
 18. What fraction of a day is 3 hr. 20 min. ? 
 
 19. Eeduce 960 minutes to hours. 
 
 20. How many minutes in a day ? 
 
 256. Written Exercises. 
 
 1. What decimal of a ton is 3 pounds ? 
 
 Pounds are changed to tons by dividing by 2000. 
 3 lb. = jfa T. s .0015 T. Ans. 
 
 2. What fraction of an hour is 12 min. 30 sec. ? 
 
 12 min. 30 sec. = 12 J min. =^ihr. = M, hr. = &. 
 
 60 
 
 3. Eeduce -£% of a day to minutes. 
 
 ft day = (3V x 24) hr. = (^ X*£ X ^) min. Cancel. 
 
 4. Eeduce .03125 day to minutes. 
 
 5. What decimal of a day is 9 minutes? 
 
 6. What will be the cost of 15 T. 500 lb. coal at $ 7 per 
 ton? 
 
 7. When coal is $5 per ton, how many tons and pounds 
 can be bought for $ 18.75 ? 
 
 8. Change 2 ft. 7 in. to the fraction of a yard. 
 
 2 ft. 7 in. = 2 T V ft. = ?i yd. = ft yd. , Ans. 
 
 Note. — An expression such as -J3 i s called a complex fraction. It 
 
 o 
 
 indicates the division of 2^ by 3 ; that is, f \ x |, or %\. 
 
178 Chapter Four. 
 
 9. Eeduce 3 pk. 4 qt. to the decimal of a bushel. 
 
 4 qt. = .5 pk. ; 3 pk. 4 qt. = 3.5 pk. = — bu. ; etc. 
 
 4 
 
 10. How many pecks and quarts in .9375 bushel ? 
 
 11. If .1875 of a gallon of cologne cost $ 1.125, what will 
 
 1 pint cost? 
 
 Note. — $.125 is read 12 cents 5 mills. 
 
 12. Find the cost of 42 gal. 3 qt. 1 pt. oil, at 16 cents 
 per gallon. 
 
 13. Reduce ij of a gallon to quarts and pints. 
 
 14. What part of 3 T. is 1 T. 960 lb. ? 
 
 15. A man raised 194 bu. 1 pk. of rye. He sold 129 bu. 
 
 2 pk. What fraction of his crop did he sell ? 
 
 16. 10 bu. 1 pk. of seed are packed in 8 bags. What 
 quantity is there in each bag ? 
 
 17. What decimal of a day is 15 hr. 45 min. ? 
 
 18. How many feet are there in a mile ? 
 
 COMPOUND ADDITION. 
 
 A compound denominate number expresses two or more denomina- 
 tions of the same kind. 
 
 316 T. 1816 lb. is a compound denominate number. 
 487 T. is a simple denominate number. 
 
 In adding and subtracting compound denominate numbers, 
 write units of the same denomination in the same column. 
 
 257. Add the following : 
 
 1. 18 bu. 3 pk. 7 qt. 7 qt. + 4 qt. + 6 qt. = 17 qt. = 2 pk. 1 qt. 
 
 9 bu. 2 pk. 4 qt. Write * qt. and carry 2 pk. 2 pk. + 2 pk. 
 
 14 bu. 1 pk. 6 qt. + 1 P k - + 2pk.+ 3pk. = lOpk. = 2bu.2pk. 
 
 2 pk w rite 2 pk. and carry 2 bu. 2 bu. + 14 bu. 
 
 Ans. 43 bu. 2 pk. 1 qt. 
 
 f 9 bu. + 18 bu. = 43 bu. 
 
Denominate Numbers. 
 
 16 yd. 2 ft. 9 in. 6. 12 T. 1576 lb. 
 
 17 yd. 4 in. 3 T. 980 1b. 
 
 lft. 6 in. 4761b. 
 
 11 in. 1 T. 1830 lb. 
 
 19 
 
 11 da. 5 hr. 19 min. 7. 2 wk. 5 da. 12 hr. 
 
 23 da. 40 min. 6 da. 15 hr. 
 
 17 hr. 50 min. 5 wk. 2 hr. 
 
 5 da. 20 hr. 6 min. 2 da. 19 hr. 
 
 4. 93 gal. 3 qt. 1 pt. 8. 18 mi. 100 rd. 
 74 gal. 34 rd. 
 18 gal. 1 qt. 29 mi. 
 
 2 qt. 1 pt. 6 mi. 160 rd. 
 
 5. 5 hr. 30 min. 20 sec. 9. 47 yr. 11 mo. 
 
 45 min. 33 sec. 5 yr. 9 mo. 
 
 6 hr. 11 min. 5 sec. 7 mo. 
 
 10 hr. 3 min. 30 sec. 22 yr. 5 mo. 
 
 10. 487 T., 316 T. 1816 lb., 247 lb., 43 T. 811 lb., 19 T. 25 lb. 
 
 11. 83 lb. 15 oz., 9 lb. 5 oz., 18 lb., 22 lb. 11 oz., 5 lb. 8 oz. 
 
 12. 8 hr. 15 min. 5 sec, 37 min. 52 sec, 5 hr. 48 min., 23 hr. 
 
 13. 72 gal. 3 qt. 1 pt., 17 gal. 1 qt., 2 qt. 1 pt., 90 gal. 1 pt. 
 
 14. 7 yd. 2 ft. 11 in., 19 yd. 6 in., 105 yd., 4 yd. 2 ft. 2 in., 1 ft. 
 
 15. 93 mi. 300 rd., 87 mi. 154 rd., 194 rd., 3 mi. 175 rd., 9 mi. 
 
 COMPOUND SUBTRACTION. 
 
 258. Subtract: 
 
 Change 83 yr. 3 mo. to 82 yr. 15 mo. Sub- 
 1. 83 yr. 3 mo. tract 9 months from 15 months. Write the 
 15 yr. 9 mo. remainder, 6 months, in the column of months. 
 Subtract 15 years from 82 years. 
 
 Ans. 67 yr. 6 mo. 
 
180 Chapter Four. 
 
 2. 62 mi. 84 rd. 7. 18 hr. 5 min. 
 
 19 mi. 159 rd. 40 min. 25 sec. 
 
 3. 
 
 76 T. 225 lb. 
 37 T. 1679 lb. 
 
 4. 
 
 100 lb. 
 83 lb. 4 oz. 
 
 5. 
 
 52 wk. 
 
 13 wk. 3 da. 7 hr. 
 
 6. 
 
 19 gal. 1 pt. 
 8 gal. 3 qt. 
 
 8. 
 
 16 yd. 9 in. 
 7 yd. 1 ft. 11 in. 
 
 9. 
 
 100 bu. 
 42 bu. 3 pk. 7 qt. 
 
 10. 
 
 45 da. 1 hr. 1 min. 
 6 da. 6 hr. 6 min. 
 
 11. From 27 bu. 1 pk. 5 qt. take 13 bu. 3 pk. 7 qt. 
 
 12. From 100 gal. 1 qt. take 83 gal. 2 qt. 1 pt. 
 
 13. From 22 hr. 15 min. 20 sec. take 15 hr. 45 min. 40 sec. 
 
 14. From 17 lb. 2 oz. take 13 lb. 8 oz. 
 
 15. From 100 bu. take 74 bu. 2 pk. 1 qt. 
 
 COMPOUND MULTIPLICATION. 
 
 259. Written Exercises. 
 
 Multiply 4 gal. 3 qt. 1 pt. by 3. 
 
 3 times 1 pt. = 3 pt. 3 pt = 1 qt. 
 
 4 gal. 3 qt. 1 pt. 1 pt. Write 1 pint in the column of 
 
 3 pints. 3 times 3 qt. = 9 qt. ; 9 qt. + 
 
 14 gal. 2 qt. 1 pt. Ans. the 1 <$• t0 cari 7 = 10 %*>• 10 <$- = 
 
 2 gal. 2 qt. Write 2 quarts in the 
 column of quarts. 3 times 4 gal. = 
 12 gal. ; 12 gal. + 2 gal. to carry = 14 gal. Ans. 14 gal. 2 qt. 1 pt. 
 
Denominate Numbers. 181 
 
 Multiply : 
 
 1. 13 bu. 3 pk. 6 qt. by 2. 7. 25 lb. 4 oz. by 8. 
 
 2. 25 gal. 2 qt. 1 pt. by 3. 8. 33 min. 33 sec. by 9. 
 
 3. 7 lb. 10 oz. by 4. 9. 2 pk. 7 qt. by 10. 
 
 4. 23 bu. 3 qt. by 6. 10. 3 qt. 1 pt. by 11. 
 
 5. 32 gal. 1 pt. by 7. 11. 4 yr. 6 mo. by 12. 
 
 6. 3 hr. 15 min. 15 sec. 12. 5 wk. 6 da. 12 hr. 
 by 5. by 16. 
 
 COMPOUND DIVISION. 
 
 260. Divide 54 yd. 1 ft. 4 in. by. 20. 
 
 54 yd. -T- 20 gives a quotient of 2 yd., 
 
 2 — 2_j : IB; which is written, and a remainder oi 
 
 2 yd. 2 ft. 2 m. Ans. «. . _ . ' . . Aa ' , 
 
 14 yd. Reduce 14 yd. to 42 ft., and 
 
 add 1 ft., making 43 ft. 43 ft. — 20 gives a quotient of 2 ft., which is 
 
 written, and a remainder of 3 ft. Reduce 3 ft. to 36 in., and add 4 in., 
 
 making 40 in. 40 in. -f- 20 gives a quotient of 2 in. which is written. 
 
 Divide : 
 
 1. 13 wk. by 5. 7. 17 lb. 7 oz. by 3. 
 
 2. 15 lb. 9 oz. by 3. 8. 37 bu. 3 pk. 6 qt. by 2. 
 
 3. 2 lb. 3 oz. by 5. 9. 67 yd. 2 ft. by 4. 
 
 4. 2 gal. 1 qt. by 3. 10. 33 da. 15 hr. 57 min. by 3. 
 
 5. 5 bu. by 4. 11. 561 gal. by 6. 
 
 6. 7 hr. by 6. 12. 22 hr. 20 min. 20 sec. by 4. 
 
 13. 109 gal. 1 qt. 1 pt. by 7. 
 
 14. 273 yd. 1 ft. 6 in. by 9. 
 
 15. 155 bu. 3 pk. 2 qt. by 6. 
 
 16. 180 da. 19 hr. 28 min. by 8. 
 
i8* 
 
 Chapter Four. 
 
 17. Divide 243 da. 4 hr. 2 min. by 15. 
 Dividing 243 days 
 
 15)243 da. 
 
 , 15 
 
 93 da. 
 
 _90 da. 
 
 3 da. 
 
 by 15 gives a quotient 
 of 16 days and a re- 
 mainder of 3 days. Re- 
 ducing 3 days 4 hours 
 to 76 hours and divid- 
 ing by 15 gives a quo- 
 tient of 5 hours and a 
 remainder of 1 hour. 
 Reducing 1 hour 2 minutes to 62 
 minutes and dividing by 15 gives 
 a quotient of 4 minutes and a 
 remainder of 2 minutes. Reducing 2 min- 
 utes to 120 seconds and dividing by 15 gives 
 a quotient of 8 seconds. 
 
 18. Divide 334 yd. 9 in. by 21. 
 15 yd. 2 ft. 9 in. 
 
 16 da. 5 hr. 4 min. 8 sec. 
 
 4 hr. 2 min. 
 
 4hr. 
 
 76 hr. 
 
 75 hr. 
 
 lhr. 
 
 2 min. 
 
 62 min. 
 
 60 min. 
 
 2 min. 
 
 120 sec. 
 120 sec. 
 
 21)334 yd. ft. 
 21 
 
 124 yd. 
 
 105 yd. 
 
 19 yd. 
 
 9in. 
 
 
 57 ft. 
 
 
 
 42 ft. 
 
 15 ft. 
 
 9 in. 
 
 
 
 189 in. 
 
 
 
 189 in. 
 
 19. 
 
 825 lb. by 48. 
 
 
 20. 
 
 112 T. by 25. 
 
 
 21. 
 
 43 mi. by 32. 
 
 
 22. 
 
 84 yr. by 24. 
 
 
 23. 
 
 462 bu. by 32. 
 
 
 24. 
 
 1078 yd. by 62 
 
 i 
 
 Insert the missing denomina- 
 tion, feet, with a cipher prefixed. 
 Reduce the 19 yards remainder to 
 57 feet. Reduce to 189 inches the 
 15 feet 9 inches remaining. 
 
 25. 288 hr. 9 min. by 54. 
 
 26. 863 gal. 2 qt. 1 pt. by 47. 
 
 27. 33 wk. 1 da. by 72. 
 
 28. 1138 T. 910 lb. by 81. 
 
 29. 1629 yd. 1 ft. by 96. 
 
 30. 1867 gal. 1£ pt. by 125. 
 
Denominate Numbers. 183 
 
 261. Avoirdupois Weight. Long Ton. 
 
 In selling iron, coal at the mines, ores, etc., and in calculating the 
 duties at the U. S. custom houses upon imported goods, the following 
 table is used : 
 
 28 pounds (lb.) = 1 quarter (qr.) 
 
 4 quarters = 1 hundredweight (cwt.) 
 
 20 hundredweight = 1 ton (T.) 
 
 1 cwt. = 112 lb. 1 T. = 2240 lb. 
 
 The ton of 2240 pounds is called a long ton. Unless otherwise 
 specified in a problem, the cwt. of 100 pounds and the ton of 2000 
 pounds are to be taken. 
 
 262. Oral Problems. 
 
 1. How many tons and pounds of coal in 40 bags, each 
 containing 80 pounds ? 
 
 2. If it takes 3 hr. 20 min. to hoe a row of corn, how 
 long will it take to hoe 3 rows ? 
 
 3. A man puts up 3^- pounds of tea into 4 ounce pack- 
 ages. How many packages does he make ? 
 
 4. 3 pk. 3 qt. of apples are divided among 9 children. 
 What quantity does each child receive ? 
 
 5. What part of a day is 30 minutes ? 
 
 6. If there are 2\ gallons of wine in 12 bottles, how 
 many pints are there in each bottle ? 
 
 7. What is the weight of two packages each containing 
 15 lb. 11 oz. ? 
 
 8. What part of an hour is 40 seconds ? 
 
 9. What is the rent of a house for 1 year 9 months at 
 $16 per month? 
 
 10. If 3 gal. 2 qt. 1 pt. of milk are taken from a can con- 
 taining 10 gallons, how much is left in the can ? 
 
184 Chapter Four. 
 
 11. 5 hams weigh 61 J pounds. What is the average 
 weight ? 
 
 12. There are on an average 41 pupils in a class. How 
 many are there in 14 classes ? 
 
 13. At 37^- cents per yard, how many yards can be 
 bought for $6.75? 
 
 • 6f-*.$t = ¥ + t = ¥-*-t. etc. 
 
 14. Find the cost of 16 barrels of flour at $6.12£ each. 
 
 15. $1.65 is equally divided among 15 boys. What is 
 the share of each ? 
 
 16. A floor containing 40 J square yards is 7 yards long. 
 How many yards wide is it ? 
 
 17. How many ounces in 5J pounds ? 
 
 263. Written Problems. 
 
 1. If a watch gains 1 min. 17 sec. per day, how much 
 will it gain during March and April ? 
 
 2. How many bushels, pecks, and quarts in 1449 pounds 
 of corn, weighing 56 pounds to the bushel ? 
 
 3. Eeduce 25 T. 13 cwt. 2 qr. 25 lb. to pounds (long 
 ton). 
 
 4. A chain, 97 yd. 8 in. long, contains 1000 links. Find 
 the length of one of the links. 
 
 5. A farmer sold out of 5 bushels of peas the following 
 quantities : 3 pk. 6 qt. ; 4 pk. ; 4 pk. 3 qt. ; 1 bu. 1 pk. 1 qt. 
 How much has he still to sell ? 
 
 6. Change 100,000 pounds to tons (long), cwt., qr., lb. 
 
 7. A man walks on Monday 15 mi. 161 rd. ; Tuesday, 
 10 mi. 84 rd. ; Wednesday, 19 mi. 15 rd. ; Thursday and 
 Friday, 12 mi. 121 rd. each day ; Saturday, 14 mi. 240 rd. 
 What distance per day does he average ? 
 
Denominate Numbers. 185 
 
 8. If the sun rises at 5 hr. 10 min. a.m., and sets at 6 hr. 
 42 min. p.m., how long is the day ? How many hours and 
 minutes of night ? 
 
 9. Find the duty at 1-j 2 ^ per pound on an invoice of tin 
 weighing 33 T. 7 cwt. 20 lb. (long ton). 
 
 10. An iron rod is 12 ft. 6 in. long. Prom it are cut 
 73 bolts, each If inches long. How much is left ? 
 
 11. A man rows a mile in 10 min. 30 sec. How long 
 would he take to row 27 miles at the same rate ? 
 
 12. What is the total weight in tons (long), etc., of 19 
 barrels of sodarash weighing 13 cwt. 2 qr. 10 lb. each ? 
 
 13. A man rows 51 miles in 23 hr. 5 min. and 30 sec. 
 How long does he take to row a mile ? 
 
 14. If I lost $50 by selling a lot for two-thirds of its 
 cost, what would I have lost if I had sold it for three-fourths 
 of its cost ? 
 
 15. At the rate of $2.75 per day of 8 hours, how much 
 should be given a man that works from a quarter before 8 in 
 the morning until 5 minutes past 11 in the morning ? 
 
 16. If a railroad train travels 18 miles in 40 minutes, 
 how far will it travel, at the same rate, in 1\ hours ? 
 
 17. A coal dealer buys 175 (long) tons of coal. How 
 much does he receive for it at $ 5 per ton of 2000 pounds ? 
 
 TIME BETWEEN DATES. 
 
 264. Oral Problems. 
 
 1. How many hours from 3 o'clock Saturday afternoon 
 to 9 o'clock Sunday morning ? 
 
 2. How many days from May 1 to June 1 ? 
 
 3. A boy takes a spoonful of medicine every hour. If 
 he takes the first dose at 2 o'clock, at what hour will he take 
 the sixth ? The second ? The fourth ? 
 
1 86 Chapter Four. 
 
 4. A man begins work on the morning of the 6th and 
 ends on the evening of the 11th. How much does he earn 
 at $ 3 per day ? 
 
 5. An importer receives some cases of goods numbered 
 consecutively. How many cases are there if the lowest 
 number is 29 and the highest number is 53 ? 
 
 6. How many posts 6 feet apart will be needed for a 
 fence 120 feet long. For a fence 6 feet long ? 12 feet 
 long? 
 
 7. Find the time from Jan. 1 to Jan. 31, counting the 
 first and the last day. Omitting both days. 
 
 8. How many days from July 4 to Aug. 15, inclusive ? 
 
 9. How many chapters from the 25th to the 49th, 
 exclusive ? 
 
 10. A girl begins at the 146th problem and solves all 
 those on two pages. If the last is the 172d problem, how 
 many does she solve ? 
 
 265. How many days from March 4 to Sept. 1 ? 
 
 March 4 to March 31, 27 days 
 
 Excluding March 4, there remain 
 in the month 31 — 4, or 27 days. To 
 this add the number of days in April, 
 May, June, July, and August. Since 
 March 4 is excluded, we take 1 day 
 in September, making the total 181 
 days. 
 
 In finding the time between dates, either the first or the 
 last day is excluded; that is, from the 1st to the 21st is con- 
 sidered 20 days. 
 
 April 
 
 30 
 
 May 
 
 31 
 
 June 
 
 30 
 
 July 
 
 31 
 
 Aug. 
 
 31 
 
 Sept. 
 
 1 
 
 Ans. 
 
 181 days 
 
Denominate Numbers. 187 
 
 L How many days from 
 
 11. Jan. 1 to Feb. 19? 16. Feb. 29 to April 1 ? 
 
 12. Oct. 31 to Dec. 30 ? 17. May 21 to July 4 ? 
 
 13. Sept. 30 to Dec. 16 ? 18. April 7 to May 27 ? 
 
 14. Nov. 1 to Dec. 19 ? 19. June 10 to Aug. 1 ? 
 
 15. March 16 to April 25 ? 20. July 4 to Sept. 1 ? 
 
 267. Written Problems. 
 
 Take note of leap year. 
 How many days from : 
 
 1. Feb. 6, 1903, to Oct. 1, 1903 ? 
 
 2. Oct. 14, 1903, to March 3, 1904? 
 
 3. Jan. 1, 1904, to April 19, 1904? 
 
 4. Dec. 23, 1904, to March 8, 1905 ? 
 
 5. Sept. 3, 1903, to Feb. 1, 1904 ? 
 
 6. March 16, 1904, to Dec. 25, 1904 ? 
 
 7. June 3, 1905, to Nov. 29, 1905 ? 
 
 8. Aug. 17, 1903, to Jan. 3, 1904 ? 
 
 9. April 4, 1905, to July 4, 1905 ? 
 10. May 16, 1906, to Oct. 14, 1906 ? 
 
 11. How much wages at $4 per day should a man receive 
 from Tuesday, Jan. 2, 1906, to Feb. 28, inclusive, no pay to 
 be received for Sundays or legal holidays ? 
 
 12. A man borrowed $100 April 4, and returned it Nov. 
 25. How many days' interest did he owe ? (Do not include 
 both days.) 
 
 13. May 1, 1903, fell on Friday. Upon what day of the 
 week did July 4 fall ? 
 
1 88 Chapter Four. 
 
 14. How many days does vacation last if it begins on the 
 morning of Saturday, July 2, and school commences on the 
 first Tuesday of September ? 
 
 15. A man borrows some money June 16, and agrees to 
 return it in 60 days. On what date should he pay it ? 
 
 16. A traveller starts upon a trip Aug. 24, 1904, and 
 reaches home again Feb. 10, 1905. How long is he away ? 
 
 In each of the preceding examples the difference between the dates 
 is less than a year, and the answer is required in days. When the 
 difference is more than a year, it is generally obtained by compound 
 subtraction, each month being considered as containing 30 days. 
 
 17. Find the difference in time between March 3, 1891, 
 and Jan. 1,1905. ^ f % 
 
 Writing 1905, 1st month, 1st day, we subtract 1891 3 3 
 
 1891, 3d month, 3d day. Ans. 13 yr. 9 mo. 28 da. 
 
 13 9 28 
 
 18. George Washington was born Feb. 22, 1732. How 
 old was he at the signing of the Declaration of Indepen- 
 dence, July 4, 1776 ? 
 
 19. Abraham Lincoln was -first inaugurated president 
 March 4, 1861. How long had he served at his death, 
 April 15, 1865? 
 
 20. The battle of Lexington took place April 19, 1775. 
 The treaty of peace was signed Sept. 3, 1783. How many 
 years, months, and days between the two events ? 
 
 21. How many years elapsed between the discovery of 
 America by Columbus, Oct. 12, 1492, and the landing of the 
 Pilgrims, Dec. 21, 1620? 
 
 22. General Harrison fought the battle of Tippecanoe 
 Nov. 7, 1811. He was inaugurated president 29 yr. 3 mo. 
 27 da. later. Give the date of his inauguration. 
 
Percentage. 
 
 189 
 
 23. How long was it after the treaty with England, 
 signed Dec. 24, 1814, that the Mexican treaty was con- 
 cluded, Feb. 2, 1848? 
 
 24. General Taylor died July 9, 1850. How long did he 
 live after the capture of Monterey, Sept. 24, 1846 ? 
 
 25. President Garfield was born Nov. 19, 1831. How 
 old was he at his inauguration, March 4, 1881? 
 
 26. The last battle of the Mexican War took place Sept. 
 14, 1847. The battle of Bull Run was fought 13 yr. 10 mo. 
 7 da. later. What was the date of this battle ? 
 
 27. Find the time between July 4, 1776, and Jan. 1, 1904. 
 
 
 PERCENTAGE 
 
 
 268 
 
 . Oral Exercises. 
 
 
 
 1. 
 
 Find 4% of $125. 
 
 6. 
 
 33 J % of 1 day. 
 
 2. 
 
 25% of 16. 
 
 7. 
 
 62£% of $12. 
 
 3. 
 
 6% of 200 cows. 
 
 8. 
 
 9 % of $23. 
 
 4. 
 
 1% of 150 pounds. 
 
 9. 
 
 75 % of 3 gallons. 
 
 5. 
 
 20% of 65 yards. 
 
 10. 
 
 lJ%of $400. 
 
 269 
 
 . Written Exercises. 
 
 
 
 1. 
 
 Find 6% of $576. 
 
 9. 
 
 25 % of $156. 
 
 
 $576 x .06 
 
 
 £ of $156 
 
 2. 
 
 41% of $340. 
 
 10. 
 
 1 % of $156. 
 
 3. 
 
 25 % of 1876 bushels. 
 
 11. 
 
 i% of $156. 
 
 4. 
 
 121% of 864 cows. 
 
 12. 
 
 50 % of 480 hours, 
 
 5. 
 
 50 % of 432 yards. 
 
 13. 
 
 \% of 480 hours. 
 
 6. 
 
 33^% of 576 soldiers. 
 
 14. 
 
 1% of $1420. 
 
 7. 
 
 16f % of 696 gallons. 
 
 15. 
 
 31% of $66. 
 
 8. 
 
 6J%of $4.96. 
 
 16. 
 
 7|% of 360 days. 
 
190 Chapter Four. 
 
 INTEREST. 
 
 270. Interest is the sum paid for the use of money. 
 The Principal is the sum loaned. 
 
 The Amount is the sum of the principal and interest. 
 
 In computing interest, the year is considered as composed 
 of 12 months of 30 days each. 
 
 271. Oral Exercises. 
 Find the interest on : 
 
 1. $90 for 2 months at 4%. 
 
 2. $60 for 60 days at 6%. 
 
 3. $100 for 2 yr. 6 mo. at 5%. 
 
 4. $ 120 for 30 days at 5%. 
 
 5. $300 for 90 days at 3%. 
 
 6. $100 for 1 yr. 3 mo. at 4%. 
 
 7. $50 for 3 years at 6%. 
 
 8. $100 for 2 yr. 4 mo. at 6%. 
 
 272. Find the interest on $ 63 for 4 yr. 5 mo. at 5%. 
 
 $63. 
 
 .05 
 
 Find the interest for one year by multi- $ 3 15 
 
 plying the principal, $63, by the rate, 6, . 5 
 
 expressed as hundredths. Multiply this prod- t-r 
 
 uct, $3.15, by the time expressed in years, '? 
 
 4&. * L31 + 
 
 12.60 
 Ans. $ 13.91 
 
 $ 63 is called the principal. 
 
 5 = rate. 4 yr. 5 mo. = time. 
 
 Rate 
 
 Interest = Principal x x Time (in years). 
 
 100 
 
Interest. 191 
 
 The work may sometimes be shortened by indicating the 
 operations and cancelling : 
 
 4 
 
 Find the interest on $160.50 for 3 mo. 15 da. at 6%. 
 
 $ mn X j|j X ?- = $11 4 235 = $ 2.808 + Ans. $2.81. 
 
 4 
 Note. — The divisor, 100, should be cancelled only in performing 
 the final division. 
 
 Find the interest on $69.75 for 1 mo. 17 da. at 4%. 
 $.007/75 s 47 
 
 • W-M X =gs X ^- = $ .36425. Ans. 36 cents. 
 
 Note. — The three ciphers in the dividend are cancelled by moving 
 the decimal point in the dividend three places to the left, prefixing a 
 decimal cipher. 
 
 273. "Written Exercises. 
 Find the interest on : 
 
 1. $192 for 3 yr. 7 mo. at 5%. 
 
 2. $ 60 for 2 mo. 12 da. at 4%. 
 
 3. $240 for 1 yr. 1 mo. at 6%. 
 
 4. $14.40 for 5yr. 5 mo. at 5%. 
 
 5. $36 for 77 days at 41%. 
 
 6. $99 for 21 months at 6%. 
 
 7. $ 192 for 2 yr. 4 mo. at 5%. 
 
 8. $600 from Jan. 1 to Jan. 16 at 4%. 
 
 9. $1200 from July 1, 1903, to Jan. 1, 1905, at 6%. 
 10. $57.60 from Oct. 4, 1904, to Feb. 4, 1908, at 5%. 
 
192 Chapter Four. 
 
 274. Oral Problems. 
 
 1. 16 is how many hundredths of 64 ? 
 
 2. What per cent of 25 is 5 ? 
 
 3. What part of £ is f ? 
 
 Change both to the same denominator : 16 twentieths, 15 twentieths. 
 
 4. What part of 2 lb. 1 oz. is 1 lb. ? 
 
 Change both to the same denomination : 33 oz., 16 oz. 
 
 5. Divide 4 gallons by 3 pints. 
 
 6. How many pencils at 4 mills each can be bought for 
 a dollar ? ! mill = ^ f a cent . 
 
 7. Write ^asa decimal. 
 
 8. Divide 34 by 200. 
 
 9. How many pounds in one-quarter of a ton? How 
 many pints in .25 of a bushel ? 
 
 10. Change 37^, 75^, 8^, 62j£ 6{f, to fractions of a 
 dollar ? 
 
 11. How many pounds of cheese at $0.16$ a pound can 
 be bought for $5.00? 
 
 12. An agent collected rents amounting to $300. What 
 was his commission at ^% ? 
 
 13. Find the interest of $200 for 1 yr. 3 mo. at 4%. 
 
 14. A farmer raised 50 bushels of cranberries, and sold 
 60% of them. How many bushels did he sell ? 
 
 15. What % of a number is fa of it ? 
 
 16. What would 42 pounds of butter cost at 33 \$ a 
 pound? 
 
 17. When the tax rate is $12 per $1000, what will Mr. 
 Smith's tax be if he owns $4500 worth of property ? 
 
Review. 193 
 
 18. A man pays $60 interest per year. How much, in- 
 terest does he pay in 3 yr. 7 mo. ? 
 
 19. At $45 per month, what is the rent of a house for 
 2 yr. 7 mo. ? 
 
 20. Express in per cents : \ ; \\ \ ; \ ; -J-. 
 
 275. Written Problems. 
 
 1. What is the interest on $760 for 5 months at 3±% ? 
 
 2. A merchant insures property worth $20,000 for J of 
 its value. How much does he pay, the rate for insuring 
 being 1J%? 
 
 3. What is the commission on $56*^8 worth of cloth at 
 
 4. At 3%, what is the commission on the sale of 5000 
 pounds of sugar at 5^ per pound ? 
 
 5. What will be the interest on $720 for 3 mo. 24 da. at 
 
 6. A clerk's income is $800. He pays 25% of it for 
 board, and 33 J % of the remainder for clothes. How much, 
 has he left ? 
 
 7. \°lo of my money is in my pocket, 38% is in the bank, 
 and the rest is in real estate. I have in all $ 24,000. How 
 much is in the bank and in real estate ? 
 
 8. An auctioneer sold for Mrs. Paul, on 10 % commission, 
 14 chairs at $1.75, 6 tables at $2.75, 40 yards carpet at 
 62i^ a yard, and a miscellaneous lot for $119.24. What 
 sum did Mrs. Paul receive after paying commission ? 
 
 9. How many feet in 62^% of a mile ? 
 What part of a day is 18 hr. 30 min.? 
 Reduce 9 cwt. 17 lb. to ounces. 
 
 10. If .625 of a cord of wood costs $3.75, what will .75 
 of a cord cost ? 
 
194 Chapter Four. 
 
 11. A business man's receipts for a week are $2575. 
 His average rate of profit is 5% of his receipts. What is 
 his profit for the week ? 
 
 12. A certain city had 14,250 inhabitants in 1900. The 
 population has increased 24 per cent. What is the present 
 number of inhabitants ? 
 
 13. A class has 56 pupils on register. When 14^ per 
 cent of the pupils are absent, how many are present ? 
 
 14. A merchant's sales for 1903 were $ 45,276. What 
 should be the sales for 1904 to make an increase of 16f per 
 cent ? 
 
 15. Thirty words were dictated as a spelling test. One 
 pupil received a mark of 93J per cent. How many words 
 did he misspell ? 
 
 16. A certain regiment went into battle with 1000 men. 
 Of these 5% were killed, 12% were wounded, 3% were 
 taken prisoners, and 1% were missing. How many re- 
 mained available for duty? 
 
 17. What is the duty at 35 cents per square yard on a 
 piece of cloth measuring 56 yards, 27 inches wide ? 
 
 18. A man bought a bill of goods amounting to $ 374.50, 
 with a deduction of 2 % for payment within 10 days. How 
 much does he save by paying the bill within the 10 days ? 
 
 19. A merchant places a bill of $ 840 in the hands of a 
 collector, who collects 75% of the amount. How much does 
 the merchant receive if the collector deducts 5% of the 
 amount collected, as his commission ? 
 
 20. How many pounds of bread can be made from 5 
 bushels of wheat weighing 60 pounds per bushel, if the 
 wheat loses 30 per cent in the process of grinding into flour, 
 and if the bread weighs 33J per cent more than the weight 
 of the flour used ? 
 
Measurements, 
 
 *95 
 
 SURFACES. 
 
 276. Preliminary Exercises. 
 
 1. What is the length in inches of a row of four enve- 
 lopes, each five inches long, placed end to end ? What is 
 the length in feet and inches. 
 
 3 inches 
 
 1 
 
 p 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2. What is the width in inches of four such rows, each 
 envelope three inches wide, just touching each other ? What 
 is the width in feet ? 
 
 3. How many envelopes are there ? How many square 
 inches are there in each envelope ? How many square 
 inches are covered by all of them? 
 
 4. How many envelopes 5 inches by 3 inches would 
 cover the top of a table 4 ft. 2 in. long and 2 ft. 6 in. 
 wide? 
 
 5. Draw a rectangle to represent a floor 24 feet long 18 
 feet wide. Draw rugs 6 feet long, 3 feet wide, and see how 
 many will be needed to cover the floor. 
 
 6. What is the difference between three square inches 
 and three inches square ? 
 
 7. What is the distance around a room that is 40 feet 
 by 30 feet? 
 
196 Chapter Four. 
 
 8. A garden is 12 feet long and 9 feet wide. How many 
 bunches of flowers will it furnish, if it takes 3 square feet 
 to furnish one bunch ? 
 
 9. A room is 36 feet long and 30 feet wide. How many- 
 square yards in the floor ? 
 
 10. How many yards is it around a room 15 feet long and 
 12 feet wide ? 
 
 11. How many square inches in the surface of a sheet of 
 paper 1 foot 8 inches long, 11 inches wide ? 
 
 12. How many pieces of paper 2 inches square will exactly 
 cover a slate 12 inches long, 8 inches wide ? 
 
 277. Written Problems, 
 
 1. How many boards 12 feet long, 6 inches wide will be 
 required for a floor 8 yards long, 6 yards wide ? 
 
 The floor is 24 feet long, 18 feet wide ; its area in square feet is 
 18 x 24. The area of the board in square feet is 12 x J, or 6. 
 
 Number of boards = 18 x 24 
 6 
 
 Note. — Labor is frequently saved in examples involving multipli- 
 cation and division by first indicating the operations and then using 
 cancellation. 
 
 2. How many bricks 8 inches by 4 inches will be needed 
 for a walk 24 yards long, 6 feet wide, making no allowance 
 for waste ? 
 
 Area of top surface of one brick =(8x4) square inches. Tne 
 length of the walk in inches = 24 x 3 x 12 ; width in inches = 6 x 12. 
 Area of walk in square inches = 24 x 3 x 12 x 6 x 12. Divide this 
 by 8 x 4, the number of square inches in the top surface of a brick. 
 
 Number of bricks = 24x3x12x6x12 
 8x4 
 
 Note. — It will be remembered that the divisor and the dividend 
 must be of the same denomination, square inches in this example. 
 
Measurements. 197 
 
 3. How many paving tiles \ foot square will cover a 
 hearth. 6 feet long, 3 feet wide ? 
 
 Make a diagram. 
 
 4. How many boards 12 feet long, 8 inches wide will be 
 required for a close fence 120 yards long, 6 feet high ? 
 
 5. Find the number of paving stones 9 inches by 3 
 inches, in a street 100 rods long, 10 yards wide. 
 
 6. Draw a rectangle 2 inches by 3 inches. Draw one 
 twice the size. What are the dimensions of the latter? 
 What are the dimensions of one four times the size ? 
 
 A plot 100 feet by 100 feet is how many times as large 
 as a plot 25 feet by 25 ? 
 
 7. A brick is 8 inches long, 4 inches wide, 2 inches thick. 
 How many square inches are there in the surface of the 
 widest face ? In the surface of one side ? In the surface 
 of one end ? 
 
 8. How many bricks laid on the widest face will be 
 needed for a walk 28S inches long, 96 inches wide ? 
 
 9. How many bricks laid on the side will be needed for 
 a walk 24 feet long, 8 feet wide ? 
 
 10. How many square feet are there in a roll of wall 
 paper 24 feet long, 18 inches wide ? 
 
 11. How many rolls 24 feet long, 1\ feet wide, would be 
 required to paper the ceiling of a room 45 feet long, 36 feet 
 wide, making no allowance for matching or waste ? 
 
 12. The owner of a piece of ground 200 feet wide, 300 
 feet long, divides it into lots 25 feet by 100 feet. How 
 many lots are there ? 
 
198 Chapter Four. 
 
 13. Make table of square measure : 
 
 square inches (sq. in. ) =1 square foot (sq. ft.) 
 square feet = 1 square yard (sq. yd.) 
 
 square yards = 1 square rod (sq. rd.) 
 
 160 square rods = 1 acre (A.) 
 
 acres = 1 square mile (sq. mi.) 
 
 14. There are 160 square rods in an acre. How many 
 square yards are there in an acre ? 
 
 15. Give the dimensions, in yards, of a field that will 
 contain just an acre. Of one that will contain two acres. 
 
 16. At $80 per acre what is the value of a field 80 rods 
 long, 70 rods wide ? 
 
 What will it cost to fence the field at 20^ per running 
 yard? 
 
 17. A man has a lot 100 feet by 200 feet. How many 
 square feet will he have left for a garden after he builds a 
 house 25 feet by 60 feet ? 
 
 18. One wall of a room is 24 feet long and 12 feet high. 
 There is a door in it 8 feet high, 4$ feet wide. How many 
 square yards of plastering will be needed to cover the wall ? 
 
 19. What would be the cost of painting 1800 feet of 
 fence 6 feet high at 15 cents per square yard ? 
 
 20. What is the length of a rectangular field 60 rods 
 wide that contains 60 acres ? 
 
 21. A farm is one mile square. How many 40-acre fields 
 does it contain ? 
 
 22. How many acres in a field in the shape of a triangle 
 whose base and perpendicular measure 40 rods each ? 
 
 23. How many acres are there in a triangular plot of 
 ground when the base measures 80 yards and the perpen- 
 dicular measures 60£ yards ? 
 
Measurements. 
 
 199 
 
 3 in. wide 
 
 VOLUMES. 
 278. Preliminary Exercises. 
 
 1. How many one-inch cubes can be placed on the bottom 
 of a box 4 inches long, 3 inches wide ? 
 
 2. If the box is one inch high, 
 how many will it hold ? If the 
 box is 2 inches high ? 3 inches 
 high? 
 
 Note. — A cube one inch long, one 
 inch wide, one inch high, contains a 
 cubic inch. 
 
 3. How many cubic inches in a 
 box 3 inches long, 4 inches wide, 
 1 inch high ? In a box 3 inches 
 long, 4 inches wide, 2 inches high ? 
 4 inches wide, 4 inches high ? 
 
 In a box 4 inches long, 
 
 4. If you had 24 one-inch cubes, how could you pile them 
 to make a solid with six rectangular faces ? 
 
 5. If the pile was 2 inches high, how many cubes would 
 there be in each tier ? How many square inches would the 
 lower tier cover ? 
 
 6. How could the 24 cubes be arranged to make a pile 
 3 inches high? 
 
 7. Can you give a rule for finding the number of cubic 
 inches in a box 4 inches long, 2 inches high, 3 inches wide ? 
 
 8. How many cubic inches of water would a tin box hold, 
 the dimensions of the box being 5 inches by 3^- inches by 4 
 inches ? 
 
 9. How many one-foot cubes could be placed in a cubical 
 box one yard long, one yard wide, one yard high ? 
 
200 
 
 Chapter Four. 
 
 279. A solid has three 
 dimensions : length, 
 breadth, and thickness. 
 
 The volume or contents 
 of a solid, is the space 
 it occupies, expressed in 
 cubic inches, cubic feet, 
 cubic yards, etc. 
 
 A cube is a solid hav- 
 ing six equal square 
 faces. 
 
 280. Cubic Measure. 
 
 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) 
 27 cubic feet = 1 cubic yard (cu. yd.) 
 
 281. Written Exercises. 
 
 1. How many cubic inches in a solid 3 yards long, 2 feet 
 wide, 6 inches high ? How many cubic feet ? How many 
 cubic yards ? 
 
 To find the volume in cubic inches, change 3 yards to 108 inches, and 
 2 feet to 24 inches. 
 
 Volume = (108 x 24 x 6) cubic inches. 
 Volume (in cubic feet) = (9 x 2 x $) cubic feet. 
 Volume (in cubic yards) = (3 x § X \) cubic yards. 
 
 2. How many cubic feet of air in a room 24 feet long, 18 
 feet wide, 12 feet high ? 
 
 3. Find the solid contents of a piece of timber 25 feet 
 long, 3 feet wide, 5 feet thick. Is it larger or smaller than 
 a piece 4 feet wide, 4 feet thick, and 23 ft. 6 in. long ? 
 
Measurements. 
 
 aoi 
 
 4. How many cubic yards of earth will have to be re- 
 moved in digging a cellar 18 feet wide, 55 feet long, 6 feet 
 deep ? What will be the cost at 60^ a load (1 cubic yard) ? 
 
 5. A brick is 8 inches long, 4 inches wide, 2 inches thick. 
 How many bricks are there in a pile 60 feet long, 20 feet 
 wide, 5 feet high ? 
 
 6. Find the number of bricks in a wall 24 feet wide, 48 
 feet high, 1 foot thick, making no allowance for mortar, etc. 
 
 7. How many bricks are there to a cubic foot ? 
 
 8. Allowing 20 bricks to a cubic foot when laid in mortar, 
 how many bricks will be needed for a wall 24 feet wide, 50 
 feet high, 20 inches thick ? 
 
 9. What will be the cost of building a stone wall 40 rods 
 long, 4 feet high, 1 yard thick, at $ 6.40 per perch of 24| 
 cubic feet ? 
 
 10. A cord of wood contains 128 cubic feet. If the wood 
 is cut into 4-foot lengths, what should be the other two 
 dimensions of a regular pile to hold just a cord? 
 
 11. How many cords of wood are there in a pile 24 
 feet long, 4 feet wide, 12 feet high ? 
 
 1 cord = 128 cubic feet. 
 
 282. Cubic Measure of 
 Capacity. 
 
 231 cu. in. =1 gallon 
 2150.4 cu. in. = 1 bushel 
 128 cu. ft. = 1 cord 
 
 12. Find the capa- 
 city in bushels of a bin 
 1 yd. long, 2 ft. 4 in. 
 wide, 5 ft. 4 in. high. 
 
202 Chapter Four. 
 
 The capacity of a bin, 3 4 10 
 
 tank, etc., corresponds to fifi x gg x QAj __ 
 the ooZume of the contents otg^ 3 Du * " dU Du * -***■ 
 
 of the bin or tank when full. -t »go 
 
 Write the dimensions in gaa 
 
 inches as factors, with the $* 
 
 number of cubic inches in 
 a bushel as a divisor, and cancel. 
 
 The decimal point in the divisor is moved one place to the right, 
 and a cipher is added to one of the numbers above the line. 21504 is 
 cancelled by 12, 7, 4, and 64. 
 
 13. Find the capacity in gallons of a tank 1 ft. 9 in. long, 
 1 ft. 3 in. wide, 1 ft. 10 in. deep. 
 
 21 x 15 x 22 -, rLL~i 
 
 — gal. Cancel. 
 
 14. How many gallons are there in a cubic foot ? 
 Give the answer as a mixed number ; as a mixed decimal. 
 
 15. How many cubic feet are there in a bushel ? 
 Give the answer as a mixed number; as a mixed decimal. 
 
 16. Give the width of a wagon body 18 inches high, 
 6 feet long, that will hold, when full, a cubic yard. 
 
 17. A gallon contains 231 cu. in. Give the dimensions 
 of a tin box that will hold exactly a gallon. 
 
 18. A pile of wood 40 feet long and 12 feet wide con- 
 tains 1920 cubic feet. How high is it? 
 
 19. How much will it cost to have it cut if it costs 80 cents 
 a cord ? 
 
 20. A pile of 4-foot wood is 16 feet long and 6 feet high. 
 Required the cost at $ 5.50 per cord. 
 
 21. A rectangular tank is 5 feet long, 2 feet wide, and 2 
 feet deep. How many gallons of water will it hold ? 
 
Measurements. 
 
 203 
 
 22. What is the cost of digging a cellar 21 feet long, 18 
 feet wide, and 6 feet deep, at $ .28 a cubic yard ? 
 
 23. How much will a block of granite weigh 15 feet long, 
 12 feet wide, and 9 feet thick, if 9 cubic feet weigh 72 lb.? 
 
 SURFACES OF RECTANGULAR SOLIDS. 
 283. Preliminary Exercises. 
 
 1. How many faces has a cube ? 
 
 2. What is the surface of each face of an inch cube ? 
 
 3. How many square inches are there in all the faces of 
 an inch cube ? 
 
 The accompanying diagram shows the dimensions of a piece of 
 paper that will exactly cover a square prism, whose base measures 
 4 inches by 4 inches, and whose height is 8 inches. 
 
 .9 
 
 .9* 
 
 
 .9 
 
 4 in. 
 
 J 
 
 00 
 
 4 in. 
 
 4 in. 
 
 .9 
 00 
 
 4 in. 
 
 4 in. J 4 in. 
 ■! .9 
 
 ooj °° 
 4 in. 1 4 in. 
 
 
 
 .9 
 
 4 h 
 
 4. How many square inches are there in the top face of 
 the prism ? In the bottom face ? In each of the four side 
 faces ? In the four side faces ? In the two ends ? In the 
 entire surface ? 
 
204 Chapter Four. 
 
 284. Written Exercises. 
 
 1. Make a diagram of a piece of paper that when folded 
 will just cover the six faces of a brick 8x4x2 inches. 
 How many square inches of paper would be needed ? 
 
 2. The owner of a piece of ground 600 feet long, 150 feet 
 wide, builds a fence 6 feet high around the plot. How many 
 square feet of fence are there ? 
 
 The surface of this fence may be considered as the four side faces of 
 a solid. The area in square feet = (150 x 6) + (600 x 6) + (150 x 6) 
 -f (600 x 6). The operation is shortened by adding 150, 600, 150, 
 and 600, and multiplying the sum by 6. (1500 x 6) sq. ft. = 9000 
 sq. ft., Ans. 
 
 3. A room is 24 feet long, 18 feet wide, 12 feet high. 
 Draw, touching each other, four rectangles representing the 
 four walls. Write the dimensions of each wall. 
 
 What are the dimensions of the large rectangle made up of the four 
 smaller ones? Give the area in square feet. In square yards. 
 
 4. Show by a diagram the shape of a piece of paper that 
 when folded will entirely cover a box 12 inches long, 6 
 inches wide, 4 inches high. Write the dimensions. 
 
 This is called the " development " of the box. 
 What is the area of the paper in square inches ? 
 
 5. How many square feet are there in a fence 10 feet 
 high enclosing a lot 250 feet long, 200 feet wide ? 
 
 6. Make a diagram of a room 24 feet long, 18 feet 
 wide, 12 feet high, showing the surface that is generally 
 plastered. 
 
 How many square yards of plaster will be needed for 
 the above room, making no allowance for doors, windows, 
 etc.? 
 
 7. A box is 4 inches long, 2 inches wide, and 2 inches 
 deep. How many square inches on its surface ? With the 
 pen, sketch a free-hand development of this box. 
 
Measurements. 205 
 
 8. One of the drawing models is a square prism 8 inches 
 long and 4 inches square. How many square inches on the 
 whole surface of the model ? 
 
 9. How many square yards in the walls of a room 
 12 feet wide, 15 feet long, and 9 feet high ? 
 
 10. The floor of a room is 18J feet long, 15^ feet wide. 
 How many square yards in the ceiling ? 
 
 A lot of land containing 5250 square feet is 125 feet long. 
 How wide is it ? 
 
 ANGLES, TRIANGLES, QUADRILATERALS. 
 
 285. The following may be drawn free-hand, the compasses being 
 reserved for the geometrical problems in Chapter VIII. 
 
 1. Draw two lines meeting at a point. 
 These lines make an angle. 
 
 2. Draw two lines that will make four angles. 
 
 3. Draw two lines so as to make two angles. 
 Two such angles are called supplementary anglea 
 
 4. Make two equal supplementary angles. 
 
 Equal supplementary angles are called right angles. A line making 
 a right angle with another line is said to be perpendicular to it. 
 
 5. Draw two lines so as to make one right angle. 
 
 Is the right angle made by two lines, each 10 feet long, any larger 
 than a right angle made by two lines, each 1 inch long ? 
 
 6. What is the smallest number of straight lines that will enclose 
 space ? 
 
 Draw a figure enclosed by the smallest possible number 
 of straight lines. What is its name ? Why ? 
 
 7. Make a triangle having one right angle. 
 
206 Chapter Four. 
 
 8. Can you draw a triangle having two right angles ? Why ? 
 What name is given to lines that will not meet, no matter how far 
 they are extended ? 
 
 9. An angle less than a right angle is called an acute angle. 
 Draw a triangle containing an acute angle. 
 
 10. Can you draw a triangle containing two acute angles ? 
 Three acute angles ? 
 
 11. An angle greater than a right angle is called an obtuse angle. 
 Draw a triangle containing an obtuse angle. 
 
 12. Can you draw a triangle containing three obtuse angles ? 
 Containing two ? 
 
 13. Draw a triangle with sides 2 inches, 3 inches, 4 inches, 
 respectively. 
 
 A triangle having no two sides equal is called a scalene triangle. 
 
 14. Draw a triangle having two equal sides. 
 
 This is called an isosceles triangle. The unequal side is called the 
 
 15. Draw an isosceles triangle with the base uppermost. 
 With the base on the left. On the right. 
 
 16. Draw a triangle having three equal sides (an equilat- 
 eral triangle). 
 
 17. Draw a square. Draw a rectangle 4 inches by 3 
 inches. 
 
 How many right angles in each ? 
 
 18. Draw a four-sided figure having its opposite sides par- 
 allel, but containing no right angle (rhomboid). 
 
 What kinds of angles does it contain ? How many of each ? Write 
 name in each angle. 
 
 19. Draw a four-sided figure, having all its sides equal, 
 but containing no right angle (rhombus). 
 
Measurements, 207 
 
 20. Draw a quadrilateral (four-sided figure) having only 
 two parallel sides (trapezoid). 
 
 21. Draw a quadrilateral having no parallel sides (trape- 
 zium). 
 
 22. Draw a rhombus, each side 2 inches. A square, each 
 side 2 inches. 
 
 What is the difference between them ? Which is larger ? 
 
 23. A parallelogram is a quadrilateral that has its opposite 
 sides parallel. 
 
 Name the parallelograms that have four equal sides (equilateral) , 
 Those that have four equal angles (equiangular). 
 
 24. The height of a parallelogram is called its altitude. 
 Draw a rectangle, base 3^- inches, altitude 2-^- inches. Draw 
 a rhomboid, base 3^- inches, altitude 2 J inches. Draw several 
 rhomboids of the above dimensions, all differing in shape. 
 
 25. Cut out of paper a rectangle, base 3 inches, altitude 
 2 inches. Cut out a rhomboid, base 3 inches, altitude 2 
 inches. Place one upon the other, and see how their areas 
 compare. 
 
 26. Can you calculate the number of square inches in a 
 rhomboid whose base is 3 inches and altitude 2 inches ? 
 
 27. Draw a rectangle, base 4 inches, altitude 3 inches. 
 Divide by a diagonal into two triangles. Mark in each 
 triangle its area. 
 
 28. Draw a right-angled triangle, base 4 inches, perpen- 
 dicular (altitude) 3 inches. Calculate its area. 
 
 29. Draw a rectangle, base 4 inches, altitude 3 inches. 
 From the middle point of the upper base draw lines to the 
 extremities of the lower base, making three triangles. Mark 
 in each triangle its area. 
 
 30. Draw an isosceles triangle, base 4 inches, altitude 3 
 inches, and calculate its area. 
 
208 Chapter Four. 
 
 286. Areas of Triangles and Quadrilaterals. 
 Find the areas of the following : 
 
 1. A right-angled triangle whose sides meas- 
 ure 15, 20, and 25 inches respectively. 
 
 Note. — Area of triangle = £ product of base by altitude (perpen- 
 dicular). 
 
 2. A right-angled triangle whose base measures 64 yards, 
 perpendicular 48 yards. 
 
 3. A triangle whose base measures 18 rods, altitude 13 
 rods. 
 
 4. A square whose side measures 35 feet. 
 
 Area of parallelogram = base x altitude. 
 
 97 ft. 
 
 5. A rectangle 42 yards by 37 yards. / j^ 
 
 / Is 
 
 6. A rhombus whose base is 97 feet, Z I 
 
 altitude 63 feet. 
 
 Show that the area of this parallelogram is equal to that of a 
 rectangle 97 feet by 63 feet. 
 
 7. A rhomboid, base 33 meters, altitude 28 meters. 
 
 8. A trapezoid whose paral- 
 lel sides measure 10 and 16 feet, 
 respectively, the perpendicular j 
 distance between them being 6 
 feet. E 
 
 10 ft. 
 
 10 ft. 
 
 U c 
 
 Draw this trapezoid on a scale of \ inch to the foot, and measure 
 AB, which divides the rectangle EFOH into two equal parts. AB = 
 \(FG + ED). Cut off the triangle BCD and add it to the upper half 
 of the trapezoid, so that CD will be a continuation of FO. The rec- 
 tangle thus formed should measure 13 feet by 6 feet. 
 
Review. 
 
 209 
 
 9. A trapezoid as shown 
 in the accompanying diagram. 
 
 Draw to a scale ; cut off a triangle 
 from A to the centre of CD, also 
 one from B to the centre of EF\ 
 and place these triangles above AB, so as to make a rectangle, 
 £(10 + 16) feet long and 6 feet wide. 
 
 10. A trapezium, one of 
 whose diagonals measures 42 
 yards, the perpendiculars to the 
 opposite corners measuring 18 
 yards and 16 yards, respec- 
 tively. 
 
 Area in square yards = (42 x \ of 18) + (42 x \ of 16) = 42 x \ of 
 (18 + 16). 
 
 SPECIAL DRILLS.— REVIEW. 
 
 287. Oral Exercises. 
 
 1. 463 + 157 = 463 + 100 + 60 + 7 = 
 In giving the solution at sight, the pupil says (or thinks) 663, 613, 
 
 620. 
 
 2. 256 + 184 4. 185 + 546 6. 167 + 734 
 
 3. 419 + 342 5. 668 + 193 7. 476 + 155 
 
 8. 4170 + 470 = 4170 + 400 + 70 
 
 Use no unnecessary words : 4570, 4640. 
 
 9. 1260 + 850 11. 3450 + 390 13. 5620+590 
 
 10. 2140 + 680 12. 4370 + 280 14. 6380 + 660 
 
 15. 400 — 163 = 400 - 100 - 60 - 3 = 
 Say only 300, 240, 237. 
 
2io Chapter Four. 
 
 16. 501-375 18. 650-488 20. 361-149 
 
 17. 275-137 19. 540-384 21. 455-358 
 
 22. 7310 — 6850 = 7310 - 6800 - 50 = 
 
 510, 460. 
 
 23. 8610-7680 25. 4960-4380 27. 6450-5760 
 
 24. 5000-4670 26. 2770-1890 28. 7320-6560 
 
 29. 24 X 66% = f of 24 hundred. 
 
 30. 48 x 16} 33. 24 x 62£ 36. 28 x 75 
 
 31. 32 x 37i- 34. 36 x 66| 37. 40 x 87£ 
 
 32. 49 x 25 35. 39 x 33^ 38. 88 x 12£ 
 
 39. 533£ -*- 66% = 5£ hundred -- f hundred = 16 -*- 2. 
 
 40. 337^-^-371 42. 687± -- 62| 44. 437£--87£ 
 
 41. 733J-r-33£ 43. 933|-f-66f 45. 212£-12| 
 
 288. Oral Problems. 
 
 1. How many ounces in 11^- pounds ? 
 
 2. 258 yards equal how many feet ? 
 
 3. A dealer bought 652 tons of coal and sold 476 tons. 
 How much had he left ? 
 
 4. Sold my wheat for $ 347 and my oats for $ 154. How 
 much did I receive for both ? 
 
 5. 40| yards of ribbon are cut into 7 pieces. Find the 
 length of each piece. 
 
 6. How many square yards in a floor 5£ yards long and 
 6£ yards wide ? 
 
 7. What will be the cost of 14 pounds of lard at 14^ per 
 pound? 
 
 8. At 1\$ each, how many lead pencils can I buy for 27^ ? 
 
Review. 211 
 
 9. What part of a 196-pound barrel of flour is contained 
 in a 49-pound bag ? 
 
 10. At 45^ per yard, bow much lace can be bought for 
 $1.35? 
 
 11. A woman has saved $ 833. How much more must 
 she save to have $1000? 
 
 12. What will be the cost of 16 pounds of sugar at 4f^ 
 per pound ? 
 
 13. Spent $ 2.56 for dry goods and $ 1.84 for groceries. 
 How much did I spend for both ? 
 
 14. Find the cost of 3 lb. 10 oz. butter at 32^ per pound. 
 
 15. At $.375 per yard how much ribbon can be bought for 
 
 $.75? 
 
 16. If it takes 1-J yards of cloth to make a jacket, how many 
 can be made from a piece of cloth containing 30 yards ? 
 
 17. A boy paid 35^ for the use of a boat for 3J hours. 
 What was the price per hour ? 
 
 18. If 13 pounds of raisins cost $1.69, what is the cost of 
 1 pound ? 
 
 APPROXIMATIONS. 
 289. Give an estimate of the answer : 
 
 1. If 3 T. 1988 lb. coal cost $19.97, what will be the 
 cost of 8 T. 1 lb.? 
 
 Nearly 4 tons cost nearly $20. 
 
 2. At $ 500 per year, what will be the rent of a house for 
 
 1 yr. 11 mo. 29 da.? 
 
 Nearly 2 years. 
 
 3. Find the cost of 5 barrels sugar, averaging 299 pounds 
 each, at 4ff ^ per pound. 
 
 4. What is the interest on $199.86 at 6%, for 5 ma 
 28 da.? 
 
212 Chapter Four. 
 
 5. If 11 men and 2 boys can finish a piece of work in 
 23£ days, how long will it take 23 men and 5 boys ? 
 
 6. What decimal of 639 acres is 321 acres ? 
 
 7. What will be the cost of 20,060 bricks at $ 4.90 per M ? 
 
 8. A farmer sells 5484 pounds rye at 87^ per bushel of 
 56 pounds. How much does he receive ? 
 
 9. If 19 lb. 15 oz. of tea cost $ 7.95, what will be the cost 
 of 21 lb. 1 oz.? 
 
 10. Paid freight on 1987 pounds at 70^ per cwt. How 
 much did I pay ? 
 
 11. If there are about 1\ gallons to a cubic foot, estimate 
 the number of gallons in a tank 5 feet long, 3 feet wide, 
 4 feet high. 
 
 12. If there are about \\ cubic feet in a bushel, estimate 
 the contents in bushels of a bin 5 ft. x 3 ft. x 4 ft. 
 
 13. Give the dimensions of a tank of 150 gallons' capacity. 
 
 14. Give the dimensions of a bin that will hold 100 
 bushels. 
 
 15. At 20 bricks laid in mortar to the cubic foot, give the 
 length and the height of a wall 1 foot thick that can be built 
 with a thousand bricks. 
 
 16. At $ 1 a load (1 cubic yard), give the dimensions of 
 an excavation that can be made for $ 100. 
 
 17. A cubic foot of water (about 1\ gallons), weighs 62^- 
 pounds. About what does a gallon weigh ? A pint ? 
 
 18. If iron is about 1\ times as heavy as water, about 
 what does a cubic foot of iron weigh ? 
 
 19. About what is 49f% of f 801? 
 
 20. About what will be the interest at 6 per cent on % 100 
 for 3 yr. 11 mo. 29 da.? 
 
Review. 213 
 
 FUNDAMENTAL PROCESSES. 
 
 290. 1. The sum of two numbers is 278. One of the 
 
 numbers is 89. What is the other ? 
 89 + ? = 278 
 
 2. The minuend is 583, the remainder is 249. What is 
 the subtrahend ? 583 - ? = 249 
 
 3. The subtrahend is 56, the minuend is 214. Find the 
 remainder. 
 
 4. The difference between two numbers is 84, the smaller 
 is 129. What is the larger number ? 
 
 5. The subtrahend is 176, the remainder is 92. Find 
 the minuend. 
 
 6. The multiplier is 98, the multiplicand is 809. Find 
 the product. 
 
 7. The product is 9045, the multiplier is 45. What is 
 the multiplicand? 
 
 8. The product of two factors is 1767. One of the 
 factors is 93. Find the other factor. 
 
 9. The multiplicand is 84, the product is 2100. What 
 is the multiplier ? 
 
 10. The dividend is 10,000, the divisor is 275. Find the 
 remainder. 
 
 11. The quotient is 32, the remainder is 21, the divisor 
 is 40. What is the dividend ? 
 
 40 ) ? 
 
 12. The dividend is 4263, the quotient is 203. Find the 
 divisor. 4263 _ 2Q3 
 
 ? 
 
 13. The dividend is 267, the quotient is 13, the remainder 
 is 7. What is the divisor ? 
 
 267 , o* 
 
H 
 
 Chapter 
 
 Four. 
 
 
 RATIO. 
 291. Sight Exercises. 
 
 
 - 87x25 
 * 75 
 
 63x19 
 3 * 21 
 
 5. gWil 
 
 7. 1x55 
 
 «, 74x24 
 2 * 37 
 
 A 96 x 27 
 4 ' 32 
 
 , gx42 
 
 8. ^x32 
 
 292. Written Exercises. 
 
 Indicate operations, and cancel where possible. Terms compared 
 *hould be of the same denomination. 
 
 1. If 90 tons of coal cost $472.50, what will be the cost 
 of 132 tons ? $472.50x132 
 
 90 
 
 2. If 3 lb. 4 oz. tea cost $ 1.95, what will 12 oz. cost ? 
 
 The ratio is 12 oz. to 52 oz. 
 
 3. A party of men can build 16 rd. 2 ft. of wall in 20 
 days. How long will it take them to build 4 yd. 6 in. ? 
 
 Change to inches. 
 
 4. What will be the cost of 3 bu. 2 pk. 7 qt. of oats if 7 
 bu. 1 qt. cost $4.50? 
 
 5. By travelling at the rate of 20 miles a day, a person 
 ■can complete a journey in 18 days. At what rate must he 
 travel to finish it in 15 days ? 
 
 6. How many rolls of merino, each containing 75 yards, 
 'worth $ .45 per yard, will it take to pay for 180 yards of 
 alpaca at $ .30 per yard ? 
 
 7. A merchant sold 20 hogsheads of oil, each containing 
 ■63 gallons, at $ 1.75 per gallon, and invested the proceeds in 
 table sauce in cases of 12 bottles each, worth $.31 J per 
 bottle. How many cases did he buy ? 
 
Review. 215 
 
 8. No allowance being made for mortar, how many- 
 bricks will be required to build a wall 50 feet long, 4 feet 
 high, and 1 foot 3 inches thick, each brick being 8 inches 
 long, 4 inches wide, and 2\ inches thicK '/ 
 
 9. If .1875 of a vessel cost % 273.12J, what is the value 
 of -^ of it at the same rate ? 
 
 10. What is the cost of 60.51 tons of coal, when .9 of a 
 ton costs §6.66? 
 
 REVIEW OF FRACTIONS. 
 293. Add across : 
 
 If the pupils work from their books the following examples in 
 addition and subtraction, they should be permitted to write only the 
 answers. The teacher should announce the number o* an example,, 
 not taking them in order, then the number of the next «o be worked,, 
 without giving time for the writing of unnecessary figures. 
 
 1. 13i-4-16| + 8f 6. 59f + 3£ + 4f 
 
 2. 4i + 5f + 27f 7. 7f-f-18f + 40i 
 
 3. 19i + 3f + 35£ 8. 35f + 5H + 8^ 
 
 4. 81 + 9^ + 14! 9. 3J + 9J + 25^ 
 
 5. 23f + 5J + 32^ 10. 66J + 8f + 14i 
 
 294. Subtract across : 
 
 
 
 11. 25\ -18^ 
 
 16. 
 
 68| -6{i 
 
 12. 63| -49f 
 
 17. 
 
 100£ - 62J 
 
 13. 70^-15£ 
 
 18. 
 
 56{ -37£ 
 
 14. 92f -24J 
 
 19. 
 
 83|. _43| 
 
 15. 33J -15^ 
 
 20. 
 
 42J -16} 
 
2i 6 Chapter Four. 
 
 Multiply : 
 
 When the fractions are small and the fraction in the multiplicand 
 has 1 for its numerator, business men do not change the mixed num- 
 bers to improper fractions. 
 
 In multiplying 38f by 11, the product of f by 11 is mentally reduced 
 to 8£, and \ written ; 11 eights (88), and 8 (96), 6 being written ; etc. 
 \ of 38| is 4 (written) with 6f remainder. This is reduced to ^ men- 
 tally, and its }, or ||, written.* 
 
 37} x 3} 
 
 12f X 5} 
 
 38 
 
 fxlli 
 
 1121- 
 18| 
 
 63f 
 
 426} 
 
 131} Ans. 
 
 68 Ans. 
 
 431 
 
 h Ans - 
 
 21. 
 
 48} x 4} 
 
 24. 18} x 5} 
 
 27. 
 
 45} x 2i 
 
 22. 
 
 64} . : 10} 
 
 25. 13} x 7} 
 
 28. 
 
 50} x 10} 
 
 23. 
 
 29f x 6} 
 
 26. 9fx8} 
 
 
 
 296 
 
 1. Divide: 
 
 
 
 
 29. 
 
 13)2051 
 
 
 
 
 The pupil should endeavor to work the following by short division : 
 into 20, once ; into 75, 5 times, remainder 10$ or -^ • ^ of -^ = f£. 
 
 Ans. 16ft. 
 
 30. 
 
 14)186} 
 
 37. 21)450} 
 
 44. 
 
 25)568} 
 
 31. 
 
 15)250} 
 
 38. 31)970} 
 
 45. 
 
 32)965} 
 
 32. 
 
 16)198} 
 
 39. 24)553$ 
 
 46. 
 
 36)722£ 
 
 33. 
 
 17)190£ 
 
 40. 23)466f 
 
 47. 
 
 16)366} 
 
 34. 
 
 18)200} 
 
 41. 26)290£ 
 
 48. 
 
 17)208} 
 
 35. 
 
 19)381} 
 
 42. 27)545 J 
 
 49. 
 
 21)640} 
 
 36. 
 
 22)264} 
 
 43. 33)9994 
 
 50. 
 
 22)888} 
 
Review. 217 
 
 
 REVIEW OF DECIMALS. 
 
 
 
 297. Sight Exercises. 
 
 
 
 
 Give products : 
 
 
 
 
 
 1. 360 x. 25 
 
 8. 
 
 840 x .075 
 
 15. 
 
 400 x .04 
 
 2. 560 x. 125 
 
 9. 
 
 960 x .005 
 
 16. 
 
 165 x .06| 
 
 3. 240 x. 375 
 
 10. 
 
 1200 x .001 
 
 17. 
 
 176 x .06^ 
 
 4. 400 x .625 
 
 11. 
 
 1500 x .002 
 
 18. 
 
 3300 x .00£ 
 
 5. 480 x. 75 
 
 12. 
 
 96 x .3£ 
 
 19. 
 
 880 x .12£ 
 
 6. 320 x. 875 
 
 13. 
 
 840 x .02£ 
 
 20. 
 
 105 x .8 
 
 7. 720 x. 025 
 
 14. 
 
 1500 x .06 
 
 21. 
 
 210 x .10 
 
 298. Give quotients : 
 
 
 
 
 1. 240 -h.5 
 
 8. 
 
 37 -*- .05 
 
 15. 
 
 76 -*- .04 
 
 2. 360 -i- .75 
 
 9. 
 
 48 -T- .005 
 
 16. 
 
 88 -*- .00£ 
 
 3. 45 -.125 
 
 10. 
 
 72 - .025 
 
 17. 
 
 65 -*• .12^ 
 
 4. 23 -h.25 
 
 11. 
 
 92 - .002 
 
 18. 
 
 84-^.8 
 
 5. 360 -.375 
 
 12. 
 
 93 - .03J 
 
 19. 
 
 11 + .06J 
 
 6. 100 -.625 
 
 13. 
 
 54 -v- .02£ 
 
 20. 
 
 42 -.6£ 
 
 7. 154 -.875 
 
 14. 
 
 132 -r- .06 
 
 21. 
 
 93-^.5 
 
 299. Written Exercises. 
 
 1. Find the value of (6.125 + 8.75 - 9.1235) -r- .0125. 
 
 2. Find the value of (1708.4592 - .00024) x .003. 
 
 4. Multiply 24.234 by .346, and write the result in 
 words. 
 
 5. Divide 96 ten-thousandths by 384 hundred-millionths. 
 
ai8 Chapter Four. 
 
 6. Why does the value of a decimal remain unchanged 
 when ciphers are annexed ? 
 
 7. Write : four hundred seven thousandths. 
 
 8. Write : six hundred four millionths. 
 
 9. Write in words 405.0067542. 
 
 10. Reduce to common fractions in lowest terms: 
 
 .004; .0125; 56.37$. 
 
 11. 16f x .045 = ? .324 x .33J = ? 3.406 x 1.00 = ? 
 
 12. .805 -^ .35 = ? 80.5 -f- 350 = ? Divide twenty-five 
 thousandths by 16 millionths. 
 
 13. Write in words : 
 
 .0105; 000125; 1.001105; 11.4141; .000008. 
 
 14. Reduce to common fractions : .95 ; .526. 
 
 15. From one thousand and (decimal) five thousandths 
 take eight hundred and (decimal) eight hundredths. 
 
 16. Divide eight hundredths by four thousandths, and 
 multiply the quotient by six ten-thousandths. 
 
 17. Find the product of the following factors: .064, 
 ,0032, 15,625, and 31.25. 
 
 300. Oral Eeview Problems. 
 
 1. At 20^ per quart, what will be the cost of 2 gal. 3 qt. 
 1 pt. of maple syrup ? 
 
 2. Find the cost of 4 T. 400 lb. of coal at $ 5 per ton. 
 
 3. A man puts 4 lb. 8 oz. of tea into 9-ounce packages. 
 How many packages does he make ? 
 
 4. 4 pk. 3 qt. of apples are given to some children. If 
 each child's share is 5 quarts, how many children are there ? 
 
 5. If it takes 3 hr. 20 min. to hoe a row of corn, how 
 many rows can a man do in 2 days of 10 hours each ? 
 
Review. 219 
 
 6. How many dozen eggs at 25^ a dozen must be given 
 for 100 pounds of sugar at 5fi a pound ? 
 
 7. Which would you rather have, -J of a dollar or 75^ ? 
 Why? 
 
 8. What will a gallon of molasses cost if a gill costs 2\f ? 
 
 1 gill = £ pint 
 
 9. Give the names to the results in the four simplest 
 processes in arithmetic. 
 
 10. $15 per week is how much per day ? 
 
 11. I of 72 is f of what number? 
 
 12. How many cubic feet in -| of a cubic yard ? 
 
 13. Which is the larger and how much larger, | of 130 or 
 f of 119? 
 
 14. Which is the larger and how much, ^ or f ? 
 
 15. How many cubic feet in a wall 30 feet long, 4 feet 
 high, and 2 feet thick ? 
 
 .16. Iff of a barrel of flour cost $2.13, what cost 1^ 
 barrels ? 
 
 • 17. The difference between 144 and 24 is how many 
 times 15 ? 
 
 • 18. John walked 12f miles, and Henry lOf miles. How 
 much farther did John walk than Henry ? 
 
 19. At 41^ a pint, what will 5 qt. 1 pt. of milk cost ? 
 
 20. After spending f of his money, James has $ 150 left. 
 What amount did he have at first ? 
 
 21. How many gallons in 462 cubic inches ? 
 
 22. If a boy eats f of a loaf of bread, how many boys 
 will be required to eat 10 loaves ? 
 
 23. 5 yd. cloth cost 90^ ; find the cost of f yd. 
 
 24. If | yd. of cloth costs 10^, how many yards can. 
 b© bought for 80^ ? 
 
220 Chapter Four. 
 
 25. A step is 3 feet. 2 steps are what part of a rod ? 
 
 26. 19+3 + 17 + 6 + 15 + 4=? 
 
 27. John had 85^. He bought strawberries for 22^-; 
 1 pound coffee for 30^ ; 3 sheets paper at 1^ a sheet. What 
 remained ? 
 
 28. Three-fourths of a mince pie is worth 18^, and James 
 eats ^ of a pie. What is the value of what he eats ? 
 
 29. If I have 1 pk. 2 qt. 1 pt. of meal, how many more 
 quarts must there be to make 1 bushel ? 
 
 30. Charles caught 12 fish, worth 41 ^ each, in four 
 hours. His time was worth 12^ an hour. Gain or loss, and 
 how much ? 
 
 31. How many times would a dish holding f of a pint 
 have to be filled to measure 9 quarts ? 
 
 32. If 5 chairs cost $ 80, what will 12 chairs cost ? 
 
 33. How many hours from 4 a.m. to 8 p.m. ? 
 
 34. Eeduce ff to lowest terms. 
 
 35. Add -J- to |, and take the sum from 5. 
 
 301. "Written Eeview Problems. 
 
 1. What part of 6 hr. 54 min. are 3 hr. 15 min. ? 
 
 2. If a man walks at the rate of 3 mi. 96 rd. per hour, 
 how far will he walk in 3 hr. 20 min. ? 
 
 3. What is one-ninth of 28 bu. 3 pk. 7 qt. ? 
 
 4. Three men buy a house for $ 1200. A furnishes 
 $600; B, $400; C, $200. They sell the house for $1500. 
 How much money should each receive ? 
 
 5. If 5 T. 1000 lb. of coal cost $30.25, how much will 
 be paid for 7 T. 320 lb. ? 
 
 6. At 25^ per hour, how much should a man receive 
 that works 8 hours and 36 minutes ? 
 
Review. 221 
 
 7. If 2 lb. 4 oz. of tea cost $1.35, what will be the cost 
 of 11 lb. 12 oz. ? 
 
 8. How many square inches in a paving tile 6 inches 
 square ? How many square inches in a rectangle 4 feet by 
 3 feet ? How many paving tiles 6 inches by 6 inches would 
 cover a surface 4 feet by 3 feet ? 
 
 9. A man buys a house and lot for $3000. He pays -f 
 of the amount in cash and the remainder after 1 year, 4 
 months, with 5% interest. Find the amount of the second 
 payment. 
 
 10. Find four-ninths of 28 bu. 3 pk. 7 qt. 
 
 11. (fof D + (fof 0-Cftcf 2) = ? 
 
 10 tiia A Qf4 f __* 
 
 I of 15 1J x 11 ' 
 
 13. AU.8f+f+f+.A,tt 
 
 14. Find the value of 728 - 1 - \ - £ - \. 
 
 15. 1}.X <&.-**) X& 
 
 16. Eeduce ^ of f of £ of ^^ to a decimal. 
 
 17. A person owning ^ of a factory sells 75 per cent of 
 his share for $ 1710. What is the value of the whole fac- 
 tory? 
 
 18. Find f of 2 da. 5 hr. 40 min. 
 
 19. If a piece of cloth is 20 yards long and f yard broad, 
 how broad is another piece which is 12 yards long and con- 
 tains as many square yards as the first ? 
 
 so. Simplify |l±l| x ^. 
 
 * 21. If 7 men can do a piece of work in 10 J days, how long 
 will it take 8 men and 5 boys to do the same work, each boy 
 doing one-half as much as a man ? 
 
222 Chapter Four. 
 
 22. A farmer drew to market three loads of wheat, 
 weighing respectively 2873 pounds, 3027 pounds, and 2911 
 pounds. At 93^ per bushel (60 pounds), how much did he 
 receive for the three loads ? 
 
 23. How many acres of land are there in a rectangular 
 farm J of a mile long and f of a mile wide? (1 square 
 mile = 640 acres.) 
 
 24. Eeduce ^""^ * to a simple fraction. 
 
 25. The sum of two numbers is 15f, and one of them is 
 9^5". Find the other number. 
 
 26. If 3 be added to both terms of the fraction -f, will 
 the value be increased or diminished, and how much? 
 
 27. Make and solve a problem to illustrate reduction 
 descending; one to illustrate reduction ascending. 
 
 28. How is the value of a fraction changed by increasing 
 its denominator? Why? 
 
 29. Add % hours, 20| minutes, and 49.2 seconds. Ex- 
 press the answer in minutes and seconds. 
 
 30. What fractional part of 31^ is 12|? 
 
 31. In a hotel the weekly wages of the clerk are $ 15, of 
 the cook $ 7.50, of the porter $ 9, of the waiter $ 3, of the 
 hostler $6, and of the errand boy $4. Find the average 
 wages paid. 
 
 32. A man was born May 24, 1832. What is his age 
 to-day? 
 
 33. A grocer's bill for $ 84.36 is paid 8 months 15 days 
 after it becomes due, with interest at 5%. How much is 
 
 34. Find the cost of 7 lb. 11 oz. of cheese at 13^ per 
 pound. 
 
Review. 223 
 
 35. Find the cost of digging a cellar 30 feet long, 15 feet 
 wide, and 5 feet deep, at 30^ per cubic yard. 
 
 36. John Smith bought of Clark and Jones, 
 
 4 lb. 13 oz. beefsteak @ 21^ per lb. 
 12 lb. of bacon @ 12j£ 
 
 Make a properly receipted bill of the above, dated at the 
 time and place of this lesson. 
 
 37. Find the cost o 2315 pounds of coal at $ 5.75 per ton. 
 
 38. Write 1249 in 1 oman notation. 
 
 39. Given the dividand 807 and the quotient 34 J, find 
 the divisor. 
 
 40. What will it cost to fill a jug, which contains 2310 
 cubic inches, with vinegar at 7 cents a quart ? 
 
 (1 gal = 231 cu. in.) 
 
 41. Mrs. C. B. Jones bought of Cole, Steele, & Co., of 
 Indianapolis, as follows: Nov. 12, 1904, 23 yards of muslin 
 @ 16|^; 45 yards of sheeting @ 12^ ; Dec. 7, 12 yards 
 of silk @ $1,621^; 8 handkerchiefs @ 45^; 2 pairs kid 
 gloves @ $ 1.371 ; 6 neckties @ 75^. Make out and receipt 
 the above bill. 
 
 42. If a boy bought f of a bushel of nuts for $2.00, and 
 sold them for 12^ a quart, what was his gain ? 
 
 43. Reduce -^ of an inch to the fraction of a rod. 
 
 44. Eeduce 35 quarts to the fraction of a barrel (31 J 
 gal.). 
 
 3450 cubic feet to cubic yards. 
 
 45. Put the following in the proper form of a bill, find 
 the amount of the bill, and receipt it : 
 
 David Wilson bought of Harry Lloyd, June 10, 1904, 
 7 pounds of oatmeal at 6^ a pound ; 10 pounds of sugar at 
 7$j a pound ; 14 pounds of ham at 13£^ a pound ; 3 brooms 
 at $ 2.25 a dozen. 
 
224 Chapter Four. 
 
 46. A family uses 2 quarts of milk a day. At 24/ a 
 gallon, what does the milk cost for May and June ? 
 
 47. From March 3d to Sept. 19th is how many days? 
 Do you include one of the days mentioned, or both of them, 
 or neither of them ? 
 
 48. How many minutes from 8.10 a.m. to 9.25 p.m. 
 
 49. Subtract 40 rd. 3 yd. 2 ft. from 81 rd. 1 yd., and 
 multiply the remainder by 10. Work by compound sub- 
 traction and multiplication, and get an answer that contains 
 no fraction. 
 
 50. Draw and divide a figure so as to show how many 
 square feet in a rectangle that is 5 feet long and 3 feet 
 wide. Draw and divide a figure so as to show how many 
 square inches in a surface that is 4 inches square. These 
 drawings are to be free-hand, and made with your pen. 
 
 51. Keduce 7 months and 15 days to the decimal of a 
 year (360 days). 
 
 52. Eeduce .32175 of 1 ton to whole numbers of lower 
 denominations. 
 
 53. If the perimeter of a square is 10 rods, what is the 
 area? 
 
 Find the area of a field, whose parallel sides measure 
 20 and 30 rods, respectively, the perpendicular distance 
 between them being 15 rods. 
 
 54. Bought 5 bushels of berries for $ 5 and sold them at 
 «8> .20 a quart. How much did I gain ? 
 
 65. From a tract of land 15 rods square I sold 65 square 
 rods. What was the value of the remainder at $ 20 an acre ? 
 
 66. What is the cost of fencing a lot 9 rods square at 
 $ .12 a foot? 
 
Review. 225 
 
 57. How many square yards are there in the walls of a 
 rocm 20 feet long, 18 feet wide, and 9 feet high ? 
 
 5S. What must I pay for the laying of a sidewalk 6 rods 
 long and 5 feet wide at $ .45 a square yard ? 
 
 59. How much will it cost to plaster a room 18 feet long, 
 15 feet wide, and 9 feet high, at $ .17 a square yard, deduct- 
 ing 108 square feet for doors and windows ? 
 
 60. Mr. Thompson has a field, around which he wishes to 
 build a tight board fence. The field is 50 rods long and 45 
 rods wide. The fence is to be 4J feet high. At 3JP a square 
 foot, what will be the cost of the fence ? 
 
 61. . A man having $ 100 went to market. He sold 10 
 bushels of potatoes at 80^ per bushel, 2 tons of hay at 
 $ 15 per ton, and 25 bushels of oats at 45^ per bushel. He 
 bought 15 barrels of flour at $ 4.50 per barrel, and 12 yards 
 of broadcloth at $ 4.75 per yard. How much money did he 
 have left? 
 
 62. Cost of a pile of wood 10 feet long, 4 feet wide, and 
 ^ feet high, at $ 7.50 a cord ? 
 
 I wish to pile 60 cords of wood in such a manner that it 
 will be 4 feet wide and 6 feet high. How long must it be ? 
 
 63. Find the interest of 1 263.75 for 1 year, 3 months, 
 20 days, at 6%. 
 
 64. At $ 17.625 a ton, how many tons of hay can be pur- 
 chased for $ 95 ? 
 
 65. Mr. Ames owns \\ of an acre of land. Mr. Jones 
 owns -| as much, which is \ of what Mr. Brown owns. 
 What part of an acre does Mr. Brown own? 
 
 66. Four men built a barn. A worked 2 days ; B, 6 days \ 
 C, 8 days; and D, 12 days. They received $84 What 
 was each man's share? 
 
226 Chapter Four. 
 
 67. A man has 768 hens, which is \ more than he had last 
 year. How many had he then ? 
 
 68. Two trains are 87£ miles apart and running toward 
 each other, one at the rate of 50f miles an hour, and the 
 other at the rate of 20f miles an hour. How far apart will 
 they be in half an hour ? 
 
 69. If 35 men earn $ 87.50 in 1 day, how much will 50 
 men earn in 10 days ? 
 
 70. Multiply 9008 by 7080, and divide the product by 
 600. 
 
 71. What is the difference between 69 x 58.8 and 291 -*- 
 0.97? 
 
 72. Find 6\% of 19,712 miles. 
 
 62i% of 2768 yards. 
 9^% of 11,223,344 pounds. 
 
 73. What is the interest of $ 150 for 2 yr. 8 mo. 15 da., 
 at 6% per annum. 
 
 74. Add : 25,037.45 ; 8,712.23 ; 9050.37 ; 815.25; 91,017.16 ; 
 419.19; 2035.75; 15,025.55; 7079.13; 14026.27. 
 
 75. Add : 87.27 ; 43.75 ; 72.50 ; 39.75 ; 64.04; 58.94; 95.83 ; 
 26.37; 75.96; 50.83; 39.49; 97.08; 62.62. 
 
 76. A lot of land containing 5250 square feet is 125 feet 
 long. What is the perimeter ? 
 
 77. A man spent ^ of his money for a house, -^ for 
 furniture, -g^j- for horses, and ■§• to build a church. What 
 part of his money had he left ? 
 
 78. Bought 10,752 cubic feet of wood at $8J a cord. 
 What did it all cost? 
 
 70. Change * ? to a simple fraction. 
 80. 9| times £ of 56| is how much ? 
 
Review. 
 
 227 
 
 81. What is the cost of digging a cellar 27 feet square 
 and 9 feet deep at 25^ a cubic yard. 
 
 82. How many yards of fence will be needed to enclose 
 the plot of ground shown in the following diagram ? 
 
 5 rods 
 
 4 rods 
 
 to 
 
 
 3 
 
 1 
 
 H 
 
 
 19 rods 
 
 TO 
 
 
 
 83. The above field was originally a rectangle, but the 
 owner sold one piece 5 rods by 3 rods, and a second piece 3 
 rods by 7 rods. How many square rods did it contain at 
 first ? What is its present area ? 
 
 84. Calculate the number 
 of square yards in the field 
 shown in the accompanying 
 diagram. 
 
 24 yds.. 
 
 € 
 
 24 yds. 
 
 85. A man buys a piece of 
 ground 300 feet long, 150 feet 
 wide. He builds a house, 50 feet by 30 feet, and a shed 12 
 feet by 13 feet. How many square feet will he have left 
 for a garden ? 
 
228 
 
 Chapter Four. 
 
 86. The owner of a piece of ground 250 feet long, 200 feet 
 wide, takes 10 feet from each side to make a gravel walk, 
 and uses the remainder for a garden. Give the dimensions 
 of the garden and its area in square feet? How many- 
 square feet in the whole piece of ground? How many 
 square feet are taken up by the walk ? 
 
 87. How many square feet of flagging would be required 
 for a sidewalk 10 feet wide outside a lot 250 feet long, 200 
 feet wide ? 
 
 88. If a piece of carpet is 27 inches wide, and contains 
 48 square yards, how long is it ? 
 
 89. I have bought 24 yards of dress goods, 27 inches 
 wide. How many square yards does the piece contain ? 
 
 How many yards of lining 32 inches wide will contain 
 the same number of square yards ? 
 
 24 yards long. 
 
 ? yards long. 
 
 I yd. 
 
 18 sq. yd. 
 
 18 sq. yd. 
 
 I yd. 
 
CHAPTER V. 
 
 PAGES 
 
 Percentage 229 to 276 
 
 Finding Percentage, Base, Rate ; Commission, Insur- 
 ance, Duties, Taxes, Profit and Loss, Commercial 
 Discount, Interest, Partial Payments, Bank Discount, 
 Interest by Aliquot Parts. 
 
 Denominate Numbers 277 to 291 
 
 Reduction Descending and Ascending, Addition, Sub- 
 traction, Multiplication, Division, Review. 
 
 Review of Simple Numbers 291 to 309 
 
 PERCENTAGE. 
 
 302. Preliminary Exercises. 
 
 Per cent means hundredths. Seven per cent means seven 
 hundredths, y^j-, or .07. It is written 7%. 
 
 How many hundredths of a number is one half of it? 
 J = how many hundredths ? \? $7 f? -f? 
 
 What per cent of a number is the half of it? J? J? £ ? 
 
 i? i? A? A? A? A? t**? A*? 
 
 What per cent of a number is f of it? J? £? f ? 
 
 A? A? A? A? A*? 
 
 303. 1 per cent of a number is equal to what fraction of 
 it? 3%? 5%? 9%? 10%? 15%? 20%? 25%? 
 30%? 40%? 50%? 60%? 75%? 90%? 
 
 304. What fractions are equal to the following ? 
 12$%? 16|%? 33|%? 37$%? 6J%? 62$%? 66$%? 
 
 87J% ? $% ? i% ? 2$% ? $% ? 
 
 305. 3 times a number is what per cent of it ? 2 J times ? 
 li times ? 4$ times ? 
 
 229 
 
230 Chapter Five. 
 
 306. Oral Exercises. 
 
 1. Find 37J% of $24. 
 
 37J % of $ 24 = f of $ 24, or $ 9. Ans. $ 9. 
 
 2. 6% of 150 bushels. 
 
 1% of 150 bushels = 1.6 bushels = \\ bushels; and 6% is 6 times 
 1£ bushels, or 9 bushels. Ans. 9 bushels. 
 
 3. 81% of 300 horses. 
 
 81 % of 100 horses = 81 horses ; of 300 horses it is 3 times 81 horses, 
 or 243 horses. Ans. 243 horses. 
 
 In examples 2 and 3 the pupil should be led to see that he can point 
 off two decimal places in the multiplicand instead of in the multiplier ; 
 without changing the result. The above analyses are suggestive merely. 
 The form given in the third example is to furnish an explanation for 
 the use of 3 as a multiplier. 
 
 4. Find 37|% of 1 gallon. 
 
 37 1 % of 1 gal. = f gal. = 3 pt. = 1 qt. 1 pt., Ans. 
 
 5. Find 121% of 1 gallon 15. 66f% of 66 horses 
 
 6. 371% of $24 16. l'6f%oflyard 
 
 7. 33 J % of 81 cows 17. 81% of $300 
 
 8. 6% of 150 pounds 18. 2^% of 80 sheep 
 
 9. 4% of 125 bushels 19. 40% of $2.50 
 
 10. 62£% of 1 peck 20. 20% of 65 rods 
 
 11. 4}% of $200 21. 10% of 15 pounds 
 
 12. 99% of 200 gallons 22. 3J% of $60 
 
 13. \% of $640 23. £%of$72 
 
 14. \% of 800 yards 24. lJ%of$96 
 
 The skilful teacher will appreciate the importance of rapid work, 
 and will gradually shorten the time to be given to a class for the solu- 
 tion of an oral example. She will also vary her methods of conducting 
 the recitation, so as to keep up the interest of the pupils. 
 
Percentage. 
 
 231 
 
 TO FIND THE PERCENTAGE. 
 
 307. "Written Exercises. 
 
 1. Find 6% of $91.50. 
 Multiply the base, $91.50, by the rate, 6, ex 
 
 pressed as hundredths. The result, •$ 5.49, is 
 called the percentage. 
 
 $91.50 
 x.06 
 $5.4900 Arts. 
 
 To find the percentage, multiply the base by the rate expressed 
 as hundredths. 
 
 2. 331% of $28.80. 
 
 While the rule is the same, to multiply $ 28.80 
 by .38^, the pupil should not fail to change 33£ 
 hundredths to one-third. 
 
 3. \% of $1240. 
 
 i°/o = jhj' Divide by 800 by cancelling the 
 two ciphers in the divisor and making two deci- 
 mal places in the dividend. 
 
 3 )$ 28.80 
 
 $9.60 Ans. 
 
 12.40/ 
 
 $1.55 Ans. 
 
 4. 41% of $92.40. 
 
 
 
 $92.40 x .04£. 
 
 
 
 5. 450% of $92.40. 
 
 
 
 $92.40 x 4.5. 
 
 
 
 6. 12% of $37.50 
 
 14. 
 
 860% of $38 
 
 7. 20% of $51.60 
 
 15. 
 
 \% of $2496 
 
 8. 1400% of $89.70 
 
 16. 
 
 25% of $52.36 
 
 9. 12i% of $73.28 
 
 17. 
 
 60% of $33.30 
 
 10. 131% of $27.60 
 
 18. 
 
 8% of $19.50 
 
 11. 6|% of $25.60 
 
 19. 
 
 6|% of $47.40 
 
 12. 3£% of $47.40 
 
 20. 
 
 12% of $62.50 
 
 13. 5£% of $29.50 
 
 21. 
 
 4i% of $ 71.50 
 
232 Chapter Five. 
 
 22. 40% of $28.30 26. 75% of $59.20 
 
 23. 160% of $39.40 27. 87|%of$392 
 
 24. 84% of $23.75 28. 93f % of $496 
 
 25. 66f% of $825 29. |f% f$496 
 
 Suggestion. — The teacher»should have a preliminary sight lesson 
 on these examples before giving them out for written solution. 
 
 TO FIND THE BASE OR THE RATE. 
 
 308. Preliminary Exercises. 
 
 1. 40 is one-half of what number ? 
 
 2. 40 is .5 of what number? 
 
 3. 40 is 50% of what number ? 
 
 4. 40 is what part of 80 ? 
 
 5. 40 is what decimal of 80 ? 
 
 6. 40 is how many hundredths of 80 ? 
 
 7. 40 is what per cent of 80 ? 
 
 8. 26 is what per cent of 65 ? 
 
 26 is ff of 65. The fraction § £ equals £, or 40 hundredths. Ans. 
 40 per cent. 
 
 9. 26 is 40 per cent of what number ? 
 
 If 40 hundredths of a number is 26, the number equals 26 divided 
 by 40 hundredths, or 26 -f- .40. Ans. 65. 
 
 To find the base, divide the percentage by the rate expressed 
 as hundredths. To find the rate, divide the percentage by the 
 base, expressing the result in hundredths. 
 
 Another method of finding the base or the rate is suggested 
 in the illustrative examples on the next page, which give 
 young pupils an introduction to the equation, a powerful 
 instrument in mathematical investigation. 
 
Percentage. 233 
 
 . Written Exercises. 
 
 1. What per cent of 65 is 26 ? 
 
 This means, how many hundredths of 65 will equal 26 ? which may 
 be expressed in the following form : 
 
 65 x — = 26. 
 100 
 
 The rate being required, the foregoing may be written as follows : 
 
 65x — = 26, or ^ = 26. 
 100 100 
 
 This is called an equation. To solve the equation, that is, to obtain 
 the value of r, the general method is to clear the equation of the frac- 
 tion by multiplying both sides by the denominator of the fraction, 100. 
 This gives 65 r = 2600, or 65 times r equals 2600. r, therefore, is equal 
 to 2600 divided by 65. Ans. 40 per cent. 
 
 Proof. — 65 x 40 hundredths = 26. 
 
 2. 40 per cent of what number equals 26 ? 
 
 & X i°- = 26, or ^ = 26. 
 100 ' 100 
 
 Clearing of fractions, 40 b = 2600 ; b = 65, Ans. 
 
 Proof. — 40 % of 65 = 65 x .40 = 26. 
 
 3. 75 per cent of a number is 42. What is the number ? 
 
 78% = I 
 
 bx— = 42, or ^=42. 
 100 4 
 
 Clearing of fractions, 3 6 = 168 ; b = 56, Ans. 
 
 4. What number is 15 per cent of 84 ? 
 
 p = 15 hundredths of 84. 
 
 5. 24 is 18 per cent of what number ? 
 
 6. 27 per cent of a number is 81. What is the number ? 
 
234 Chapter Five. 
 
 7. A boy spelled correctly 20 words of 25 given out 
 What per cent of the- words did he spell correctly ? 
 
 Note. — 25 is the base, 20 is the percentage ; required the rate. 
 
 8. 132 is 120 per cent of what number ? 
 
 9. -J- per cent of a number is 23. What is the number ? 
 
 10. f = what per cent of -f ? 
 
 i x _?! = §. Cancelling, -!-=-?. 
 5 100 5 b 125 5 
 
 Clear of fractions by multiplying both terms of the equation by 125. 
 
 Prove the correctness of your answer. 
 
 310. To clear an equation of fractions, multiply both terms 
 of the equation by the least common denominator of the 
 fractions. 
 
 11. i is what per cent of f ? 
 
 12. f is what per cent of £ ? 
 
 5 100 4 ' ' ' 125 4 
 
 13. 3J is what per cent of § ? 
 
 14. What per cent of $ 389.50 is $ 124.64 ? 
 
 15. $ 174.04 is 95% of what sum of money ? 
 
 16. f% of a number is 81. What is the number ? 
 
 ifa of 6 = 81. 
 
 17. 984 is 133£% of what number ? 
 
 18. What number increased by 33£% of itself equals 
 984? 
 
 Let n represent the number. 
 
 Then n + - = 984 ; i. e. — = 084. 
 
 Clearing of fractions, 4 n = 984 x 3 = 2952. n = 738, Am. 
 Proof. — 738 + 33$ % of 738 = 738 + 246 = 984. 
 
 19. What number increased by 25% of itself equals 85? 
 
Percentage. 23 5 
 
 311. Oral Exercises. 
 
 1. 3 is what part of 6 ? 
 
 2. 3 is what decimal of 6 ? 
 
 3. 3 is how many hundredths of 6 ? 
 
 4. 3 is what per cent of 6 ? 
 
 5. 6 is what per cent of 3 ? 
 
 6. What number is 50% of 6? 
 
 7. 3 is 50% of what number? 
 
 8. 2 is what % of 100 ? 
 
 9. 2 is what % of 200 ? 
 
 10. What number is 5% of 100 ? 
 
 11. What % of 20 is 1? 
 
 12. 4 is what % of 200? 
 
 13. 3 is \°lo of what number ? 
 
 14. What per cent of 9 is 20J ? 
 
 ~ = 20^ ; 9 b = 20£ hundred ; b = 2\ hundred = 225, Ans. 
 J.0U 
 
 15. What number, increased by ^ of itself, equals 10 ? 
 
 16. What number, increased by 25 % of itself, equals 20 ? 
 
 17. 65 diminished by 5% of itself equals what ? 
 
 18. Buying price $ 100, selling price $ 112.50. Gain % ? 
 
 19. Cost $ 80, profit 20%. Selling price ? 
 
 20. What principal will give $ 30 yearly interest at 6% ? 
 
 21. A man had $ 600 in the bank. He drew out 16f per 
 cent of it. How many dollars remained in the bank ? 
 
 22. A lost 40 per cent of his money, and had $ 750 left. 
 How much had he at first ? 
 
236 Chapter Five. 
 
 23. If I am compelled to lose 12^% on damaged goods, 
 how must I sell those that cost me $ 5.60 ? 
 
 24. A man put $15, which was 16|% of his month's 
 salary, in the bank. What was his month's salary ? 
 
 25. If each boy eats 60% of a loaf of bread, how many 
 boys will eat 6 loaves ? 
 
 Note. — In the solution of the foregoing oral problems, pupils 
 should not be compelled to use the method suggested for the written 
 exercises. 
 
 REVIEW. 
 312. Written Problems. 
 
 1. A man receives from a bank 4% a year as interest 
 on money he has in the bank. If his interest for a year is 
 $ 60, how much money has he in the bank ? 
 
 2. A city had a population of 4500 at the end of 1903. 
 The population at the end of 1904 was 1080 greater. What 
 per cent did the population increase during the year ? 
 
 1080 = what per cent of 4500 ? 
 
 3. A person who sold an article for 25% more than its 
 cost, received $ 85 for it. What was the cost ? 
 
 Cost + £ cost = $ 85. 
 
 4. A person receives $ 45 annual interest on $ 1000. 
 What rate per cent does he receive ? 
 
 5. A farmer sold 16J per cent of his sheep, and had 75 
 remaining. How many had he at first ? 
 
 6. A clerk has an income of $1100 per annum. He 
 pays 20 per cent of it for board, 1£ per cent for washing, 
 2 per cent for incidentals, 15 per cent for clothing, 9 per 
 cent for other expenses, and loses in various ways 50 per 
 cent of the amount then remaining. What sum does he 
 have left ? 
 
Percentage. 237 
 
 7. What per cent of a school is boys, and what per cent 
 girls, there being 640 of the former and 560 of the latter ? 
 
 8. What per cent of 9.075 is 24.2 ? 
 
 9. How large a sale must a merchant make, at a profit 
 of 15 %, that his gain may be % 3750 ? 
 
 10. A coal dealer bought 25,784 tons of coal at % 5 a ton. 
 He sold 40% of it at % 7, 20% of it at $8.50, and the remain- 
 der at % 4.50. How much did he gain ? 
 
 11. A man shipped 600 barrels of flour, and lost 16f% of 
 it by storm ; he sold 75% of the remainder. What per cent 
 of the whole remained ? 
 
 12. 66f % of 200 bushels is 2£% of how many bushels ? 
 
 13. If corn selling for 21^ a bushel more than cost gives 
 a profit of 30%, what did it cost? 
 
 14. \ + I of a number is what per cent of it? 
 
 15. A boy deposited $ 15 in bank. This was 30 per cent 
 of what he had in bank before making this deposit. What 
 had he there after this deposit ? 
 
 16. A man can do a certain work in 18f days. What per 
 cent of it can he do in 6| days ? 
 
 17. A man spent 30 per cent of his money for clothes, 20 
 per cent for rent, and had $ 75 left. What rent did he pay ? 
 
 18. What is the difference between £ per cent of $ 15,000 
 and 50 per cent of $ 15,000 ? 
 
 19. A pole extended into the mud 5f feet; 33^% of its 
 length was in the river and 25% of it in the air. What was 
 the length of the pole ? 
 
 20. There were 984 patients in a certain hospital, classi- 
 fied as follows : 369, pulmonary diseases ; 246, nervous dis- 
 eases ; 123, diseases of heart ; and 246, various other diseases. 
 Give the per cent of each class. 
 
238 Chapter Five. 
 
 APPLICATIONS OF PERCENTAGE. 
 
 313. Commission. — The term per cent occurs in many- 
 business transactions. A person who sells goods for another 
 receives a certain per cent of the amount he obtains for the 
 goods, as a commission. A person who buys goods for 
 another is paid a commission, which is a certain per cent of 
 the cost of the goods. A person who collects money for 
 another is paid a commission of a certain per cent of the 
 amount collected. 
 
 Commission is a percentage paid to an agent for his services. 
 
 314. Insurance. — The owner of property who desires to 
 be insured for a definite sum pays some per cent of this 
 sum for the insurance. The amount he pays is called the 
 premium. The document given by the insurance company 
 as a receipt is called a policy. It states the agreement of 
 the company to pay the owner of the property a sum equiva- 
 lent to the loss sustained, provided that it does not exceed 
 the sum for which the owner is insured. Thus, the owner 
 of a house valued at $ 5000 may insure it against fire for 
 $4000. If the house is injured to the extent of $4000 or 
 less, the owner receives from the company the amount of 
 the loss. 
 
 Insurance is a contract by which one party agrees to pay to 
 another a specified sum in case of loss or damage. 
 
 Note. — The teacher should show pupils an insurance policy, and 
 read the contract made by the company as expressed therein. 
 
 315. Duties. — The United States government collects from 
 the importers of certain classes of goods a stated percentage 
 of the value of the goods. This charge is called a duty. 
 
 Duties are taxes on imported goods. 
 
 Note. — Some duties are based upon a certain rate per square yard, 
 per pound, etc. 
 
Applications of Percentage. 
 
 *39 
 
 316. Taxes. — For the expenses of maintaining a city, 
 property owners pay a certain percentage of the valuation 
 of their property as determined by the proper officials. The 
 money thus collected from the owner is called a tax. The 
 value fixed by the authorities is called the assessed value, 
 which is generally somewhat less than the actual value. 
 
 A tax is a sum of money levied on persons or property for 
 public purposes. 
 
 Note. — In many places the tax rate is fixed at so many thou- 
 sandths of the assessed value. 
 
 The following ten oral and twenty written problems in- 
 volve no new principles. The general formula b x -^- =p 
 
 is applicable to each of them. The accompanying statement 
 
 shows the base on which the percentage is calculated in 
 
 certain classes of examples; also the name given to the 
 
 percentage. 
 
 Base 
 
 JYalue of goods bought 
 
 [ or sold, etc 
 
 f Sum for which property 
 is insured .... 
 Assessed value of prop- 
 erty 
 
 Duties .... Value of goods imported Duty 
 
 Commission . 
 
 Insurance 
 
 Taxes 
 
 Percentage 
 Commission 
 Brokerage 
 
 Premium 
 Taxes 
 
 317. Oral Problems. 
 
 1. An agent collected a bill, and sent to his employer 
 the amount, less 2J% commission. If his commission was 
 $ 1.60, how much did he remit to his employer ? 
 
 2. My house, worth $ 12,000, is insured for J of its value, 
 at \°lo' What premium do I pay ? 
 
24° Chapter Five. 
 
 3. A man collected a bill of $ 300 for me, at |% com- 
 mission. How much was his commission ? 
 
 4. Mr. Eastman collects bills for me, and I pay him 
 12 J%. He pays over to me §56. How much did he 
 collect ? 
 
 5. What is the premium for insuring $ 3600 on my house 
 at |% ? 
 
 6. What will it cost to insure a house worth $ 5000, at 
 \°/o premium ? 
 
 7. Eind the duty at 35% on goods valued at $ 2000. 
 
 8. My taxes for 1904 are $ 175. The rate is If per cent. 
 
 What is the assessed value of my property ? 
 
 9. My agent collects the yearly rent of my house, and 
 retains $ 15, the amount of his commission at- 2\ per cent. 
 Eor how much does the house rent per year ? 
 
 318. Written Problems. 
 
 1. How much insurance does a man receive for $ 12.50 
 when the rate is 2|% ? 
 
 2. An importer paid duties amounting to $386.75. If 
 the duty was 25% of the cost of the goods, what was their 
 cost? 
 
 3. A collector deducts 2J% commission, and returns to 
 his employer $ 745.68. How much did he collect ? 
 
 Let x represent the sum collected. Then 2£% of se, or — , will repre- 
 
 of) ~ 40 
 
 sent the commission ; and x , or -^—, will represent the amount 
 
 returned to the employer. 
 
 ^ = 745.68. 
 40 
 
 Clearing of fractions : 39s = 29,827.20. 
 
 a; = 764.80. Ans. $764.80. 
 
 It will be noted that only abstract numbers are used in an equation, 
 the denomination being supplied in the answer. 
 
Applications of Percentage. 241 
 
 4. The tax rate of a certain city is lf% upon the 
 assessed value of property. If this value is 75% of the 
 actual value, how much taxes does Mr. Smith pay upon a 
 house and lot, the actual value of which is $ 24,000 ? 
 
 5. The tax on an assessment of $8500 is $48.45, 
 Eequired the rate on $ 1000 of assessment. 
 
 6. Find the amount of an agent's sales, when his com- 
 mission at 5 per cent amounts to $ 37.65. 
 
 7. An agent buying wheat is offered a commission of 4^ 
 per bushel, or one of 4^- per cent, and he chooses the former. 
 The average price paid per bushel is 91^. Does he gain 
 or lose by his choice, and how much per bushel ? 
 
 8. A commission of $121.29 was charged for selling 
 $ 1866 worth of goods. What was the rate of commission ? 
 
 9. A man insured his house for $ 6500, his store for 
 $3500, and his goods for $7000, at £%. What did his 
 insurance come to? 
 
 10. If a piece of property is taxed $ 28.60, at a tax rate 
 of -f of one per cent, what is the assessed value of the 
 property ? 
 
 11. A house valued at $ 24,000 was insured for two-thirds 
 of its value, at f %. What is the premium ? 
 
 12. An agent collected 20% of an account of $ 750, charg- 
 ing 4% commission. What was his commission, and what 
 sum should he have paid over ? 
 
 13. Paid $27 for an insurance policy on my house. If 
 the rate is f %, for how much is my house insured ? 
 
 14. My agent collected 80 per cent of a debt of $ 4500, 
 and charged 1\ per cent commission. What amount should 
 he pay me ? 
 
242 Chapter Five. 
 
 15. A farmer bought 6 cows through an agent. He sent 
 $ 525.30 to pay for the cows and a commission of 3%. How 
 much did each cow cost ? 
 
 16. What will be a broker's commission, at 2|%, for sell- 
 ing a farm of 673 acres @ $52 per acre ? 
 
 17. If the tax rate is $13.80 on $1000, what is the 
 assessed value of property that pays a tax of $ 144.90 ? 
 
 18. A house is insured for -| of its value at -J%. The 
 annual cost (premium) is $ 8.40. What is the value of the 
 house ? 
 
 Let x represent the value. Then — ^ x — , or — — , will represent 
 4 . . 3 800 1200 
 
 the premium. 
 
 7 x 
 The equation becomes — — = 8.40. 
 
 19. What will be the taxes on a house worth $48,000 
 and assessed at -| of its value, the tax rate being $ 18.50 per 
 $ 1000 of assessed value ? 
 
 20. A commission merchant receives 2\°fo commission for 
 buying grain for a customer. The cost of the grain and his 
 commission amount to $ 4223. How much does the grain 
 cost? 
 
 Let x represent the cost of the grain ; — will be the commission. 
 
 21. An importer paid $134.40 duties on imported goods 
 valued at $ 384. Find the rate. 
 
 22. What is the duty in United States money on glass 
 ware valued at 1500 francs, the rate being 60%, and the 
 franc being worth 19.3 cents ? 
 
 23. Find the duty on a gross of scissors, valued at $2.50 
 per dozen, the rate being 75 cents per dozen and 25% on 
 the value. 
 
Profit and Loss. 243 
 
 PROFIT AND LOSS. 
 
 In determining the rate per cent of gain or loss on goods 
 sold, the buying price of the goods is taken as the base. 
 
 319. Oral Problems. 
 
 1. What is the gain per cent on sugar bought at 5 cents 
 per pound and sold at 6 cents per pound ? 
 
 Profit \f, which is one-fifth of buying price, or 20%. 
 
 2. By selling a house for $ 3500, 1 lose $ 500. What is 
 my loss per cent ? 
 
 The loss, $ 500, is what per cent of the cost, $ 4000 ? 
 
 3. By selling a lot for $ 1000, Mr. Jones loses 20 per 
 cent. What did the lot cost ? 
 
 The selling price, $ 1000, is four-fifths of the cost. 
 
 4. Find the cost of an article which was sold for $ 60, 
 at a loss of 70%. 
 
 5. If I buy a dozen pencils at 2^ each, and sell at 3fi 
 each, what is the gain per cent ? 
 
 6. A saddle was sold for $ 18, which was 12 J- % more 
 than the cost. How much did it cost ? 
 
 7. What % is gained on goods sold at double the 
 cost? 
 
 8. Sold flour at a profit of $ 2, and gained 25</ . What 
 was the cost per barrel ? 
 
 9. What is the % of gain, when boots which cost $ 2 a 
 pair are sold for $2.50? 
 
 10. What per cent is lost in buying potatoes at 80^ a 
 bushel, and selling them at 60 ^ a bushel ? 
 
 11. If I buy butter at 30^ a pound, how much per cent 
 do I gain by selling it at 36 j a pound ? 
 
244 Chapter Five. 
 
 320. Written Exercises. 
 
 Find the profit or the loss, and the selling price : 
 
 1. Cost $1876; gain 15%. 
 
 Gain = 15 per cent of $ 1876. Selling price = cost + gain. 
 
 2. Cost $36.75; loss 20%. 
 
 3. Cost $1012.50; gain 16|%. 
 
 4. Cost $875; loss 5%. 
 
 5. Cost $934.56; gain 12£%. 
 
 ^ Find the profit or the loss per cent. 
 
 6. Cost $600; selling price $618. 
 
 $18= ?%of $600. 
 
 7. Cost $ 1203; selling price $802. 
 
 8. Cost $86.20; selling price $ 73.27. 
 
 9. Cost $908.40; selling price $1090.08. 
 
 10. Cost $84; selling price $78.75. 
 
 11. Selling price $78.75; loss $5.25. 
 Note. —Cost = $78.75 + $5.25 = $84 
 
 12. Selling price $ 150; gain $ 25. 
 Use the cost ($ 150 - $25) as the hase. 
 
 13. Selling price $831.25; loss $43.75. 
 
 14. Selling price $ 1051.38 ; gain $ 116.82. 
 
 15. Selling price $843.75; loss $168.75. 
 
 Find the cost, and the profit or loss : 
 
 16. Selling price $ 468.75 ; gain 25%. 
 
 5x 
 
 Representing the cost by x, the selling price is —-• 
 
 ^ = 468.76. 
 
Profit and Loss. 245 
 
 17. Selling price $73.84 5 loss 20%. 
 
 18. Selling price $1646.08; gain 33 \% 
 
 19. Selling price $204; loss 15%. 
 
 20. Selling price $66.30; gain 4%. 
 
 21. A man buys a horse for $ 275, and sells it at a profit 
 of 20 per cent. How much does he gain ? 
 
 22. A cow is sold for $75, on which the profit is $15. 
 What is the gain per cent ? 
 
 23. A lot is sold for $ 960, which is 20 per cent more than 
 it cost. Find the cost of the lot. 
 
 24. Tea that costs 32 f per pound is sold for 48 £ What 
 is the gain per cent ? 
 
 25. A man buys a horse for $ 175 and sells it for $200. 
 What per cent does he gain ? 
 
 26. What per cent was lost on a horse that cost $200, 
 and that was sold at a loss of $ 25 ? 
 
 27. What is the selling price of dress goods costing ?&\i 
 per yard, on which a profit of VZ\ per cent is made ? 
 
 28. Sold a coat for $ 33.60, thereby losing 16 per cent. 
 What was its cost ? 
 
 29. How much did I gain on a house for which I paid 
 $8760, my profit being 2\ per cent? 
 
 30. A man paid for a house $4500, and for repairs $ 150, 
 and then sold it for 18% above the entire cost. What did 
 he receive for it ? 
 
 31. To make 15 per cent profit, what must goods be 
 marked that cost 96 cents per yard ? 
 
 32. Goods costing 96 cents per yard are marked at 25% 
 advance, what per cent is gained if they are sold 10 % below 
 the marked price ? 
 
 33. Find the per cent of profit on apples bought at $ 1.25 
 per bushel, and sold at 25 cents per half peck. 
 
246 
 
 Chapter Five. 
 
 COMMERCIAL DISCOUNT. 
 
 321. Wholesale dealers in certain classes of goods allow 
 to purchasers of large quantities a deduction from the prices 
 printed in their catalogues. This is called a trade discount 
 or commercial discount. A discount for prompt payment is 
 also frequently allowed. The following bill contains a trade 
 discount and a discount for cash : 
 
 Philadelphia, Jan. 17, 1904. 
 The Ocean Bathing Suit Co. 
 
 Terms: Cash less 5 per cent. Sold to Mr. J. H. HAAREN. 
 
 12% doz. Suits 
 
 918 
 less 15% 
 
 Cash less 5°J 
 
 Rec'd Paym% 
 Jan. 17, 1904, 
 
 0. B. S. CO. 
 
 per M. M. 
 
 §225 
 
 — 
 
 
 S3 
 
 75 
 
 
 $191 
 
 25 
 
 9 
 
 56 
 
 $181 
 
 
 
 69 
 
 1. Make out a bill for 16 gross of roman candles at 
 $26.75 per gross, less 60%. 
 
 2. On a bill of goods amounting to $583.40, a discount 
 of 5% is given for cash. What is the amount paid ? 
 
 3. Sept. 1, 1905, Mr. Maxwell bought tea amounting to 
 $ 1876.50. If 5% is deducted for payment within ten days, 
 how much should he pay if he paid the bill Sept. 9 ? 
 
 4. What will be the cost of 15 cases cocoa @ $13.20 
 each, less 20% ? 
 
Commercial Discount. 247 
 
 5. Bought 5 gross of essence of lemon at 50^ per doz., 
 less 5%. What is the amount of my bill ? 
 
 6. Find the cost of 15 cases of chloride of lime, 50 lb. 
 per case, at 9|-^ per pound, less 15%. 
 
 7. Find the cost of a wagon, the catalogue price of 
 which is $750, the discount being 30%. 
 
 8. What will be the cost of goods amounting to $ 1837.60, 
 on which there is allowed a discount of 17^ % ? 
 
 9. Find the net cost of 1630 yd. silk, invoiced at $ 1.10 
 per yard, less 16% discount. 
 
 Note. — The amount previous to the deduction of the discount is 
 known as the gross amount. The net amount or the net cost is the 
 sum actually due after the deduction of the discount. 
 
 10. What is the cost, in francs, of 843.72 meters silk, at 
 5.75 francs per meter, less 12% ? 
 
 11. What will be the net cost of a bill of plated ware 
 amounting to $ 84.75, on which a discount of 33^ and 
 10% is allowed? 
 
 $ 84.75 
 1 ! no ok When more than one discount is given, 
 
 56.50 
 
 each successive discount is based on the re- 
 mainder left after the deduction of the previ- 
 
 less A- 5.65 
 
 . 1 °_ ous discount. 
 
 Ans. $ net. 
 
 Note. — The mark % is generally written only after the last rate. 
 
 12. Find the difference between $ 390 less 43^% discount, 
 and $ 390 less 33J and 10% discount. 
 
 13. An army fought two battles. In the first it lost 15 
 per cent, and in the second 20 per cent of the original num- 
 ber, after which it mustered 19,500 men. What was the 
 original strength of the army ? 
 
248 Chapter Five. 
 
 14. Find the net cost of 18,500 bags at $ 4.40 per M, less 
 60 and 10 and 5%. 
 
 15. What is the net cost of a lot of musical instruments 
 amounting to $ 1875.60, on which a discount of 10, 5, and 
 2\% is allowed? 
 
 16. What would be the net cost of the same articles, if 
 the discount were 2^-, 5, and 10% ? 
 
 17. Find the net cost of the same, at 17|-% discount. 
 
 18. Which is the better discount for the buyer, 40 and 
 10% or 30 and 20 ? What will be the difference on a bill 
 of $ 100 ? 
 
 19. $100 less 33 J and 10% discount is equal to what? 
 What per cent discount is 33 J and 10 % equal to ? 
 
 20. $ 100 less 10 and 33J% is equal to what ? 
 
 Note. — The pupil will note that the result is the same as in 
 Problem 19. 
 
 21. A man marks an article $1.50, and sells it at a dis- 
 count of 25% from the marked price. If the article cost 
 him 90 ^, what is his gain per cent ? 
 
 22. John Jasper & Co. sold the following goods. Make 
 out the bill, less 50 and 10 and 10 and 10% discount. 
 
 500 ^-pound bags at $ 1.00 per M. 
 1500 -i-pound bags at 1.20 per M. 
 3000 1-pound bags at 1.60 per M. 
 5500 Impound bags at 1.70 per M. 
 2000 2-pound bags at 2.00 per M. 
 
 Note. — In making out large numbers of bills clerks have no time 
 for unnecessary words. The first item would be written as follows : 
 
 600 \ lb. Bags $1. .50 
 
 the words " at " and M per M " being considered unnecessary. 
 
Commercial Discount. 249 
 
 322. Oral Problems. 
 
 1 . A piano, marked $ 800, is sold at a discount of 25 and 
 10%. What is the selling price ? 
 
 2. Bought goods amounting to $600, less 5% for cash. 
 What is the net cost of the goods ? 
 
 3. What single discount is 50 and 10% equal to? 
 Taking f 100 as a base, 50 % discount deducts f 50 and leaves $ 50. 
 
 10 % deducts $ 5, leaving $ 45. The total deduction is $ 55 ; the 
 single equivalent discount is 55 %. 
 
 4. What single discount is 30 and 30% equal to? 
 Representing the base by x, the first discount is 30 % of x, or — — , 
 
 leaving — -. The second discount is T 3 7 of — — , which is —^r. The 
 
 Qfk 91 x,^ 
 
 two discounts are —p-r and ■——, which make a total of — — , or 51 % of 
 ., , 1UO 1U0 100 
 
 the base. 
 
 5. Paid $729 for goods, on which 10% was allowed. 
 What was the " gross " price ? 
 
 - 6. How much will be paid for 12 doz. bottles flavoring 
 extract, at 60^ per dozen, less 10% ? 
 
 7. What is the "list " price of an article for which I paid 
 $ 48, after a discount of 25% was deducted ? 
 
 Note. — The " list " price, " gross " price, or " catalogue " price ia 
 the price before the deduction of discounts. 
 
 8. What is the net price of an article catalogued at $ 880, 
 on which there is a discount of 75% ? 
 
 Note. — A discount of 75 % from the list price means that the net 
 price is 25 % of the list price. Instead, therefore, of finding 75 % of 
 $ 880 and deducting it from $ 880, the pupil should shorten the work 
 by taking 25% of $880. 
 
 9. 75 is 25% more than what number? 
 
 10. Find the cost of a wagon " catalogued " at $ 700, the 
 discount being 30%. 
 
 Note. — The cost is 70 % of $ 700. 
 
250 Chapter Five. 
 
 INTEREST. 
 
 323. Preliminary Exercises. 
 
 1. A farmer, needing $1000 to purchase additional land, 
 borrows the money, agreeing to repay it at a given time 
 with 6% of the sum for each year he has the use of it. 
 This annual payment of $ 60 for the use of $ 1000 is called 
 interest. The $ 1000 is called the principal. If the borrower 
 repays the $ 1000 at the end of two years and also $ 120 
 interest, the total payment of $ 1120 is called the amount. 
 
 2. What is the interest on $1000 at 6% for 6 months ? 
 
 3. What is the amount of $1000 for 3 years at 6% ? 
 
 4. What is the interest on $ 1000 for 1 month at 6% ? 
 
 5. Taking 30 days to a month, find the interest on 
 $1000 for 15 days at 6%. 
 
 324. "Written Exercises. 
 
 1. Find the interest on $ 750 at 6% for 2 years 6 months. 
 
 $750 Principal 
 The interest for 1 year A « ^ 
 
 .„,,,.,., X .06 Rate. 
 
 is found by multiplying the gKnA 
 
 • • 1 atca v *i. Interest for 1 yr. $45.00 
 
 principal, $750, by the rate, 01 • 
 
 6 hundredths, and this prod- mn * ? Time * 
 
 uct by the number of years, 
 
 2*. 
 
 22.50 
 
 Interest for 2 yr. 6 mo. $ 112.50 
 The foregoing may be expressed by the formula : 
 
 Principal x ^^ x Time (in years) = Interest 
 100 
 
 It is suggested 
 
 that the work be ^^ 3 
 
 arranged in this $ ±_ 5 $2^5 = fmM ^ 
 
 manner, so that it rr J[fifl JJ 2 
 
 may be shortened 2 
 
 by cancellation. 
 
Interest. 251 
 
 2. What is the interest on $84.75 at 4% for 3 months 
 6 days ? 3 months 6 days = qq days = 96 year 
 
 ooO 
 
 The 100 in the divisor is can- .0565 i aa 
 
 celled by removing the decimal $.84/75 X =7777 X 07577 = $ .9040 
 
 point in the principal two places qa 
 
 to the left. Ans ^ go centg 
 
 3. Find the interest on $394.50 for 2 years 7 months 
 
 * **%. 477 
 
 $.394/50x ? xW $188.1765 
 
 24 days at 4|%. 4?? 
 
 4 
 
 = $47,041+. ^ns. $47.04. 
 
 The time is readily changed to days : 720 + 210 + 24 = 954. 
 
 This is expressed in years by placing 360 in the divisor, i.e. below 
 the line. The three ciphers in the divisor are cancelled by moving 
 the decimal point in the principal three places to the left. 
 
 In calculating interest, take 30 days to a month, 12 months to a 
 year. 
 
 Find the interest on 
 
 4. $308 at 5% for 20 days. 6. $720 at 7% for 21 days. 
 
 5. $360 at 5% for 33 days. 7. $1000 at 5% for 8 days. 
 
 8. $94.43 at 7% for 2 mo. 3 da. 
 
 9. $464.75 at 6% for 8 mo. 12 da. 
 10. $400 at 41% for 1 yr. 1 mo. 1 da. 
 
 325. Amount = Principal + Interest 
 Find the amount : 
 
 1. $ 813, from April 19, 1902, to March 4, 1907, at 6%. 
 The time is found by compound subtraction. 1907 3 4 
 4 yr. 10 mo. 15 da. 1902 4 19 
 
 .271 4 10 15 
 
 Interest = S-TO/x g x 1755 = % m M 
 
 m x m 
 2 
 
 Amount = $ 813 + $ 237.80 = $ 1050.80, Arts. 
 
252 Chapter Five. 
 
 2. $ 960, from Jan. 1, 1903, to Dec. 21, 1904, at 4%. 
 
 3. $ 27.84, for 3 yr. 6 mo. 9 da., at 6%. 
 
 4. $ 48.90, for 17 da., at 6%. 
 
 5. $ 144, for 2 yr. 5 da., at 3f %. 
 
 6. $ 834.76, for 15 mo. 27 da., at 4J%. 
 
 7. $ 5760, for 1 yr. 5 mo. 29 da., at 5%. 
 
 8. 9 2346.50, for 7 yr. 13 da., at 3%. 
 
 9. $ 1892, for 3 yr. 5 mo., at 7%. 
 
 10. $ 150.40, for 1 yr. 2 mo. 3 da., at 6%. 
 
 326. Interest-bearing Demand Notes. 
 
 A promissory note is a written agreement to pay a stated 
 sum of money after a given time or on demand to a certain 
 person with or without interest. The person signing the note 
 below, James Dunne, is called the maker; the person in whose 
 favor it is drawn, Charles C. Wise, the payee. If the latter 
 wishes to transfer it to James H. Tully, he writes on the 
 back of the note : Pay to the order of James H. Tully ; and 
 underneath he signs his name. This is called an indorse- 
 ment in full. By merely signing his name on the back, 
 which is called an indorsement in blank, Mr. Wise makes it 
 payable to any person holding it. The effect of an indorse- 
 ment is also to make the indorser liable in the event of the 
 maker refusing to pay. 
 
 1. San Francisco, Jan. 7, 1902. 
 
 On demand, I promise to pay Charles C. Wise, or order, 
 Seven Hundred Sixty-five T 4 ^j- Dollars, value received, with 
 interest at 6 per cent. 
 
 $765 T Vtf. James Dunne. 
 
 • How much money will be required to pay the above note, 
 with interest, July 15, 1903 ? 
 
Interest. 253 
 
 2. A demand note, dated Sept. 25, 1902, with interest at 
 8% from date, is paid Jan. 2, 1905. How much was due, 
 the face of the note being $ 750 ? 
 
 3o Find the amount due March 4, 1904, on a note for 
 $ 365.84, dated May 20, 1902, with interest from date at 7%. 
 
 4. Find the amount necessary, Oct. 16, 1906, to pay a 
 note of $ 1240, with interest at 6% from Aug. 15, 1902. 
 
 5. An interest-bearing note for $ 87.60 is dated April 3, 
 1900. How much is due on it for principal and interest 
 Jan. 2,1908? Rate 4^%. 
 
 327. Oral Problems. 
 
 If these are first used as sight problems, an opportunity will be 
 afforded to develop different methods for solving many of them. 
 
 1. Find the interest on $ 300, for 1 yr. 7 mo., at 4%. 
 
 $ 12 per year is how much for 7 months ? 
 
 2. On $ 60, for 33 days, at 6%. 
 
 $ 3.60 for 360 days is how much for 33 days ? 
 
 3. On $ 120, from Jan. 1, 1903, to July 1, 1904, at 5%. 
 
 4- How long will it take $ 100 to produce $ 15, interest 
 at6%? 
 
 5. At what rate per cent will $ 50 produce $ 6 in 2 years ? 
 
 6. What is the interest on $ 300, at 6%, from Feb. 1 to 
 Feb. 21 ? 
 
 7. What part of a year is 72 days ? 
 
 8. Find the interest at 4%, for 90 days, on f 150. 
 
 9. On 1 240, for 36 days, at 5%. 
 
 10. What is the amount of $ 200, for 3 yr. 1 mo., at 6% ? 
 
 11. How long will it take $ 1 to make $ 1 interest at 5% ? 
 
 12. How long will it take any sum to double itself at 6% ? 
 
 13. How long will it take $ 14.90 to double itself at 4% ? 
 
254 Chapter Five. 
 
 PARTIAL PAYMENTS. 
 
 328. United States Eule. 
 
 . When the maker of an interest-bearing note pays a portion 
 of the debt represented by the note, the money is applied in 
 the first place to the payment of the interest, then to the 
 reduction of the principal. 
 
 1. Duluth, Minn., Jan. 5, 1902. 
 
 On demand, I promise to pay to the order of James F. 
 McGee Three Hundred Dollars, value received, with interest 
 at 7 per cent. 
 
 $300^. J. Eandolph Page. 
 
 Payments: May 20, 1902, $100; Oct. 30, 1902, $100; 
 March 6, 1903, $50. 
 
 How much was due Jan. 5, 1904 ? 
 
 Find amount of $300 Jan. 5, 1902, to first payment May 20, 1902, 
 
 4 mo. 15 da. (by compound subtraction), $307.88 
 
 Deduct first payment, 100.00 
 
 Balance May 20, 1902, $207.88 
 
 Interest on $ 207.88 to Oct. 30, 5 mo. 10 da., 6.47 
 
 Amount, $214.35 
 
 Less second payment, 100.00 
 
 Balance Oct. 30, 1902, $114.35 
 
 Interest on $ 114.35 Oct. 30 to March 6, 4 mo. 6 da., 2.80 
 
 Amount, $117.15 
 
 Less third payment, 50.00 
 
 Balance March 6, 1903, $67.15 
 
 Interest on $67.15 March 6 to Jan. 5, 9 mo. 29 da., 3.90 
 
 Due Jan. 6, 1904, $71.05 
 
 Find the amount of the principal to the time when the pay- 
 ment or the sum of two or more payments equals or exceeds the 
 interest. 
 
 From this amount deduct the payment or sum of payments. 
 
 Use the balance then due as a new principal, and proceed as 
 before. 
 
Partial Payments. 2$$ 
 
 2. How much is due June 3, 1905, on a demand note for 
 $1200, with interest at 6%, dated June 3, 1902, bearing 
 indorsements of payment of $ 500, Sept. 18, 1903 ; $ 600, 
 Jan. 3, 1904? 
 
 Note. — Anything written on the back of a document is called an 
 indorsement. Payments made are usually written on the back of the 
 notes. 
 
 3. A demand note for $ &00, bearing interest at 5 %, was 
 given Feb. 18, 1902. A payment of $ 250 was made May 
 28, 1903 ; one of $ 150 was made Oct. 8, 1903. How much 
 is due Jan. 23, 1905 ? 
 
 4. A note for $2000, with interest at 7%, was dated 
 April 15, 1901. Indorsements were made as follows : $ 50, 
 Sept. 20, 1901 ; $ 100, May 26, 1902 ; $ 1000, June 20, 1903. 
 How much is due Dec. 27, 1904 ? 
 
 Face of note, $2000.00 
 
 Interest from April 15 to Sept. 20, 1901, 5 mo. 5 da., 60.28 
 
 Amount due Sept. 20, 1901, $2060.28 
 
 If the $ 50 payment were deducted, and interest computed 
 on the balance, $2010.27, the maker would be charged in- 
 terest on $ 10.27 more than the face of the note, and this the 
 law does not allow. Interest is taken on $ 2000 until next 
 payment, May 26, 1902, 8 mo. 6 da., 95.67 
 
 Amount due May 26, 1902, $2155.95 
 
 As the two payments are not large enough to meet the 
 interest now due, the interest is again computed on the 
 original $2000 from May 26, 1902, to June 20, 1903, 1 yr. 
 24 da., 149.33 
 
 Amount of $ 2000 from April 15, 1901 to June 20, 1903, $ 2305.28 
 Less $ 50 + $ 100 + $ 1000 (three payments), 1150.00 
 
 Balance due June 20, 1903, $ 1155.28 
 
 Interest on $ 1155.26 to Dec. 27, 1904, 1 yr. 6 mo. 7 da., 122.87 
 Due Dec. 27, 1904, $ 1278.16 
 
256 Chapter Five. 
 
 1 
 
 5. Albany, N.Y., March 5, 1903. 
 One year after date, I promise to jpay John Harrigan, or 
 
 order, Nine Hundred Dollars, value received, with interest 
 at six per cent. 
 
 $ 900^. Andrew T. Sullivan. 
 
 Indorsed as follows : June 5, 1903, $ 10 ; Sept. 5, 1903, 
 $ 50 ; Jan. 5, 1904, $ 120. What was due March 8, 1904 ? 
 
 6. Alexandria, La., June 19, 1903. 
 On demand I promise to pay to the order of George H. 
 
 Dotzert,Two Thousand Four Hundred Fifty-four -ftjfo Dollars, 
 value received, with interest at 6 per cent. 
 
 $2454£fc Charles W. Lyon. 
 
 The following payments were made: July 5, 1903, $450; 
 Sept. 18, 1903, $ 700 ; Oct. 25, 1903, $ 300. Find the amount 
 due Jan. 2, 1904. 
 
 329. In the United States courts, and in those of some of the states, 
 interest for a portion of a year is taken by days, upon the basis of 365 
 days to the year. To make the work easier for the pupils, however, 
 the year of 360 days should be used in the examples given, and the 
 time between dates should be found by compound subtraction. 
 
 330. Merchants' Eule. 
 
 The merchants' rule is frequently used where all the 
 payments are made within a year. 
 
 The interest is computed on the face of an interest- 
 bearing note from its date until settlement, and interest is 
 allowed on all credits from their payment until settlement. 
 
 The exact number of days is taken, and the interest is 
 computed on the basis of 360 days to the year. 
 
 Boston, Mass., June 19, 1903. 
 
 On demand, I promise to pay Charles R. Buttrick, or 
 order, Two Thousand Four Hundred Fifty-four -^ Dollars, 
 value received, with interest at 6 per cent. 
 
 $ 2454^. John J. P. Fagan. 
 
ne, 
 
 82454.75 
 
 
 80.60 
 
 
 $2535.35 
 
 $ 200.00 
 
 
 6.03 
 
 
 450.00 
 
 
 11.78 
 
 
 700.00 
 
 
 12.37 
 
 
 300.00 
 
 
 3.45 
 
 1683.63 
 
 Partial Payments. 257 
 
 The following payments are endorsed on the note : July 5, 
 1903, $200; July 29, 1903, $450; Sept. 18, 1903, $700; 
 Oct. 25, 1903, $ 300. 
 
 Find the amount due Jan. 2, 1904. 
 
 If no payments had been made, there would be due, 
 
 And interest from June 19 to Jan. 2, 197 days, 
 
 Total due, 
 
 The credits are: Payment July 5, 1903, 
 
 Interest on $ 200, July 5 to Jan. 2, 181 days, 
 
 Payment July 29, 1903, 
 
 Interest on $450, July 29 to Jan. 2, 157 days, 
 
 Payment Sept. 18, 1903, 
 
 Interest on $ 700, Sept. 18 to Jan. 2, 106 days, 
 
 Payment Oct. 25, 1903, 
 
 Interest on $ 300, Oct. 25 to Jan. 2, 69 days. 
 
 Balance due, $ 851.72 
 
 Find the amount of an interest-bearing note at the time of 
 settlement. 
 
 Find the amount of each credit from its time of payment to 
 the time of settlement ; subtract their sum from the amount of 
 the note. 
 
 331. Written Exercises. 
 
 1. A note for $ 500, with interest at 6%, is dated July 
 25, 1904. Payments are made: $ 100, Sept. 18 ; $ 200, Feb. 
 5, 1905. How much is due April 1, 1905? 
 
 2. Find amount due Sept. 15, 1903, on a demand note for 
 $ 1875, with interest at 6 %, dated Jan. 18, 1903. Payments 
 of $ 1000 and $ 500 were made March 30 and June 17, 
 respectively. 
 
 3. June 12, 1904, Robert Colgate bought goods amounting 
 to $ 600. Dec. 31, 1904, he paid $ 300 ; April 5, 1905, $ 200 ; 
 June 1, 1905, he settled the account. How much did he pay 
 on that date, if he Is charged 6 % on the purchase from its 
 date, and is allowed 6 % interest on his payments ? 
 
258 
 
 Chapter Five. 
 
 4. John C. Kelley loaned Chas. R Robertson $ 500, Sept. 
 1, at 6 %. Payments of $ 200 each were made Oct. 1 and 
 Nov. 1. How much is due Dec. 1 ? 
 
 Dr. 
 
 Horace E. Dresser 
 
 Or. 
 
 1905. 
 
 
 
 
 
 1905. 
 
 
 
 
 
 Feb. 
 
 5 
 
 To merchan- 
 dise, 
 
 840 
 
 00 
 
 Mar. 
 
 9 
 
 By cash, 
 
 500 
 
 00 
 
 Dec. 
 
 31 
 
 To interest to 
 date, 
 
 
 
 Sept. 
 Dec. 
 
 M 
 
 13 
 31 
 it 
 
 By cash, 
 
 By interest to 
 
 date, 
 By cash, 
 
 200 
 
 00 
 
 
 
 
 
 5. Find the amount paid in settlement of the foregoing 
 account, Dec. 31, 1905. Interest 6 %. 
 
 6. A merchant's books show the following debits : Feb. 13, 
 merchandise, $725.00; April 14, merchandise, $603.00. 
 The credits are : April 5, cash, $ 600 ; Aug. 29, cash, $ 300. 
 How much is due Oct. 5, interest 6 % ? 
 
 332. Oral Exercises. 
 
 1. If I sell for $4.50 a book which cost me $3, what 
 per cent do I gain ? 
 
 2. What is the interest of $ 200, for 90 days, at 3% ? 
 
 3. One acre of corn yields 80 bushels, and another acre 
 20% more. What does the second acre yield ? 
 
 4. What will it cost to fence a garden 10 rods long and 
 6 rods wide, at $ 1 a rod ? 
 
 5. In a certain school 40 pupils are present and 10 are 
 absent. What per cent are absent ? 
 
 6. What is the difference between a floor 40 feet square 
 and two others each 20 feet square ? 
 
Miscellaneous. 259 
 
 7. What is the interest of $ 12, for 1 yr. 4 mo., at 6%? 
 
 8. If 2^- pecks of berries cost one dollar, what would 3 
 quarts cost at the same rate ? 
 
 9. Bought 5 bushels nuts at a dollar a peck, and got 5% 
 off for cash. How much did I pay for the nuts ? 
 
 333. Written Problems. 
 
 1. Gold coin contains 90 per cent gold, 9 per cent silver, 
 1 per cent copper. Find the quantity of each metal in 50 
 double-eagles ($ 20), each containing 516 grains. 
 
 2. A, B, and C buy a farm. A pays $8700, B pays 
 $7200, C pays $4100. What per cent of the purchase 
 money does each furnish? 
 
 3. The one-cent pieces weigh 48 grains. How many 
 dollars would weigh 120 pounds avoirdupois (7000 grains to 
 pound) ? 
 
 4. If a person lends me $ 250 for 8 months, for how 
 long ought I to lend him $400 as an equivalent? 
 
 5. Goods costing $8 are sold at an advance of 20 per 
 cent. The marked price is $ 12, What per cent reduction 
 is made on the marked price ? 
 
 6. There are 5 boys whose heights are 4 ft. 9 in., 5 ft. 
 1 in., 4 ft. 5 in., 3 ft. 11 in., and 4 ft. 4 in., respectively. 
 What is their average height? 
 
 7. In the written number 185.4, the number expressed by 
 the first two (left-hand) figures is how many time the value 
 expressed by the second two figures ? 
 
 8. Express decimally, and also as a common fraction, the 
 value of each of the following: 115 per cent; ^ of 1 per 
 cent ; |~| of 1 per cent. 
 
 9. M bought ^ of a manufacturing business for 
 $3517.85, and N bought -^ of the same business at the 
 same rate. How much did N's interest cost him? 
 
260 Chapter Five. 
 
 334. To find Principal, Kate, or Time. 
 
 1. What principal will produce $ 2.88 interest in 8 months 
 at 4i% ? 
 
 The interest on $ 1 at 4£% for 8 months is $ 1 x gfo x T 8 2 , or $.03. 
 Since $.03 is produced by $ 1 (at the given rate for the given time), 
 $2.88 will require a principal of as many dollars as $ .03 is contained 
 times in $2.88, or 96. Ans. $96. 
 
 Proof. $ 96 x gfa x A = 1 2 - 88 - 
 
 2. What principal will amount to $ 98.88 in 8 months 
 at 4£% ? 
 
 The amount of $ 1 at 4£ % for 8 months is $ 1.03. If an amount of 
 $1.03 is produced from a principal of $1, an amount of $98.88 will 
 be produced from a principal of as many dollars as $ 1.03 is contained 
 times in $ 98.88, or 96. Ans. $ 96. 
 
 To find the principal, divide the given interest (or amount) 
 by the interest (or amount) o/ $ 1 at the given rate for the given 
 time. 
 
 The following is an algebraic method of solving No. 1 : 
 
 (1) Let x represent the required principal. 
 
 (2) The interest will be x X — X — , or ££. 
 v " 200 12 100 
 
 (3) £^- = 2.88. • 
 v J 100 
 
 (4) Clearing of fractions, 3 x = 288. 
 
 (6) Dividing, x = 96. Ans. $96. 
 
 The following is an algebraic method of solving No. 2 : 
 
 (1) Let x represent the required principal. 
 
 (2) The interest will be x x — X — , or ^ 
 K J 200 12' 100 
 
 (3) The amount will be x + — , or 1^. 
 v ' 100 100 
 
 (4) 12^ = 98.88. 
 v J 100 
 
 (6) Clearing of fractions, 103 x = 9888. 
 
 (6) Dividing, x = 96. Ans. $96 
 
Interest. 261 
 
 3. At what rate per cent will 1 723.60 produce $ 36.18 
 interest in 1 yr. 1 mo. 10 da. ? 
 
 The interest on $723.60 at 1 % for 1 yr. 1 mo. 10 da. is $723.60 
 x jfa x Iff, or $8.04. Since $8.04 is produced by a rate of 1%, 
 $36.18 will require a rate of as many per cent as $8.04 is contained 
 times in $ 36. 18, or 4J. Ans. 4% %. 
 
 Proof. $ 723.60 x rfo X fff = $36.18. 
 
 4. At what rate per cent will $ 723.60 amount to f 759.78 
 in 1 yr. 1 mo. 10 da. ? 
 
 Find the interest by subtracting the principal $723.60 from the 
 amount $ 759.78, and proceed as in No. 3. 
 
 To find the rate, divide the given interest by the interest at 
 1% on the given principal for the given time. 
 
 The following is an algebraic solution of Nos. 3 and 4 : 
 
 (1) Let x represent the rate. 
 
 (2) The interest will be 723.60 x — X — , or 8.04 x. 
 K J 100 360 
 
 (3) 8.04 a; = 36.18. 
 
 (4) Clearing of decimals, 804 x = 3618. 
 
 (5) Dividing, x = 4£. Ans. ±\ %. 
 
 5. In what time will $ 85.50 produce $ 8.17 interest at 
 4%? 
 
 The interest on $ 85.50 at 4 % for 1 year is $ 85.50 x T £ 7 , or $ 3.42. 
 Since $3.42 is produced in 1 year, $8.17 will require as many years as 
 $3.42 is contained times in $8.17, or 2^. 
 
 Ans. 2 T 7 j years, or 2 yr. 4 mo. 20 da. 
 Proof. $ 85. 50 x ^ x 2 T \ 
 
 = $85.50 x^xtf = $8.17. 
 
 6. In what time will $ 85.50 amount to $ 93.67 at 4% ? 
 'Find the interest by subtracting the principal $85.50 from the 
 
 amount $ 93.67, and proceed as in No. 5. 
 
 To find the time, divide the given interest by the interest for 
 i year on the given principal at the given rate. 
 
262 Chapter Five. 
 
 335. The following is an algebraic solution of Nos. 5 
 and 6: 
 
 (1) Let x represent the time in years. 
 
 (2) The interest will be 85.50 x T fo x x, or 3.42 x. 
 
 (3) 3.42 x = 8.17. 
 
 (4) Clearing of decimals, 342 x = 817. 
 (6) Dividing, x = 2^. 
 
 Ans. 2 T 7 j years, or 2 yr. 4 mo. 20 da. 
 
 336. The algebraic method consists (1) in representing 
 the unknown quantity (principal, rate, or time) by x ; (2) find- 
 ing the interest, by multiplying principal by rate by time ; 
 (3) forming an equation, by making this product equal to 
 given interest ; (4) solving the equation. 
 
 337. Written Exercises. 
 Find rate, time, etc. 
 
 1. Principal, $ 2000 ; time, 3 yr. ; interest, $ 300. Rate? 
 
 2. Principal, $1800; rate, 4%; interest, $144. Time? 
 
 3. Time, 8 mo. ; rate, 4|% ; interest, $ 2.88. Principal? 
 
 4. Principal, $ 38 ; time, 2 yr. ; amount, $ 40.28. Rate ? 
 
 5. Principal, $140; rate, 3£% ; time, 3 mo. 15 da. 
 Interest ? 
 
 6. Amount, $39.60; rate, 4%; time, 2 yr. 6 mo. 
 Principal ? 
 
 7. Amount, $484.15; rate, 3J% ; principal, $460. 
 Time ?   
 
 8. Principal, $ 39.60; rate, 4% ; time, 1 yr. 7 mo. 15 da. 
 Amount ? 
 
 9. Time, 8 yr. ; rate, 3% ; amount, $ 6200. Principal ? 
 
 10. Principal, $ 7548 ; time, 3 mo. 5 da. ; interest, $ 119.51. 
 Rate? 
 
 11. Principal, $ 9000 ; rate, 4% ; interest, $ 632. Time ? 
 
 12. Time, 2 yr. 3 mo. 20 da. ; rate, 5% i amount, $160.60. 
 Principal ? 
 
Interest. 263 
 
 13. Principal, $ 756; rate, 3\% ; time, 3 yr. 4 mo. 20 da. 
 Interest ? 
 
 14. Principal, $120; time, 1 yr. 2 mo. 15 da.; interest, 
 $ 4.35. Rate ? 
 
 15. Amount, $ 97.57 ; rate, 4% ; interest, $ 7.57. Time ? 
 
 16. Time, 3 yr. 8 mo. 19 da. ; rate, 4J% ; amount, $ 93.39. 
 Principal ? 
 
 17. Principal, $ 1848 ; rate, 3£%; time, 4 yr. 9 mo. 25 da. 
 Amount ? 
 
 18. Kate, 5% ; time, 4 yr. 6 mo. 23 da. ; interest, $ 16.43. 
 Principal ? 
 
 338. Oral Exercises. 
 
 1. In what time will $100 amount to $109, at 6% 
 interest ? 
 
 2. At what rate will $ 200 produce $ 16 interest in 
 2 years ? 
 
 3. What principal will produce $ 12 interest in 3 years, 
 at4%? 
 
 4. In what time will $ 300, at 4 % , produce $ 29 interest ? 
 
 5 . In what time will $ 170 produce $ 1.70 interest, at 5 % ? 
 
 6. In what time will $ 360 produce $ 3.60 interest, at 4% ? 
 
 7. In what time will $ 725 produce $7.25 interest, at 6%? 
 
 8. In what time will $ 45 produce 45 f interest, at 4£%? 
 
 9. In what time will $ 72 produce $ 1.44 interest, at 6 % ? 
 
 10. Find the interest on $84 for 144 days, -at 5%. 
 
 11. Find the interest on $125, at 5%, for 2 months 
 12 days. 
 
 12. At what rate will $ 64 produce 64 i interest in 80 days ? 
 
 13. At what rate will $ 40 produce $ 1.20 interest in 
 6 months ? 
 
 14. A certain principal produces $ 120 interest at 6%. 
 What would be the interest if the rate were 4% ? 
 
264 Chapter Five. 
 
 339. Written Review Exercises. 
 
 1. What number increased by 16% of itself equals 1276? 
 
 2. A capitalist sends a commission merchant $8670 to 
 invest in cotton and to include commission at 2%. How 
 much does the commission amount to ? 
 
 3. A joiner worked on Monday 9 hr. 45 min., on Tuesday 
 and Wednesday 10 hr. 45 min. each day, on Thursday and 
 Friday 10 hr. 15 min. each day, and on Saturday 6 hr. 45 min. 
 What was the average length of his day's work ? 
 
 4. Thirty-two clerks are to distribute 36,000 letters on 
 a certain day. Half of the clerks are experienced men and 
 half of them new men. If each experienced man does twice 
 as much as a new man, how many letters will be distributed 
 by one of each kind ? 
 
 5. Sold my house and farm of 94£ acres for $ 12,300. 
 Allowing $ 7000 for the house, what did I receive per acre 
 for the land ? 
 
 6. A commission merchant receives $ 1071 to invest in 
 oats at 30^ per bushel and to cover his commission at 2% 
 for buying. How many bushels of oats does he purchase ? 
 
 Should the commission merchant deduct 2 % of $ 1071, or 2 % of the 
 cost of the oats ? 
 
 7. What is the total weight of 4 hogsheads of sugar, 
 weighing respectively 936J, 1025^, 846$, and 987-^ pounds, 
 deducting tare at 10 per cent ? 
 
 8. The product of three factors is 3289; two of the 
 factors are 23 and 11. What is the third factor ? 
 
 9. A man received $2.75 per day, exclusive of Sundays, 
 during 1903. He paid $73 for clothing for himself and 
 family, $ 15 per month rent, $ 1.10 per day for provisions, 
 $ 8 per month for fuel and light, and 25 ^ per day for other 
 expenses. How much had he left at the end of the year ? 
 
Bank Discount. 26$ 
 
 BANK DISCOUNT. 
 
 340. Thomas Tierney, wishing to borrow three hundred 
 
 dollars from The Borough Bank, draws up the following 
 
 promissory note : 
 
 Denver, May 16, 1903. 
 
 Three months after date I promise to pay to the order 
 of myself Three Hundred Dollars, value received, at The 
 Borough Bank. 
 
 $300 T Vo. Thomas Tierney. 
 
 As the note now stands, it is payable to Thomas Tierney. 
 He transfers by indorsing it ; that is, by writing his name 
 on the back of the note. The effect of this indorsement is 
 to transfer the note to the holder, in this case, the bank. 
 As a bank requires at least a second person as a security 
 for the payment of the loan, Mr. Tierney gets Herman A. 
 Metz to indorse it also. By this indorsement, Mr. Metz 
 agrees to pay the note if Mr. Tierney does not pay it at 
 maturity, August 16. 
 
 The Borough Bank thereupon pays over to Thomas 
 Tierney, or places to his credit on the books of the bank, 
 the face of the note less the interest for 92 days, $300 
 — $ 4.60, or $ 295.40. This interest taken in advance is 
 called bank discount. The sum turned over to Thomas 
 Tierney is called the proceeds. 
 
 Face of note, $300.00 
 
 Discount 92 days, 4.60 
 
 Proceeds, $ 295.40 
 
 To find the bank discount, compute the interest on the face 
 of the note from the date of discount to the date of maturity. 
 To find the proceeds, deduct the discount from the face. 
 
 Note. — The usage of banks varies in different parts of the country, 
 and the teacher should inform herself as to the local practice. 
 
i66 Chapter Five. 
 
 341. Written Exercises. 
 
 Find the discount at 6 % on the following : 
 
 1. A 30-day s note for $75. 
 
 2. 15-day s note for $ 183.60. 
 
 3. 60-days note for $275.40. 
 
 4. 20-days note for $ 96. 
 
 5. 4-months note for $ 336. 
 Face of note — bank discount = proceeds. 
 
 Find the proceeds, at 7%, on 
 
 6. A 6-months note for $ 180. 
 
 7. A 3-months note for $ 36.90. 
 
 8. A 24-day s note for $ 795.60. 
 
 9. A 90-day s note for $ 180. 
 10. A 72-days note for $ 1000. 
 
 342. Find the discount, at 6%, on 
 
 11. A 1-month note for $600, dated Feb. 6, 1904. Due 
 March 6, 29 days. 
 
 12. A 2-months note for $ 240, dated July 17, 1903. 
 
 13. A 3-months note for $ 360, dated April 8, 1904. 
 
 14. A 4-months note for $ 84, dated Dec. 24, 1905. 
 
 15. A 6-months note for $ 172.60, dated March 4, 1903. 
 
 16. A 60-days note for $ 240, dated July 17, 1904. 
 
 17. A 90-days note for $ 360, dated April 8, 1903. 
 
 In each of the preceding examples, it has been assumed that the 
 note has been presents for discount the day on which it was made. 
 
 In some of the following examples, the notes are discounted at a 
 later date, and the term of discount is to be ascertained ; that is, the 
 time between the date of discount and that of maturity. 
 
 The term of discount of a 30-day s note dated May 1, and discounted 
 May 19, is the time from May 19 to May 31, 12 days. 
 
Bank Discount. 267 
 
 343. In the following examples, find (a) date of maturity; 
 (b) term of discount ; (c) discount ; (d) proceeds. 
 
 Note. — The pupil that works without thinking, frequently finds the 
 
 difference in time between the two dates given in the problem and uses 
 
 this as the term of discount. The time between the dates given below 
 
 shows in each case the time the note was not in the possession of the 
 
 bank. 
 
 Dated. Face. Time. Discounted. Kate. 
 
 18. July 16,1902; $ 87.60; 30 days ; August 11, 1902 ; 6% 
 
 This note is due 30 days after July 16, which is August 15. If the 
 bank discounts it August 11, 4 days before it is payable, it deducts 
 4 days' interest, which is 6 cents. 
 
 Ansicers — Date of maturity Aug. 15, 1902. 
 Term of discount 4 days. 
 Discount 6 cents. 
 
 Proceeds $87.54. 
 
 19. Date, Sept. 9, 1902; face, $ 124. 18; time, 4 months; 
 discounted, Nov. 18, 1902; rate, *8%. 
 
 20. Date, Dec. 5, 1902; face, $504.60; time, 30 days; 
 discounted, Dec. 12, 1902; rate, 7%. 
 
 21. Date, Nov. 14, 1903; face, $72.36; time, 3 months; 
 discounted, Dec. 20, 1903; rate, 6%. 
 
 22. Date, Oct. 30, 1903; face, $234; time, 90 days; 
 discounted, Jan. 5, 1904; rate, 6%. 
 
 23. Date, Jan. 2, 1904; face, $95.90; time, 2 months; 
 discounted, Feb. 13, 1904; rate, 6%. 
 
 24. Date, Aug. 5, 1904; face, $164; time, 60 days; 
 discounted, Aug. 31, 1904; rate, 8%. 
 
 25. Date, Feb. 27, 1904; face, $83.20; time, 100 days; 
 discounted, March 9, 1904; rate, 6%. 
 
268 Chapter Five. 
 
 DISCOUNT OF INTEREST-BEARING NOTES. 
 
 344. Written Problems. ^ „_ -. - M «,_. 
 
 Brooklyn, N.Y., Oct. 16, 1904. 
 
 Sixty days after date I promise to pay to the order of 
 John Karst, One Hundred Forty-eight -ffo Dollars, value 
 received, with interest at 6%. 
 
 $148-/^. Daniel Kelly. 
 
 1. Find the proceeds of the above note if discounted 
 Dec. 1, 1904, at 6%. 
 
 At maturity, Dec. 15, 1904, there is due $148.50 with $1.49 in- 
 terest for sixty days, a total of $ 149.99. If it is discounted Dec. 1, 
 14 days before maturity, the bank deducts 14 days' interest on 
 $ 149.99, which is 35 cents, and pays over to John Karst the proceeds, 
 $ 149.64, Arts. 
 
 To find the bank discount of an interest-bearing note, com- 
 pute the interest on the amount due at maturity from the time 
 of discount to the date of maturity. 
 
 2. Find the proceeds of, a 90-day s note for $ 175, bearing 
 interest at 6%, discounted 33 days after date, at 6%. 
 
 3. Find the proceeds of a 60-days note for $ 350, bearing 
 interest at 6%, discounted at 6%, 10 days after date. 
 
 4. Find the proceeds of a 3-months note for $840, 
 bearing interest at 7%, discounted at bank 47 days before 
 maturity, at 8%. 
 
 5. A 4-months note for $720, dated March 17, 1905, 
 bearing interest at 6%, is discounted at 7%, May 10. What 
 are the proceeds ? 
 
 6. The following note was discounted at 6%, Sept. 19, 
 1904. Find the proceeds. 
 
 Milwaukee, Wis., June 30, 1904. 
 
 Four months after date I promise to pay Thomas Cacciola, 
 or order, Five Hundred Dollars, value received, with interest 
 at 6 per cent. 
 
 $500^. George H. Greene. 
 
Bank Discount. 269 
 
 345. To find the face of note, rate of discount, or term. 
 
 1. The discount at 6% on a note having 84 days to run, 
 is $ 10.50. Find the face of the note. 
 
 The discount on $1 for 84 days at 6% = $1 x y-Jfo X flfo or $.014. 
 If $.014 is the discount on a note for $ 1, $10.50 will be the discount 
 on a note for as many dollars as $.014 is contained times in $10.50, or 
 $750. Ans. $750. 
 
 2. The proceeds of a note having 84 days to run, dis- 
 counted at 6%, are $739.50. Find the face of the note. 
 
 The discount on $1 for 84 days at 6% is $.014 ; the proceeds are 
 $1 -$.014, or $.986. If $.986 are the proceeds of a note for $1, 
 $739.50 will be the proceeds of a note for as many dollars as $.986 is 
 contained times in $739.50, or $750. Ans. $750. 
 
 To find the face of a note, divide the given discount {or pro- 
 ceeds) by the discount (or proceeds) of $1 for the given term 
 at the given rate. 
 
 3. The discount on a note for $ 750 having 84 days to run 
 is $ 10.50. What is the rate of discount ? 
 
 The discount on $750 at 1% for 84 days is $750 x T J 7 x /fa, or $1.75. 
 If $1.75 is produced by a rate of 1%, $10.50 will be produced by a rate 
 of as many per cent as $1.75 is contained times in $10.50, or 6%. 
 
 To find the rate of discount, divide the given discount by the 
 discount at 1 % on the given sum for the given term. 
 
 4. The discount at 6% on a note for $750 is $10.50. 
 How many days has the note to run? 
 
 The discount on $750 at 6% for 1 day is $750x^X3-^, or $.125. 
 If $.125 is the discount for 1 day, $10.50 will be the discount for as 
 many days as $.125 is contained times in $10.50, or 84 days. 
 
 To find the term of discount, divide the given discount by the 
 discount for 1 day on the given sum at the given rate. 
 
 346. Note. — To solve by the algebraic method, use x to represent 
 the unknown quantity. 
 
270 Chapter Five. 
 
 347. Written Exercises. 
 
 1. Three-months note; face, $108; rate, 6%. Find 
 proceeds. 
 
 2. 90-days note ; face, $ 360 ; discount, $ 6.30. Find rate. 
 
 3. Proceeds, $717.60 ; rate, 5% ; face, $ 720. Find term. 
 
 4. Discount, $ J 1.20 ; rate, 7% ; term, 48 days. Find face. 
 
 5. 15-days note ; face, $ 1560 ; rate, 6%. Find discount. 
 
 6. Term, 20 days; face, $158.40; proceeds, $157.96. 
 Find rate. 
 
 7. Eate, 7%; discount, $2.10; face, $150. Find term. 
 
 8. Two-months note ; discount, $ 14.70 ; rate, 7%. Find 
 face. 
 
 9. For what amount must a 60-day s note be drawn so 
 that the proceeds will be $ 300 when the rate of discount is 
 8 per cent ? 
 
 10. A note for $ 120 was discounted at a bank March 15, 
 1905. What is the date of the maturity of the note, the 
 proceeds being $119.52 and the rate of discount 6 per cent ? 
 
 11. Find the proceeds of a 6-months note for $875 drawn 
 Jan. 2, 1906, and discounted at 6 per cent 35 days after that 
 date. 
 
 12. A merchant bought 300 barrels of flour at $4.75 per 
 barrel, cash, and sold it for $5 per barrel, taking in payment 
 a 60-days note for the amount. If he has the note dis- 
 counted immediately at a bank, at 7 per cent, what does 
 he gain by the transaction ? 
 
 13. What will be the face of a 30-day s note, the proceeds 
 of which when discounted at a bank at 6% will pay for 
 3000 bushels corn at 49|^ per bushel ? 
 
 14. The proceeds of a note for $1200, due March 15, 
 1904, and discounted at 6%, were $1184.80. When was it 
 discounted ? 
 
Interest. 271 
 
 INTEREST BY ALIQUOT PARTS. 
 
 348. Written Exercises. 
 
 1. Find the interest on $387.45, for 2 yr. 8 mo. 18 da., 
 
 at7 %- $387.45 x .07. 
 
 $27.1215 interest for 1 yr. 
 
 27.1215 interest for 1 yr. 
 
 6 mo. = \ yr. 13.5607 interest for 6 mo. 
 
 2 mo. = \ (of 6 mo.) 4.5202 interest for 2 mo. 
 15 da. = | (of 2 mo.) 1.1301 interest for 15 da. 
 
 3 da. = $ (of 15 da.) .2260 interest for 3 da. 
 
 Ans. $73.68 interest for 2 yr. 8 mo. 18 da. 
 
 2. Find the interest on $432.90, at 6%, for 1 yr. 7 mo. 
 
 12 da " $432.90 x .06. 
 
 interest for 1 yr. 
 6 mo. = \ yr. interest for 6 mo. 
 
 1 mo. = \ (of 6 mo. ) interest for 1 mo. 
 10 da. = \ (of 1 mo.) interest for 10 da. 
 
 2 da. = \ (of 10 da.) interest for 2 da. 
 
 interest for 1 yr. 7 mo. 12 da. 
 
 3. Find the amountf of $874.16, at 5%, for 1 yr. 9 mo. 
 
 4 da * $874.16 principal. 
 
 5 % = ^ • 43.708 interest for 1 yr. 
 
 6 mo. = \ yr. interest for 6 mo. 
 
 3 mo. = \ (of 6 mo.) interest for 3 mo. 
 3 da. = ^ (of 3 mo.) interest for 3 da. 
 1 da. —\ (of 3 da.) interest for 1 da. 
 
 amount for 1 yr. 9 mo. 4 da. 
 
 4. What is the amount of $95.72, for 3 yr. 6 mo. 20 da., 
 
 at5 % ? $95.72 principal. 
 
 10 % = ^ 9. 572 interest for 2 yr. 
 
 1 yr. = \ (of 2 yr.) 4.786 interest for 1 yr. 
 
 6 mo. = \ yr. interest for 6 mo. 
 
 20 da. = ? of 6 mo. ___^___ interest for 20 da. 
 
 amount for 3 yr. 6 mo. 20 da. 
 
0.J2 Chapter Five. 
 
 5. Interest of $1806.45, at 4%, for 1 yr. 7 ino. 25 da, 
 
 1 yr., 6 mo., 1 mo., 15 da., 5 da., 5 da. 
 
 6. Interest for 10 mo. 29 da., at 4%, on $380.40. 
 
 $380.40 x .04. 
 $15.2160 interest for 1 yr. 
 1 mo. = ^ yr. interest for 1 mo. ) deduct from interest 
 
 1 da. = ^ mo. interest for 1 da. ) for 1 yr. 
 
 interest for 10 mo. 29 da. 
 
 7. Amount, at 6%, of $125.73, for 2 yr. 10 mo.. 4 da. 
 
 8. Interest on $84.66, at 7%, for 1 yr. 4 mo. 12 da. 
 
 9. Interest, at 5%, for 4 yr. 2 mo. 7 da., on $250. 
 
 10. Amount of $1000, at 6%, for 33 days. 
 
 349. When the time is less than a year, the following 
 facts should be remembered: 
 
 6% for a year is 1 per cent for 60 days. 
 5% for a year is 1 per cent for 72 days. 
 4£% for a year is 1 per cent lor ? days. 
 4% for a year is 1 per cent for ? days. 
 
 11. Find the interest for 81 days, at 5%, on $876.40. 
 Since 5% for a year is 1% for 72 days, we have : — 
 
 72 days' interest is 1% of principal, or $8,764 
 9 days' interest is $ of 72 days, or 1.095 
 
 $9.86 interest for 81 days. 
 
 12. Amount of $954, at 4%, for 4 mo. 10 da. 
 
 Principal $954.00 
 3 months' interest = 1% 9.54 
 1 mo. = $ (of 3 mo.) 3J8 
 
 10 da. = |(of 1 mo.) 
 
 amount for 4 mo. 10 da. 
 
Interest. 273 
 
 13. Interest of $1874, at 4£%, for 93 da. 
 
 80 days = 1% 
 10 days 
 
 2 days 
 
 1 day 
 
 14. Interest of $753.20, at 5%, for 158 days. 
 
 72 da., 72 da., 12 da., 2 da. 
 
 15. Amount of $1234.50, for 193 days, at 6%. 
 
 60 da., 120 da., 12 da., 1 da. 
 
 16. Find the proceeds of a 90-day s note, for $873.60, 
 
 at 6 ^* Face $873.60 
 
 60 da. 8.7361 ^ J 
 
 y Deduct. 
 30 da. 4. 368 J 
 
 $860.50 proceeds. 
 
 17. Find the discount on a 3-months note, for $1596, 
 
 at 6%. 
 
 18. What are the proceeds of a 6-months note, for $ 785, 
 discounted at 6%. 
 
 19. Find the interest on $484.40, for 1 yr. 3 mo. 17 da., 
 
 at 7%. 
 
 20. Find the amount of $ 683, for 3 yr. 4 mo. 11 da., at 
 
 350. N.B. — Do not use unnecessary figures. 
 
 21. Principal, $360; 5% ; 3 yr. 7 mo. 18 da. Interest? 
 
 22. Principal, $ 613 ; 4|% ; 157 da. Amount ? 
 
 23. Principal, $1774; 3|% ; 17 mo. 23 da. Interest? 
 
 24. Principal, $ 875; 6% ; 2 yr. 3 mo. 1 da. Amount? 
 
 25. Principal, $976; 7%; 325 da. Interest? 
 
274 Chapter Five. 
 
 351. By the time of a note is meant the number of days, etc., for 
 which it is drawn. In these four examples the note is discounted the 
 day it is made. 
 
 26. Face of note, $ 254 ; time, 30 days ; 7%. Proceeds ? 
 
 27. Face of note, $515; time, 6 months; 5%. Dis- 
 count ? 
 
 28. Face of note, $493; time, 60 days; 8%. Proceeds? 
 
 29. Face of note, $717; time, 15 days; 6^%. Discount? 
 
 352. Find the exact number of days. Take 360 days to year. 
 
 30. Principal, $1836.50; 6%; Jan. 2 to Dec. 1. Amount? 
 
 31. Principal, $1295.70; 7%; March 8 to April 9. 
 Interest ? 
 
 32. Principal, $ 765.90; 4% ; Oct. 1 to Dec. 17. Interest? 
 
 33. Principal, $275.84; 51%; May 9 to July 3. Amount? 
 
 353. By the term of a note is meant the number of days it has to 
 run after it has been discounted. 
 
 34. Face of note, $100; term, 60 days; 7%. Discount? 
 
 35. Face of note, $ 200 ; term, 90 days; 6£%. Proceeds? 
 
 36. Face of note, $300; term, 24 days; 5J%. Discount? 
 
 37. Face of note, $400; term, 117 days; 8%. Proceeds? 
 
 354. In examples 38-41, inclusive, find the time by compound sub- 
 traction. 
 
 38. Principal, $25.83; 6%; Jan. 14, 1902, to Sept. 5, 
 1904. Interest ? 
 
 39. Principal, $47.96; 5%; Feb. 6, 1903, to Aug. 1, 
 
 1906. Amount ? 
 
 40. Principal, $85.30; 7% ; March 25, 1904, to Jan. 13, 
 
 1907. Interest ? 
 
 41. Principal, $75; 4%; April 15, 1900, to Feb. 6, 
 1907. Amount? 
 
Review. 275 
 
 REVIEW. 
 355. Oral Problems. 
 
 1. Out of 500 pupils, 50 are absent. What is the per 
 cent of attendance ? 
 
 2. A can do a piece of work in 4 days ; B can do it in 
 
 4 days. In what time can A and B do it, if they work 
 together ? 
 
 3. What is the interest of $ 1500, for 60 days, at 6% ? 
 
 4. In a certain class \ of the pupils are under 10 years, 
 -J- of them are between 10 and 12, and the rest are over 12. 
 What per cent are over 12 years ? 
 
 5. If a bushel of English walnuts costs $ 1.60, what will 
 6 quarts cost ? 
 
 6. A man put 5 gal. 2 qt. of syrup into bottles holding 
 2 quarts each. How many bottles did it require ? 
 
 7. If I of a yard of cloth costs £ of a dollar, what will J 
 of a yard cost ? 
 
 8. If 9 pounds of sugar cost 48 ^, what will 12 pounds 
 cost? 
 
 9. What is the difference between f of 8§ and f of 4 J ? 
 
 10. How many eggs, at the rate of 15 for 25 cents, can be 
 bought for 60^? 
 
 11. A merchant's receipts are $1200; his gain is 20 per 
 cent. What part of his receipts is profit ? 
 
 12. If 3 men earn $72 in 8 days, how many dollars will 
 
 5 men earn in 11 days ? 
 
 13. If a dealer loses 25% by selling a horse for $ 225, 
 what per cent would he gain or lose by selling the horse for 
 $325? 
 
 14. If A can do a piece of work in 2 days, B in 3 days, 
 and C in 4 days, in what time can they do it, working to- 
 gether ? 
 
276 Chapter Five. 
 
 356. Written Problems. 
 
 1. A man sold 18 barrels sugar, each containing 306 
 pounds ; 21 barrels, each containing; 297 pounds ; 5 barrels, 
 each containing 291 pounds. What is the average weight 
 per barrel ? 
 
 2. Three men engage in a business venture. One fur- 
 nishes if 3000, another furnishes $ 5000, a third furnishes 
 $ 4000. They gain $ 1800. What is each one's share of 
 the profit ? 
 
 What part of the money did the first furnish ? What part of the 
 profit should he receive ? 
 
 3. Three ounces is what per cent of 5 pounds ? 
 
 4. What is the product of \ of f of 15|. State the re- 
 sult in decimals. 
 
 5. What is 87£% of $896? $896 is 87£ % of what 
 sum? 
 
 6. What number is that which, diminished by 2 J, will 
 leave 2^- 
 
 7. How long will 200 pounds flour last 18 persons if 
 each person is allowed If pounds per day ? 
 
 8. If f of I of a ship cost $ 84,000, what is £ of it worth ? 
 
 9. The dividend was $ 4689.036, the quotient .027, what 
 was the divisor? 
 
 10. Harry Hedge earns $ 12 a week. He pays $ 4.25 for 
 board, $0,625 for car fare, $0,375 for library fees, and 
 $ 4.875 for other expenses. In how many weeks would he 
 save $ 97.50. 
 
 11. For how long must $450 be at interest, at five per 
 cent per annum, to amount to $ 481.62 ? 
 
 12. Divide 320 acres of land among A, B, and C, so that 
 A shall have 15 acres more than B, and C shall have 27 
 acres more than B. 
 
Denominate Numbers. 277 
 
 DENOMINATE NUMBERS. 
 
 357. Inductive Exercises. 
 
 1. Change 1 yd. 1 ft. to inches. 
 
 2. Change 1 yd. 1 ft. 1 in. to inches. 
 
 3. Change 49 inches to yards, feet, etc 
 
 4. Change 49 pints to gallons, etc. 
 
 5. Add 4 lb. 8 oz. and 4 lb. 8 oz. 
 
 6. From 9 pounds take 4 lb. 8 oz. 
 
 7. Multiply 4 lb. 8 oz. by 2. 
 
 8. Divide 9 pounds by 2. 
 
 9. Divide 9 pounds by 4 lb. 8 oz. 
 
 10. How many inches in J yd. ? 
 
 11. How many feet and inches in .75 yd. ? 
 
 12. 75 per cent of a yard = ? 
 
 13. What fraction of a yard is 27 inches ? 
 
 14. Change 2 ft. 3 in. to the decimal of a yard. 
 
 15. 1 ft. 6 in. is what per cent of 2 feet? 
 
 16. Multiply 9 yd. 18 in. by 7. 
 
 17. From 18 lb. 6 oz. take 9 lb. 12 oz. 
 
 358. Troy Weight. 
 
 24 grains (gr.) = 1 pennny weight (pwt.) 
 20 pennyweight = 1 ounce (oz.) 
 12 ounces = 1 pound (lb. ) 
 
 Troy weight is used in weighing gold, silver, precious stones, etc. 
 
 359. English Money. 
 
 12 pence (<f.) =1 shilling (s.) 
 20 shillings = 1 pound (£) 
 
 A farthing is a quarter of a penny. 
 
278 Chapter Five. 
 
 REDUCTION DESCENDING. 
 
 360. Change 43 yd. 2 in. to inches. 
 
 Write 43 yd. ft. 2 in., inserting the 3 ft. 12 in. 
 
 missing denomination, feet. Above ft. 43 yd. ' f t. 2 in. 
 write the number of feet in a yard, and T20 ft. 1550 in 
 
 above the 2 in. the number of inches in a A 1 ~~ n . 
 
 foot. Since there are 3 feet in a yard, in 
 
 43 yards there are 43 times 3 feet, or 129 feet. This is written in the 
 column of feet. Since there are 12 inches in a foot, in* 129 feet there are 
 129 times 12 inches, or 1548 inches. Add 2 inches, making 1550 inches, 
 which is written in the column of inches, and cancel 129 feet. 
 
 In working this example, 3 and 12 are used as the multipliers instead 
 of 43 and 129. At the time the 9 of 129 is multiplied by 12, the 2 is 
 added in, the pupil saying 12 nines are 108, and 2 are 110, writing the 
 ; 12 twos are 24, and 1 1 are 35, writing the 5 ; etc. 
 
 361. Written Exercises. 
 Change : 
 
 1. 4 yards 2 feet 8 inches to inches. 
 
 2. 2 miles 46 rods 3 yards to yards. 
 
 3. 3 pecks 5 quarts 1 pint to pints. 
 
 4. 6 bushels 3 pecks 6 quarts to quarts. 
 
 5. 2 gallons 3 quarts 1 pint to pints. 
 
 6. 7 gallons 1 quart 1 pint to pints. 
 
 7. 4 ounces 12 pennyweights 3 grains to grains, 
 
 8. 2 pounds 16 pennyweights 14 grains to grains. 
 
 9. 6 pounds 9 shillings 7 pence to pence. 
 
 10. 8 pounds 18 shillings 4 pence to pence. 
 
 11. 3 wk. 4 da. 13 hr. to hours. 
 
 12. I of a week to hours. 
 
 13. -J of a mile to yards. 
 
 14. .25 of a rod to inches. 
 
Denominate Numbers. 279 
 
 REDUCTION ASCENDING. 
 
 362. Change 1550 inches to yards, feet, etc. 
 
 Above 1550 in. write 12 in., the num- 3 ft. 12 in. 
 
 ber in a foot. Dividing 1550 inches by 729 ft. 1550 in. 
 
 12 inches we obtain the quotient 129, the 43 y^ ft. 2 in. 
 number of feet, and 2 inches remainder. A . , . 
 
 w * *u > a • a 1 m ^ ns - 43 yd- 2 in. 
 
 Write the remainder in the column of ■* 
 
 inches and 129 ft. to the left of 1550 in. Reduce 129 ft. to yards, 
 
 writing the result, 43 yd., as shown above, and cancelling 129 ft. 
 
 363. "Written Exercises. 
 Change : 
 
 1. 4530 feet to rods, yards, etc. 
 
 2. 6324 yards to miles, rods, etc. 
 
 3. 244 pints to bushels, pecks, etc. 
 
 4. 467 quarts to bushels, pecks, etc. 
 
 5. 923 pints to gallons, quarts, etc. • 
 
 6. 785 pints to gallons, quarts, etc. 
 
 7. 543 pennyweights to pounds, etc. 
 
 8. 175 grains to pennyweights, etc. 
 
 9. 625 pence to pounds, shillings, etc. 
 
 10. 836 shillings to pounds, etc. 
 
 11. 8423 min. to days, hours, etc. 
 
 12. 2348 inches to yards, etc. 
 
 ADDITION OF DENOMINATE NUMBERS. 
 
 8 in. + 10 in. + 5 in. = 23 in. = 1 f t 
 11 in. Write 11 in. and carry 1 ft. 
 
 1 f t. + 1 ft. + 2 f t. = 4 f t. = 1 yd. 1 ft. 
 Write 1 ft. and carry 1 yd. 
 16 yd. 1ft. 11 in. Ana. 1 yd. +4 yd. + 11 yd. = 16 yd. Write 
 
 16 yd. 
 
 364. 
 
 Find the sun 
 
 Hyd. 
 
 2 ft. 
 
 8 in. 
 
 
 1ft. 
 
 10 in. 
 
 4 yd. 
 
 Oft. 
 
 5 in. 
 
28o Chapter Five. 
 
 365. Written Exercises. 
 
 Find sums : 
 
 1. 8 mi. 44 rd. 3 yd. 4. 243 gal. 2 qt. 1 pt. 
 
 6 mi. 298 rd. 4 yd. 168 gal. 3 qt. 1 pt. 
 
 67 rd. 1 yd. 1 qt. 1 pt. 
 
 27 rd. 
 
 3 yd. 
 
 2 ft. 
 
 5. 4 1b. 
 
 10 oz. 
 
 14 pwt. 
 
 3rd. 
 
 2 yd. 
 
 1ft. 
 
 3 1b. 
 
 9oz. 
 
 16 pwt. 
 
 78 rd. 
 
 4 yd. 
 
 2 ft. 
 
 lib. 
 
 11 oz. 
 
 7 pwt. 
 
 8bu. 
 
 3pk. 
 
 5qt. 
 
 6. 8 oz. 
 
 9 pwt. 
 
 21 gr. 
 
 16 bu. 
 
 2pk. 
 
 3qt. 
 
 3oz. 
 
 11 pwt. 
 
 6gr. 
 
 4bu. 
 
 3pk. 
 
 7qt. 
 
 
 17 pwt. 
 
 23 gr. 
 
 •3. 
 
 SUBTRACTION OF DENOMINATE NUMBERS. 
 
 From 35 yd. 1 ft. 4 in. Since 8 in. is greater 
 
 Take 19 yd. 2 ft. 8 in. than 4 in -> w © must use 
 
 15 yd. 1 ft. 8 in. Ans. l ft ' 4 ,n - or 16 in " M 
 
 the minuend. 16 in. — 
 
 8 in. = 8 in. As the minuend now contains ft., 1 yd. is taken from 
 
 35 yd. Changing the yard to 3 ft., and deducting 2 ft., leaves 1 ft, 
 
 U yd. - 19 yd. = 15 yd. 
 
 867. "Written Exercises. 
 Find differences : 
 
 1. 
 
 183 rd. 4 yd. 
 68 rd. 5 yd. 
 
 1ft. 
 2 ft. 
 
 2. 
 
 91 mi. 83 rd. 
 26 mi. 122 rd. 
 
 2 yd. 
 4 yd. 
 
 3, 
 
 3 pk. 1 qt. 
 1 pk. 4 qt. 1 
 
 pt. 
 
 4. 29 gal. 2 qt. 
 
 28 gal. 3 qt. 1 pt. 
 
 5. 
 
 6. 
 
 8 1b. 
 6 1b. 
 
 3 oz. 8 pwt. 
 8 oz. 10 pwt. 
 
 £24 
 £3 
 
 6s. 3d. 
 9s. Sd. 
 
Denominate Numbers. 281 
 
 MULTIPLICATION OF DENOMINATE NUMBERS. 
 
 368. Multiply 34 yd. 2 ft. 9 in. by 7. 
 
 244 yd. 1 ft. 3 in. 
 
 7 times 9 in. = 63 in. = 5 f t. 3 in. Write 3 in. 7 times 2 ft. = 
 14 ft. Carry 5 ft., making 19 ft., or 6 yd. 1 ft. Write 1 ft. Multiply 
 34 yd. by 7, adding in 6 yd. when the 4 is multiplied. 
 
 369. Written Exercises. 
 Find products : 
 
 1. 6 mi. 24 rd. 4 yd. by 9. 6. 3 gal. 1 qt. 1 pt. by 32. 
 
 2. 36 rd. 4 yd. 2 ft. by 12. 7. 8 lb. 4 oz. 12 pwt. by 10. 
 
 3. 24 bu. 3 pk. 6 qt. by 14. 8. 16 oz. 12 pwt. 20 gr. by 4. 
 
 4. 2 pk. 3 qt. 1 pt. by 36. 9. £4 12s. 6d. by 20. 
 
 5. 11 gal. 2 qt. 1 pt. by 8. 10. £28 16s. 9d. by 7. 
 
 DIVISION OF DENOMINATE NUMBERS. 
 
 370. Divide 244 yd. 1 ft. 3 in . by 7. 
 
 34 yd. 2 ft. 9 in. Ans. 
 
 The quotient of 244 yd. divided by 7 is 34 yd., with a remainder of 
 6 yd. Reducing 6 yd. to ft. and adding in 1 ft., the dividend is 19 ft. 
 19 ft. 4- 7 = 2 ft. with 5 ft. remainder. 5 ft. 3 in. = 63 in. 63 in. -*- ? 
 = 9 in. 
 
 371. "Written Exercises. 
 Find quotients : 
 
 1. 44 mi. 124 rd. 2 yd. by 8. 
 
 2. 14 yd. 1 ft. 9 in. by 21. 
 
 3. 37 bu. 1 pk. 2 qt. by 6. 
 
 4. 12 bu. 3 pk. 6 qt. by 18. 
 
 5. 7 gal. 3 qt. 1 pt. by 3. 
 
282 Chapter Five. 
 
 6. 3 gal. 1 pt. by 5. 
 
 7. 28 lb. 10 oz. 16 pwt. by 24. 
 
 8. 4 oz. 10 pwt. 3 gr. by 9. 
 
 9. £ 24 l»7s. 4d. by 16. 
 10. £ 3 7s. 6d. by 10. 
 
 372. Divide 244 yd. 1 ft. 3 in. by 34 yd. 2 ft. 9 in. 
 
 In dividing one concrete number by another concrete number, the 
 divisor and the dividend must be of the same denomination. Thus, 
 to divide $ 2 by 25^, we change the dividend to cents, 200 cents -f- 25 
 cents, or the divisor to dollars, $ 2 -f- $ \. • The quotient is 8, an abstract 
 number ; that is, 25 cents is contained in 200 cents 8 times. 
 
 244 yd. 1 ft. 3 in. = 8799 in, ; 34 yd. 2 ft. 9 in. = 1257 in. 8799 in. 
 -T- 1257 in. = 7, Ans. 
 
 The result would be the same if we divided 733£ ft. by 104£ ft., or 
 244& yd. by 34H yd. 
 
 373. Written Exercises. 
 Find quotients : 
 
 1. 4 mi. 36 rd. 1 yd. by 6 rd. 3 yd. 
 
 2. 88 rd. 2 yd. 2 ft. by 8 rd. 4 yd. 2 ft 
 
 3. 21 bu. 2 pk. 4 qt. by 1 bu. 3 pk. 4 qt 
 
 4. 15 bu. 1 pk. by 3 pk. 6 qt. 1 pt. 
 
 5. 60 gal. 1 pt. by 4 gal. 2 qt. 1 pt. 
 
 6. 16 gal. 3 qt. 1 pt. by 2 qt. 1 pt. 
 
 7. 17 lb. 11 oz. 10 pwt. by 8 lb. 11 oz. 15 pwt. 
 
 8. 1 lb. 2 oz. 18 pwt. by 3 oz. 14 pwt. 12 gr. 
 
 9. £24 16s. 8d by £18 12s. 6<f. 
 10. £2 14s. 3d. by £ 8 2s. 90. 
 
Denominate Numbers. 283 
 
 374. Oral Problems. 
 
 1. What will be the cost of 3 lb. 7 oz. of tea, at 64^ per 
 pound? 
 
 2. How many feet in 2\ rods ? 
 
 3. At 37-^ per peck, what shall I receive for 4 bushels 
 of potatoes ? 
 
 4. What will be the cost of a ton of hay at 97 \$ per 
 cwt. ? 
 
 5. If slate pencils cost 2 mills each, how many can be 
 bought for $4? 
 
 6. At $5.00 per ton, how many pounds of coal can be 
 bought for Iff 
 
 7. Find the cost of 3 T. 480 lb. coal at $5 per ton. 
 
 8. At $5 per ton, how many tons and pounds of coal 
 can I buy for $10.80? 
 
 9. Find the cost of 4 yd. 1 ft. of ribbon, when 2 yd. 2 ft. 
 cost 40 cents. 
 
 10. In 2| pecks, how many quarts ? 
 
 11. How many hours in | of a day ? 
 
 12. 1.25 pecks are how many quarts ? 
 
 13. At $12 per ounce, what is f of a pound of gold 
 worth ? 
 
 14. How many feet in a quarter of a mile ? 
 
 15. How many tablespoons, each weighing 2 ounces, can 
 be made from 2 lb. 10 oz. of silver ? 
 
 375. Written Problems. 
 
 1. What will be the cost of 150 yards silk at 3/6 per 
 yard ? 3/5 _ 3s ^ t rea( i t h ree an( j sixpence. 
 
 2. If £ 1 = $ 4.8665, what will be the cost in U. S. money 
 of 75 books at 18 pence each ? 
 
284 Chapter Five. 
 
 3. A merchant sells 37 coats at £ 3 5s. each, less 10%. 
 What is the amount of his bill in English money ? 
 
 4. Find 25% of £ 183 14s. 8d. 
 
 5. A silver dollar weighs 412J grains. How many ounces 
 of pure silver are there in 1000 silver dollars if the coin is 
 y 9 ^ pure silver ? 
 
 6. The wheels of an engine being 16 ft. 8 in. in circum- 
 ference, and the number of revolutions 150 per minute, how 
 far does it go in an hour ? Give answer in miles and rods. 
 
 7. What fractional part of 30 rd. 5 yd. 1 ft. is 8 rd. 4 yd. 
 2ft.? 
 
 8. What decimal part of a mile is 39.27 yd. ? 
 
 9. 3 bu. 1 pk. 5 qt. is what per cent of 20 bu. 1 pk. 6 qt. ? 
 
 10. If a letter-carrier in delivering letters takes 47,520 
 •steps in a day, each step averaging 20 inches, how many 
 miles does he walk ? 
 
 11. 43 gal. 3 qt. 1 pt. alcohol are sold for $ 70.20. What 
 is the price per gallon ? 
 
 12. After taking out 15% of the grain in a bin, there re- 
 mained 40 bu. 3 pk. 5 qt How many bushels were there at 
 first? 
 
 13. A merchant bought 51 tons 17 cwt. 3 qr. 25 lb. of 
 wool, and sold 27 tons 4 cwt. 2 qr. 27 lb. Of the remainder, 
 one-half was lost by fire. How much had he left ? 
 
 28 lb. = 1 quarter ; 4 quarters = 1 cwt. 
 
 14. An invoice of wool weighs 32 tons 17 cwt. 2 qr. 11 lb. 
 State the value in £ s. d. f at lOd. sterling per pound. 
 
 1 ton = 2240 lb. 
 
 15. How many minutes in February, 1904 ? 
 
Denominate Numbers. 285 
 
 16. If a locomotive runs 25 mi. 48 rd. in 50 minutes, 
 how far will it run in 12 hours ? 
 
 Give answer in miles and decimals of a mile. 
 
 17. I wish to put 111 bu. 2 pk. 4 qt. of grain into 47 
 bags. What quantity must each contain ? 
 
 18. If a river current carries a raft of lumber at the rate 
 of 4 miles 180 rods per hour, how long will it take the raft 
 to float 365 miles? 
 
 19. Bought 28,500 pounds of hay at $ 12^ a ton, and sold 
 it at $ 0.87^ per hundredweight. What was the gain ? 
 
 376. 1 pound Troy = 5760 grains. 
 
 1 pound Apothecaries' = 5760 grains. 
 
 1 pound Avoirdupois = 7000 grains. 
 How many grains in a Troy ounce ? In an Avoirdupois ounce ? 
 
 1. Find the value of a dozen silver spoons, each weigh- 
 ing 3 oz. 5 pwt, at $ 1.20 per oz. 
 
 2. A gold chain weighs 384 grains. What is its cost at 
 $1.15 per pwt? 
 
 3. Add 4 lb. 6 oz. 18 gr., 5 oz. 9 pwt., 3 lb. 20 gr., and 
 9 lb. 11 oz. 15 pwt. 5 gr. 
 
 4. How many spoons, each weighing 2 oz. 18 pwt., can 
 be made from 5 lb. 9 oz. 12 pwt. silver ? 
 
 5 . What fraction of a pound Avoirdupois is a pound Troy? 
 What per cent of an ounce Avoirdupois is a Troy ounce ? 
 
 6. What is the value, at $ 1.60 per oz. Troy, of a silver 
 pitcher weighing 4 lb. 8 oz. Avoirdupois ? 
 
 7. At 60^ per ounce, what is the value of the silver con- 
 tained in a half-dollar, which weighs 192.9 grains, ^ being 
 pure silver? 
 
 8. What per cent of a lb. Avoirdupois is a Troy pound ? 
 
286 Chapter Five. 
 
 MISCELLANEOUS. 
 
 377. Oral Problems. 
 
 1. If 4 books cost $ 1.25, what will a dozen cost ? 
 
 2. If 3 pounds of sugar cost 16|-^, what will be the cost 
 of 50 pounds? ip 0undcosts6 ^,etc. 
 
 3. If 48 pounds of tea cost $ 20, what will 12 pounds 
 COSt ? 12 pounds will cost \ of $ 20. * 
 
 4. Bought 17 yards of cloth for $ 30. How many yards 
 could I have bought for $ 90 ? 
 
 5. If 36 men do a piece of work in 105 days, how long 
 will it take 72 men to do it ? 
 
 6. If 7 railway trucks weigh 14 tons, how much would 
 29 trucks weigh ? 
 
 7. How long will it take 8 horses to plough a field, if 3 
 horses can do it in 8 days ? 
 
 8. What is the height of a steeple that casts a shadow 
 of 300 feet, if an 8 foot pole casts a shadow of 12 feet. 
 
 9. If 18 men mow 90 acres of grass in 5 days, how many 
 acres will 36 men mow in 5 days ? In 10 days ? 
 
 10. If 60 yd. carpet } yard wide will cover a floor, how 
 many yards f yard wide will be required ? 
 
 378. Written Problems. 
 
 1. A piece of cloth, measured with a yard measure that 
 is 1 inch too short, appears to be 25 yards long. What is 
 its true length ? 
 
 2. Exchanged 40 yd. muslin, worth 10J^ per yard, for 
 15 yards linen. What is the value of the linen per yard ? 
 
Miscellaneous. 287 
 
 3. If 3 men or 6 women can do a piece of work in 56 
 days, in what time will 1 man and 2 women working together 
 doit? 
 
 4. If 5 men can do as much- in a day as 8 boys, how 
 long will it take 32 boys to finish a piece of work which 15 
 men can do in 12 days ? 
 
 5. If $ 100 gain $ 4 in 1 year, what will $350 gain in 
 Z\ years ? 
 
 6. If 48 horses in 10 days consume 180 bushels oats, 
 how many bushels will 32 horses consume in 10 days ? In 
 12 days ? In 15 days ? 
 
 7. If 5 men mow 45 acres of grass in 6 days, in how 
 many days will 12 men mow 90 acres ? 
 
 379. If 5 men mow 45 acres in 6 days, 
 
 1 man will- mow 45 acres in 6 days x 5. 
 
 1 man will mow 1 acre in 6 days x 5 . 
 
 45 
 
 12 men will mow 1 acre in 6 days x 5 . 
 
 45 x 12 
 
 12 men will mow 90 acres in 6 days x 5 x 90 . 
 
 45 x 12 
 
 Cancelling, Mays x 5* ?0 = 5 days, Ans. 
 
 I 
 
 380. In practice, the work is somewhat shortened. Since the 
 number of days is required, we write the given number of days last, 
 with a line underneath. 
 
 5 men mow 45 acres . days. 
 1 man mows 1 acre I " x 5 x rr. 
 12 men mow 90 acres J 45 x 12 
 
 If 5 men do the work in a certain time, 1 man will require 5 times 
 as many days. We place 5 in the num«rator (as a multiplier). To 
 
288 Chapter Five. 
 
 cut 1 acre, he will take ^ of the time required to cut 45 acres. Place 
 45 in the denominator (as a divisor). 
 
 12 men will take T ! 2 of the time 1 man requires. Place 12 in the 
 denominator. To cut 90 acres will require 90 times as long. Place 90 
 in the numerator. 
 
 8. If 12 horses eat 60 bushels of oats in 6 days, how 
 many bushels will 24 horses eat in 3 days ? 
 
 Make bushels the last term. 
 
 12 horses in 6 days eat 1 bu# 
 1 horse in 1 day eats I 60_ 
 24 horses in 3 days eat J 
 
 9. If 24 men use 240 pounds of beef in 2 weeks, how 
 many pounds will 18 men use in 8 weeks ? 
 
 24 men in 2 weeks use 240 lb. 
 
 10. If 6 printers can print 1656 books in 9 days, how 
 many books will "15 printers print in 10 days ? 
 
 11. How much will it cost to feed 520 sheep for 36 days, 
 if it costs $ 128 to feed 160 sheep 48 days ? 
 
 12. In what time will 8 masons build a wall 84 feet long, 
 working 1.0 hours a day, if 12 masons build a wall 96 feet 
 long in 8 days, working 8 hours a day ? 
 
 13. How much money must I lend for 1 year and 
 3 months, when the rate of interest is 5 per cent, in return 
 for $ 60 lent me for 9 months, which I borrowed at 4 per 
 cent? 
 
 14. If 27 men build 54 rods of wall in 6 days, how many 
 rods will 32 men build in 9 days ? 
 
 15. If 50 men can do a piece of work in 90 days, working 
 8 hours a day, in how many days will 72 men do it, working 
 10 hours a day ? 
 
Miscellaneous. 289 
 
 16. If f 350 earns $ 42 interest in 3 years, how much 
 will $ 225 earn in 5 years ? 
 
 17. If a wall 34 feet high could be built by 68 men in 
 15 days, how many men could build a wall 32 feet high in 
 8 days ? 
 
 18. If a ship's crew of 500 men have provisions to serve 
 for 48 days, at the rate of 27 ounces a day for each man, 
 how many men will the same provisions serve for 60 days, 
 allowing each man 30 ounces a day ? 
 
 19. How many hours a day must 9 men work so that 
 they may do as much in 16 days as 12 men can do in 15 days 
 of 8 hours each ? 
 
 20. If 30^ is paid for 6 lb. 14 oz. of bread, when wheat 
 is 85^ per bushel, what should be paid for 23 lb. 12 oz., 
 when wheat is 99^ per bushel ? 
 
 21. If 3 men can do as much work as 7 boys, how long 
 will it take 28 boys to do as much work as 16 men can do in 
 24 days ? 
 
 22. A crew of 16 men have provisions for 36 days, allow- 
 ing 20 ounces to each man per day. After sailing 10 days 
 they pick up 10 shipwrecked sailors. How long will the 
 provisions then last at the rate of 16 ounces per man ? 
 
 23. If A can do a piece of work in 4 days, and B can do 
 the same work in 5 days, how many days will it take both, 
 working together ? 
 
 A can do \ of the work in one day, and B \ of it. Together they 
 can do \ + £, or ^ in one day. If they do 9 twentieths in one day, 
 to do 20 twentieths, or the whole work, will require (20 -?- 9) days, or 
 2| days. 
 
 24. If one man can do a piece of work in 24 days, and 
 another man can do it in 48 days, how long will it take both, 
 working together ? 
 
290 Chapter Five. 
 
 APPROXIMATIONS. 
 
 Pupils should be drilled to take a broader view of their work, by 
 estimating the probable result before taking a pencil. In this way 
 many absurd answers might be avoided. 
 
 381. Give approximate answers at sight : 
 
 1. Find the interest of $ 150, at 4%, from Jan. 1, 1903, 
 to Dec. 30, 1905. (Nearly 3 years.) 
 
 2. What is the weight, at 57 J lb. per cubic feet, of a 
 cake of ice 4 ft. by 2 ft. by 1 J ft ? (Nearly 60 lb. per cubic 
 feet.) 
 
 3. Find the amount of goods sold, the commission at 
 2£% being $ 11.75. (About 3%.) 
 
 4. What % of 497 is 249 ? 
 
 5. What % of 3ff is lift? 
 
 6. Cost of 19,987 ft. boards at % 30.05 per M ? 
 
 7. How much will be paid for 4 barrels sugar, each 
 containing 299 pounds, at 5^ per pound ? 
 
 8. 18.0327 -j- 4.5026. 
 
 9. 83«-3ff 
 
 10. 74 A. 155 sq. rd. land at $ 79 per acre ? 
 
 11. 487| is what per cent of 960 ? 
 
 12. If 17 bu. 37 lb. of corn cost $8.75, what will 52 
 bushels cost ? 
 
 13. About how many cords of wood in a pile 25 feet long, 
 4 feet wide, 5 feet high ? 
 
 14. How many bushels (1\ cu. ft.) can be placed in a bin 
 6 feet long, 5 feet wide, 4 feet high ? 
 
 15. How many acres in a field 52 rods long, 30 rods wide? 
 
 16. About how many yards are there in the side of a 
 square field containing 1 acre (4840 square yards) ? 
 
Review of Simple Numbers. 291 
 
 REVIEW OF SIMPLE NUMBERS. 
 
 382. Written Exercises. 
 6748 
 
 X 427 After multiplying by 7, the pupil multiplies this 
 
 47236 latter product by 6 tens, which gives him the product 
 
 283416 by 42 tens. In this way one line is saved. 
 
 2881396 
 
 383. Find products: 
 
 1. 3,925 x 328 6. 31,265 x 164 
 
 2. 12,345 x 273 7. 5,763 x 426 
 
 3. 2,087 x 287 8. 87,093 x 486 
 
 4. 20,308 x 142 9. 6,905 x 364 
 
 5. 4,321 x 189 10. 64,271 x 357 
 3289 
 
 832 First multiply by 800, by placing the first figure 
 
 26312 of the product by 8 in the hundreds' place. Multiply 
 
 105248 tn i s Dv ^> writing the first figure in the units' place. 
 
 2736448 
 
 11. 4008 x 214 16. 6352 x 927 
 
 12. 8736 x 742 17. 2781 x 525 
 
 13. 3764 x 327 18. 9060 x 1166 
 
 14. 1087 x 848 19. 6329 x 618 
 
 15. 8319 x 416 20. 2345 x 1272 
 
 21. Multiply 6984 by 25. ± of 698400. 
 
 22. 4327x75 
 
 23. 3762 X 62%. Multiply 376,200 by f 
 
 24. 5796 x 62£ 27. 7154 x 87£ 
 
 25. 8383 x 12£ 28. 6419 x 33J 
 
 26. 3428 x 37J 29. 6208 x 66$ 
 
292 Chapter Five. 
 
 REVIEW OF FRACTIONS. 
 
 384. "Written Exercises. 
 
 
 7854 xf 
 
 9365 xf 
 
 1963£ Deduct J. 
 
 1170| Deduct f 
 
 5890J Ans. 
 
 8194f ^tis. 
 
 Multiply 6578 by 9|. 
 
 
 65,780 = 10 times number. 
 
 2,192} = i 
 
 number (deduct). 
 
 63,587J Ans. 
 
 
 385. Find products: 
 
 
 1. 176 x H 
 
 11. 4844 x9£ 
 
 2. 273 x H 
 
 12. 8960 x 8J 
 
 3. 4554 x £ 
 
 13. 3245 x 7J 
 
 4. 1001 X ff 
 
 14. 9060 x 11£ 
 
 5. 3243 x i 
 
 15. 658 x 99£ 
 
 6. 6776 x | 
 
 16. 658 x 99f 
 
 7. 2307 x £ 
 
 17. 725 x 119| 
 
 8. 7284 x J 
 
 18. 347 x 79f 
 
 9. 5631 x A 
 
 19. 418 x 89£ 
 
 10. 9657 x H 
 
 20. 543 x 49J 
 
 386. Written Exercises. 
 
 Note. — Do not use too many figures. 
 
 1. Add£,2i,f f. 
 
 2. Divide each of the following fractions by 6 : 
 
 3. Keduce J- of -jSj- of -fa of 2-J to a simple fraction. 
 
 4. 38$-21|f 40$ -18$ 
 
Review of Fractions. 293 
 
 5. What fraction of £ 1 18s. 9d. is 5s. 6d. ? 
 
 6. Multiply 24£ by f of f 
 
 7. What is the greatest common divisor of 657 and 1168 ? 
 the least common multiple of 12, 16, 20, 30 ? 
 
 8. What must be taken. from 8^ to leave 3^ ? 
 
 9. Eeduce fff.and Mf to their lowest terms. 
 
 10. Which is the greatest and which is the least, -J- of ^, -J 
 of f, and 2i of -ft? 
 
 11. What must be added to 3ft- to make 5f ? 
 
 12. Add f of a week, f of an hour, ft- of a minute. 
 
 13. How much is 9 times each of the following fractions? 
 
 ™„ o „ - ** TT' «* tr 
 
 14. 3<ty-*-fof7. 
 
 15. ft + fof ft + fof f. 
 
 16. What part of a 10-acre field is 4 A. 100 sq. rd. ? 
 
 17. What is the least number that will contain each of 
 the numbers 6, 15, 18, and 20 ? 
 
 18. What must be multiplied by 4£ to produce 16\ ? 
 
 19. What is the value of tilt? 
 
 20. What quantity must be divided by 4^ to produce 8| ? 
 
 21. Find the value of 
 
 gt±j 
 
 22. How much is i_JI of 3 da. 15 hr. 32 min. ? 
 
 23. Eeduce ft- mile to rods. 
 
 24. Add I, f, 5£. Subtract 4ft from the sum. 
 
 25. Multiply f of 5J- by 7f Divide the result by If 
 
294 Chapter Five. 
 
 REVIEW OF DECIMALS. 
 
 387. Written Exercises. 
 
 1. Express as decimals fifo, y^, and ^. 
 
 2. .395 + 86.7 + 209.0043 + .81 + 3.075 + 27. 
 
 3. Divide 34,020.072 by 5.309. 570 -s- .005 = ? 
 
 4. Multiply 80.037 by 10. Seventy-three hundred-thou- 
 sandths by one hundred. .2054 x 1000 = ? 
 
 5. Subtract 48.8067 from 53.07. .0539 x 26.08 = ? 
 
 6. The smaller of two numbers is 8.5307, and their sum 
 is 25.07. Find the larger number. 
 
 7. Express .39, 6.175, .00036, and 74.0005 as common 
 fractions (or mixed numbers). 
 
 8. Divide .826 by 100 ; 543.71 by 10,000 ; and fifty-nine 
 ten-thousandths by one thousand. 
 
 9. Find the difference between 9.84 and 38.005, and the 
 continued product of 83.09, .734, and 5.007. 
 
 10. Eeduce 6 shillings 9 pence to the decimal of a pound 
 sterling. 
 
 11. Express as decimals seven hundredths, forty-three 
 ten-thousandths, and ninety-one millionths. 
 
 12. Change y 1 ^, 8^, y^, and jfa into decimals. Find 
 their sum. 
 
 13. Express .42796 as a common fraction, and the sum of 
 ft, dhr> and. y$ Jfo- as a decimal. 
 
 14. 3.009 x .07 x .0907. 
 
 15. Divide .0075 by .15, and .00044408 by .0112. 
 
 16. Divisor, 403.6 ; quotient, 2.709. Dividend ? 
 
 17. What is the value of ° 35 * QQ56 ? 
 
 .00007 
 
 18. Change 69 rods to the decimal of a mile. 
 
 19. Change .4285 month (30 days) to days, hours, etc. 
 
Special Drills. 295 
 
 SPECIAL DRILLS. 
 388. Give sums: 
 
 1. 856 + 256 = 856 + 200 + 50 + 6 
 
 The pupil says (or thinks) only 1056, 1106, 1112. 
 
 2. 576 + 425 4. 749 + 312 6. $6.73 + $3.94 
 
 3. 685 + 599 5. 567 + 658 7. $8.27 + $4.89 
 
 Give remainders : 
 
 8. 1244 - 655 = 1244 -600-50-5 
 
 Think only 644, 594, 589. 
 
 9. 1021-576 11. 1040-312 13. $12.00 -$8.73 
 10. 1264-685 12. 1322-643 14. $11.05 -$2.69 
 
 Give products : 
 
 15. 24 x 21 = 20 times 24 + 24 = 480 + 24 
 
 Say only 480, 504. 
 
 16. 33 x 21 18. 41 x 41 20. 31 x 31 
 
 17. 22x31 19. 32x41 21. 44x21 
 
 Give sums : 
 
 22. 425 + 99 = 425 + 100-1 
 
 Say only 525, 524. 
 
 23. 576 + 99 24. 999 + 425 25. $8.68 + $4.99 
 
 Give remainders : 
 
 26. 565-99 = 565-100 + 1 
 
 Say only 465, 466. 
 
 27. 743-99 28. 1230-999 29. $12.13 -$4.99 
 
 Give products : 
 
 30. 27x99 = 100 times 27 -27 = 2700 -27 
 
 31. 36x99 32. 24x99 33. 98x99 
 
2<)6 Chapter Five. 
 
 389. Oral Eeview Problems. 
 
 1. What will be the cost of 48 yards of cloth at 87-^ per 
 yard? 
 
 2. A horse was sold for $ 80, which was J of the cost. 
 How much was lost on the horse ? 
 
 3. How many yards of carpet 27 inches wide will be 
 needed to cover a floor containing 48 square yards ? 
 
 4. Paid $3.45 for groceries, $1.50 for dry goods, and 
 99 ^ for sundries. What is the total ? 
 
 5. From a chest containing 25 J pounds of tea, 8£ pounds 
 were sold. How many pounds remain ? 
 
 6. What would be the cost of 2 bushels blueberries at 
 5^ per quart? 
 
 7. 83 J yards of cloth are divided into 9 pieces. How 
 many yards are there in each piece ? 
 
 8. I buy hardware to the amount of $6.37. I give the 
 storekeeper two $5 bills. How much change should I 
 receive ? 
 
 9. What will be the cost of 24 yards of calico at 4|^ per 
 yard? 
 
 10. What should I pay for 19 baseballs at $1.25 each? 
 
 11. At $ 1.87-J per yard, what will be the cost of 120 yards 
 of silk ? 
 
 12. For $120, how many yards of silk can I buy at 
 $1.87£ per yard? 
 
 13. What is the interest of $300, for 30 days, at 6 per 
 cent? 
 
 14. What will 18 oranges cost at 35^ per dozen ? 
 
 15. At 4f ^ per yard, how many yards of calico can I buy 
 for 95^? 
 
Review. 297 
 
 16. How many square yards are there in a field 41 yards 
 long, 42 yards wide ? 
 
 17. If I pay 15^ for 3J yards of muslin, what is the price 
 per yard ? 
 
 18. How many acres of land are there in two farms con- 
 taining, respectively, 347 and 495 acres ? 
 
 19. At 87£^ each, how many baseballs can be bought for 
 $56? 
 
 20. How much will be paid for 21 pounds butter, at 28/ 
 per pound ? 
 
 21. Paid 23/ for calico, 27/ for ribbon, and 48/ for 
 collars. What was the amount of my bill ? 
 
 22. A farmer had 95 sheep. He sold 39, and 17 died. 
 How many had he left ? 
 
 23. What will be the cost of 16 baseballs, at 49/ each ? 
 
 24. How much paint will there be in 27 casks, each con- 
 taining 75 pounds ? 
 
 25. A man divided a 429-acre farm into plots of 13 acres 
 each. How many such plots were there ? 
 
 26. There are 900 men in a certain regiment. How many 
 companies of 75 men each are in the regiment ? 
 
 27. Find the cost of 136 pounds sal-soda, at \j per lb. 
 
 28. At 19J/ per yard, what will be paid for 64 yards 
 gingham ? 
 
 29. How many square inches in a sheet of paper 10-J- 
 inches long by 4£ inches wide ? 
 
 30. If 2| yards of cloth are needed for a jacket, how 
 many jackets can be made from 18| yards? 
 
 31. How many yards around a field 96 yards long, 75 
 yards wide ? 
 
298 Chapter Five. 
 
 32. What will be the area, in square rods, of a triangle 33 
 rods base, altitude 42 rods ? 
 
 33. How many acres in 4960 square rods ? 
 
 34. How many feet in a mile ? 
 
 35. I paid $16.25 for cloth at $1.25 per yard. How 
 many yards did I buy ? 
 
 36. Half a number -f ^ of the same number = 85. What 
 is the number ? 
 
 37. I mix 4 pounds of coffee costing 20^, with 6 pounds 
 costing 25^. What is the mixture worth per pound ? 
 
 38. A tailor makes up 99 yards of cloth into trousers, 
 using 2| yards per pair. How many pairs of trousers does 
 he make ? 
 
 39. At 60^ per pound, what will be the cost of a chest of 
 tea weighing 45 pounds ? 
 
 40. A man owns a strip of land with a frontage of 576 
 feet. How many lots 18 feet front can he make ? 
 
 41. A can do a piece of work in 5 hours, B in 7 hours. 
 How long will it take both working together ? 
 
 42. At what rate will $ 300 gain $ 24 in 2 years ? 
 
 43. What sum of money will gain $30, in 2 yr. 6 mo., at 
 6%? 
 
 44. If a staff i2 feet long casts a shadow of 3 feet, what 
 is the length of a pole that casts a shadow of 27 feet at the 
 same time ? 
 
 45. If 20 men can perform a piece of work in 8 days, 
 how many men will it take to do the same work in 5 days? 
 
 46. An agent receives $8200 to invest after deducting 
 his commission of -fa of the amount invested. What is the 
 agent's commission? 
 
 47. A lot is sold for $1200, at a loss of 20 per cent 
 What part of $ 1200 is the loss ? 
 
Review. 299 
 
 390. Written Problems. 
 
 1. A rug costs $ 20. It is sold at a profit of 20%. The 
 selling price is 20% below the marked price. How much is 
 received for the rug ? What is the marked price ? 
 
 2. What price must cloth, which cost $ 2 per yard, be 
 marked so that a profit of 20 % will be made when the cloth 
 is sold at 20 % less than the marked price ? 
 
 3. A coal bin is 6 feet long and 4 feet wide. How deep 
 must it be to contain 5 tons of stove coal, if one ton occupies 
 36 cubic feet of space ? 
 
 4. A man walking at the rate of 3 mi. 96 rd. per hour 
 will walk how far in 3 hr. 16 min. ? 
 
 5. If a merchant pays 6\tf per yard for muslin, and sells 
 the same for 7\ $ per yard, what is his gain per cent ? 
 
 6. Make and solve a problem illustrating the application 
 of percentage to the finding of an agent's commission. 
 
 7. Multiply eight hundred (units) and forty-six ten- 
 thousandths by three thousand forty millionths. 
 
 8. What is the interest on $ 128.40, for 1 yr. 5 mo. 17 da. 
 at 6 per cent ? 
 
 9. A regiment of 940 men, during the war, lost 532 of 
 their number by death and 125 by desertion. What was 
 the percentage of loss in each case, and what per cent 
 remained for service ? 
 
 10. A merchant sold a lot of damaged sugar at a loss of 
 25 per cent, receiving $ 1972.65. How much did the sugar 
 cost him ? 
 
 11. What is a pile of wood 15 feet long, 10£ feet high, 
 and 12 feet wide worth, at $ 4^- per cord ? 
 
 (1 cord = 128 cu. ft.) 
 
 12. Add the greatest and the least of the three fractions 
 H> h if 5 an ^ divide the sum by the remaining fraction. 
 
joo Chapter Five. 
 
 13. Multiply 82 ten-thousandths by 7 and 5 hundredths, 
 and divide the product by 705 millionths. 
 
 14. Find the cost of 96 feet of pine lumber at $ 25 per M, 
 and 1650 laths at $ 3 per M. 
 
 15. A horse costing $ 160 is sold for $ 180. What is the 
 gain per cent ? What is the loss per cent when a horse 
 costing $ 180 is sold for $ 160 ? 
 
 16. A merchant sold 600 barrels of flour for $ 3450, at 
 a loss of 4J per cent. What did the flour cost him per 
 barrel ? 
 
 17. How long would it take a person to count a million 
 silver dollars, at the rate of 100 a minute, and working 
 8 hours a day? 
 
 18. Find the number of days from March 2, 1903, to 
 August 11, 1903. 
 
 19. Find the interest on a note for $ 250, dated Jan. 21, 
 1904, and paid May 30, 1904, at 6 %. 
 
 20. Divide 22.5 by 51.75, and express the result in the 
 form of a fraction. 
 
 21. By the census of 1890, the population of a certain 
 city was 26,275. By the census of 1900, its population was 
 31,530. Find the per cent of increase. 
 
 22. Each of two boys bought 100 apples for a dollar. 
 The first boy sold his, 4 apples for 5^ ; the second sold his, 
 5 apples for 6$. Which boy gains the more per cent? 
 How much more ? 
 
 23. A quantity of coal was bought for $900. For what 
 must it be sold to gain 33J % ? 
 
 24. By selling a house for $5760, a man gained on the 
 cost 25 %. What was the cost ? 
 
 25. Change to other methods of expression, J, -J-, .37^, J, 
 .16J. 
 
Review. 301 
 
 26. A note of $ 1260, dated July 5, 1904, was paid June 7, 
 1906, with interest at 8%. What was the amount paid ? 
 
 27. A flock of sheep has been increased by 250% of its 
 number, and now numbers 1050. What was the original 
 number ? 
 
 28. Bought a house for $ 6240, and sold it so as to gain 
 35%. What did I sell it for ? 
 
 29. Sold goods at a loss of 20%, an actual loss of $ 57.50. 
 What was the first cost ? 
 
 30. The milk from a herd of 25 Jersey cows, sold at 6 $ 
 a quart, amounted in one summer to $2025. How many 
 quarts were sold, and what was the average quantity from 
 each cow ? 
 
 31. A woman has three children. She pays for each $ 15 
 a year for having his clothes made, $ 1.50 a month for his 
 mending, and $ 0.35 a week for his washing. How much 
 could she save in a year if she knew how to wash, make 
 clothes, and mend ? 
 
 32. A farmer exchanged 340 bushels of corn worth 75^ 
 per bushel, for barley worth $ 1 per bushel, and oats worth 
 50 ^ per bushel. How many bushels of each did he receive, 
 the quantity of barley and oats being equal ? 
 
 33. A pole stands \ in the mud, f in the water, and 32 ft. 
 in the air. How long is the pole ? 
 
 34. Bought flour for $ 8.25, and sold it for $ 9. What is 
 the per cent of gain ? 
 
 35. Bought flour for $ 9 and sold it for $ 8.25. What is 
 the per cent of loss ? 
 
 36. If two-thirds of a yard of silk can be bought for $f, 
 how many yards can be bought for $ 3| ? 
 
 37. A drover sold 250 sheep for $1150, which was 15% 
 more than they cost. What was the cost of each sheep ? 
 
302 Chapter Five. 
 
 38. Find a common divisor of 72 and 90. 
 
 39. How many feet of paper, 18 inches wide, will paper 
 the sides of a room 16 feet by 14 feet, and 10 feet high, de- 
 ducting 174 square feet for doors and windows ? 
 
 40. Find the sum of fa $, f§, -^, ^-, in decimals, correct 
 to fourth place. 
 
 41. The dividend is 9876, the quotient is 87, the remain- 
 der is 45. Find the divisor. 
 
 42. Change .03125 to a common fraction in smallest 
 terms. 
 
 43. Bought a hogshead of sugar containing 848 pounds for 
 $ 38.16, and paid $ 4.24 freight and cartage. At what price 
 per pound must it be sold to gain 20 % ? 
 
 44. To f of f add \ of fa, and reduce to lowest terms ; 
 multiply the sum so obtained by If, and reduce to a mixed 
 number ; from the product subtract f , and reduce to lowest 
 terms ; divide the remainder by 5, and convert the quotient 
 into a decimal fraction; add 1.1 ; multiply by 2.5; subtract 
 .9 ; and divide the remainder by .007. 
 
 45. A can weigh a certain quantity of goods in 15 days 
 by working 7 hours a day. How long will it take him to 
 do the same work by working 9 hours a day ? 
 
 46. In an example in division the remainder is 14, the 
 divisor is 16, and the quotient is 18. What is the dividend ? 
 
 47. Solve by cancellation : 
 
 How many pieces of cotton cloth, each piece containing 
 42 yards, at 9-J- ^ per yard, can be bought for 14 firkins of 
 butter, each containing 56 pounds, at 19^ per pound ? 
 
 48. What must be the depth of a bin which is 4 ft. wide 
 and 6 ft. long, to contain 40 bushels oats ? 
 
 49. A farmer sold 9875 pounds hay at $ 12£ per ton, and 
 took in part payment 5000 feet of boards at $11 per 
 thousand. How much remained due him ? 
 
Review. 
 
 303 
 
 50. Bought 80 barrels of flour at $ 6 per barrel, paying 
 for freight $ 30. At what price must I sell it per barrel to 
 gain 30 % on the total cost ? 
 
 51. What is the amount of $ 720.50, for 3 yr. 5 mo. 19 
 da., at 6 per cent ? 
 
 52. Three men buy a house for $ 2500. A pays $ 500, 
 B pays $ 900, C pays f 1100. They rent it for $ 250. 
 What is each one's share of the rent ? 
 
 53. If 12.875 acres of land cost $ 1030, what will 4.75 
 acres cost ? 
 
 54. Write three-fourths of one per cent, first as a pure 
 decimal, and again as a common fraction. 
 
 55. If a man paid $18f for a load of hay weighing 1£ 
 tons, what would he pay at the same rate for f of a ton ? 
 
 56. If 11 weavers in 9 days weave 1584 yards, what will 
 1 man do in 1 day ? 6 men in 7 days ? 
 
 57. What is the exact interest of $ 500, for 100 days, at 
 8 per cent ? (Take 365 days to the year.) 
 
 58. Divide the product of 8| and llf by their difference. 
 
 59. A merchant bought 340 bushels of potatoes at 80^ 
 per bushel ; 20 per cent of them proved worthless, and were 
 thrown away. He sold the remainder at $ 1.10 a bushel. 
 What did he gain or lose ? 
 
 60. Divide eighty-four and eighty-four hundredths by 
 forty-eight thousandths. 
 
 61. How much money in silver dollars, 41 2 J- grains each, 
 will weigh 165 pounds Avoirdupois, 7000 grains to the 
 pound ? 
 
 62. What is the amount of f 1395, at 4 per cent, for 
 7 mo. 24 da. 
 
 63. A coal dealer buys 150 tons of coal, 2240 pounds 
 each, at $ 4.50 per ton. He sells it at f 4.75 per ton, giving 
 2000 pounds to the ton. What is his profit ? 
 
304 Chapter Five. 
 
 64. What is the value of (J of f of 3f +8£)-^(10£-7f£) ? 
 
 65. How many bushels of grain will fill a bin 8.5 feet 
 long, 4.25 feet wide, and 3| feet deep ? 
 
 66. Three workmen receive $ 283.50 for doing a piece of 
 work. One worked 32 days, the second worked 53 days, the 
 third worked 41 days. What is the share of each ? 
 
 67. A man bought silverware for $ 120, and sold it for 
 $ 250 less 33£ and 10 per cent. What was his profit per 
 cent? 
 
 68. What is the interest on f 356.75, at 4 per cent, for 
 3 yr. 5 mo. 14 da. ? 
 
 69. A note for $ 600, drawn Jan. 16, payable 4 months 
 after date, is discounted March 25 at a bank, at 6 per cent. 
 What are the proceeds ? 
 
 70. A dry-goods merchant sells goods 12J^ per yard more 
 than their cost, and realizes a profit of 8 per cent. What is 
 the cost per yard ? 
 
 71. A man bought 396 acres of land for $40,293. He 
 sold 150 acres at $ 120 per acre, 134 acres at $ 80 per acre, 
 and the remainder at cost. Did he gain or lose, and how 
 much ? 
 
 72. If 44f yards of calico cost $ 1.99, how much must be 
 paid for 80 yards ? 
 
 73. Divide the sum of 75 thousandths and 75 ten-thou- 
 sandths by the difference between 75 hundredths and 75 
 tenths. 
 
 74. What number divided by 320 gives 47 for quotient 
 and 163 for remainder? 
 
 75. In a schoolroom there are 35 pupils and a teacher. 
 The room is 30 feet long, 20 feet wide, and 15 feet high. 
 How many cubic feet of air space has each person ? 
 
 76. A merchant sold a quantity of flour for $ 282, losing 
 6 per cent. How much money did he lose ? 
 
Review. 305 
 
 77. I bought 2500 bushels of wheat at 80^ per bushel, 
 and sold it for 84 f per bushel, on a note for 60 days, which 
 I had discounted immediately at a bank, at 6 %. How much 
 did I gain ? 
 
 78. A merchant bought 84 yards of linen at 55^ per yard, 
 and 105 yards of muslin at 20^ per yard. He sold all the 
 linen at 40^ per yard. What must he charge per yard for 
 the muslin in order to make up exactly his loss on the linen ? 
 
 79. A fruit dealer bought a lot of oranges for $ 240. He 
 sold \ of them for \ of the entire cost ; \ of the remainder 
 for I of the entire cost ; \ of what then remained for \ of 
 the entire cost ; and the final remainder for \ of the entire 
 cost. What was his gain or loss? 
 
 80. The owner of 165 shares of gas stock sold them at 
 $ 25 per share, and with the proceeds purchased two lots, 
 32 feet by 115 feet, and 30 feet by 105 feet, respectively, 
 and had just $ 27 left. What was the price per square foot 
 of the lots? 
 
 81. *A man purchased a house, paying for it in four pay- 
 ments as follows : on the first payment \ of the purchase 
 price; on the second payment \ of the remainder; on the 
 third payment f of what then remained due; and on the 
 last payment $ 2000. What was the full amount paid for 
 the house ? 
 
 82. Find the difference between the greatest common 
 divisor of 480 and 520, and the least common multiple of 
 5, 6, 15, and 20. 
 
 83. Find the value of a pile of wood 40 feet long, 8 feet 
 wide, and 4 ft. 6 in. high, at $5.50 a cord. 
 
 84. A cargo of flour was bought for $690. For what 
 must it be sold to gain 66f % ? 
 
 85. Find the sum of all the prime numbers to 50. 
 
306 Chapter Five. 
 
 86. If A and B can mow a field in seven days, and A, B, 
 
 and C mow it in five days, for $ 25, what ought C to receive ? 
 
 87. To f of a score add f of a dozen, and from the sum 
 subtract f of a hundred. What is the remainder ? 
 
 88. What must be the length of a load of wood that is 
 4 feet wide and 5 J feet high to contain 2 cords ? 
 
 89. Bought a hogshead of molasses containing 128 gallons, 
 at 65 $ a gallon ; paid 80 ^ for cartage, and lost 16 gallons 
 by leakage. At what price per gallon must the remainder 
 be sold to gain one-fifth of the entire cost ? 
 
 90. What is the least number that will exactly contain 
 48, 20, 21, 24 ? 
 
 91. Sold 50 sofas for $2250. 25 of them were sold at a 
 gain of 20 per cent, and 25 at a loss of 20 per cent. What 
 was the gain or loss on the transaction ? 
 
 92. Bought a number of eggs, and sold 11 of them for 
 what 18 cost me. What was my gain per cent ? 
 
 93. A bookseller wishes to mark up the price of a book 
 which he is now selling for $2, so that he can deduct 15 per 
 cent, and yet receive the present price. What must be the 
 marked price ? 
 
 94. What is the difference between .75 divided by 75, 
 and 75 divided by .75 ? 
 
 95. A watch that loses 35 seconds in an hour was set 
 right at noon on Monday. What time did it show at 6 p.m. 
 the following Thursday ? 
 
 96. Mr. A. sold a horse for $ 240, which was 20 per cent 
 less than he asked for it, and his asking price was 20 per 
 cent more than the horse cost him. What was the cost of 
 the horse ? 
 
 97. Three quarts dry measure is what per cent of a 
 bushel ? 
 
Review. 307 
 
 98. What will it cost to carpet an office room measuring 
 21 feet in length, and 18 feet in width, the carpeting being 
 § yard wide, and costing $ 1.35 per lineal yard ? 
 
 99. A physician accepts, in payment of a bill, a note for 
 $275.75, due in one year and three months, with interest 
 at 7 per cent. What amount will be due at maturity ? 
 
 100. At what rate will $1500 amount to $1684.50, in 
 2 yr. 18 da. ? 
 
 101. How shall I mark goods that cost me $.96 a yard, 
 in order to abate 15% and still make 15% ? 
 
 102. What will it cost to insure a factory valued at 
 $21,000, at£%, and the machinery valued at $15,400, 
 at f % ? 
 
 103. In what time will $750 gain $195 interest, at 4% ? 
 
 104. What is the rate per cent when the amount of $500 
 is $593.75, for 2 yr. and 6 mo. ? 
 
 105. What principal will gain $360 in 5 yr. 4 mo., at 
 
 4£%? 
 
 106. Bought 480 barrels of flour, at $4.50 a barrel, and 
 sold it for $2880. Find the gain per cent. 
 
 107. By selling a house for $10,304, a man gained 15% 
 on the cost. What was the cost ? 
 
 108. A man, dying, left -J of his estate to his wife, § of 
 the remainder to his son, and 'the remainder to his daughter, 
 who received $5000. What was the value of the estate, 
 and what was the son's share ? 
 
 109. What is the interest of $10, for 10 yr. 10 mo. 10 da., 
 at 10 per cent ? 
 
 110. If it takes one man 1\ days to do a piece of work, 
 how long will it take 3 men to do 2J times as much ? 
 
308 Chapter Five. 
 
 111. A grocer pays 18^ per pound for coffee, and roasts 
 it, losing 10% of the weight in the process. What must he 
 charge per pound for the roasted coffee in order to make a 
 profit of 20% ? 
 
 112. A merchant bought 48 bales of cotton, and then sold 
 the lot for $2008.80, losing 7%. What was the cost per 
 bale ? 
 
 113. What is the cost of sawing a pile of wood 20 feet long, 
 4 feet wide, and 6 feet high, at $ 1.20 a cord ? 
 
 114. After increasing the wages of his workmen 33^%, a 
 manufacturer paid them $2.60 a day. What did he pay 
 them before ? 
 
 115. What should a bookseller charge for a book for 
 which he paid at the rate of $ 54 a dozen, that he may make 
 20% on the cost? 
 
 116. What is the per cent profit or loss when a hundred 
 logs which cost $ 65 are sold at 78 ^ each ? 
 
 117. A man spent ^-, and invested in his business ^, of 
 his income. He deposited the remainder, $ 1850, in a bank. 
 What was his income ? 
 
 118. Sold a horse for $ 322, and thereby lost 8%. W T hat 
 should I have sold it for to gain 15% ? 
 
 119. Bought a horse for $340; paid $60 for keeping 
 him, and then sold him for $ 540. What per cent was 
 gained ? 
 
 120. John bought Yl\ pounds of sugar at h\f a pound, 
 spending 25% of his money. How much had he at first ? 
 
 121. When 10.25 bushels of wheat cost $ 12.71, what will 
 7J bushels cost ? 
 
 122. Mr. Jones paid $ 15.12 for the use of a sum of money 
 for 1 yr. 6 mo., at 5%. What was the sum ? 
 
 123. What were the proceeds of a note for $ 725.14, due 
 July 7, discounted at a bank June 20, at 8% ? 
 
Review. 309 
 
 124. After Mr. Jones had spent 37|% of his money, he 
 found that he then had enough to buy 80 pounds of rice at 
 6i$ a pound. How much could he have bought with the 
 whole of his money ? 
 
 125. On the 10th day of November, 1899, you lent 
 William Rogers $ 864.50. How much does he owe you to- 
 day, the rate of interest being 4-J % ? 
 
 126. A man bought wheat for $10,867, and sold it at a 
 gain of 4|%. What did he receive for it ? 
 
 127. Divide three million by six thousand, and multiply 
 the quotient by .024. 
 
 128. How much must I have invested at 5 % that my in- 
 come may be $ 2880 per year ? 
 
 129. Add these across, placing the totals in the space in- 
 dicated ; then add the totals : 
 
 Totals. 
 
 14,305 
 
 10,702 
 
 18,346 
 
 37,946 
 
 43,865 
 
 17,387 
 
 22,324 
 
 17,437 
 
 18,438 
 
 3,741 
 
 22,972 
 
 25,960 
 
 13,849 
 
 67,431 
 
 34,965 
 
 12,674 
 
 32,905 
 
 1,468 
 
 15,607 
 
 27,865 
 
 32,476 
 
 18,430 
 
 33,301 
 
 18,695 
 
 19,898 
 
 13,460 
 
 27,686 
 
 23,492 
 
 13,852 
 
 26,973 
 
 130. If 1998, or 27 per cent, of the inhabitants of a town 
 are voters, how many inhabitants has the town ? 
 
 131. Ten cows were sold for $ 690, at a gain of 15 per 
 cent. For how much per head on the average should they 
 have been sold to gain 20 per cent ? 
 
 132. Find the interest of $ 575.50, for 1 yr. 10 mo. 15 da., 
 at 5%. 
 
CHAPTER VI. 
 
 PAGES 
 
 Ratio and Proportion 310 to 328 
 
 Ratio, Proportion, Partitive Proportion, Partnership, 
 Compound Proportion. 
 
 Involution and Evolution 328 to 338 
 
 Square Root, Applications of Square Root, Cube Root. 
 
 Mensuration 339 to 357 
 
 The Circle, Areas of Circles, Areas of Triangles, Areas 
 of Quadrilaterals, Surfaces of Prisms and Cylinders, 
 Surfaces of Pyramids and Cones, Volumes of Prisms 
 and Pyramids, Volumes of Cylinders and Cones, Sur- 
 face of Sphere, Volume of Sphere, Circular Measure. 
 
 Longitude and Solar Time 358 to 363 
 
 Standard Time, Solar Time. 
 
 Review Problems 363 to 366 
 
 Miscellaneous, Oral, Written. 
 
 Stocks and Bonds 367 to 372 
 
 Domestic Exchange 373 to 377 
 
 Sight Drafts, Time Drafts, Bills of Exchange. 
 
 Interest 378 to 380 
 
 Compound Interest, Annual Interest. 
 
 Metric System 380 to 384 
 
 Review Problems 384 to 414 
 
 Special Drills, Review of Fractions, Review of Denomi- 
 nate Numbers, Review of Commercial Discount, Review 
 of Interest, Review of Bank Discount, Exact Interest, 
 Miscellaneous — Oral and Written. 
 
 RATIO. 
 
 391. Ratio is the relation which one number has to another 
 of the same kind. 
 
 The sign of ratio is the colon (:). 
 
 The ratio of 3 to 6 is expressed 3 : 6. 
 
 The colon (:) is the sign used in France and Germany to indicate 
 
 division as well as ratio. 
 
 310 
 
Ratio. 311 
 
 ). The terms of the ratio are the numbers compared, the 
 first being called the antecedent, and the second the conse- 
 quent. Both terms constitute a couplet. 
 
 The ratio of 3 to 6 is obtained by dividing the antece- 
 dent by the consequent ; 3 : 6 means f , which is equal to £. 
 
 393. Oral Exercises. 
 Find the ratio of : 
 
 1. 175 to 700. m = h Ans. 
 
 2. $ 36.50 to $ 18.25. §|^? = 2. Ans. 
 
 % 18.25 
 
 Note. — The quotient is abstract. 
 
 3. 6 pecks to 5 bushels. 6 pecl f = A- Ans. 
 
 Note. — The antecedent and the consequent must be like numbers, 
 
 4. 1 19 to $ 95. 6. 7 tenths to 3 fifths. 
 
 5. 20 mills to 1 dollar. 7. 3 quarts to 4 gallons. 
 8. 1 gallon to 500 cubic inches. 
 
 394. Written Problems. 
 
 1. One line is 3 rd. 4 yd. long; the length of another is 
 5 rd. 1 ft. Find the ratio of the first to the second. 
 
 The antecedent 3 rd. 4 yd. is to be divided by the consequent 
 
 5 yd. 1 ft. As the divisor and the dividend must be like numbers, 
 
 both terms of the couplet 
 
 are reduced to feet. The 3 rd. 4 yd. = 61^ ft. _ 123 j^ 
 
 division is indicated by writ- * ro -* 1 "• °3 J ft. 167 
 
 ing the antecedent above the 
 
 consequent as a fraction. The concrete fraction — 2 — 1 \ s changed to 
 
 83* ft. 
 
 the abstract complex fraction —   , which is reduced to a simple f rac- 
 
 88| 
 
 tion by multiplying both terms by 2, giving | £^ for the result. 
 
 Make the antecedent and the consequent like numbers, and 
 divide the former by the latter. 
 
312 Chapter Six. 
 
 2. M walks in 1 hr. 47 min. as far as N walks in 2 hr. 
 ■3 min. What is the ratio of M's speed to N's ? 
 
 In this example is required the ratio of M's speed to N's. The 
 antecedent is, therefore, M's speed, and the consequent is N's speed. 
 As the distance walked is not given, x may be used to represent the 
 number of feet or yards or miles walked by M in 107 minutes, and by 
 N in 123 minutes. -^- will represent the distance walked by M in 
 
 1 minute, or M's speed, and -^- , N's speed. The ratio of M's speed 
 
 to N's will be JL + JL or — x— • Cancelling x in each, the 
 107 123' 107 3 & 
 
 result is {ffi , or 1^. Ans. 
 
 3. One candle lasts 4 hr. 20 min.; another lasts 3 hr. 
 15 min. Find the ratio of the first to the second. 
 
 4. A pound of coffee costs 25 J f ; 1 pound of sugar costs 
 &A ^ What is the ratio of price of sugar to that of coffee ? 
 
 5. P earns in 19f days as much as Q in 18| days. What 
 is the ratio of Q's daily earnings to P's ? Of P's to Q's ? 
 
 6. One wheel makes 600 revolutions in 8^ seconds; a 
 second makes 300 revolutions in 3\ seconds. What is the 
 ratio of the speed of the first wheel to that of the second ? 
 
 7. The circumference of a circle is 12.5664 feet, and its 
 radius is 2 feet. What is the ratio of the diameter to the 
 circumference ? 
 
 8. One train goes 40 miles in 50 minutes ; another goes 
 24 miles in a half hour. What is the ratio of the speed of 
 the second to that of the first ? 
 
 Find the number of miles each goes in an hour. 
 
 9. One window is 6 ft. 8 in. by 4 ft. 2 in.; a second is 
 4 ft. 8 in. by 2 ft. 1 in. What is the ratio of the area of the 
 second to that of the first ? 
 
 (4f X 2^) + (6f x H) 
 
Ratio. 3 I 3 
 
 10. A mother is now 35 years old, and her son is 3 years 
 and 6 months old. Fourteen months ago what was the 
 ratio of the mother's age to that of her son ? 
 
 11. A farm costing $ 4750 was sold for $ 5750. What is 
 the ratio between the profit and the cost ? 
 
 12. A man can do a piece of work in 4^ days. What 
 part of it can he do in a day and a half ? What decimal ? 
 What per cent ? 
 
 13. What is the ratio between a ton of 2000 pounds and 
 one of 2240 pounds ? 
 
 395. Oral Problems. 
 
 1. One line is a rod long; another is 5|- ft. long. What 
 is the ratio of the first to the second ? 
 
 2. What is the ratio of 7 hours to one day ? 
 
 3. A pound of coffee costs 30^, of sugar 6^. What is 
 the ratio of their respective prices ? 
 
 4. A walks in 4 hours as far as B in 5. What is the 
 ratio of A's speed to B's ? 
 
 5. E earns in 6 days as much as D earns in 8 days. 
 Find the ratio of E's daily earnings to D's. 
 
 6. One wheel makes 300 revolutions in 2 minutes ; the 
 second requires only 1J minutes to make the same number. 
 Find the ratio of the number of revolutions made by the 
 first wheel in 1 minute to the number made by the second 
 wheel in the same time. 
 
 7. A circle whose diameter is 1 foot has a circumference 
 of 3| feet. What is the ratio of the diameter to the circum- 
 ference ? 
 
 8. One train goes 40 miles an hour ; a second goes 45 
 miles an hour. What is the ratio of the speed of the first 
 to that of the second ? 
 
314 Chapter Six. 
 
 17 
 21" 
 
 _51 
 
 
 18 _ 
 
 36 
 
 
 ? 
 
 "70* 
 
 
 ? 
 24 = 
 
 57 
 
 '72* 
 
 
 $16 7 
 
 marks 
 
 ? 
 
 21 marks 
 
 5 + 
 
 22 = 
 
 ?-f-88. 
 
 PROPORTION. 
 
 396. Preliminary Exercises. 
 
 1. A = l. 
 
 16 64 
 2 18 - 36 
 
 3. »..£ 
 
 13 65 
 
 • lj^l? 
 ' 3bu. $24* 
 
 3qt 1 = 30^ > 
 lgal. ?** 
 
 11. 6 horses -j- ? horses = $ 600 -r- $ 900. 
 
 12. lft. -5-? yd. = 15^-*- 90*. 
 
 13. 1 qt. lpt. -*-l pt. = ?jt + 4? 
 
 14. li+i^JH-f. 
 
 15. 2.8 -r- .4 = .14 -h #. 
 
 397. Two equal ratios form a proportion. 
 
 The ratio of 3 to 9 is \, which is also the ratio of 13 to 39. 
 This may be expressed -| = Jf , or 3 : 9 = 13 : 39. Substitut- 
 ing a double colon (: :) for the sign of equality (=), we 
 have the following proportion : 
 
 3 : 9 : : 13 : 39. 
 
 This is read, 3 is to 9 as 13 is to 39. 
 
 In the foregoing proportion, 3 and 13 are the antecedents, and 9 
 and 39 are the consequents. 
 
 398. The first and the last term of a proportion constitute 
 the extremes ; the second and the third the means. 
 
 In the following proportion 
 
 5:15:: 9:27 
 6 and 27 are the extremes, 15 and 9 are the means. 
 
Proportion. 315 
 
 The foregoing proportion may be written 
 
 ft-* 
 
 Multiplying each of these two fractions by the product 
 of the denominators, 15 x 27, we have 
 
 5x ?5x27 ^ 9 x!5xgT 
 
 u » 
 
 Cancelling, 5 x 27 = 9 x 15. 
 
 In the same way it may be shown that in any proportion 
 the product of the numbers in the extremes is equal to the 
 product of the numbers in the means. 
 
 399. Written Exercises. 
 Find the missing term. 
 
 1. 3:44.-:5:z. 
 
 As the product of the extremes is equal to the product of the means, 
 3 multiplied by x is equal to 4f multiplied toy 5 ; i.e. 3 x = 4$ x 5. 
 jc, therefore, is equal to ? x • This reduces to ^, or 8. Ans. 
 
 To find an extreme, divide the product of the means by 
 the other extreme. 
 
 2. %:#■■ ■.*■■& 
 
 The product of the means |f x X equals the product of the extremes 
 | x J£. x is equal, therefore, to $ x || h- |£» Inverting the divisor, 
 we have | x £J x jft. Cancel. 
 
 To find a mean, divide the product of the extremes by the 
 other mean. 
 
 3. 3% + 16 = i + x. 6. ?:19::28:76. 
 
 4. 5:7:: 121 : Xm 7. a; : 15 : : 4 : f 
 
 5. 3-*-a = 12---20. 8. a:£::2:7. 
 
316 Chapter Six. 
 
 9. f:*::J:f 12. *fftill>**|; 
 
 10. f:f:j«:2J> 13. SB : 9 : : 4 : a;. 
 
 11. l:|::lf:aj. 
 
 14. 1 lb. 1 oz. : 2 lb. 4 oz. : : 17^ : »#. 
 
 15. 3qt. 1 pt.-r-l gal. = »^-f-80^. 
 
 16. 4 bottles : x bottles = 6 pints : 15 pints. 
 
 17. x men : 9 men = 16 acres : 36 acres. 
 
 400. Oral Problems. 
 
 1. If 9 eggs cost 25 /, what will 3 dozen cost? 
 Explanation. — 3 dozen, or 36, will cost 4 times as much as 9 ; 4 
 
 times 25 j* = $1. 
 
 2. If 7 pounds of flour cost 23^, what will be paid for 
 49 pounds ? 
 
 3. For $5 1 can get 12 straw hats. How many can I 
 get for $20? 
 
 4. A wheel makes 75 revolutions in 5 minutes. How 
 many does it make in an hour ? 
 
 5. $100 principal gives $6 interest. How much will 
 be the interest of $ 450 principal ? 
 
 6. A merchant pays 75^ freight on 125 pounds of mer- 
 chandise. How much will be the freight on 1000 pounds at 
 the same rate ? 
 
 7. A locomotive goes 3 miles in 4 minutes. How far 
 does it go in an hour ? 
 
 8. 4 horses can eat a certain quantity of hay in 10 
 months. How long will it last 20 horses ? 
 
 9. 12 men can do a piece of work in 15 days. How long 
 will 36 men require ? 
 
 10. 15 yards cost 270 cents. What will be the cost of 5 
 yards ? 
 
Proportion. 317 
 
 401. Written Problems. 
 
 1. If 9 cows cost $267, what will be the cost of 36 at 
 the same rate ? 
 
 The ratio of the cost, $ 267 : $ x, must be the same as the ratio of the 
 number of cows, 9 : 36. Making the proportion, we have 
 
 9 : 36 : : 267 : x. 
 
 mu t 9 267 X 36 
 
 Therefore, x = - — • 
 
 9 
 
 Cancelling, x = $ 1068. Ans. 
 
 2. 7 barrels of sugar cost $ 104.32. Find the cost of 42 
 barrels at the same rate. 
 
 3. A wheel makes 248 revolutions in 5 minutes. How 
 many does it make in 1 hour 20 minutes ? 
 
 Make the required number of revolutions the fourth term. The 
 proportion will then be as follows : 
 
 5 minutes : 80 minutes : : 248 revolutions : x revolutions. 
 
 _ 248 revolutions x 80 
 5 
 
 4. A locomotive goes 2.8 miles in 4 minutes. How far 
 does it go in an hour ? 
 
 5. From 9 pounds of yarn are made 42 yards of dress 
 goods. How many yards can be made from 165 pounds of 
 yarn? 
 
 How many pounds of yarn are needed for 196 yards of 
 goods ? 
 
 6. If 17 men receive $ 357 for a week's work, how much 
 should 24 men receive ? 
 
 7. If 17 men take 27 days to finish some work how long 
 would it take 51 men ? 
 
 Note. — The work done by 51 men would be fy of the work done 
 by 17 men. The time required by 51 men would be £| of the time re- 
 quired by 17 men. 
 
3 18 Chapter Six. 
 
 8. When a sura of money is divided among 48 persons, 
 each receives $ 27.50. How much would each receive if the 
 same sum were divided among 16 persons ? 
 
 9. For $ 85 I can purchase 238 yards of dress goods. 
 How many yards can I purchase for $ 5 ? 
 
 10. A can do a piece of work in 6 days ; B can do it in 7 
 days. If B's wages are $ 2.10 per day, how much should 
 A receive per day ? 
 
 11. If for 7s. 6d. I can buy 9 pounds of raisins, how many 
 pounds can I buy for £ 56 16s. ? 
 
 12. A quantity of provisions would last a ship's crew 20 
 days, allowing each man 2 lb. 4 oz. daily. What should 
 each man be allowed so as to make the provisions last 4 
 days longer ? 
 
 24 days : 20 days : : 36 ounces : x ounces. 
 
 13. If 40 men are able to do a piece of work in 10 hours, 
 how many extra men must be employed to finish it in 8 
 hours ? 
 
 8 hours : 10 hours : : 40 men : x men. The number of extra men is 
 x-40. 
 
 14. If it requires 40 yards of carpet 2 ft. 9 in. wide to 
 cover a floor, how many yards of carpet 2 ft. 6 in. wide 
 would be needed ? 
 
 15. How long will it take a train to go 112 miles, at the 
 rate of 46 miles in 1 hr. 20 min. 30 sec. ? 
 
 16. If a beam 5 ft. 6 in. long, 10 inches wide, and 8 inches 
 thick weighs 924 pounds, find the length of another beam 
 of the same material which weighs 3024 pounds, and whose 
 end is a square foot. 
 
Partitive Proportion. 319 
 
 PARTITIVE PROPORTION. 
 
 402. Preliminary Exercises. 
 
 1. Coin silver consists of 9 parts silver and 1 part cop- 
 per. What is the ratio of the weight of the silver in a dime 
 to the weight of the coin ? 
 
 2. What is the ratio of the weight of the copper to the 
 weight of the coin ? 
 
 3. How many ounces of copper are there in a bar of 
 coin silver weighing 90 ounces ? How many ounces of pure 
 silver ? 
 
 403. Partitive proportion is the process of dividing a num- 
 ber into parts proportional to given numbers. 
 
 404. Written Problems. 
 
 1. Divide 180 into parts proportional to 2, 3, and 4. 
 
 If the parts were 2, 3, and 4, the whole number would be 2 + 
 
 2 3 + 4, or 9. The ratio of the whole to the first part must be 9 to 
 
 3 2 ; of the second, 9 to 3 ; of the third, 9 to 4. These ratios give 
 
 4 rise to the proportions indicated. 
 
 9:2::180:z. .\z = 40. 
 9:3:: 180: y. .'.?/ = 60. 
 9 : 4 : : 180 : z. .'. z = 80. Ans. 40, 60, and 80. 
 
 2. Gunpowder is composed of 15 parts of saltpeter, 2 of 
 sulphur, and 3 of charcoal, mixed together. How many 
 pounds of each are there in 72 pounds of powder ? 
 
 15 In a mixture of 15 lb. -f 2 lb. + 3 lb., or 20 lb., there will be 
 
 2 15 lb. saltpeter; hence, the ratio of the whole weight to the 
 
 3 weight of the saltpeter is 20 lb. to 15 lb., etc. 
 
 20 : 15 : : 72 : number of pounds of saltpeter. 
 20 : 2 : : 72 : number of pounds of sulphur. 
 20 : 3 : : 72 : number of pounds of charcoal. 
 
320 Chapter Six. 
 
 3. A bankrupt surrenders property worth $1287 for the 
 benefit of three creditors to whom he owes $750, $1125, 
 and $ 1245, respectively. How much should each creditor 
 receive ? 
 
 4. A had on storage in a warehouse 2400 bales of cotton, 
 B 1500 bales, and C 1100 bales. After a fire that destroyed 
 all distinguishing marks, the damaged cotton was sold for 
 $ 10,000. How should this sum be divided ? 
 
 5. Our standard gold coin consists of 900 parts gold, 90 
 parts silver, 10 parts copper. What is the quantity of each 
 metal in 50 pounds of coin ? 
 
 6. Two men hire a pasture for $45. Cme puts in 15 
 cows ; the other puts in 12 cows. What should each pay ? 
 
 7. A and B hire a boat for 50 days, paying $30. A 
 uses it 27 days; B uses it 23 days. How much should 
 each pay? 
 
 8. Three farmers together paid $54 for threshing their 
 grain. A threshed his crop of 900 bu. ; B threshed his crop 
 of 828 bu. ; C 672 bu. What did each pay ? 
 
 9. A and B contract to haul a pile of lumber for $ 105. A 
 furnishes 3 teams, and B 4 teams. How much does B 
 receive ? 
 
 10. Three merchants shipped a cargo of iron by sea. A 
 gent 180 tons, B sent 105 tons, C sent 315 tons. During a 
 storm the sailors were obliged to throw overboard 180 tons 
 to save the vessel. Assuming that the cargo should sustain 
 one fourth of the loss, what portion of the loss should each 
 merchant sustain? 
 
 11. Divide 90 into two parts which shall be to each other 
 as 9 to 1. 
 
Partnership. 321 
 
 • PARTNERSHIP. 
 
 405. Written Problems. 
 
 1 . B and C gain by trade $ 182. What is the gain of 
 each, B having put in $300, and C $400 ? 
 
 The total investment is $ 700. The ratio of the total invest- 
 300 ment to B's investment is 700 to 300. This should be the ratio 
 400 of the total profit to B's share, etc. 
 700 : 300 : : $ 182 : B's share. 
 700:400::$182:C's share. 
 
 Make proportions whose antecedents in each case are the 
 total investment and the total profit, the consequents being the 
 investment of one partner and his share of the profit. 
 
 2. A, B, and C invest $ 720, $ 340, and $ 960, respectively. 
 The profits are $ 101. What is each one's share ? 
 
 3. A, B, and C buy a house for $7500. A furnishes 
 $2000; B, $2500; C, the remainder. The yearly rent, less 
 expenses, is $ 576. To what amount is each entitled ? 
 
 4. M and N entered into partnership. M puts $ 200 into 
 the business for 5 months, and N $ 300 for 4 months. They 
 gained $ 132. Find the share of each. 
 
 An investment of $ 200 for 5 months is equivalent 
 200 x5 = 1000 to an investment of $ 1000 for 1 month ; an invest- 
 300 X 4 = 1200 ment of $ 300 for 4 months, to $ 1200 for 1 month. 
 2200 : 1000 : : $ 132 : M's share. 
 2200 : 1200 : : $ 132 : N's share. 
 In ascertaining the ratio of the whole capital to the share con- 
 tributed by each, $1000 and $1200 are taken as representing the 
 shares of each in a total capital of $2200. 
 
 Multiply each partner's share of the capital by the time it is 
 in the busiiiess, and consider the products, respectively, as the 
 sums contributed by the partners. 
 
 Note. — This mode of ascertaining a partner's share of profits or 
 losses is based upon the assumption that the agreement of the partners 
 does not provide for a different division. 
 
322 Chapter Six. 
 
 5. X and Y rent a field for $ 32. X puts in 8 horses for 
 6 months, and Y 10 horses for 8 months. How many dol- 
 lars should each pay ? 
 
 8 horses for 6 months = how many for one month ? 
 10 horses for 8 months = how many for one month ? 
 
 6. Three men hire a pasture for $ 84. One puts in 15 
 cows for 12 weeks; the second puts in 20 cows for 6 weeks; 
 the third puts in 18 cows for 10 weeks. What amount 
 should each pay ? 
 
 7. Four men hire a pasture field together. The first 
 pastures 4 cows 18 weeks ; the second, 5 cows 12^ weeks ; 
 the third, 11 cows 6J weeks ; the fourth, 9 cows 16 weeks. 
 What part of the rent should each pay ? 
 
 8. Two men hire a pasture for f 420. A puts in 300 
 sheep for 5 weeks, and B puts in 450 sheep for 6 weeks. 
 What should each pay ? 
 
 9. A, B, and C enter into partnership. A puts in $500 
 for 4 months, B $400 for 6 months, and C $800 for 3 
 months ; they gain $ 340. Find each man's share of the 
 gain. 
 
 10. A partnership is formed between A with a capital of 
 $ 1500 and B with a capital of $ 2500. Six months there- 
 after they take in C with a capital of $ 4000. How should 
 a profit of $ 3500 be divided at the end of the year ? 
 
 11. A and B form a partnership. A furnishes $ 2000, B 
 $ 3000. After a year A furnishes an additional $ 1000. 
 At the end of 2 years the business is disposed of for $ 7100. 
 How much should each receive ? 
 
 Suggestion. — A receives his $3000 and how much of the profits ? 
 Should he receive as much as B, who had $ 3000 in the business the 
 whole time ? 
 
20: 
 
 :30 
 
 
 18: 
 4 
 
 :27 
 : 5: 
 
 men men 
 
 :72 : x 
 
 15: 
 
 : 3 
 
 
 9: 
 
 :10 
 
 
 Compound Proportion. 323 
 
 COMPOUND PROPORTION. 
 
 406. A compound proportion is one in which either ratio is 
 compound. 
 
 407. Written Problems. 
 
 1. If 72 men dig a ditch 20 yd. long, 1 ft. 6 in. broad, 4 
 ft. deep, in 3 days of 10 hours each, how many men would 
 be required to dig a ditch 30 yd. long, 2 ft. 3 in. broad, and 
 5 ft. deep, in 15 days of 9 hours each ? 
 
 Since the number of men is required, 72 men 
 is made the third term of the proportion. Con- 
 sidering the length alone, the ratio of 72 men to 
 the required number would be equal to the ratio 
 of 20 feet to 30 feet. Considering the width, 
 the ratio would be 18 inches to 27 inches. Con- 
 sidering the depth, the ratio would be 4 feet to 5 feet. Considering 
 the number of days, the ratio would be 15 days to 3 days. Con- 
 sidering the number of hours per men 
 
 day, the ratio would be 9 hours to 72 X 30 X 27 X 5 X 3 X 10 
 10 hours. Dividing the product of 20 X 18 X 4 X 15 X 9 
 
 the means by the product of the ex- 
 tremes, the number of men is found to be 46. 
 
 Place the number required as the fourth term, making the 
 like number the third term. Arrange the couplets, considering 
 the effect of each separately on the result. Divide the product 
 of the means by the product of the extremes. 
 
 2. If 45 horses eat 1\ tons of hay in 30 days, how many 
 tons should last 84 horses 56 days ? 
 
 3. If 4 men, working 8 hours per day, can mow a meadow 
 in 3 days, how many men, working 9 hours per day, can mow 
 a meadow three times as large in 4 days ? 
 
 4. If 10 men, working 8 hours per day, can build a cer- 
 tain wall in 6 days, how many hours a day must 12 men 
 work to build the same wall in 4 days ? 
 
324 Chapter Six. 
 
 5. If 108 men can build a fort in 12£ days of 12 J hours 
 each, in how many days can 84 men build it by working 10£ 
 hours daily ? 
 
 6. What will it cost to transport 1000 pounds of mail 
 matter 1000 miles, at $ 1 per 100 pounds per 100 miles ? 
 
 7. If 12 men can do a piece of work in 20 days, what 
 number of men will be required to do four times as much 
 work in a fifth part of the time ? 
 
 8. If 14 men can mow 168 acres in 12 days of 8 hr. 15 
 min. each, how many acres can 20 men mow in 11 days of 
 7 hr. 48 min. each ? 
 
 9. If 5 needlewomen can do a piece of work in 11 days 
 of 9 hours each, how long will it take 3 needlewomen to do 
 two such pieces, supposing them to work 10J hours each 
 day? 
 
 10. A employs a capital of $2500 in business, and at the 
 end of 3 years takes into partnership B, who furnishes 
 $4000. Four years later they are joined by C, with a cap- 
 ital of $ 5000. At the end of 12 years from the commence- 
 ment of the business the profits, amounting to $ 15,000, are 
 divided. What amount should each receive ? 
 
 A's money is in the business how many years ? B's how many 
 years ? C's how many ? 
 
 11. A and B rented a field for a year for $175. A put 
 in 6 horses for the whole time; !B put in 5 horses for 11 
 months and 3 horses for 5 months. How much of the rent 
 had each to pay ? 
 
 12.' A field of grain was to be cut down by 40 men in 10 
 days. Eight of the men, however, failed to come. How 
 long did it take the others to do the work ? 
 
Review. 325 
 
 REVIEW. 
 408. Oral Problems. 
 
 1. How many weeks will 4-J- tons of coal last Mrs. 
 Bright, if she uses ^ of a ton each week ? 
 
 2. I can buy 2 pairs of shoes for 12 shillings. How 
 many pairs at the same rate can I buy for £ 3 ? 
 
 3. If two-thirds of your age is 8 years and 4 months, 
 how old are you ? 
 
 4. 5 quarts equal what decimal of a peck ? 
 
 5. What is the cost of 700 pounds of coal at $ 7 a ton ? 
 
 6. How much would you pay for 2| yards of cloth at 
 37£ bayard? 
 
 7. In what time will f 50, at 6%, give $ 18 interest? 
 
 8. If I buy an article for $ 75 and sell it for $ 50, what 
 is my loss per cent ? 
 
 9. By selling an article for $9, a man gained 25%. 
 How many dollars would he have gained if he had sold the 
 article at an advance of 50 % over cost ? 
 
 10. How many quarts of peanuts in 1 bushel and 3 
 pecks ? 
 
 11. What would be the cost of 120 books at 660 each ? 
 
 12. Change 66,321 mills to dollars. 
 
 13. $120 is i per cent of what number of dollars ? 
 
 14. In what time will $ 50 doubie itself at 8% ? 
 
 15. If $ 1 is paid for insuring a piano worth $ 500, what 
 is the rate of insurance ? 
 
 16. Into how many lots, containing f of an acre each, can 
 8 acres be divided ? 
 
 17. A man lends $ 1200 at 6%, and 1500 at 5%. ! What 
 is the difference in the amount of yearly interest due on 
 each? 
 
326 Chapter Six. 
 
 18. A man owning § of a ship sold § of his share. What 
 part of the ship did he still own ? 
 
 19. How many rings, each 2 pwt. 12 gr., can be made 
 from \ pound of gold ? 
 
 20. Find the number of square inches on the surface of 
 a block 10 inches long by 4 inches wide by 3 inches thick. 
 
 409. Written Problems, 
 
 1. A traveller walked 23\ miles the first day, 3 J miles 
 more the second day than the first, and 3J miles more the 
 third day than the second. How far did he walk in the 
 three days ? 
 
 2. Multiply 63.15 by 1.04; divide the product by 6.25, 
 and subtract the quotient from 11 
 
 3. How many bricks, 8 inches long and 4 inches wide, 
 will be needed to make a sidewalk 20 feet long and 4 feet 
 wide? 
 
 4. If it costs $ 10.24 to carry 1500 pounds 356 miles, what 
 will it cost to carry 2700 pounds 890 miles ? 
 
 5. A house rents for $ 30 a month, and the owner pays 
 $ 75 a year for taxes and repairs. What is the value of the 
 house, if his net profit is 5 per cent per annum ? 
 
 6. A loaned B a sum of money at 4£ per cent interest per 
 annum. At the end of 18 months B paid the debt, principal 
 and interest, in all $ 1814.75. What was the sum borrowed ? 
 
 7. If a 5-months note for $ 760, dated March 13, is dis- 
 counted at a bank May 23, the rate being 7 per cent a year, 
 what will be the proceeds ? 
 
 8. A grocer bought 40 gallons of maple syrup at the rate 
 of 4 gallons for $ 6, and sold it at the rate of 5 gallons for 
 $ 8. What was the whole gain, and the gain per cent ? 
 
Review. 327 
 
 9. Two pictures were sold for $ 99 each. On one there 
 was a gain of 10% ; on the other a loss of 10%. Was there 
 a gain or a loss on the sale of both, and how much ? 
 
 10. New York, Jan. 1, 1904. 
 
 One year after date I promise to pay J. Edward Swans- 
 ton Eight Hundred Dollars for value received, with interest. 
 
 $800 T °oV Kufus L. Scott. 
 
 Indorsed as follows: Apr. 1, 1904, $10; July 1, 1904, 
 $35; Nov. 1,1904, $100. 
 What was due Jan. 1, 1905 ? (Merchant's Eule.) 
 
 11. What is the difference on a bill of $780, between a 
 discount of 40% and a discount of 35 and 5% ? 
 
 12. How many cords in a pile of wood 42 feet long, 
 12 feet high, and 8 feet wide ? Find its cost at $ 6.35 per 
 cord. 
 
 13. What principal, on interest for 2 yr. 6 mo. at 4%, 
 will gain $850? 
 
 14. What is the cost of insuring a house, worth $ 25,000, 
 for | of its value at 1£% ? 
 
 15. At 9fi a cubic foot what will be the cost of a block 
 of stone 9 ft. long, 4 ft. wide, and 5 ft. 6 in. thick ? 
 
 16. If a steeple 150 feet high casts a shadow of 275 feet, 
 how long a shadow will be cast by a man 6 feet tall, at the 
 same time of day ? 
 
 17. The tax to be raised in a certain town is $1350. 
 The taxable property is valued at $ 108,000 What is the 
 tax on one dollar ? 
 
 18. Mr. Fox buys one-fifth of an acre of land for $21.78. 
 For how much a square foot must he sell it to gain 20% ? 
 
328 Chapter Six. 
 
 19. What is the cost of covering a floor 16£ ft. long, 
 12 ft. wide, with oil-cloth 1% yd. wide, at 75^ a yard ? 
 
 20. The edges of a large cubical box are 5 feet long. 
 How many square feet of paper will cover the outside of 
 the box? 
 
 21. A field 110 yards long and 44 yards wide contains an 
 acre. What is the area of a field 220 yards long and 88 
 yards wide ? Of one 440 yards long and 176 yards wide ? 
 
 110 x 44 : 220 x 88 : : 1 acre : x acres. 
 
 22. If a steel bar 12 feet long, 4 inches broad, and 2\ 
 inches thick weighs 480 pounds, what is the weight of 
 another steel bar 18 feet long, 3 inches broad, and 2 inches 
 thick ? 
 
 23. At a certain hour a pole 6 feet high casts a shadow 
 measuring 4 ft. 2 in. Calculate the height of a steeple 
 whose shadow at the same hour is 104 ft. 2 in. 
 
 24. If 7 men receive $ 126 for 5 weeks' work, how much 
 should they receive for 9 weeks' work ? 
 
 25. If 76 boards, each 14 feet long and 10 inches wide, 
 are worth $19.76, how much would 50 such boards be 
 worth ? 
 
 INVOLUTION. 
 
 410. Involution is finding any power of a number. 
 
 A power of a number is the product obtained by using the 
 number a certain number of times as a factor. 
 
 2 is the first power of 2. 2 x 2, or 4, is the second power 
 of 2. 2 x 2 x 2, or 8, is the third power of 2, etc. 
 
 411. The second power of a number is called its square; 
 the third power is called its cube. 
 
Involution. 329 
 
 412. The power of a number is indicated by writing a small 
 figure, called an exponent a little to the right of the upper 
 part of a number. 
 
 The square of 2 is written 2 2 . 
 
 The cube of 2 is written 2 3 . 
 
 The fourth power of 2 is written 2*. 
 
 5 2 = 25, 12 2 = 144. 
 
 What is the square of 4 ? Of 6? Of 7? Of 9? Of 10? 
 Of 11? 
 
 Square 13. 15. 21. 16. 19. 14 2 =? 17 2 *=? 54 2 = ? 
 33 2 =? 
 
 413. The square of 25 = (20 + 5) x (20 + 5). 
 
 20+5 
 
 20+5 
 Multiplying by 20 20 2 + 20 x 5 
 
 Multiplying by 5 20 x 5 + 5 2 
 
 20 2 + 2 (20 x 5) + 5 2 = 400 + 200 + 25 = 625. 
 
 414. The square of the sum of two numbers is equal to 
 the square of the first -f- twice the product of the first by 
 the second -f- the square of the second. 
 
 . 13 2 =(10 + 3) 2 = 10 2 +-2(10x3)+3 2 =? 
 18 2 = (10 + 8) 2 = 100 +• 160 +• 64 = ? 
 27 2 = (20 + 7) 2 = 400 + 280 + 49 = ? 
 
 415. Oral Exercises. 
 
 
 
 
 
 Square : 
 
 
 
 
 
 
 1. 19. 
 
 4. 26. 
 
 7. 51. 
 
 10. 32. 
 
 13. 
 
 27. 
 
 2. 22. 
 
 6. 31. 
 
 8. 61. 
 
 11. 42. 
 
 14. 
 
 33. 
 
 8. 24. 
 
 6. 41. 
 
 9. 23. 
 
 12. 52. 
 
 15. 
 
 43. 
 
23° Chapter Six. 
 
 EVOLUTION. 
 
 416. Evolution is finding any root of a number. 
 A root is one of the equal factors of a number. 
 
 The square root of a number is one of its two equal factors. 
 
 The square root of 4 is 2 j of 9 is 3 ; of 16 is 4 ; of 25 is 5c 
 
 417. Give the square root of 36. Of 64. Of 81. Of 121. 
 Of 49. Of 100. Of 144. 
 
 418. The sign of a square root is -y/. 
 
 V81 = 9. Vl2l = ?. V25 = ?. V49 = ?. 
 
 SQUARE ROOT. 
 
 419. Find the square root of 169. 
 
 10 2 = 100. 20 2 = 400. The square root is between 10 and 20 ; it is, 
 therefore, 10 + a second number. 
 
 169 = 10 2 + 2 (10 x second) + second 2 . 
 
 169 = 100 + 20 x second + second 2 . 
 
 20 x second + second 2 = 69. 
 
 From this it appears that the second number is 3, since 
 
 20 x 3 + 3 2 = 69. 
 
 420. It may be shown in this way : 
 
 10 (first number) 
 169 
 10 2 = 100 
 Trial divisor — twice 10 20) 69(3 second number) 
 
 60 
 9 
 S* = 9 
 Ans. 10 + 3 = 13. 
 
Square Root. 331 
 
 421. Find the square root of 2116. 
 
 40 (first number) 
 2116 
 40 2 1600 
 
 40 x 2 = 80, trial divisor ) 516(6 second number) 
 
 480 
 
 36 =6 2 
 Arts. 46. 
 
 Instead of multiplying the trial divisor by the second number, and 
 then ascertaining whether the remainder is the square of the second 
 number, the second number is added to the trial divisor and this sum 
 is multiplied by the second number. 
 
 In practice, the work is shortened by omitting the ciphers. 
 
 40 (first number) 4 6 Ans. 
 
 2116 21'16 
 
 1600 16 
 
 (2 x 40) + 6 = 86) 516(6 second number) 86) 516 
 
 516 516 
 
 First, point off in periods of two figures, commencing at 
 units. Find the greatest square in the first period, and place 
 the root in the quotient. Subtract the square from the 
 first period and bring down the next period. Multiply the 
 quotient figure by 2, and use it as a trial divisor. Place 
 the second figure in the quotient and annex it also to the trial 
 divisor. Multiply the figures in the trial divisor by the second 
 quotient figure. Bring down the next period, and proceed as 
 before until the square root is found. 
 
 422. "Written Exercises. 
 Extract the square root : 
 
 1. 196. 4. 1225. 7. 2809. 10. 6889. 
 
 2. 324. 5. 1764. 8. 3721. 11. 8281. 
 
 3. 676. 6. 1936. 9. 5184. 12. 9025. 
 
33 2 Chapter Six. 
 
 423. Find the square root : 
 
 Note. — Extract the square root of each term separately. 
 
 1- jfc 4. ifr. 7. #ft. 
 
 3- itt- 6- t¥A- 9- «H» 
 
 Note. — Before extracting the square root of the following, reduce 
 the mixed numbers to improper fractions. 
 
 10. 12£. 12. 2ff. 14. 156J. 
 
 11. 11J. 13. lOfH- 15. 264 T V 
 
 424. Find the square root of 425,104 
 
 6 5 2 
 42'51'04 
 12|5 651 
 
 130|2 26 04 ^Ln«. 652. 
 
 Jn finding any figure of the root after the first, we multiply 
 the other figure or figures by 2 for a trial divisor. 
 
 425. Find the square root of 20,857,489. 
 
 4 5 6 7 
 
 
 20'85'74'89 
 
 8|5 
 
 4 85 
 
 90|6 
 
 60 74 
 
 912|7 
 
 6 38 89 Ans. 4567. 
 
 Find the square root of 
 
 
 1. 64,516. 
 
 4. 71,824. 
 
 2. 73,441. 
 
 5. 141,376. 
 
 3. 18,769. 
 
 6. .702244. 
 
Square Root. 333 
 
 4. 74. 
 
 5. 350. 
 
 
 V£6=V£60 
 
 
 1.89+ 
 3.60'00 
 
 2.8 
 
 1 
 
 2.60 
 
 
 2.24 
 
 3.69 
 
 .3600 
 
 
 .3321 
 
 426. Written Exercises. 
 
 Find the square roots to two decimal places : 
 
 1. 7. 2. 14. 3. 38. 
 
 6. Find the square root of 3.6. 
 
 Note. — Commence at the units and 
 point off two places to the right as well 
 as to the left, annexing a decimal cipher, 
 if necessary. 
 
 7. 6.4. 8. .121. 9. .144. 10. .196. 11. .225 
 
 APPLICATIONS OF SQUARE ROOT. 
 
 427. Written Problems. 
 
 1. How many inches in the side of a square table top 
 containing 529 square inches ? 
 
 2. The surface of a square piece of board contains 3 sq. 
 ft. 97 sq. in. What is the length of one side in feet and 
 mcnes . (Reduce area to square inches.) 
 
 3. How many rods long is a square field containing 90 
 acres ? How many yards of fence would be needed to 
 enclose it ? 
 
 4. Land surveyors use a measure called a chain. What 
 is its length in rodsj 10 square chains being equal to an 
 acre ? What is the length in feet ? 
 
 It is subdivided into 100 " links." Find the length of a 
 link in inches and decimal. 
 
 5. The surface of the six equal faces of a cube is 1350 
 square inches. What is the length of each edge of the 
 cube ? 
 
334 
 
 Chapter Six. 
 
 428. Preliminary Exercises. 
 
 1. Carefully construct a right-angled triangle, base, 4 
 inches, perpendicular, 3 inches. Measure the hypotenuse. 
 
 Take the square of the length of each side and endeavor 
 to show the relation between the square of the hypotenuse 
 and the squares of the other two sides. 
 
 2. Construct a right-angled triangle, base, 3 inches, per- 
 pendicular, 1J inches. Measure the hypotenuse, and see if 
 the relation between this hypotenuse and the other two 
 sides of this triangle is the same as that found in the 
 other triangle. 
 
 3. A right-angled triangle has a base 12 inches long ; its 
 perpendicular is 3J inches. What is the length of the hy- 
 potenuse ? 
 
 4. The hypotenuse of a right-angled triangle is 25 inches ; 
 its perpendicular is 7 inches. What is the base ? 
 
 6. The base of a right-angled triangle is 12 feet; the 
 hypotenuse is 13 feet. Find the perpendicular. 
 
 429. Draw a right-angled triangle (Fig. 1). Upon each 
 side construct a square (Fig. 2). From the upper portion of 
 the largest square (7, cut a right-angled triangle of the same 
 
 /m 
 
 A 
 
 B 
 
 
 Fig. 2 
 
 Fig. 8 
 
 Fig. 4 
 
 
 
 
 V^ 
 
 
 B 
 
 m/ 
 
 </ 
 
 
 
 
 
 
 C 
 
 
 
 A 
 
 Fig. 5 
 
 dimensions as those of the original triangle m. Cut another 
 triangle of the same dimensions from the left-hand portion 
 (Fig. 3). Place one of these triangles below the remainder 
 
Square Root. 335 
 
 of the square G, and the other at the right, as in Fig. 4, and 
 the resulting polygon will be exactly equal in surface to the 
 two squares A and B (Fig. 5). 
 
 430. The square described on the hypotenuse of a right- 
 angled triangle is equal to the sum of the squares described 
 on the other two sides. 
 
 431. Written Exercises. 
 
 Find the missing side of each of the following ten right- 
 angled triangles : 
 
 1. Base, 15; perpendicular, 8 ; hypotenuse,?. 
 
 Square of hypotenuse = 15 2 + 8 2 
 = 225 + 64 
 = 289. 
 Hypotenuse = V289, or 17. 
 
 To find the hypotenuse, extract the square root of the sum 
 of the squares of the other sides. 
 
 2. Base, 35 ; perpendicular, ? ; hypotenuse, 37. 
 
 P 2 + P 2 = H* ; 
 35 2 + P 2 = 37 2 ; 
 1225 + P 2 = 1369; 
 P 2 = 144; 
 P=12. 
 
 To find the base or the perpendicular, extract the square root 
 of the square of the hypotenuse diminished by the square of the 
 given side. 
 
 3. Base, ?; perpendicular, 15; hypotenuse, 39. 
 
 4. Base, 20 ; perpendicular, 21 ; hypotenuse, ?. 
 
2^6 Chapter Six. 
 
 5. Base, ?; perpendicular, 45 ; hypotenuse, 53. 
 
 6. Base, 56; perpendicular. ?; hypotenuse, 65. 
 
 7. Base, 55; perpendicular, 48; hypotenuse, ?. 
 
 8. Base, ?; perpendicular, 14; hypotenuse, 50. 
 
 9. Base, 63; perpendicular, ?; hypotenuse, 65. 
 
 10. Base, 112; perpendicular, 15; hypotenuse, ?. 
 
 11. Find the area in acres of a right-angled triangle, the 
 length of the sides being 24 rods, 7 rods, 25 rods. 
 
 12. A courtyard 84 feet by 36 feet is to be paved with 
 flag-stones measuring 6 feet by 3 feet. How many stones 
 will be needed ? What will be the cost of the work at 
 $ 1.25 per square yard ? 
 
 13. Find the length of the fourth ™ rd. 
 side of the following piece of ground. «g 
 
 How many yards in the perimeter? 5 
 How many acres does it contain ? 100 rd. 
 
 14. Find the perimeter in rods of the 
 field shown in the diagram. 
 
 1 chain = 66 feet. 
 Note. — A right angle contains 90 degrees. 
 
 15. What is the length of the diagonal of a rectangular 
 field 90 yards wide, 120 yards long ? 
 
 16. The dotted line in the accom- 
 panying diagram indicates a path through 
 the field. How many yards are saved 
 by taking the path instead of following 
 the road ? 
 
 *o 
 
 16 chains. 
 
 36 chains. 
 
 Koud. 
 
 17. Find the length (in rods and a decimal) of the 
 diagonal of a square 40-acre field. 
 
Cube Root. 337 
 
 CUBE ROOT. 
 
 432. To cube a number is to employ it three times as a 
 factor. 
 
 The cube of 4, written 4 3 , is 4 x 4 x 4, or 64. 
 Find the cube of 1, 9, 6, 3, 5, 8, 2, 7. 
 
 To find the cube root of a number is to find one of the 
 three equal factors of the number. 
 
 The cube root of 343, written ^343, is 7. 
 
 The cube of 25, 20 -f 5, is equal to the following : 
 We have seen (Art. 413) that 
 
 (20 + 5) 2 = 20 2 + 2x20 x5 + 5 2 
 
 Multiplying by 20 + 5 we have 
 
 Product by 20 = 20 3 + 2 x 20 2 x 5 + 20 x 5 2 
 
 Product by 5 = 20 2 x 5 + 2 x 20 x 5 2 + 5 8 
 
 (20 + 5)3 = 20 3 + 3 x 20 2 x 5 + 3 x 20 x 5 2 + 6* 
 which may be written in this way, 
 
 203+ [(3 x 202) + (3 x 20 x 5) + 5 2 ] x 5. 
 
 433. Extract the cube root of 15,625. 
 
 "We see by inspection 9(\ _±_K 
 
 that the cube root is 
 between 20 and 30 ; 
 that is, 20 + a second 
 number. Subtract from 
 15,625 the cube of 20, 
 8,000. The remainder, 
 7,625, is equal to the 
 second number multi- 
 plied by the sum of three times the square of the first (1,200), etc. 
 Using 1,200 as a trial devisor, the second number is seen to be 6 or less. 
 
 Taking 5 as the second number, we add to the 1,200 three times the 
 product of the first and second (300), and the square of the second 
 (25), making a total of 1,525. Multiplying this sum by the second 
 number, we get 7,625, which is equal to the difference between 15,625 
 and 8,000. The second number is, therefore, 5, and the cube root of 
 15,625 is 25. 
 
 
 
 15,625 
 
 (20) 3 = 
 
 
 8,000 
 
 3 x20 2 = 
 
 : 1,200 
 
 7,625 remainder 
 
 X 20 x 5 = 
 
 : 300 
 
 
 5 2 = 
 
 25 
 
 
 
 1,525 
 
 7,625 
 
338 Chapter Six. 
 
 ■#110,592 -#658,503 
 
 40 + 8 8 7 
 
 110,592 658'503 
 
 40^ = 64,000 8 8 = 512 
 
 3x40 2 =4,800 46,592 3 x 80 2 =19,200 146,503 
 
 3 x 40 x 8 = 960 3 x 80 x 7 = 1,680 
 
 8 2 = 64 7 2 = 49 
 
 5,824 46,592 20,929 146,503 
 
 Ans. 48. Ans. 87. 
 
 In the last example we point off three places, beginning at the right, 
 and find the greatest cube in the first period, placing its cube root as 
 the first figure of the answer. 
 
 434. Find the cube root of the following : 
 
 1. 2,197 6. 238,328 11. ffffy 
 
 2. 9,261 7. 421,875 12. 3.375 
 
 3. 32,768 8. 551,368 13. 1£|£ 
 
 4. 68,921 9. m 14. 1^ 
 
 5. 148,877 10. 4-ffi 15. 5m 
 
 435. Find the cube root of 9,938,375. 
 
 When the root contains 2 15 
 
 more than two figures, con- 9'938'375 
 
 tinue, as shown in the accom- g 
 
 panying example, taking for 3 X 20 2 = 1 200 1938 
 
 divisor three times the square 3 v 20 X 1 = 60 
 
 of the first two figures con- -. 2= _ 11 261 
 
 sidered as tens, plus three — - — — - — — ; 
 
 times the produet of the first » X 210 ° = 132 300 677 375 
 
 two figures considered as tens 3 X 210 X 5 — 3 150 
 
 by the third figure, plus the ^ 2 *% *° 
 
 square of the third figure. 135 475 677 375 
 
 436. Find the value of the following: 
 
 1. ^1,442,897 3. ^3,723,875 5. #12.977875 
 
 2. </l,906,624 4. ^/39,651,821 6. #66.923416 
 
Measurements. 
 
 339 
 
 .\*«**f***+ 
 
 MENSURATION. 
 THE CIRCLE. 
 
 437. A circle is a plane figure whose boundary is at all points 
 equally distant from the centre. 
 
 The curved line that forms the 
 boundary is called the circumfer- 
 ence. 
 
 The diameter is any straight 
 line drawn from one point of the 
 circumference to another and pass- 
 ing through the centre. 
 
 The radius is any straight line 
 drawn from the centre to the cir- 
 cumference. 
 
 438. Preliminary Exercises. 
 
 1. Cut out of stiff paper a circle whose diameter measures 
 3^ inches. Mark a point A on the circumference, and roll 
 
 2? 
 
 the circle on a plane surface along the line MN. Make a 
 mark at X where the point A touches the line at the begin- 
 ning of its revolution, and at Y where A touches the line at 
 the end. Measure the distance XY, which is the length of 
 the circumference of the circle. 
 
 2. If the distance between X and Y is 11 inches, what is 
 the ratio of the diameter of a circle to the circumference ? 
 
 3. Draw several diameters in the circle you have cut out. 
 Measure each. How do the diameters compare ? 
 
 4. What is the ratio between the diameter of a circle and 
 the radius ? 
 
340 Chapter Six. 
 
 439. "Written Exercises, 
 
 Find the circumference of a circle whose diameter is 3% 
 
 inches. 
 
 3| inches x 3.1416 = 10.9956 inches. Ans. 
 
 The ratio between the circumference of a circle and its diameter has 
 been ascertained to be 3.1416. 
 
 Process. — To find the circumference of a circle multiply the 
 diameter by 3.1416. 
 
 1. Find the circumference of a circle whose diameter is 
 25 feet. 
 
 2. The circumference of a circle measures 39.27 feet. 
 What is the diameter ? 
 
 3. The radius of a circle is 8 yards. Find its circumfer- 
 ence. 
 
 4. The diameter of a bicycle wheel is 28 inches. How 
 far does a bicycle go during 10 revolutions of the wheel ? 
 
 AREAS OF CIRCLES. 
 
 440. Preliminary Exercises, 
 
 1. Divide a circle whose diameter measures 3J inches into 
 sixteen equal parts by cuts passing through the centre in 
 each case. Arrange the parts as shown in the accompanying 
 
 figure : 
 
 B O 
 
 2. When the diameter measures 3J inches, what is the 
 length of AB ? What part of the diameter is AB ? 
 
Measurements. 341 
 
 3. What part of the circumference is embraced between 
 A and D ? 
 
 4. When the given circle is divided into a very great 
 number of parts, what will be the length of AD ? Of AB ? 
 
 5. When the number of parts is extremely great, ABCD 
 becomes a rectangle. Find its area. 
 
 The number of square inches (feet, etc.) in the area of a circle is 
 obtained by multiplying one-half the number of inches (feet, etc.) in 
 the diameter by one-half the number in the circumference. 
 
 This may be expressed by the formula : 
 
 Area of circle = \ diameter x £ circumference. 
 
 As the circumference equals diameter x 3.1416, £ circumference 
 equals radius x 3.1416. Multiplying by £ diameter, or radius, we 
 
 Area of circle = square of radius x 3.1416. 
 
 To find the area of a circle multiply the square of the radius 
 by 8.1416. 
 
 441. "Written Exercises. 
 
 1. What is the area of a circle whose radius is 36 feet? 
 1 sq. ft. x 36 2 x 3.1416 = 1296 sq. ft. x 3.1416. 
 
 2. Find the area of a circle whose diameter is 50 yards. 
 
 3. What is the area of a circle whose circumference is 
 
 10 feet? 1Q 5 
 
 Diameter = ; Radius = 
 
 3.1416 3.1416 
 
 B* = 5_x_5 ; J?2 X 3,1416= 5x5x3.1416 ^^ 
 
 3.1416x3.1416' 3.1416x3.1416 
 
 4. Calculate the area of a circle whose radius is 1 inch. 
 Of a circle whose radius is 2 inches. What is the ratio of 
 the two areas ? 
 
34* 
 
 Chapter Six. 
 
 5. What is the ratio between the area of a circle whose 
 radius is 1 inch and that of a circle whose radius is 3 
 inches ? 
 
 Indicate operations and cancel. 
 
 6. How many square yards are there in a circular walk, 
 the radius, AB, of the inner edge of walk 
 being 10 feet, and that of the outer edge, 
 AC, being 15 feet ? 
 
 Find the difference between the area of a circle 
 of 15 feet radius, and that of a circle of 10 feet 
 radius. 
 
 7. A circular flower bed 20 feet in diameter is surrounded 
 by a walk 5 feet wide. How many square feet of surface 
 does the walk contain ? 
 
 If you have to subtract 100 times 3.1416 from 225 times 3.1416, how 
 can you shorten the work ? 
 
 8. How many square inches are there in the surface of a 
 frame 3 inches wide, around a looking-glass 
 6 inches in diameter ? 
 
 Area = ? x 3.1416. 
 
 9. What is the ratio between the sur- 
 face of the above frame and that of the 
 looking-glass ? 
 
 Indicate operations and cancel. 
 
 10. What is the radius of a circle whose area is 153.9384 
 square yards ? 
 
 11. Find the radius of a circle whose area is 314.16 square 
 inches. 
 
 12. Find the area of a circle whose circumference is 
 15.708 feet. 
 
Measurements. 343 
 
 AREAS OF TRIANGLES. 
 
 442. Written Problems. 
 
 1. What is the area of a triangle whose sides measure 15, 
 16, and 17 inches, respectively ? 
 
 15 
 16 
 
 From the half sum of the three sides subtract 
 each side separately. The square root of the con- 
 17 tinued product of the half sum and the three 
 
 2 )48 remainders will be the area. 
 
 9J. _ 1 K _ Q 
 
 2 4 _ 16 Z s V24 x 9 x 8 x 7 = Vl2^96 = 109.98 
 24 — 17 = 7 Ans. 109.98 square inches. 
 
 2. Find the area in square feet of a triangle whose sides 
 measure 35 feet, 84 feet, 91 feet. 
 
 3. Find the area of a triangle whose sides measure 21, 28, 
 and 35 rods, respectively. 
 
 4. In the following field, AB meas- 
 ures 39 rods; BC, 52 rods; CD, 25 
 rods ; AD, 60 rods ; and the diagonal, 
 AC, 65 rods. Find the area of the 
 field in square rods. 
 
 Find the area of each triangle separately. 
 
 5. Find the area of an isosceles triangle whose base is 30 
 yards, its equal sides measuring 25 yards. 
 
 6. What is the altitude of an isosceles triangle, base, 64 
 feet, equal sides, 68 feet ? Find its area. 
 
 7. Find the area of an equilateral triangle, each side 
 being 6 feet. 
 
 8. Find the area of a right-angled triangle, base, 42 feet, 
 hypotenuse, 70 feet. 
 
 First ascertain the length of the perpendicular. ' 
 
 9. Find the area of an isosceles triangle, altitude, 48 feet, 
 equal sides, 50 feet. 
 
344 
 
 Chapter Six. 
 
 AREAS OF QUADRILATERALS. 
 
 443. Written Problems. 
 
 1. Find the area of a square whose diagonal is 150 rods. 
 Suggestion. — Calling one side of the square S, we have S 2 + S* 
 
 = 150 2 , 150 being the hypotenuse of an isosceles right-angled triangle, 
 the other sides being the sides of the square. S 2 is the required area. 
 Do not find the length of S. 
 
 2. Find the area of the rhomboid (Fig. 1). 
 
 Note. — The altitude is the perpendicular of a right-angled triangle 
 having a base of 7 rods and a hypotenuse of 25 rods. 
 
 3. Of the rectangle (Fig. 2). 
 
 40 rd. 80 yd. 
 
 \ h 
 
 -s 5 
 
 Fig. 1. 
 
 Fig. 2. 
 
 4. Of the rhombus (Fig. 3). 
 
 5. Of the trapezoid (Fig. 4). 
 
 25 rd. 
 
 CO rd. 
 
 60 
 
 40 
 
 Fig. 3. 
 
 Fig. 4. 
 
 6. Of the trapezium (Fig. 5). 
 
 7. Of the rhombus (Fig. 6). 
 
 30 yd. 
 
Quadrilaterals. 345 
 
 8. Find the altitude AB of the preceding triangle (Fig. 7). 
 
 (First find the area.) 
 
 9. Find the diagonal (in rods) of the square whose area 
 is 5 acres. 
 
 10. Find the area of a regular hexagon, composed of six 
 equilateral triangles, each side being 6 inches (Fig. 8). 
 
 Note. — A plane figure bounded by straight lines is called a poly- 
 gon. A three-sided polygon is called a triangle. A four-sided polygon 
 is called a quadrilateral. A hexagon is a six-sided polygon. A reg- 
 ular polygon is one having all its sides and all its angles equal. The 
 square is the only regular polygon of four sides. 
 
 Fig. 9. Fig. 10. 
 
 11. What is the area of the circle circumscribed about the 
 above hexagon (Fig. 9) ? 
 
 12. What is the area of the square inscribed in a circle 
 whose diameter is 10 feet (Fig. 10) ? 
 
 13. Find the area of a square circumscribing a circle 
 whose diameter is 10 feet, and give the ratio of its area to 
 that of the inscribed square. 
 
 14. Find the perimeter of a rectangle 80 yards long, the 
 diagonal of which measures 100 yards. 
 
 15. A square piece of ground containing 40 acres is 
 divided into 4 square fields of 10 acres each. How many 
 rods of fence will be needed to enclose all the fields ? 
 
 16. The area of a triangular plot is 480 square yards. 
 Two of the sides are equal in length, and the third measures 
 32 yards. Find the perimeter. 
 
346 
 
 Chapter Six. 
 
 SURFACES OF PRISMS AND OF CYLINDERS. 
 
 Note. — The pupils should first examine a number of prisms, right 
 and oblique, regular and irregular, triangular, quadrangular, pentago- 
 nal, etc. Right and oblique cylinders should also be at hand. 
 
 444. A prism is a body bounded by plane faces, two of which 
 are equal and parallel polygons, the remaining faces being 
 parallelograms. 
 
 A C H Q 
 
 E 
 
 F 
 
 Fig. 1. 
 
 s^^> 
 
 Fig. 3. 
 
 The two parallel faces of a prism are called its bases. The 
 remaining faces taken together constitute its convex surface. 
 
 In Fig. 1, ABC and DEF are the bases ; in Fig. 2 the bases are 
 GHIJ and KLMN\ in Fig. 3, OPQB 8 and TUVWX. 
 
 The sides AB, CE, etc., GH, IN, etc., QB, OT, etc., are called 
 
 445. Prisms may be either right or oblique. The convex 
 surface of a right prism consists of rectangles. 
 
 Fig. 1 is a right prism ; Fig. 2 is an oblique prism. 
 
 Note. — When a prism is spoken of, a right prism is meant unless 
 the word oblique is used. 
 
 The altitude of a prism is the perpendicular distance be- 
 tween the bases. 
 
 AD, BF, or CE is the altitude in Fig. 1. GYis the alti- 
 tude in Fig. 2. 
 
 446. The number of sides in each base determines the 
 name as triangular (Fig. 1), quadrangular (Fig. 2), pentagonal 
 (Fig. 3), etc. 
 
Surfaces. 
 
 347 
 
 A quadrangular prism whose 
 "bases are parallelograms is called 
 a parallelopipedon. Fig. 4 is an 
 oblique parallelopipedon. Fig. 5 
 is a right parallelopipedon. Any 
 two opposite faces of a parallelopi- 
 pedon may be considered the bases. 
 
 k \ 
 
 
 1 
 
 1 
 1 
 
 
 \[_ 
 
 \ 
 
 Fig. 4. 
 
 Fig. 5. 
 
 447. When the bases are regular polygons, the prism is 
 said to be regular. 
 
 Fig. 1 is a right regular triangular prism j Fig. 2 is an oblique irreg- 
 ular quadrangular prism. 
 
 448. A cylinder is a body having two circular parallel plane 
 faces, and one curved face. 
 
 (TZD 
 
 The plane faces are the bases, 
 face constitutes the convex 
 
 The curved 
 
 Fig. 6. 
 
 surface. 
 
 Cylinders, like prisms, are either right 
 or oblique. The altitude of a cylinder is 
 the perpendicular distance between the 
 
 & 
 
 Fig. 7. 
 
 £2 
 
 Fig. & 
 
 449. Written Problems. 
 
 Note. — The pupils should be encouraged to make 
 cardboard models of the forms studied. 
 
 1. Find the convex surface of a square 
 prism, one side of its base being 4 inches and 
 its height 6 inches. Draw the development. 
 
 Note. — The convex surf ace is the surface exclusive 
 of the bases. 
 
 2. Find the convex surface of a triangular 
 prism, each side of whose base measures 4 
 inches and whose altitude is 6 inches. Draw 
 the development. 
 
348 Chapter Six. 
 
 3 . Find the convex surface of an hexagonal -=^^^ 
 prism, each side of its base being 4 inches and 
 
 its altitude 6 inches. Draw the development. 
 
 4. Can you show that the convex surface 
 
 of a prism is found by multiplying the perim- |||H j^J 
 eter of the base by the altitude (height) ? 
 
 5. Find the convex surface of a cylinder, |j| ||| 
 the diameter of its base being 4 inches and its 
 
 height 6 inches. 
 
 To find the convex surface of a right prism 
 (or cylinder) multiply the perimeter (circumfer- 'llillljjj^^ 
 ence) of the base by the height. 
 
 6. How do you find the entire surface of a prism or 
 cylinder ? 
 
 Note. — The entire surface is the surface including the bases. 
 
 7. What is the entire surface of a cube whose side is 
 7 inches ? Of a cube whose side is 12 inches ? 
 
 8. The entire surface of a cube is 216 square inches. 
 What is the length of one side ? 
 
 Suggestion. — Calling the length of one side L, the area of each 
 face will be £ 2 , and of the six faces, 6 L 2 . Then, 6 L 2 = 216. 
 
 9. The convex surface of a cube is 144 square inches. 
 Find the entire surface. 
 
 How many faces in the convex surface ? 
 
 10. Find the entire surface of a square prism, one side of 
 whose base measures 4 inches, and whose altitude is 6 inches. 
 
 Entire surface = convex surfaces + areas of bases. 
 
 11. The convex surface of a square prism is 600 square 
 feet, the altitude is 15 feet. What is the length of one side 
 of the base ? 
 
 12. The entire surface of a square prism is 1650 square 
 inches. One side of the base measures 15 inches. What is 
 its convex surface ? What is its altitude ? 
 
 Convex surface = entire surface — area of bases. • 
 
Surfaces. 
 
 349 
 
 SURFACES OF PYRAMIDS AND CONES. 
 
 450. A pyramid is a body whose 
 convex surface is made up of triangles 
 having a common vertex, the base of the 
 pyramid being a polygon. 
 
 Pyramids are either right or oblique; 
 regular or irregular ; triangular, quadrangular, pentagonal, etc. 
 
 In a right pyramid, each of the triangles that make up 
 the convex surface is isosceles. When, 
 in addition, the pyramid is a regular 
 one, these triangles will be equal to 
 each other. 
 
 The altitude of any of these equal 
 triangles constitutes the slant height of 
 a right regular pyramid. The altitude of 
 the prism is measured by a line drawn 
 from the apex to the centre of the base. 
 
 AG is the slant height of the square pyramid, Fig. 3. AF is its 
 altitude. 
 
 451. The cone is a body having 
 a single circular base, and a curved 
 convex surface sloping to the apex. 
 
 In the right cone, Fig. 4, HI is the 
 )j slant height, and HK is the altitude. 
 LO is the altitude of the oblique cone, 
 Fig. 4. Fig> 5# 
 
 Fig. 3. 
 
 452. Written Problems. 
 
 1. The convex surface of a square pyramid 
 consists of how many equal triangles ? Find 
 the convex surface when one side of its base 
 measures 4 inches and its slant height (AX) 
 6 inches. 
 
 2. Draw the development. 
 
35° 
 
 Chapter Six. 
 
 To find the convex surface of a pyramid (or cone) multiply 
 the perimeter (circumference) of the base by one-half the slant 
 height. 
 
 3. Find the entire surface of the above pyramid. 
 Entire surface = convex surface + area of base. 
 
 4. Calculate the entire surface of a square pyramid whose 
 slant height is 18 inches, the area of its base being 144 
 square inches. 
 
 5. Draw the developed convex surface of a 
 cone, the diameter of whose base is 4 inches, 
 and whose slant height is 6 inches. 
 
 Calculate the convex surface. 
 
 6. How many square inches of paper would 
 be required to cover the side and the base of a cone 6 inches 
 in diameter at the base, and having a slant height of 10 
 inches ? 
 
 7. Calculate the slant height of a cone whose 
 altitude is 12 inches, the diameter of its base 
 being 10 inches. What is its convex surface ? 
 
 Note. — The slant height is the hypotenuse of a 
 right-angled triangle, the other sides measuring 12 in. 
 and 5 in., respectively. 
 
 8. What is the convex surface of a cone, 
 the diameter of whose base is 6 inches, and its slant height 
 10 inches ? Draw the development. 
 
 6 in. 9. A semicircular 
 
 piece of paper 6 inches 
 
 in diameter is folded 
 
 into a hollow cone 
 
 (without overlapping). 
 What will be the diameter AB of the mouth 
 of the cone (the base) ? What will be the slant height BC? 
 
Volumes. 351 
 
 VOLUMES OF PRISMS AND OF PYRAMIDS; OF CYLINDERS 
 AND OF CONES. 
 
 453. Written Problems. 
 
 Suggestion. — Have the pupils construct of cardboard a hollow 
 square prism of convenient size, and a pyramid having base and alti- 
 tude respectively equal to those of the prism. Let them use sand or 
 water to ascertain how many times the contents of the pyramid must 
 be taken to exactly fill the prism. 
 
 Volume of prism or cylinder = area of base x altitude. 
 
 Volume of pyramid or cone = area of base x \ altitude. 
 
 1. Find the volume of a square pyramid, the area of the 
 base being 9 square feet and the altitude 6 feet. 
 
 1 cu. ft. x 9 x i of 6. 
 
 2. What is the volume of a square pyramid whose alti- 
 tude is 12 inches, one side of the base being 10 inches ? 
 
 3. The base of a prism is a triangle whose sides measure 
 3, 4, and 5 inches respectively. Find the solidity, its alti- 
 tude being 10 inches. 
 
 4. The base of a prism 19 feet high is a rectangle whose 
 sides are 9 feet and 13 feet. How many cubic yards does it 
 contain ? 
 
 5. Find the volume of a prism whose bases are equi- 
 lateral triangles, each side being 4 feet, and the height of 
 the prism being 12 feet. 
 
 6. How many cubic feet are 6 ft. 
 there in a stone roller 6 feet long, 
 8 feet in circumference ? 
 
 7. Find the volume of a cone 
 whose altitude is 18 meters, diame- 
 ter of base 6 meters. 
 
35* 
 
 Chapter Six. 
 
 SURFACE OF SPHERE. 
 
 454. A sphere is a body all 
 points on whose surface are equally 
 distant from the centre. 
 
 The distance from the centre 
 
 to the surface is called the Ffc— 
 
 radius of the sphere. The di- 
 ameter is a line running between 
 two points on the surface and 
 passing through the centre. 
 
 CG } CK, CD, CF, and CJ are radii ; AD and FG are diameters. 
 
 455. If a sphere be cut through at any part, the cut sur- 
 face will be a circle. When the cutting plane passes 
 through the centre of the sphere, the circle is called a great 
 circle; other circles are called small circles. 
 
 FXGG is a great circle ; HYIB and JLEZ are small circles. 
 
 A 
 
 456. Take a wooden hemisphere and drive a tack into the 
 centre of its curved surface. Commencing at the tack, care- 
 fully wind a waxed cord about the curved surface, in the 
 way a boy winds a top. When this surface is exactly cov- 
 ered, cut the cord. 
 
 Wind the same cord around a tack driven into the plane 
 surface of the base of the hemisphere, pressing it closely to 
 the surface. When the latter is entirely covered, just one- 
 half of the cord will be used. 
 
The Sphere. 353 
 
 As a hemisphere is made by passing the cutting plane 
 through the centre of the sphere, its base is a great circle 
 of the sphere. 
 
 The above experiment shows that the surface of the hemi- 
 sphere is equal to that of two great circles of the same sphere. 
 
 457. The surface of a sphere is equal to that of four great 
 circles. 
 
 Since the surface of a great circle of the sphere is R 2 x 
 3.1416, the surface of the sphere is 4 R 2 x 3.1416 = D 2 x 
 3.1416. 
 
 To find the surface of a sphere, multiply the square of the 
 diameter by S.lJf.16. 
 
 458. Written Problems. 
 
 1. Find the surface of a sphere whose radius is 1 inch. 
 
 2. The diameter of a sphere is 2 inches. Find its surface, 
 
 3. What is the surface of a sphere whose circumference 
 is 6.2832 inches ? 
 
 4. At 10 cents a square foot, what will be the cost of gild- 
 ing a sphere 12 inches in diameter ? 
 
 5. Find the ratio between the surface of a sphere 1 foot 
 in diameter, and the convex surface of a cylinder 1 foot high, 
 the diameter of the base 1 foot. 
 
 6. What is the ratio between the surface of a sphere 
 1 foot in diameter, and the entire surface of a cylinder 
 1 foot high, the diameter of the base 1 foot ? 
 
 7. Find the surface of a sphere whose circumference is 
 20 inches. 
 
 8. What is the ratio between the surfaces of two spheres 
 whose diameters are 1 inch and 2 inches, respectively ? 
 
 9. Find the ratio between the surfaces of two spheres 
 whose diameters are 2 feet and 13 feet, respectively. 
 
354 
 
 Chapter Six. 
 
 VOLUME OF SPHERE. 
 
 459. Cut up a sphere (a round potato, for instance) into a number 
 of small pieces, passing the knife in each case through the centre of the 
 
 sphere. Each piece is a solid, having for its base a portion of the sur- 
 face of the sphere, and for its altitude the radius of the sphere. 
 
 When the pieces become very numerous, the base of each may be 
 considered a plane and the solid a pyramid. The volume of each 
 
 pyramid is equal to the base x | altitude ; and the total volume of all, 
 which is the volume of the sphere, is equal to the total surface of all 
 the bases, which is the surface of the sphere, multiplied by | altitude, 
 that is, | radius, or £ diameter. 
 
 Surface of sphere = D 2 x 3.1416, 
 therefore, volume of sphere = D 2 x 3.1416 x £ D = 
 
 ^X 3.1416. 
 
 To find the volume of a sphere, multiply one-sixth of the cube 
 of the diameter by 8.1J/.16. 
 
 460. Written Problems. 
 
 1. Find the volume of a sphere whose radius is 3 inches. 
 
 1 cu. in x 36 x 3.1416. 
 
 2. If the diameter of a sphere is 3 inches, what is its 
 volume ? 
 
The Sphere. 355 
 
 3. What is the ratio between the volumes of two spheres 
 whose diameters are one foot and two feet, respectively ? 
 
 4. Find the ratio between the volume of a sphere 1 foot 
 in diameter, and that of a cube whose side is 1 foot. 
 
 5. The radius of a sphere is 18 inches. What is the cir- 
 cumference of a great circle ? The surface ? The volume ? 
 
 6. What is the weight of an iron cannon-ball 12 inches in 
 diameter, considering the weight of a cubic foot of water as 
 1000 ounces, and considering iron 7.5 times as heavy as 
 water ? 
 
 7. Find the ratio between the volume of a sphere 4 inches 
 in diameter, and that of a cylinder 4 inches in altitude, 
 diameter of base 4 inches. 
 
 Note. — Indicate the volume of each, and cancel. 
 
 8. A man has a cubical block of hard wood, its side 
 measuring one foot, which he wishes made into a sphere one 
 foot in diameter. What decimal part of the block is cut 
 away? 
 
 The volume of the sphere is about what fraction of the 
 volume of the cube ? 
 
 MISCELLANEOUS. 
 
 461. Written Problems. 
 
 1. If a piece of cloth is 20 yards long and f yards broad, 
 how broad is another piece of cloth 12 yards long that con- 
 tains as many square yards as the former ? 
 
 2. An iron beam 16 feet long, 2\ feet broad, and 8 inches 
 thick, weighs 1280 pounds. What is the length of a similar 
 beam whose breadth is Z\ feet, thickness 7£ inches, and 
 weight 2028 pounds ? 
 
 3. What will it cost to carpet a room 22^ feet long by 
 15| feet wide with carpet 2 \ feet wide, costing $1.50 per 
 yard? 
 
3S& Chapter Six. 
 
 4. What is the length of a box 6| feet wide and 1\ feet 
 high, that will exactly contain 12 boxes 4£ feet long, 2>\ feet 
 wide, and 2\ feet deep ? 
 
 5. What is the value, at $120 per acre, of a square field 
 whose side is 35.25 chains? 
 
 10 square chains = 1 acre. 
 
 6. Find the capacity, in bushels, of a bin 22 feet long, 
 14 feet wide, 12 feet high ? 
 
 7. How many gallons will a tank hold, its dimensions 
 being 4 ft. 1 in. by 3 ft. 8 in. by 2 ft. 3 in. ? 
 
 8. How many square yards are there in the walls and 
 the ceiling of a room 21 feet long, 18 feet wide, 12 feet 
 high? 
 
 9. A tank 5% feet by 6 feet by 7 feet can be emptied by 
 two pipes, one of which discharges 9 gallons per minute and 
 the other 7 gallons per minute. How long will it take 
 each to empty the tank ? How long will it take both 
 together ? 
 
 10. A parlor is 18 feet long, 15 feet wide. Make a dia- 
 gram, showing how carpet 27 inches wide can be laid without 
 cutting the carpet lengthwise. Which would be the better 
 way to lay carpet 30 inches wide in the above room ? 
 
 11. Calculate the number of running yards of carpet 30 
 inches wide needed for the floor of the above room, including 
 4 i yards wasted in matching the pattern. 
 
 Find the cost of carpeting the room at 95 cents per 
 running yard for carpet, 5 cents per square yard for lining, 
 and 10 cents per running yard for sewing and laying. 
 
 12. A room is 18 feet wide, 24 feet long, and 9 feet high. 
 There are two doors 4 feet wide, 7-J- feet high ; two windows 
 4 feet wide, 6 feet high ; and a fireplace 5 feet square. How 
 many square feet of plastering will there be on the walls 
 
Miscellaneous. 357 
 
 and ceiling, deducting for a baseboard 12 inches wide ? How 
 many running feet of baseboard will be needed ? 
 
 Draw "development" of the above room, showing the four 
 walls and the ceiling, and locating the doors, the windows, 
 and the baseboard. 
 
 Do not use baseboard where it is not required. 
 
 13. At the rate of $1400 for a pile of lumber 25 feet 
 long, 20 feet wide, 10 feet high, what is the value of a pile 
 50 feet long, 40 feet wide, 20 feet high ? 
 
 14. If it costs $ 14 to paint the walls and the ceiling of a 
 room 25 feet long, 20 feet wide, and 10 feet high, what will 
 it cost to paint the walls and the ceiling of a room 50 feet 
 long, 40 feet wide, and 20 feet high ? 
 
 15. Measure accurately the interior dimensions of a quart 
 or a pint cup, and calculate its volume. 
 
 Note. — How many cubic inches in a quart, liquid measure ? 
 
 462. Circular Measure. 
 
 60 seconds (") 1 minute (*). 
 60 minutes 1 degree (°). 
 
 360 degrees 1 circle. 
 
 16. If the equatorial circumference of the earth is 25,000 
 miles, how many miles apart are two places on the equator, 
 the distance between them being 20° ? 
 
 20° = J; circle. 
 
 17. What is the length of a degree on a circle whose 
 diameter is 18 feet ? 
 
 18. The 60th parallel of latitude is a circle one-half as 
 long as the equator. How many miles due east of Christi- 
 ania is St. Petersburg, both situated on this parallel, the 
 former being 10° east of Greenwich, and the latter 30° 
 east? 
 
358 Chapter Six. 
 
 LONGITUDE AND SOLAR TIME. 
 
 Note. — This topic should be taught in connection with the study 
 of Mathematical Geography. The globe should be used to show the 
 pupils that all places on the same meridian have the same solar time, 
 that a difference in longitude of 15 degrees produces a difference in 
 time of 1 hour, and that the more easterly of two places has the later 
 time. 
 
 463. Preliminary Exercises. 
 
 1. The difference in time being 1 hour for each 15 degrees, 
 find the difference in longitude between two cities differing 
 in solar time 3 hours. 
 
 2. Two places differ in longitude 60 degrees. What is 
 their difference in solar time ? 
 
 3. London is 75° east of Philadelphia. When it is 1 
 o'clock at Philadelphia, what is the time at London ? 
 
 4. When it is 2 p.m. at London, what is the time at Phila- 
 delphia ? 
 
 5. How many degrees of longitude correspond to a 
 difference of 3 hr., 40 min. in solar time? 
 
 6. What is the difference in longitude between Phila- 
 delphia, 75° west longitude, and St. Petersburg, 30° east 
 longitude ? 
 
 7. Washington is in 77° west longitude, and uses "stand- 
 ard time," that is, the time of 75° west longitude. What 
 is the difference between the correct time at Washington 
 and its clock time ? 
 
 8. A town in 84° west longitude uses standard time, 
 that of 90. What is the correct time when the clocks are 
 striking 12, noon ? 
 
 9. Chicago is 87° 35' west of Greenwich. Is it earlier 
 or later than noon at Chicago when it is noon at Greenwich ? 
 Why? 
 
Standard Time. 
 
 359 
 
 STANDARD TIME. 
 
 464. In 1883, the railroads of the United States adopted 
 a system of dividing the country into four time sections, 
 each of 15° longitude. The 75th meridian west of Green- 
 wich, which passes between New York and Philadelphia, 
 was selected as the starting-point. The section governed 
 by the time of this meridian, called eastern time, included 
 the territory between the Atlantic coast and a line drawn 
 through Detroit, Pittsburg, Wheeling, Parkersburg, Hunt- 
 
 ington, Bristol, Tenn., Augusta, G-a., and Charleston, these 
 cities being the termini of important railroads. Central 
 time is governed by the time of the 90th meridian, and is 
 used by the section west of Detroit, etc., to Bismarck, North 
 Platte, Dodge City, etc. The next section which takes the 
 time of the 105th meridian, called mountain time, extends 
 to Helena, Ogden, and the western boundary of Arizona. 
 The rest of the country to the Pacific Ocean takes the time 
 of the 120th meridian, called Pacific time. 
 
360 Chapter Six. 
 
 SOLAR TIME. 
 465. Written Exercises. 
 
 1. Find the difference between the sun time of London 
 and that of Chicago, longitude 87° 35' west of London. 
 
 A difference of 15 degrees of longitude makes a difference of 1 hour ; 
 of 16 minutes of longitude, a difference of 1 minute ; of 15 seconds of 
 longitude, a difference of 1 second. 
 
 If 1 degree of longitude made a time difference of 1 hour, the differ- 
 ence in time between London and Chicago would be 87 hr. 35 min. ; 
 as it takes 15 degrees to make a 
 
 difference of an hour, the difference 15 )87 hr. 35 min. 
 of time between London and Chicago 5 hr. 50 min. 20 sec. 
 
 is ^ of 87 hr. 35 min. Dividing, there- 
 fore, 87 hr. 35 min. by 15, we get the time difference as 5 hr. 50 min. 
 20 sec. 
 
 To find the time difference, divide the longitude difference 
 etopressed as hours, minutes, and seconds, by 15. 
 
 t. When it is midnight at London, what is the sun time 
 at Chicago ? 
 
 Since the more easterly place has the later time, it is 5 hr. 50 min. 
 20 sec. before midnight at Chicago. 12 hr. (p.m.) — 5 hr. 50 min. 20 
 sec. = 6 hr. 9 min. 40 sec. p.m. Ans. 
 
 3. Two places differ in longitude 37° 18V What is their 
 difference in solar time ? 
 
 4. Find the difference in longitude between two places 
 differing in solar time 3 hr. 44 min. 
 
 Multiply 3° 44' by 15. 
 
 To find the longitude difference, multiply by IS the time 
 difference expressed as degrees, minutes, and seconds. 
 
 5. Find the difference in sun time between two places in 
 longitude 74° 31' and 93° 14' west of Greenwich, respec- 
 tively. 
 
Solar Time. 361 
 
 6. When it is noon at a place 11° east of Greenwich, it is 
 1 : 30 p.m. at another place. Find the longitude of the latter 
 place. 
 
 Note. — Owing to the general use of standard time by civilized 
 countries, problems in longitude and time have no practical value 
 except for navigators. The following problems should be worked only 
 after more important topics have been completed. 
 
 Note. — The word "time" in the following problems means 
 "mean solar time." 
 
 7. Given the longitude of A as 95° east, and that of B as 
 74° east, and the time at A as 1:30 p.m., to find the time at B. 
 
 Since the latitude of B has no bearing upon its time, both places 
 may be located upon the same line running east and west. 
 
 Time difference = ? hours. 
 
 Time=? Time 1:30 p.m. 
 
 B A 
 
 West 1 1 1 East 
 
 0° 74° 95° 
 
 Longitude difference = 21°. 
 
 Locate the prime meridian (that of 0°), then the meridians of 74° 
 and 95° east. Mark above the last two the names of the places, B and 
 A. Write above A its given time, 1 : 30 p.m. 
 
 To find the time at B, we must find the difference of time between 
 B and A. The difference in longitude is 95° - 74° = 21°. The dif- 
 ference in time is 21 hours -f- 15. 
 
 Note. — Remember that the more easterly of the two places has the 
 later time. 
 
 8. A is situated in 71° west longitude, B in 107° west 
 longitude. What time is it at B, when it is noon at A ? 
 
 Time difference = ? 
 Time ? 12 m. 
 B A 
 
 West 1 1 1 East 
 
 107° 71° 0° 
 
 Longitude difference = ? 
 
362 Chapter Six. 
 
 9. Find the longitude of B, whose time is 8 : 10 : 30 a.m., 
 when it is 7 : 15 a.m. at A, whose longitude is 156° 48' west. 
 
 Time difference = ? 
 7:15 a.m. 8:10:30 a.m. 
 
 A B 
 
 West 1 1 1 East 
 
 156° 48' Longitude = ? 0° 
 
 Longitude difference = ? 
 
 Since B has the later time, its location is east of A. The difference 
 in time, being nearly an hour, shows the difference in longitude to be 
 nearly 15°. Find the exact difference. Is it to be added to 166° 48' 
 or subtracted from it, to give the longitude of B ? 
 
 10. When it is 2:40 a.m. at A, in 57° 24' west longitude, 
 it is 10 a.m. at B. Find the longitude of B. 
 
 Time difference = 7\ hours. 
 
 2:40 a.m. 10 a.m. 
 
 A B 
 
 West 1 1 1 East 
 
 57° 24' 0° Longitude = ? 
 
 Longitude difference = 15° x7| = 110°. 
 
 If we go 110° eastward from A, we shall reach the prime meridian 
 after going how many degrees and minutes ? How many more degrees 
 and minutes must we travel to reach B? Is B in east or in west 
 longitude ? 
 
 11. When it is noon at B, what is the time at A, the 
 former being in longitude 44° east, and the latter in longi- 
 tude 57° west ? 
 
 Time difference = ? 
 Time = ? 12 m. 
 
 A B 
 
 West 1 1 1 East 
 
 67° 0° 44° 
 
 Longitude difference = 101°. Why ? 
 
Solar Time. 
 Find the longitude or the time : 
 
 363 
 
 Longitude of A. 
 
 12. 63° east 
 
 Time at A. 
 9 A.M. 
 
 13. 57° 25' east ? 
 
 14. 156° 48' west 3:15 p.m. 
 
 15. ? 
 
 16. 2° 15' west 
 
 17. 27° 10' east 
 
 18. ? 
 
 19. 74° 56' west 
 
 20. 4' 30" east 
 
 21. ? 
 
 11:42 a.m. 
 6:53 a.m. 
 9 
 
 4:10 p.m. 
 
 3:50 a.m. 
 
 8:47 a.m. 
 
 10:30 p.m. 
 
 Longitude of B. 
 
 54° east 
 83° 20' east 
 ? 
 
 56° 25' west 
 67°: 48' east 
 27° 10' west 
 18° 4' east 
 9 
 
 90° 15' west 
 32° 30' east 
 
 Time at B. 
 9 
 
 1 : 45 p.m. 
 
 4:10 p.m. 
 
 1:27 p.m. 
 
 9 
 
 12 m. 
 11:30 a.m. 
 
 11 A.M. 
 9 
 
 6:48 p.m. 
 
 REVIEW. 
 466. Oral Problems. 
 
 1. At what per cent will $12, in 3 yr. 4 mo., amount to 
 $14? 
 
 2. What will be the cost of a building lot 100 feet long 
 and 50 feet wide at 50 ^ a square foot ? 
 
 3. A horse was sold for $90, at which price 12^% was 
 gained. What per cent would have been gained by selling 
 him for $100? 
 
 4. What is the premium for insuring $6000 on my 
 house at 1\% ? 
 
 5. How many cubic inches in a ten-inch cube ? 
 
 6. Bought 2 chairs at $1.25, one wash-tub for $1.50, 
 1 table for $3.00, and 5 dozen glasses at 48^ a dozen. Gave 
 a ten-dollar bill in payment. How much change did I 
 receive ? 
 
364 Chapter Six. 
 
 7. My desk is 1£ feet long, and 1 foot wide. How many 
 inches around it ? 
 
 8. If a man spends 50^ a day during April, May, and 
 June, what does he spend in the three months ? 
 
 9. A grocer bought 15 barrels of flour at $5 a barrel. 
 At what price must he sell them to gain $ 36 ? 
 
 10. Seven-eighths of James's vacation will be equal to 
 seven-ninths of yours ; yours will be 63 days. How many 
 will his be ? 
 
 11. A man sold two cows for $30 each. On one he 
 gained 25% ; on the other he lost 25 %. Did he gain or lose, 
 and how much ? 
 
 12. What principal, in three years and 4 months, at 6%, 
 will give $40 interest? 
 
 Note. — To the following ten problems the wrong answers are very 
 frequently given. 
 
 13. Sold a horse for $ 250, losing $ 50. What is the loss 
 per cent ? 
 
 14. If 3 boys solve 3 problems in 3 minutes, how long 
 will it take 6 boys to solve 6 problems ? 
 
 15. Two boys go fishing; one brings 7 cakes for lunch, 
 the other brings 5 cakes. A third boy joins them at noon, 
 and pays 12^ for his share of the meal. How should the 
 first two divide the money received ? 
 
 16. If 100 per cent is gained by selling an article for $ 1, 
 how much would be gained by selling it for $ 2 ? 
 
 17. A boy had a slate 5 inches by 7 inches. He buys 
 one twice as large. Give the dimensions of the new slate. 
 
 18. A man wishes to put up on the front of his lot a 
 fence 30 feet long. If the posts are 6 feet apart, how much 
 will they cost at 25^ each? 
 
Review. 365 
 
 19. One-half the money taken in by a newsboy is profit. 
 What per cent does he make ? 
 
 20. 50 per cent of a number multiplied by 30 per cent of 
 the same number equals 60. What is the number ? 
 
 21. Three-fourths per cent of a number is 90. What is 
 the number ? 
 
 22. An importer receives some cases of goods numbered 
 consecutively. How many cases are there, if the number of 
 the first is 28, and of the last 75 ? 
 
 467. Written Problems. 
 
 1. What is the profit on 9 boxes of oranges, each con- 
 taining 20 dozen, bought at $ 1.10 per hundred and sold at 
 the rate of 18 for 25^? 
 
 2. How long will it take a train to go 176 miles at the 
 
 rate of 3520 feet per minute ? 
 
 i 
 
 3. If .0375 of an acre of land is worth $ 9, what is -^ 
 
 acre worth ? 
 
 4. At £ 1 Is. Id. per barrel, how many barrels of flour 
 can be bought for £ 161 17s. 6d. ? 
 
 5. If 580 tiles, each 6 inches square, will cover a certain 
 area, how many tiles, each 4 inches long and 3 inches wide, 
 will be needed to cover the same area ? 
 
 6. A man receives $ 1500 commission on his yearly 
 sales. What is the amount of his sales if he is allowed \ 
 per cent commission ? 
 
 7. At what rate per cent will $360 produce $3.06 in- 
 terest in 2 mo. 12 da. ? 
 
 8. Find the square root of 25.00400016. 
 
 9. What will be the capacity, in gallons, of a tank 9 ft 
 long, 6 ft. 8 in. wide, and 6 ft. 5 in. deep ? 
 
366 Chapter Six. 
 
 10. What decimal multiplied by 312.5 will give the sum 
 of I, ^ t> -09375, and 2.46 ? 
 
 11. A dealer bought a lot of coal $ 4.95 per ton. What 
 was the total cost if he gained $ 142.50 by selling it at 
 $ 5.25 per ton ? 
 
 12. Find the value of ?i±iA_ 1 f 64. 
 
 1| x 3 ¥ T 
 
 13. The front wheel of a wagon measures 13 feet in cir- 
 cumference. What is the distance travelled in miles, rods, 
 yards, etc., when the wheel has made 527 revolutions ? 
 
 14. Write in words .349, 300.049, $h% 300^^. 
 
 15. If a bar of silver weighing 4 lb. 6 oz. 12 pwt. is worth 
 £6 14s. 2d., what is the value (in English money) of a 
 similar bar weighing 7 lb. 9 oz. 12 pwt. ? 
 
 16. A and B form a partnership. A furnishes $5000; 
 B, $ 10,000. During the year A draws $ 1500 of the profits 
 and B draws $1000. At the end of the year the entire 
 business is disposed of for $ 20,000. What amount should 
 each receive ? 
 
 17. What per cent is gained on an article bought for 20 
 per cent less than its value and sold for 20 per cent more 
 than its value ? 
 
 18. A person loans $ 750 to M and $ 1200 to N at the 
 same rate. From the latter he receives half-yearly $ 9 more 
 interest than from the former. What is the annual rate of 
 interest ? 
 
 19. A 4-months note for $ 375, drawn March 19, was dis- 
 counted at a bank June 4. Find the proceeds. Eate, 6%. 
 
 20. M can do a piece of work in 4 days, N can do it in 5 
 days, O in 6 days. How long will it take the three to- 
 gether to do the work ? 
 
 1 day + (£ + | + $). Analyze. 
 
Stocks. 367 
 
 STOCKS. 
 
 468. Some undertakings, such as the construction of a 
 railroad, the building and equipment of a factory, the de- 
 velopment of a mine, and the like, require more money than 
 any individual may care to risk. It then becomes necessary 
 to secure the cooperation of a number of persons. 
 
 The people of a certain town desire to build a street rail- 
 road, the construction and equipment of which will require 
 $50,000. The projectors organize a company. If it is 
 desired to interest people of small means, the required capi- 
 tal may be divided into shares of $10 each, making the total 
 number of shares 5000. If the shares are fixed at $100 
 each, there will be 500 shares. 
 
 To every purchaser of shares, a certificate is issued, coun- 
 tersigned by the officers of the company, setting forth the 
 amount of capital, the total number of shares, and the num- 
 ber issued to the holder of the certificate. 
 
 At certain fixed periods, quarterly, semi-annually, or an- 
 nually, the directors of the company determine what part of 
 the profits shall be distributed to the stockholders, the 
 remainder being reserved for new cars, extension of the 
 road, etc. The profits thus distributed are called dividends. 
 
 469. Written Problems. 
 
 1. A company is organized with a capital of $50,000, 
 divided into shares of $ 100 each. What part of the stock 
 is held by the owner of 10 shares ? 
 
 2. If dividends of $2000 are distributed at the end of 
 six months, how much should the holder of 10 shares 
 receive ? 
 
 3. The company announces the dividend as a certain per 
 cent of the capital. What per cent dividend is declared in 
 this case ? To what per cent per year is it equal ? 
 
368 Chapter Six. 
 
 4. Mr. H. has $4500 in the savings bank, on which he 
 receives 4 per cent interest. He gives this amount for 30 
 shares of the stock. What price does he pay per share? 
 What per cent of the par value ? 
 
 5. If the next semi-annual dividend is 4%, how much 
 more income does Mr. H. receive from the stock than he 
 would obtain from the savings bank ? 
 
 6. What per cent has Mr. H. received for six months on 
 his investment of $4500 ? 
 
 7. If Mr. H. sells the 30 shares at $164.50 per share, 
 how much more does he receive for it than it cost him ? 
 
 470. Stocks are generally bought and sold by brokers, 
 who charge, as a rule, J% of the par value for buying or for 
 selling. The prices of the stocks as given in the news- 
 papers are generally a percentage of the par value. Thus, 
 the New York quotation of Pennsylvania E..R. on March 
 4, 1903, is 151-J-. This means that the shares of the Penn- 
 sylvania R.R. sold for $50x1.51^, or $75.75, the par 
 value being $50. The Philadelphia papers of the same 
 date, however, quote the stock at 75f , it being the practice 
 in that city to give the price per share. 
 
 471. "Written Exercises. 
 
 1. Find the cost of 240 shares Anaconda Copper Mining 
 Co., par value $25, at 134|, brokerage \%. 
 
 Cost = $25 x 240 x (1.34$ + .00J). 
 
 To find the cost, multiply the face value of the given number 
 of shares by the rate plus the brokerage. 
 
 2. How much brokerage is paid by the buyer of 275 
 shares bank stock, par value $100, brokerage £%? 
 
 | % of $100 x276. 
 
Stocks. 369 
 
 3. Paid $11,445 for 120 shares Cleveland, Cincinnati, 
 Chicago, & St. Louis, par value $100, brokerage £%. What 
 was the value of the stock per share ? 
 
 The brokerage on 120 shares, par value $ 100, is £% of $ 12,000, or 
 $15. The cost of the stock is, therefore, $11,430. Dividing by the 
 number of shares gives the value per share. 
 
 ($ 11,445 - } % of [$ 100 x 120]) -s- 120. 
 
 4. Bought 150 shares Evansville and Terre Haute at 
 69f, brokerage \%, paying for it $5212.50. What is the 
 par value per share ? 
 
 The cost of each share is $ 5212.50 -*- 150. Divide this cost by the 
 rate, including the brokerage, .69| + .00$. 
 
 ($ 5212.50 -r- 150) + (.69f + .00$). 
 
 5. A broker sells for a customer 200 shares stock, par 
 value $25, at 102 J. If he retains -$% brokerage, how much 
 does he pay over to the former owner of the stock ? 
 
 6. A man buys 60 shares bank stock, par value $ 100, at 
 450, no brokerage. If the annual dividend is 18%, what is 
 his income therefrom ? What per cent does he receive on 
 his investment ? 
 
 Note. — Dividends are based upon the par value. 
 
 7. A manufacturing corporation makes $20,000 a year 
 over all expenses. The stock consists of 4000 shares, par 
 value $ 50. What rate of dividend can be declared ? 
 
 What per cent on his investment does a man receive who 
 has bought his stock at 175, no brokerage ? 
 
 8. A capitalist bought 360 shares stock, par value $ 25, 
 at 168£. He paid therefor, including brokerage, $ 15,176.25. 
 What was the rate of brokerage ? 
 
 9. A broker sold 250 shares, par value $100, at 107f. 
 He deducted brokerage and paid over the proceeds, amount- 
 ing to $ 26,875. Find the amount of the brokerage and the 
 rate per cent. 
 
370 Chapter Six. 
 
 10. A woman invests $ 35,050 in stock at 175, brokerage 
 J%. If the annual dividends are 7^-%, what is her income 
 from the investment ? 
 
 11. Which investment will pay better, one in a gas com- 
 pany paying 6% dividends annually, their stock selling at 
 150, the other in a bank paying 7% dividends annually, 
 stock selling at 175 ? 
 
 12. What annual dividend should be declared on railroad 
 stock bought at 125, so that the buyer will receive 4% per 
 annum on his investment ? What semi-annual dividend ? 
 
 13. What will be the cost of 17 shares of canal stock, par 
 value $ 50, at 93 J, and 143 shares gas stock, par value $ 10, 
 atl02f? 
 
 Note. — An examination of the prices of stocks as given in the 
 newspapers will show that the rate of dividends constitutes but one 
 consideration influencing buyers. The following prices were offered 
 March 4, 1903, f cr stocks of four banks, respectively, each of which 
 paid 6 per cent dividends annually ; 185, 245, 390, 685. Purchasers of 
 shares of the last three banks evidently hoped for larger dividends in 
 the immediate future. 
 
 . The values of bonds depend in the first instance upon the character 
 of the corporation issuing them, then upon the rate of interest and the 
 length of time before redemption. United States bonds bring the 
 highest prices, as buyers have no fear of the failure of the government 
 to keep its promises. The following are the prices obtained at the last 
 sales reported to March, 1903 : 
 
 
 Date of 
 
 
 
 Rate. 
 
 Redemption. 
 
 Price Paid. 
 
 Last Sale. 
 
 U. S. 2's 
 
 1930 
 
 108f 
 
 Nov. 14, 1902 
 
 U. S. 4's 
 
 1907 
 
 no* 
 
 Feb. 4, 1903 
 
 U. S. 4's 
 
 1925 
 
 136 
 
 Feb. 26, 1903 
 
 U. S. 5's 
 
 1904 
 
 103 
 
 Feb. 23, 1903 
 
 The 5 per cent bonds, although bearing the highest rate of interest, 
 bring only 103, as they will be redeemed at par a little more than a 
 year after they are bought. The purchaser, who paid $ 103 for a bond, 
 will receive for it in 1904 only $100, with about $5 interest, his net 
 profit for the year being $2 on an investment of $ 103. 
 
Bonds. 371 
 
 BONDS. 
 
 472. A bond is a form of interest-bearing note issued by a 
 corporation. 
 
 A coupon bond is one containing certificates of interest 
 which are cut off and presented for payment as interest 
 becomes due. A 10 years' U. S. coupon bond has 40 cou- 
 pons, one for each quarter-year's interest. Upon each is 
 engraved the date when due, and the sum payable, which 
 is $ 10 in the case of a $ 1000 f our-per-cent bond. 
 
 A registered bond contains no coupons, a check for the 
 interest being mailed to the owner, whose name is registered 
 on the books of the corporation. 
 
 473. Written Exercises. 
 
 1. A railroad company needing more money to extend its 
 road, issues bonds bearing interest at 4%. If these bonds 
 are sold at 95, what rate of interest on the money invested 
 does the owner of a bond receive ? 
 
 For each $95 invested the owner receives $4 interest. The rate is 
 4 -4- .95. 
 
 To find the rate on the investment, divide the rate of interest 
 by the rate paid for the bond, including brokerage, if any. 
 
 2. Find the cost of 20 one thousand dollar bonds at 120£, 
 brokerage §-%. 
 
 3. If the foregoing bonds bear interest at the rate of 6%, 
 what is the annual income ? What rate per cent annually 
 is received on the sum invested ? 
 
 4. A man desires to secure an annual income of $ 650 for 
 his daughter. What is the face value of 5% bonds necessary 
 to produce this income? What will be the cost of 5% 
 bonds of Denver & Rio Grande at 107, brokerage l%? 
 
372 Chapter Six. 
 
 5. A person desirous of obtaining a semi-annual income 
 of $900 is offered Central Pacific 4's at 99 J, Chicago & Alton 
 3's at 83 J, or Western Union 4£'s at 104 J, no brokerage in 
 any case. Find the difference between the smallest and the 
 largest outlay necessary to secure the desired income from 
 these bonds. 
 
 Note. — 4's means bonds paying 4 per cent interest per year. 
 
 6. How much money must be invested in the U. S. 2's to 
 yield a quarterly income of $225, bonds selling at 108|, 
 brokerage \°/ ? 
 
 7. An owner of 6 per cent bonds sells them at the market 
 quotation of 118, and invests the proceeds in 4£ per cent 
 bonds. The latter investment yields him the same income as 
 the former. What did he pay per hundred for the 4^- per 
 cent bonds, no brokerage ? 
 
 8. A, having a farm of 109 acres, which rents for $ 681.25 
 above taxes, etc., sells the same for $200 per acre, and 
 invests the proceeds in U. S. 2's @ 108£%, brokerage |%. 
 Will his yearly income be increased or diminished, and how 
 much? 
 
 9. What is the difference in the rate of income obtained 
 from an investment in U. S. 2's at 109|, and one in U. S. 4's 
 at 137f, brokerage \% in each case? 
 
 Note. — In calculating the rate of interest in the foregoing exam- 
 ples, the time of the redemption of the bonds is omitted from consid- 
 eration. In the following example, however, the term of the bond is 
 made an element in the computation. The holder of it has received 
 $30 in interest, and he is paid $100 for the bond. Ignoring the mat- 
 ter of compound interest, the question becomes: At what rate will 
 $ 104 amount in 6 years to $ 130 ? * 
 
 10. Mr. Tower pays $104 for a $100 five per cent bond. 
 At the end of six years the bond is redeemed at par. What 
 rate of interest does he receive on his investment of $ 104 ? 
 
Exchange. 373 
 
 DOMESTIC EXCHANGE. 
 
 474. Arthur S. Somers, of Memphis, Tenn., wishes to pay- 
 John R. Thompson, of The City of New York, $ 3475.86. If 
 Mr. Somers sends a check, drawn on his Memphis bank, 
 Mr. Thompson will be charged a certain sum by his New 
 York bank for collecting the amount of the check, and he 
 will thus receive somewhat less than the sum due him. Mr. 
 Somers, therefore, buys from J. E. Washington, a Memphis 
 banker, who has funds in a New York bank, the following 
 
 SIGHT DRAFT. 
 
 f 3475 J£6^. Memphis, Tenn., Aug. 9, 1904. 
 
 At sight, pay to the order of John R. Thompson Three 
 Thousand Four Hundred Seventy-five -fa Dollars, value 
 received, and charge to the account of 
 
 To Chemical Bank, JOSEPH E. WASHINGTON. 
 
 The City of New York. 
 
 Mr. Somers is charged for this draft a premium of $ 1.50 
 
 per $ 1000; that is, he pays Mr. Washington $1001.50 for 
 
 each $ 1000. The cost of the draft is, therefore, $ 3475.86 
 
 X 1.0015, or $ 3481.07. 
 
 475. Exchange is at a premium when the cost of a sight 
 draft is greater than its face; it is at a discount when the 
 cost of a sight draft is less than its face. 
 
 476. Mr. Thompson could collect the sum due him by 
 making a draft on Mr. Somers as follows : 
 
 TIME DRAFT. 
 $ 3475_8 T 6 r . New York, Aug. 9, 1904. 
 
 At three days' sight pay to the order of The National 
 Bank of Commerce Three Thousand Four Hundred Seventy- 
 five and ■££$ dollars, value received, and charge to the 
 account of JoHN R# Thompson. 
 
 To Arthur S. Somers, 
 Memphis, Tenn. 
 
374 Chapter Six. 
 
 Mr. Thompson deposits the draft in the National Bank of 
 Commerce for collection. This bank forwards it to a Mem- 
 phis bank. The latter notifies Mr. Somers. If he wishes 
 to pay the draft at the expiration of three days, he writes 
 across the face in red ink, "Accepted," with the date, 
 " Aug. 11, 1904/' and adds his signature. Aug. 14 he pays 
 the money to the Memphis bank, which notifies the Bank 
 of Commerce, and the sum is placed to the credit of Mr. 
 Thompson, less the cost of collection. 
 
 477. A sight draft is payable upon presentation, except 
 in those states allowing " days of grace." A time draft is 
 one payable a specified number of days after acceptance. 
 In some states three additional " days of grace " are 
 allowed. 
 
 478. Written Exercises. 
 
 1. Find the cost of a Boston draft on New York for 
 $ 1875, at 12 ^ discount per $ 1000. 
 
 Face $ 1875. 
 
 Discount $ 1875 x .00012 .225 
 
 $1874.775 
 Ans. $ 1874.78. 
 
 To find the cost of a sight draft, add the premium to the 
 face, or subtract the discount from the face. 
 
 2. What will a St. Louis merchant pay for a draft on 
 New York for $ 2460.53, at 50 ^ premium per $ 1000 ? 
 
 3. At \°lo premium, find the cost of a sight draft for 
 $1843.60. 
 
 4. At 75^ discount per $1000, how much will cost a 
 sight draft on Milwaukee for $946.75? 
 
 5. Paid $ 632.18 for a sight draft on Milwaukee. What 
 was the face of the draft, the discount being ■&%? 
 
Exchange. 375 
 
 BILLS OF EXCHANGE. 
 
 479. Bills of exchange are either domestic or foreign. A 
 domestic bill of exchange is called a draft, the term bill of 
 exchange being generally applied only to foreign bills. 
 
 480. Fred Johnston owes John Ahern & Co., of London, 
 £ 180 17s. 6d. He buys from John Cottier & Brother a 
 bill of exchange drawn on their London correspondent. 
 The bill is drawn in duplicate, one being sent by Mr. 
 Johnston to John Ahern & Co., and the other being retained 
 by the former to send in case of the loss of the first. When 
 either is paid the other becomes of no value. 
 
 The following is the form of the first of a set of exchange. 
 
 Exchange for £ 180 17s. 6d. New York, Dec. 14, 1903. 
 
 Sixty days after sight of this First of Exchange (Second 
 unpaid), pay to the order of John Ahern & Co., One Hun- 
 dred Eighty Pounds Sterling, Seventeen Shillings Six 
 Pence, value received, and charge the same to account of 
 
 To James Lennon & Co., John Cottier & Brother. 
 London. 
 No. 39. 
 
 Upon receipt of this bill, John Ahern & Co. present it 
 for acceptance. They receive the money sixty days there- 
 after. 
 
 481. Written Exercises. 
 
 1. Find the cost of the above bill at $4.87 per pound. 
 
 £200 = $974.00 
 20 = 97.40 
 £ 180 = $ 
 10s. = 2.435 ££ 
 5s. = 
 2s. 6d. = 
 
376 Chapter Six. 
 
 2. What is the cost of a cable transfer of £251 lis. 9&, 
 at $4.88£ per pound? 
 
 £250 = ^1221.25 \ of £1000 
 
 1 = 
 
 10s. = 
 
 Is. = 
 
 6d.= 
 
 3d. = 
 
 The newspapers give quotations of foreign exchange for sight and 
 60-day bills, also for cable transfers. 
 
 482. The New York quotations for French exchange give the 
 number of francs for $1. 
 
 Paris cable transfers 5. 16 \ @ 5.15f. 
 
 Paris bankers' 60 days 5.18f@5.18|. 
 
 Paris bankers' sight 5.16| @ 5. 16 J. 
 
 The quotations for German exchange give the value in U. S. money 
 of 4 Reichmarks (or marks). 
 
 Reichmarks (4) 60 days 95J @ 95J. 
 
 Reichmarks (4) sight 95| @ 95$. 
 
 3. Find the cost of a sight bill on Paris for 1000 francs, 
 at 5.16J francs for $ 1. 
 
 4. Find the cost of a 60-day bill of exchange on Berlin 
 for 1874.35 marks, at 95J fi for 4 marks. 
 
 5. What will be the face in marks of a sight bill of ex- 
 change on Berlin that can be bought for $ 1000, at 95 J ^ for 
 4 marks ? 
 
 6. A New York merchant pays $ 1637.50 for a 60-day 
 bill on Paris. What is the face of the bill, the rate of ex- 
 change being 5.18J francs for $ 1 ? 
 
Exchange. 377 
 
 7. At $4.88 per pound, what will be the face of the 
 sight bill on London that can be bought for $ 1500 ? 
 
 18750 * £307 7s., etc. 
 
 g|| = 18750 61 )18750 
 
 i& 61 _450 
 
 to £23 remainder 
 
 20_ 
 460s., new dividend 
 
 8. Bought goods in London amounting to £ 437 5s. 10c?. 
 less 4%. How much do I pay in Boston for a sight bill of 
 exchange at $ 4.88^ to settle the account ? 
 
 9. What will be the cost in Chicago for a 60-day bill on 
 Paris that will pay for the following articles ? Rate, 1 franc 
 -10} A 
 
 18 pieces silk, 44 meters each, at 25 francs per meter, 
 less 1\% 
 
 3 pieces of cloth, 50 meters each, at 20 francs per meter, 
 less 5%. 
 
 Packing charges, 60.50 francs. 
 
 10. I wish to send a sight bill of exchange on Berlin in 
 payment of the following invoice : 
 
 4 cases musical instruments, amounting to 3598.60 marks, 
 less 10, 5, and 2\%. 
 
 Freight to Hamburg, 165 kilos, at 4.80 marks per kilo. 
 At 95J ^ for 4 marks, what will be the cost of the bill of 
 exchange ? 
 
 11. If the rate of exchange is 50^ discount per $1000, 
 what is the face of the sight draft on Boston, that can be 
 bought in New York for $ 1000 ? 
 
 Note. — $ 999.50 in New York will buy a sight draft on Boston for 
 $1000. 
 
 12. When the premium is $1.25 per $1000, Mr. Brown 
 pays $ 1634.04 for a draft on Louisville. What is the face 
 of the draft? 
 
378 Chapter Six. 
 
 COMPOUND INTEREST. 
 
 483. Compound Interest is interest on the principal and on 
 the unpaid interest, which is added to the principal at regu- 
 lar intervals. The interest may be compounded annually, 
 semi-annually, or quarterly, according to agreement. 
 
 Compound interest is allowed by savings banks. It is 
 not collectible on notes, mortgages, or the like. 
 
 484. "Written Exercises. 
 
 1. Find the amount of $375, for 1 year, at 6%. Con- 
 sidering this as a new principal, find the amount for a year, 
 same rate. Find the amount of this last principal for 3 
 months. 
 
 2. What is the amount of $ 375, for 2 yr. 3 mo., at 6%, 
 compound interest ? 
 
 3. What is the amount of $ 375, for 2 yr. 3 mo., at 6%, 
 the interest compounded semi-annually ? 
 
 Principal, $375. 
 
 3% 11.25 6 months' interest 
 
 386.25 Amount 6 months. 
 
 3% 11.5875 6 months' interest. 
 
 Amount 1 year, 
 3% 6 months' interest. 
 
 Amount 1£ years, 
 etc., etc., etc. 
 
 4. Find the compound interest on $ 375, for 2 yr. 3 mo., 
 at 6 per cent, compounded semi-annually. 
 
 Note. — To find the compound interest, deduct $ 375 from the 
 amount for 2 yr. 3 mo. 
 
 5. What is the amount of $ 100, at compound interest, 
 for 3 years, interest at 6%, compounded annually ? 
 
Annual Interest. 379 
 
 ANNUAL INTEREST. 
 
 When the maker of a note fails to keep his contract to pay interest 
 annually, the laws of some states, including Michigan, permit the col- 
 lection of simple interest on the deferred payments of interest. 
 
 485. Written Problems. 
 
 1. Find the amount due June 1, 1908, on the following 
 note, no payments of principal or interest having been 
 
 Detroit, Mich., June 1, 1904. 
 
 Tour years after date, without days of grace, I promise to 
 pay to the order of Daniel W. Lawler, Six Hundred Dollars, 
 value received, with annual interest at six per cent. 
 
 $600^. George Oxj^ard. 
 
 Principal, $600.00 
 
 Interest, 4 years, at 6 %, 144.00 
 
 3 years' interest, at 6 %, on the 1st year's interest, $ 36, 6.48 
 2 years' interest, at 6 %, on the 2d year's interest, $ 36, 
 
 1 year's interest, at 6 %, on the 3d year's interest, $ 36, 
 
 Amount due June 1, 1910, $ 
 
 Find the interest on the principal for the entire time, and on 
 each annual interest for the time it remained unpaid. The 
 sum of the principal and all the interest is the amount due. 
 
 2. Find the amount due, at 5%, for 5 years, on a note 
 for $ 1200, annual interest being unpaid. 
 
 3. The maker of a note for $ 900, with annual interest 
 at 7%, makes the first and the second interest payments 
 when due. How much will he owe at settlement, 6 years 
 after the date of the note ? 
 
 4. Find the difference between the amount due at 6% for 
 3 years on a note for $ 300, annual interest unpaid, and the 
 amount of the same sum placed at compound interest for 
 the same time at the same rate. 
 
380 Chapter Six. 
 
 5. What is the amount of a note for $ 720, at 4 years, at 
 4|%, annual interest unpaid after the first year? 
 
 6. Find the amount due March 1, 1906, on a note for 
 $500, dated March 1, 1900, with interest at 6%, annual 
 interest unpaid after the third year. 
 
 METRIC SYSTEM. 
 
 486. The metric system, which is used in nearly all the 
 countries of continental Europe, is based upon the meter. 
 The length of the meter is one ten-millionth part of the 
 length of the meridian from the equator to the poles — 
 about 39.37 inches. 
 
 The subdivisions of the meter are denoted by the Latin 
 prefixes milli (tuW)» centi (y-J-^), deci (y 1 ^). For the multi- 
 ples, the Greek prefixes deka (10), hecto (100), kilo (1000), 
 and myria (10,000) are used. 
 
 487. It will be noticed, in the table below, that small 
 letters are used for the abbreviations of the Latin prefixes 
 of the subdivisions, and capital letters for the Greek pre- 
 fixes of the multiples. The following is the table of 
 
 488. Measures of Length. 
 
 10 millimeters (mm) 1 centimeter (cm) 
 
 10 centimeters 1 decimeter (dm) 
 
 10 decimeters 1 meter (m) 
 
 10 meters 1 dekameter (Dm) 
 
 10 dekameters 1 hectometer (Hm) 
 
 10 hectometers 1 kilometer (Km) 
 
 10 kilometers * 1 myriameter (Mm) 
 
 The units of this table in common use are the centimeter, the meter, 
 and the kilometer. 
 
 Long distances are expressed in kilometers. The thickness of wire 
 is given in millimeters. 
 
Metric System. 381 
 
 489. Written Problems. 
 
 1. What will be the cost in francs of 380 m 75 of dress 
 goods at 2 f 60 per meter ? 
 
 380 m 75 is read 380 meters 75 centimeters. It is also written 
 380.75 m, but the first method is the more common one in Europe. 
 2 f 60 is read 2 francs 60 centimes. A period (.) is not used after the 
 abbreviations of meter, liter, franc, etc. 
 
 2. How many square meters in a piece of carpet 26 m 50 
 long, 85 cm wide ? 
 
 3. How many square meters in a circle whose diameter 
 is 15 meters ? 
 
 4. An are is a surface 10 meters long, 10 meters wide. 
 How many ares in a field 135 meters long, 69 meters wide ? 
 
 5. Find the area in ares of a right-angled triangle whose 
 base is 245 meters, hypotenuse 875 meters. 
 
 6. A stere is a cubic meter. What will be the cost, at 
 8 f 50 per stere, of a pile of wood 10 meters long, 1 meter 
 wide, 3.25 meters high ? 
 
 7. A cube one decimeter each way contains a liter (1), 
 which is the principal unit of dry and liquid measure. 
 
 How many liters' capacity has a tank 10 m 50 long, 8 m 
 wide, 6 m 50 high ? 
 
 Change each dimension to decimeters. 
 
 8. How many bottles, each containing 1 75, can be filled 
 from a hogshead containing 222 1 ? 
 
 9. How much will be received for 36 bags of beans, each 
 containing 68 liters, at 1 mark 25 per dekaliter ? 
 
 10. A liter of water weighs a kilogram (1000 grams). 
 How many kilos of oil would a tank contain, its dimensions 
 being 5 meters by 4 meters by 3 meters, the weight of the 
 oil being 92% of the weight of water? 
 
 11. Assuming the length of the meter as 39.37 inches, 
 what is the length of the kilometer in yards ? 
 
382 Chapter Six. 
 
 . 490. Measures of Surface. 
 
 100 sq. mm = 1 sq. cm 
 100 sq. cm = 1 sq. dm 
 100 sq. dm = 1 sq. m = 1.196 sq. yd. 
 
 491. The square meter is the principal unit of surfaces, 
 such as walls, ceilings, floors, etc. 
 
 100 centiares (ca) = 1 are (a) = 119.6 sq. yd. 
 100 ares = 1 hectare (Ha) = 2.47 acres. 
 
 The are is the principal unit of surface of small plots of 
 land. The area of a farm is expressed in hectares, of a 
 country in square kilometers. 
 
 492. Measures of Volume. 
 
 1000 cu. mm = 1 cu. cm 
 1000 cu. cm = 1 cu. dm 
 1000 cu. dm = 1 cu. m = 35.316 cu. ft. 
 
 The principal unit is the cubic meter. 
 
 493. The stere (cubic meter) is used for measuring wood. 
 10 decisteres (dst) = 1 stere (st) = 35.316 cu. ft. 
 
 10 steres = 1 dekastere (Dst). 
 
 The stere is the only unit used. 
 
 494. Dry and Liquid Measures. 
 
 10 milliliters = 1 centiliter. 
 
 10 centiliters = 1 deciliter. Dry. Liquid. 
 
 10 deciliters = 1 liter (1) = .908 qt. = 1.057 qt. 
 
 10 liters = 1 dekaliter = 1.135 pk. = 2.642 gal. 
 
 10 dekaliters = 1 hectoliter = 2.837 bu. = 26.417 gal. 
 
 10 hectoliters = 1 kiloliter. 
 
 10 kiloliters = 1 myrialiter. 
 
 The liter and the hectoliter are the principal units. 
 
Review. 383 
 
 495. Table of Weight. 
 
 10 milligrams (mg) = 1 centigram. 
 
 10 centigrams = 1 decigram. 
 
 10 decigrams = 1 gram (gr). 
 
 10 grams = 1 dekagram. 
 
 10 dekagrams = 1 hectogram. 
 
 10 hectograms = 1 kilogram (kilo) = 2.2046 lb. 
 
 10 kilograms (Kg.) = 1 myriagram. 
 
 10 myriagrams = 1 quintal. 
 
 10 quintals = 1 tonneau (ton). 
 
 The kilo is the ordinary unit. Heavy articles are sold by the 
 tonneau. 
 
 496. Written Exercises. 
 
 1. The Eiffel tower is 300 meters high. What is its 
 height in feet? 
 
 2. The Danube is 2600 kilometers long. Find its length 
 in miles. 
 
 3. A bottle filled with water weighs 1.170 kilos; the 
 weight of the bottle is 420 grams. What is the capacity 
 of the bottle in liters? 
 
 4. Find the weight in kilos of 15 liters of olive oil, which 
 weighs .915 time as much as water. 
 
 5. A rectangular field 123 meters long, and 85.5 meters 
 wide, yielded 13.25 hectoliters of wheat per hectare. The 
 wheat weighed 84.350 kilos per hectoliter and sold for 23.50 
 francs per 100 kilos. What sum did the crop bring ? 
 
 6. What will be the cost in francs of papering a room 
 5 m 42 long, 4 m 18 wide, and 3 m 10 high, at 1 f 20 per 
 square meter ? 
 
 7. Calculate the expense of building a wall 14 m 50 long, 
 7 m 80 high, m 22 thick, of bricks m 22 long, m 11 
 wide, m 06 thick, the bricks costing 58 francs per thou- 
 sand and the labor, etc., 32 f 80 per cubic meter. 
 
384 Chapter Six. 
 
 8. Find the profit on a pile of wood 20 meters long, 4 
 meters wide, 8 meters high, bought at 12 francs per stere, 
 and sold at 4 francs per 100 kilos, the weight of the wood 
 being .42 times the weight of water. 
 
 9. A liter of wheat weighs 760 grams. When ground it 
 produces 89 per cent flour and 11 per cent bran. Find the 
 weight of the flour that can be made from the wheat con- 
 tained in a bin 2 m 60 long, 2 m 40 wide, and 1 m 50 deep. 
 Find the value of the wheat at 4 f 85 per double dekaliter. 
 
 10. If sea water contains -£$ of its weight of salt, how 
 many hectoliters of sea water should be evaporated to obtain 
 100 kilos of salt, a liter of sea water weighing 1.026 kilos ? 
 
 REVIEW. 
 497. Oral Problems. 
 
 1. If I yard costs $4.50, what will £• yard cost? 
 
 2. If 3 men can do a piece of work in 4 days, how long 
 will it take 24 men to do it ? 
 
 3. What principal at interest for 5 years, at 6 per cent, 
 will produce $ 12, simple interest ? 
 
 4. A stack of hay will keep a cow 20 weeks, or a horse 
 15 weeks. How long will it keep them both ? 
 
 Note. — What part will each eat in a week ? What part will both 
 eat in a week ? 
 
 5. How many days from May 16 to July 5 ? 
 
 6. Sold a cow for $24, losing thereby 40% of the cost 
 price. Had I sold her for 33^% advance on the cost, what 
 should I have received for her ? 
 
 7. What will 460 pounds of tea cost at $ .48 per pound ? 
 
 8. If 12 ounces of bread are destroyed in making a gill of 
 whiskey, how much will be destroyed in making a gallon ? 
 
 4 gills = 1 pint 
 
Review. 385 
 
 9. If the weight of air is 15 pounds on the square inch, 
 what is it on the square foot ? 
 
 10. Seven is three-fifths of what number ? 
 
 11. What is the value of 960 pounds of wheat at $1.05 
 per bushel of 60 pounds ? 
 
 12. At what rate per cent will $400 make $37.50, simple 
 interest, in 1 yr. 3 mo. ? 
 
 13. What is the brokerage on $10,400, at If % ? 
 
 14. What will 3280 feet of lumber cost @$25 per 
 thousand? 
 
 15. A and B are partners; A puts in -^ of the stock, and 
 B the remainder ; B's gain is $ 1400. Find A's gain. 
 
 16. What is the difference in the longitude of two places 
 whose difference in sun time is two hours and three minutes? 
 
 17. A room is f as wide as it is long. Its length is 20 
 feet. How many square feet are there in the floor ? 
 
 18. If 5 yards of cloth cost 90^, what will f of a yard 
 cost? 
 
 19. An agent insured a house for me at a commission of 
 \°/ . His commission was $15. For how much was the 
 house insured ? 
 
 20. A gold-digger who had 3 pounds of gold dust, lost 9 
 ounces. What per cent was left ? 
 
 498. "Written Problems. 
 
 1. What number must be added to the sum of -|, -J, and 
 \\ to make 5^? 
 
 2. Find the interest on $ 2320, for 5 months and 21 days, 
 at the rate of 7 per cent a year. 
 
 3. Find the interest on $640, from Sept. 3, 1904, to 
 Oct. 30, 1905, at 6 per cent per annum. 
 
386 Chapter Six. 
 
 4. At compound interest, what will $ 200 amount to in 
 1 year and 3 months, at 6 per cent, interest compounded 
 semi-annually ? 
 
 5. A man drew out of the bank f of his money, and ex- 
 pended 30% of 50% of this for 936 bushels of wheat at 
 $ 0.87J a bushel. What sum had he left in bank ? 
 
 6. A house that cost $ 14,500 rents for $ 1189. What 
 per cent does it pay on the investment ? 
 
 7. If 4 men dig a ditch 24 rods long in 20 days, how 
 long a ditch can 5 men dig in 8 days ? 
 
 8. For what sum must a 60-day note be written to yield 
 $ 294.75 at a bank, discounting at 6% ? 
 
 9. An agent receives $ 5616 for silk he has purchased 
 and his commission on it at 4%. How many yards did he 
 purchase at $ 1.50 per yard ? 
 
 10. What will be the proceeds of a 60-day note for $ 500, 
 dated June 4, 1904, and discounted at a bank July 1, 1904, 
 at 6% ? 
 
 11. At what rate will $142 gain $21.30 interest in 3 
 years ? 
 
 12. What is the duty, at 50^ a pound and 30% ad valorem, 
 on 700 yards of French broadcloth, invoiced at $1.25 per 
 yard, and weighing 1 \ pounds per yard ? 
 
 13. What will be the amount, at compound interest, of 
 $ 340, at 8%, for 1 yr. 3 mo., the interest compounded semi- 
 annually ? 
 
 14. If I lose 10% by selling goods at 18^ a yard, for 
 what must they be sold to gain 20% ? 
 
 15. I sold 24 J % of my estate, or $ 1372 worth. I am 
 worth, in addition to my real estate, $14,000. How much 
 am I worth in all ? 
 
Fractions. 387 
 
 REVIEW OF FRACTIONS. 
 
 499. Oral Exercises. 
 Give products : 
 
 1. 84 x 24 = 25 times 84 - 84 = 2100 - 84. 
 
 2. 48x24. 4. 48x49. 6. 84x74. 
 
 3. 24x36. 5. 84x49. 7. 48x74. 
 
 8. 84 x 241 = 25 times 84 - £ of 84 = 2100 - 42. 
 
 9. 48x241 11. 48x24f. 13. 48 x 24f 
 10. 36 x 241 12. 36 x 24f. 14. 36 x 24|. 
 
 15. 48 x 361 = 371. times 48 - 48 = (| of 4800) - 48. 
 
 16. 48xllJ. 17. 48x86£. 18. 48 x 37f 
 
 Give quotients : 
 
 1. 36 -*-}. 7. 18I-T-3. 13. 12}-* If 
 
 2. 36-i-|. 8. 20£-r-4. 14. 16£-*-lf 
 
 3. 36-2}. 9. 17!- -5- 5. 15. 13^-j- 3f 
 
 4. 36 -i- f 10. 191 -J- 6. 16. 14} + If 
 
 5. 36 -s-lf 11. 16J-5-7. 17. 15} -j-2f 
 
 6. 36-r-lJ. 12. 17J-J-8. 18. 17£-i-3f 
 
 500. Written Exercises. 
 Find products : 
 
 1. 648 x}. 9. 792x25. 
 
 2. 976 x if 10. 457x16. 
 
 3. 1648 x87f . 11. 1864x250. 
 
 4. 2592 x9if 12. 983x51. 
 
 5. 2416x875. 13. 1576 x 62f 
 
 6. 874x99. 14. 176x23f. 
 
 7. 848x125. 15. 1128x875. 
 
 8. 375x999. 16. 895 x 44f 
 
388 Chapter Six. 
 
 501. Written Exercises. 
 
 1. Divide the sum of 6f and 1-| by the difference be- 
 tween 2\ and 3£. 
 
 2. What is the difference between the sum of % and | 
 and the product of f and fa ? 
 
 3. What is the product of the sum and the difference of 
 4| and 6J ? 
 
 4. Subtract § of J from fj ; and find the value of fa of 
 16s. 6cZ. 
 
 5. Add 7f , } of ft, I of 7| , and «. 
 
 6. Keduce f of a square rod to the fraction of an acre, 
 and find the value of ^ of a ton in pounds and ounces. 
 
 7. Keduce T 6 ^^ to its lowest terms, and Q * "~ t to its 
 simplest form. ¥ + s 
 
 8. Add |j f, f, and J; multiply the sum by fa; and 
 subtract the product from 1. 
 
 9. Find the value of 9^ meters at 4-J francs per meter. 
 
 10. Divide 2\ by 3£, and add the quotient to fa. 
 
 11. Multiply 2 fa by 16f , and divide the result by 1\ of 2f 
 
 12. Keduce 7s. 6c?. to the fraction of a pound, and 7 hr. 12 
 min. to the fraction of a day. 
 
 13. Keduce to its simplest form - + * 5 ^. 
 
 14. Add together £ £ and ^ of 5| shillings. 
 
 15. What fractional part of 7 A. 127 sq. rd. is 5 A. 
 81 sq. rd. ? 
 
 16. What must be added to f of j- to make it equal to 
 AofSf? 
 
 17. \ of a number is 148. What is the number ? 
 
 18. If 4 of a field is worth $325, what is the field worth ? 
 
 19. If J of a house is worth $4900, what is the value 
 of i? 
 
Denominate Numbers. 389 
 
 REVIEW OF DENOMINATE NUMBERS. 
 
 502. Written Exercises. 
 
 1. Change 43 yards to rods and a fraction. 
 
 2. Change 43 yards to rods and yards. 
 
 43 yards -4- 5| yards gives the number of rods. 
 
 3. Change 43 yards to rods, yards, and feet. 
 
 4. Change 43 yards to rods, yards, feet, and inches. 
 
 5. Change 72 yards to rods, etc. 
 
 6. Change 66 yards to rods. 
 
 Change to rods, yards, etc. : 
 
 7. 49 yards. 11. 1836 inches. 
 
 8. 147 feet. 12. 1837 inches. 
 
 9. 1764 inches. 13. 52 yards. 
 10. 8Jf rods. 14. 49J yards. 
 
 503. Change to rods, etc. : 
 
 15. 1483 inches. 18. 2796 inches. 21. 3453 inches. 
 
 16. 984 inches. 19. 1121 inches. 22. 1278 inches. 
 
 17. 1345 inches. 20. 1470 inches. 23. 1576 inches. 
 
 504. Add: 
 
 24. 4 rd. 3 yd. 1 ft. 25. 5 rd. 4 yd. 2 ft. 
 
 9 rd. 4 yd. 2 ft. 5 yd. 1 ft. 
 
 3 rd. 1 ft. 6 in. 6 rd. 1 yd. 
 
 26. From 8 rd. 1 ft. take 2 rd. 2 ft. 
 
 27. Find the difference between 3 rd. 1 yd. 1 ft. and 16 rd. 
 
 28. Multiply 5 rd. 4 yd. 2 ft. by 4. 
 
 29. Multiply 11 rd. 2 ft. by 10. 
 
 30. Divide 30 rd. 5 yd. 2 ft. by 8. 
 
 31. Divide 34 rd. 2 yd. by 9. 
 
390 Chapter Six. 
 
 REVIEW OF COMMERCIAL DISCOUNT. 
 
 505. Oral Exercises. 
 
 When the list price is $ 1, what is the net price after the 
 deduction of each of the following discounts ? 
 
 1. 30 and 20%. 4. 50 and 10%. 
 The net price after a deduction ~ ^q j 2Q ol 
 
 of 30 % is 70 % of .$ 1 , or 70 f. De- 
 ducting 20 % of 70 f leaves 80 % of 6.10 and 5 % . 
 
 W ' or66 ^- 7. 20 and 20%. • 
 
 2. 40 and 10%. m 001 , 1|A-I 
 
 7 8. 334 and 10%. 
 
 The net price is G0% of 90% of 6 
 
 $1, or 54% of |1. 9. 20 and 15%. 
 
 3. 25 and 40%. 10. 30 and 15%. 
 
 506. What single discount is equal to each of the follow- 
 ing double discounts ? 
 
 11. 30 and 30%. 15. 40 and 30%. 
 
 The net price is 70% of 70% of ^ ^ ^ ^ 
 
 list price, or 40% of list price. 
 
 The discount is, therefore, 100% 17, 40 an d 5%. 
 
 -49%, or 51%. 
 
 ™ -, *** 18 - 50 and 20%. 
 
 12. 20 and 25%. ' 
 
 13. 25 and 20%. 19 ' 40 and 15%. 
 
 14. 15 and 30%. 20. 50 and 15%. 
 
 Find the single discount equal to each of the following : 
 
 21. 50 and 20 and 10%. 
 
 22. 40 and 25 and 20%. 
 
 23. 10 and 10 and 10%. 
 
 24. 30 and 20 and 10%. 
 
Interest. 391 
 
 507. Written Exercises. 
 
 Which is the better discount for the buyer ? 
 
 1. 30 and 20%, or 40 and 10%. 
 
 30 and 20% off = 70% of 80% net, or 56% net. 40 and 10% off s 
 60% of 90% net, or 64% net. The latter is the better for the buyer. 
 
 2. 50 and 10%, or 40 and 20%. 
 
 3. 20 and 20%, or 30 and 10%. 
 
 4. 20 and 15%, or 30 and 5%. 
 
 5. 30 and 15%, or 25 and 20%. 
 
 6. 30 and 30%, or 50 and 10%. 
 
 7. 40 and 30%, or 20 and 50%. 
 
 8. 40 and 5%, or 30 and 15%?. 
 
 9. 20 and 50%, or 60 and 10%. 
 10. 40 and 15%, or 30 and 25%. 
 
 REVIEW OF INTEREST. 
 
 508. Six Per Cent Method. 
 
 Interest is the product of the principal by the rate ex- 
 pressed as hundredths by the time in years and fraction. 
 The usual method is to perform the operations in the above 
 order. When the rate is 6%, some prefer to first multiply 
 the rate by the time, and to use this as a multiplier of the 
 principal. 
 
 In finding the product of the rate by the time, advantage 
 is taken of the fact that 6 is a factor of 12 and 30. Six per 
 cent a year is -J per cent a month and -fa per cent a day. 
 
 When the rate is a different per cent, the interest is first 
 obtained at 6 per cent by this method, and from this result 
 the interest is calculated for the given rate. 
 
39* 
 
 Chapter Six. 
 
 509. Find the 
 7 mo. 19 da. 
 
 $2874.35 
 
 •218* 
 
 .47905+ 
 22.99480 
 28.7435 
 574.870 
 6)$ 627.08735 
 $104.5145+ 
 
 3f 
 
 313.5435 
 78.3859 
 
 $391.9294 
 $391.93 Ans. 
 
 interest on $2874.35 at 3f% for 3 ft. 
 
 6 % for 1 year is for 3 years .18 
 
 \ % for 1 month is for 7 months .035 
 
 B V/ for 1 day is for 19 days .003$ 
 
 6 % for 3 yr. 7 mo. 19 da. is .218 \ 
 
 Multiplying the principal by .218$ gives the 
 interest at 6 % for 3 yr. 7 mo. 19 da. 
 
 Dividing this product by 6 gives the interest 
 at 1 %. Multiplying the quotient by 3f gives the 
 interest at 3f %. 
 
 To find the interest at 6 per cent, multi- 
 ply the principal by 6 times the number of 
 years and J the number of months as hun- 
 dredths, together with % the number of days 
 as thousandths. 
 
 510. Written Exercises. 
 Find the interest at 6% on: 
 
 1. $ 1428 for 1 yr. 4 mo. 6 da. 
 
 2. $ 372.50 for 2 yr. 6 mo. 24 da. 
 
 3. $ 1875 for 3 yr. 9 mo. 18 da. 
 
 4. $ 240 for 4 yr. 7 mo. 15 da. 
 
 5. $ 92.75 for 5 yr. 4 mo. 8 da. 
 
 6. $ 817.80 for 10 mo. 19 da. 
 
 The interest at 6 % plus $ of itself gives the interest at 7 %. 
 The interest at 6 % minus $ of itself gives the interest at 5 %. 
 The interest at 6 % plus $ of itself gives the interest at 8 %. 
 The interest at 6 % minus $ of itself gives the interest at 4 %. 
 The interest at 6 % plus \ of itself gives the interest at 1\ %. 
 The interest at 6 % minus \ of itself gives the interest at 4 \ %. 
 
Interest. 393 
 
 511. "Written Exercises. 
 Find the amount : 
 
 1. $1875.25 for 3 yr. 5 mo. 15 da., at 4|%. 
 
 2. $487.50 for 1 yr. 10 mo. 25 da., at 6%. 
 
 3. $1206.84 for 2 yr. 1 mo. 16 da., at 5%. 
 
 4. $595.00 for 7 yr. 7 mo. 7 da., at 7%. 
 
 5. $763.25 for 8 mo. 11 da., at 4%. 
 
 6. $685.70 for 19 mo. 5 da., at 51%. 
 
 7. $1563.00 for 3 mo. 20 da., at 5%. 
 
 8. $998.45 for 87 da., at 4|%. 
 
 9. $2575.50 for 149 da., at 3%. 
 
 10. $693.27 for 214 da., at 2£%. 
 
 Find the principal, rate, or time : 
 
 11. Principal, $240; interest, $32.04; time, 2 yr. 11 mo. 
 
 18 da. Eate ? 
 
 12. Eate, 6% ; amount, $717.40; time, 3 yr. 3 mo. 4 da. 
 Principal ? 
 
 13. Principal, $360; rate, 3% ; interest, $48.87. Time? 
 
 14. Principal, $288; rate, 2£%; amount, $307.22. Time? 
 
 15. Eate, 6%; interest, $13.10; time, 4 mo. 11 da. 
 Principal ? 
 
 16. Principal, $270; amount, $273.27; time, 3 mo. 
 
 19 da. Eate ? 
 
 17. Eate, 4£% ; interest, $ 25.11 ; principal, $ 360. Time? 
 
 18. Interest, $50.22; time, 3 yr. 1 mo. 6 da. ; rate, 
 Amount ? 
 
394 Chapter Six. 
 
 REVIEW OF BANK DISCOUNT. 
 
 512. Written Exercises. 
 
 Find face of note, term of discount, rate, discount, or 
 proceeds : 
 
 By the term is meant the number of days the note has to run, 
 including grace, if any. 
 
 1. Face, $600; discount, $6.30; rate, 6%. Term? 
 
 2. Term, 33 days; proceeds, $397.80; rate, 6%. Face? 
 
 3. Term, 90 days; face, $300; rate, 6%. Proceeds? 
 
 4. Term, 21 days; face, $600; discount, $2.45. Rate? 
 
 5. Term, 4 months; face, $200; rate, 6%. Discount? 
 
 6. Term, 132 days ; proceeds, $ 2689.50 ; rate, 6 % . Face ? 
 
 7. Face, $150; proceeds, $147.75; rate, 6%. Term? 
 
 8. Face, $1650; discount, $4.95; rate, 6%. Term? 
 
 9. Term, 69 days; proceeds, $469.30; rate, 6%. Face? 
 
 EXACT INTEREST. 
 
 Exact interest is used by the United States Government in its cal- 
 culations. 365 days are taken to the year. 
 
 513. Written Exercises. 
 
 1. Find the exact interest of $280 from April 14 to 
 Sept. 6 at 4%. 
 
 Time, 145 days. Ans. $280 x T ^x }f$. 
 
 2. Find the exact interest on $76.65 from March 4 to 
 Dec. 15 at 6 per cent. 
 
 3. On $ 384 at 7^ per cent for 75 days. 
 
 4. On $438 at 5% from Jan. 1 to March 15. 
 
 5. On $109.50 at 4|% for 87 days. 
 
 6. On $847.60 at 5% from April 29 to Sept. 20. 
 
 7. $584 at 3J% from May 16 to Dec. 1. 
 
 Unless "exact" or "accurate" interest is specified, use 360 days 
 to the year. 
 
Review. 395 
 
 MISCELLANEOUS. 
 
 514. Oral Review Problems. 
 
 1. A has 96 sheep ; B has 28 sheep more than A. How 
 many sheep have both ? 
 
 2. There are 56 pupils in one class, 48 in a second class, 
 and 52 in a third class. How many pupils are there in the 
 three classes ? 
 
 3. March 29 is what day of the year 1904 ? 
 
 4. How far is a man from his starting-point, if he travels 
 due east 150 miles, due west 23 miles, due east again 48 
 miles ? 
 
 5. A body falls 16 feet in the first second, three times 
 as far in the second second, five times as far in the third 
 second. How far does it fall in three seconds? 
 
 6. The base of a right-angled triangle is 12 feet, the 
 perpendicular is 16 feet. What is the hypotenuse ? 
 
 7. At $35 per month, what will be the rent of a house 
 for 16 months ? 
 
 8. A field containing 169 square rods is 13 rods long. 
 What is the perimeter ? 
 
 9. 25 packages of sugar weigh together 87£ pounds. 
 How many pounds are there in each ? 
 
 10. At 45 miles per hour, how many hours, minutes, and 
 seconds will it take a train to go 230 miles ? 
 
 11. How many years have elapsed since the invention of 
 gunpowder, 1356 ? 
 
 12. What profit is made on an article bought for $175, 
 less 12%, and sold for $200 ? 
 
 13. How many square rods in a field 71 rods long, 81 
 rods wide ? 
 
396 Chapter Six. 
 
 14. Assuming a kilo to be 2\ pounds, how many kilos 
 will be equal to 143 pounds ? 
 
 15. A degree of longitude in latitude 45° is about 70% of 
 the length of a degree on the equator. Calling the latter 
 length 69 miles, how long is a degree of longitude in latitude 
 45°? 
 
 16. At $44 per acre, how much land can be bought for 
 
 17. A number of marbles divide'd among 29 boys gives 
 each 16 marbles, and leaves a remainder of 26. How many 
 marbles are there ? 
 
 18. What is the monthly salary of a clerk who receives 
 $1500 per year? 
 
 19. How many revolutions in a mile, 5280 feet, are made 
 by a locomotive wheel 16 feet in circumference ? 
 
 20. What is the perimeter of a lot 49 feet wide, 87 feet 
 long? 
 
 21. How many bricks 8 inches by 4 inches by 2 inches 
 would make a cubic foot ? 
 
 22. 13 is one factor of 1001. Find the other two prime 
 factors. 
 
 23. What are the three equal factors of 343 ? 
 
 24. What is the square root of 1225 ? 
 
 25. At 4 J miles per hour, how long will it take a man to 
 walk 37£ miles ? 
 
 26. What will be the cost of 9 dozen hats at $1.33^ 
 each? 
 
 27. Paid 92^ for coffee, 48^ for butter, and 18^ for lard. 
 How much was my bill ? 
 
 28. I had $ 150. Spent $ 23 for a suit of clothes and $ 48 
 for tools. How much was left ? 
 
 29. What is the area of a field 36 yards by 31 yards ? 
 
Review. 397 
 
 30. 600 hours equal how many days ? 
 
 31. What is the cost of a cow if I pay $630 for 15? 
 
 32. How many ounces in 2&±- pounds ? 
 
 33. 109J pounds of sugar are divided among 4 people. 
 What is the share of each ? 
 
 34. At 1 T 9 7 ^ per pound, how many pounds of iron can 1 
 get for $ '5.70 ? 
 
 35. What is the cost of 51 tons iron at $ 17 per ton ? 
 
 36. What will be the average age of 9 boys, each 12 years 
 old, and 6 boys, each 10 years old ? 
 
 37. At 42 miles per hour, how long will it take a train to 
 go 882 miles ? 
 
 38. At 25^ per hour, what will a man earn in 18 days of 
 10 hours ? 
 
 39. What will ^be the net price of an article whose cata- 
 logue price is $20.00, the discount being 90 and 10% ? 
 
 40. A man had $ 181 in bank. What will be his balance 
 after taking out $47 and $33 ? 
 
 41. How many feet in 14 rods ? 
 
 42. 77 yards are how many rods ? 
 
 43. How many square yards are there in a floor lOf yards 
 long and 6^ yards wide ? 
 
 44. What is the cost of 372 eggs at 15^ per dozen ? 
 
 45. A man owns 3 farms containing 65 acres, 86 acres, and 
 98 acres, respectively. How many acres does he own ? 
 
 46. What is the area of a piece of glass measuring 8J by 
 6J inches ? 
 
 47. What is the value in U. S. money of 50 marks at 23^ 
 cents ? 
 
 48. How many francs will a calf cost, if 18 are worth 630 
 francs ? 
 
398 Chapter Six. 
 
 49. A man spends $ 1740 per year. What is the average 
 amount spent per month ? 
 
 50. What would 51 pounds of butter cost at 33^ ^ a pound ? 
 
 51. Mrs. Allen bought 7 chairs at $4 apiece, 2 tables at 
 $9 apiece, and a carpet for $33. She paid two f 50 bills. 
 How much change was due her ? 
 
 52. In what time will any sum of money double itself, at 
 6%? 
 
 53. Find the sum of the prime numbers as far as 12. 
 
 54. Interest of $1234, for 30 days, at 6% ? 
 
 55. Interest of f 1234, for 6 months, at 4% ? 
 
 56. Oil is worth 37^ a pint. How many pints can be 
 bought for $6? 
 
 57. Sold oranges for \$ apiece, gaining 50%. How much 
 did they cost apiece ? 
 
 58. What will be the cost of 1 pk. 1 qt. 1 pt. of nuts, at 
 10 ^ per quart ? 
 
 59. What is the value of an acre of land, at 10^ per 
 square foot ? 
 
 60. 3 desks are bought at $ 10 each, and sold for $ 45. 
 Find the rate of gain. 
 
 61. A wheelman sells his old bicycle for $25, and loses 
 16}%. How much did it cost him ? 
 
 62. How much does an agent get for buying 5 bale? of 
 goods at $ 400 each, if he receives 3% for his services ? 
 
 63. 10% of 200 is £ of what number ? 
 
 64. How old, December 1, 1903, was a boy born November 
 25,1889? 
 
 65. A man has $ 1000 in bank. What will remain after 
 he has taken out $ 478 ? 
 
 66. How many hours in the month of January ? 
 
Review. 399 
 
 67. In how many years, months, and days will $100 
 amount to $111, at 5%, simple interest? 
 
 68. What will 5 tons of granulated sugar cost, at 6 J ^ per 
 pound ? 
 
 69. What is the interest of $ 50, for 3 yr. 7 mo. 12 da., at 
 6%? 
 
 70. A farmer makes 675 gallons of cider. He has but 12 
 barrels, each of 45 gallons' capacity, to store it in. How 
 many more such barrels does he need ? 
 
 71. What will be the cost of 36 yards of cloth, at $2.75 
 per yard ? 
 
 72. Add 3794 and 2975. 
 
 73. What is the bank discount on a sixty-days note for 
 $400, at 6% ? 
 
 74. Change ^-toa decimal of three places. 
 
 75. How much wood in three piles containing, respec- 
 tively, \ of a cord, ^ of a cord, and \ of a cord ? 
 
 76. What is the percentage of gain in case of railroad 
 stock bought for $ 80 per share, and sold for $ 90 per 
 share ? 
 
 77. A dealer sold flour at a profit of 50^ a barrel, and 
 gained 10%. What was the cost ? 
 
 78. At 10 ^ a quart, what are 3 bu. 1 pk. 5 qt. of chest- 
 nuts worth ? 
 
 79. How many yards in 288 inches ? 
 
 80. What decimal of a number is -f per cent of it ? 
 
 81. If a broker buys for me 5 shares of railroad stock 
 whose par value is $ 100, what is his brokerage at \% ? 
 
 82. If I sell 10 shares of railroad stock for $ 1090, and 
 gain 9% on the cost, what was the cost? 
 
 83. What is the interest of $ 660, for 3 months, at 4% ? 
 
4-00 Chapter Six. 
 
 84. What per cent does a merchant lose by selling goods 
 at | of their cost ? 
 
 85. What principal at 6% simple interest will gain $36 
 in 1 year and 6 months ? 
 
 86. What per cent is gained on goods sold at double the 
 cost? 
 
 87. What is 8% of 50 bushels ? 
 
 88. $3000 is 11£% of my property. How much am I 
 worth ? 
 
 89. What is the interest on $ 700, for 15 days, at 6% ? 
 
 90. Bank discount on a 65-day s note for $ 1000, dis- 
 counted at date ? 
 
 91. At what rate will $ 2 gain $ 20 in 5 years ? 
 
 92. A capitalist wishes to realize 5% on money invested 
 in stock. What must be the annual dividend on stock cost- 
 ing 300, in order to produce this rate ? 
 
 93. What will be the taxes on property assessed at 
 $25,000, the rate being $16 per $1000 ? 
 
 94. Find the compound interest on $ 1000, for two years, 
 at five per cent, interest compounded annually. 
 
 95. What will be the net cost of an article marked $8, 
 on which a discount of 50, 25, and 10% is allowed ? 
 
 96. Find the "list" price of an article sold for $ 10 after 
 a discount of 50 and 50 per cent had been deducted. 
 
 97. Paid 90 ^ for an article. The discount is 25 and 25 
 per cent. What is the list price ? 
 
 98. One boy can do a certain piece of work in 2 hours, a 
 second boy requires 3 hours, a third needs 6 hours. How 
 long will it take the three working together ? 
 
 99. Sold a cow for $ 60, losing 25 % • What was the loss ? 
 100. Sold a cow for $60, gaining 25%. What was the 
 
 gain? 
 
Review. 401 
 
 101. Sold two horses at $240 apiece. On one I gained 
 20%, on the other I lost 20%. Did I gain or lose on both, 
 and how much ? 
 
 102. What is the interest of $1500, for 60 days, at 6% ? 
 
 103. How many years will it take $20 to gain $20 at 
 5 per cent simple interest ? 
 
 104. John has $60, James has $80. James has what 
 per cent more money than John ? John has what per cent 
 less money than James ? 
 
 105. I is what per cent of J ? J is what per cent of f ? 
 
 106. Two men working together can finish a piece of 
 work in 8 days ; one can do it in 12 days. How long would 
 the other take to do the work ? 
 
 107. How many yards of cloth at $3.75 per yard can be 
 bought for $90? 
 
 108. A puts $600 into business; B, $400; the profits 
 are $500. What is the share of each ? 
 
 109. Two boys hire a camera for 26 weeks, paying $5.20. 
 How much should be paid by the boy that uses it 12 weeks ? 
 
 110. New Orleans is 90° west of Greenwich. When it is 
 2 p.m. at the latter place, what is the time at New Orleans ? 
 
 111. Find the discount, at 6%, on a note for $300, that 
 has 48 days to run. 
 
 112. What will be the cost of 84 yards of cloth at 49 ? 
 a yard ? 
 
 113. Two men hire a pasture for $84. One puts in twice 
 as many head of cattle as the other. What should each pay? 
 
 114. A base-ball club won 17 games, and lost 13 games. 
 What per cent of its games did it win ? 
 
 115. What per cent of 4 is 64 ? 
 
 116. 2| is what per cent of 3£? 
 
 117. How many acres in a rectangular farm 1 mile long, 
 J mile wide ? 
 
402 Chapter Six. 
 
 118. What per cent of the "list" price is paid by a buyer 
 who receives a discount of 20 and 10 per cent ? 
 
 119. A tank is filled by two pipes, one of which can fill 
 it in 6 hours, and the other in 8. How long will it take 
 both together to fill the tank ? 
 
 120. Find the interest on $80, for 72 days, at 6%. 
 
 121. A man sold a wagon for $420, which was 16% less 
 than it cost. How much did he lose ? 
 
 122. A kilo is 2.2046 lb. How many pounds in 1000 
 kilos ? 
 
 515. "Written Keview Problems. 
 
 1. What number subtracted 88 times from 80.005 will 
 leave .013 as a remainder ? 
 
 2. At what price must an article that cost $30 be 
 marked so that after deducting 40% from the marked price, 
 30 % profit may be realized ? 
 
 3. Write a ninety-days promissory note for which you 
 should get $ 240 at the bank, discount being 6%. 
 
 4. If a horse dealer buys a span of horses at 10 per cent 
 less than their value, and sells them at 10 per cent more 
 than their value, what per cent does he make ? 
 
 5. If a boy buys peaches at the rate of 5 for 2 cents, and 
 sells them at the rate of 4 for 3 cents, how many must he 
 buy and sell to gain $ 4.20 ? 
 
 6. What is the difference between the compound interest 
 on $ 5000, for 3 years, at 5%, and on $ 10,000, for 1£ years, 
 at the same rate ? 
 
 7. A can do a piece of work in 27 days, and B in 15 
 days ; A works at it alone for 12 days, B then works alone 
 for 5 days, then C finishes the work in 4 days. In what 
 time could C have done the work by himself ? 
 
Review. 403 
 
 8. A room is 15 feet long, 10 feet broad, and 9 feet 9 
 inches high. Find the cost of painting the walls and the 
 ceiling, at Is. 9d. a square yard. 
 
 9. What is the value of a pile of wood 40 feet long, 4 
 feet wide, and 5 feet high, at $ 5.30 a cord? 
 
 10. By buying a cargo of coal at $6 per ton, and selling 
 it at $ 8 a ton, I gained $ 198. How much did I pay for it ? 
 
 11. Make out a receipted bill for the following : 325 yards 
 of silk at $ 2.25 per yard ; 296 yards of lace at $ 1.50 per 
 yard ; 480 yards of ribbon at 9 0.50 per yard ; 45 dozen 
 pairs of gloves at $ 15 per dozen pairs. 
 
 12. My dividend is 8|, quotient 94. What is the divisor ? 
 
 13. I gave away ^ and -§ of 41 bushels of chestnuts. 
 What % was left ? 
 
 14. The perimeter of a square field is 16 rods. WTiat is 
 the field worth, at 8J^ a square foot? 
 
 15. A broker's bill for cotton at 4J^ per pound and his 
 commission for buying at 21% was $1998.75. How many 
 bales of 400 pounds each did he buy, and what was his 
 commission ? 
 
 16. I sold 80 yards of broadcloth for $ 240, thereby losing 
 20% on the cost. For what should I have sold it per yard 
 to have gained 15 % on the cost ? 
 
 17. A man bought 60 casks, of 65 gallons each, for 
 $ 1542; 80 gallons leaked out. For what must he sell the 
 remainder per gallon to gain 12 J % on the cost? 
 
 18. Each of two men sold his horse for $ 180. One made 
 20%, the other lost 20% on the cost. Cost of each horse? 
 
 19. A man agrees to dig a cellar 30 feet long, 24 feet 
 wide, and 6 feet deep. What % of the work is to be done 
 when he has removed 144 cubic yards ? 
 
404 Chapter Six. 
 
 20. A man bought 672 yards of cloth at $ 1.25 a yard. 
 He sold it immediately for $ 2.25 a yard, receiving in pay- 
 ment a 60-day s note for the amount, which he had dis- 
 counted at a bank at 7%. How much money did he make ? 
 
 21. What will it cost to fill in a street 55 feet wide, 
 600 feet long, and 5| feet below grade, at 40 ^ a cubic yard ? 
 
 22. The quotient arising from the division of 6985.473 by 
 a certain number is 51, and the remainder is 68.853. What 
 is the divisor ? 
 
 23. What is the value of the following? 
 
 8flr + **-«t . 1 xftxf 
 2f + l|-3i " fxfxi 
 
 24. In going 1 mi. 94 rd. 2 yd. 1 ft., a carriage wheel 
 makes 526 revolutions. What is the circumference of the 
 wheel ? 
 
 25. On a note dated Oct. 16, 1903, for $2650, with in- 
 terest at 6 per cent, the following payments were made: 
 Jan. 28, 1904, $ 575 ; May 22, 1904, $ 25 ; and Aug. 4, 1904, 
 $ 948. 'What was due Nov. 25, 1904 ? 
 
 26. A grocer pays 18 ^ per pound for coffee and roasts it, 
 the coffee losing 10 per cent of its weight in the process. 
 What must he charge per pound for the roasted coffee in 
 order to make a profit of 20 per cent, allowing 4 per cent 
 for bad debts ? 
 
 Note. — 96% of the price he receives per pound must be 20% more 
 than the rate of 18 f for ^ lb. 
 
 27. A merchant imported from Bremen 32 pieces of linen 
 of 32 yards each, on which he paid for the duties, at 24 per 
 cent, $ 122.16, and other charges to the amount of $ 40.96. 
 What was the invoice value per yard, and the cost per yard 
 after duties and charges were paid ? 
 
Review. 405 
 
 28. A garrison of 1200 men is provisioned for 100 days. 
 At the end of 30 days, 600 men are withdrawn, and at the 
 end of 60 days, 900 men are added. How long will the 
 provisions last ? 
 
 29. What will be the result, if £ of f of 3 J- be multiplied 
 by £ of itself, and the product be divided by £ ? 
 
 30. A collector of internal revenue deposited in the treas- 
 ury $ 762,742.50, retaining 2\ per cent of the amount col- 
 lected. What amount did he collect ? 
 
 31. What is the duty on 25 tons 2 cwt. 3 qr. of iron at 
 % 8 per ton ? (1 ton = 2240 lb.) 
 
 32. An importer sold a part of a cargo of tea at 30 cents 
 a pound and made a profit of 20 per cent. What per cent 
 did he make on the remainder of the cargo, which he sold at 
 40 cents a pound ? 
 
 33. Divide % 4.14 among Thomas, Richard, and Henry in 
 such a way that Henry shall receive 3 cents for every 5 
 cents that Thomas gets, and Richard shall receive 2 cents 
 for every 5 cents that Henry gets. 
 
 34. Reduce 272 liquid quarts to dry quarts. 
 
 35. A pipe discharging 3 gallons 1 pint a minute fills a 
 tub in 4 minutes 20 seconds. Another pipe discharges 83 
 quarts a minute. If both pipes discharge together into the 
 tub, how long will they take to fill it ? 
 
 36. William Wilson sold goods to the amount of % 1000. 
 One-half of his sales showed a profit of 25 per cent on the 
 cost, and the remaining half a loss of 16| per cent on the cost. 
 Required the total cost of the goods. 
 
 37. If I sell I of my farm for f of what the farm cost me, 
 what is my gain per cent ? 
 
 38. Which is the higher rate of freight on wheat, $.16 
 per hundred or $ .10 per bushel (60 lb.), and what per cent ? 
 
406 Chapter Six. 
 
 39. Write in words : 
 
 (a) .267; (6)200.067; (c) flftj ((f) 200^fo. 
 
 40. If 40 per cent of the selling price of an article is 
 profit, what is the per cent of gain on the cost? 
 
 41. What number added to 4 J times itself will equal 60 J ? 
 
 42. Divide J by .00003J. 
 
 43. Keduce to lowest terms (a) i|-f| ; (6) |f|. 
 
 44. If 4 men eat 64 pounds of bread in 2 weeks, how 
 many pounds will 16 men eat in 7 weeks at the same rate ? 
 
 45. Divide .75 of 17f by £ of .035. 
 
 46. Find the cost of 3846 pounds of hay at $15 per ton. 
 
 47. Find the cost of plastering the walls and the ceiling 
 of a hall 72 feet long, 50 feet wide, and 22 feet high, at 18f 
 •cents a square yard, allowing 972 square feet for openings 
 •and baseboards. 
 
 48. A certain quantity of paper will make 4000 copies of 
 an octavo book (8 pages to the sheet). How many copies 
 ■of a 12mo book (12 pages to the sheet) will the same paper 
 make? 
 
 49. Find the diagonal of a square park containing 20 acres. 
 
 50. Bangor, Maine, June 24, 1904. 
 On demand, I promise to pay Joseph I. Totten, or order, 
 
 Two Thousand Five Hundred Fifteen Dollars, with interest, 
 value received. 
 
 $2515^. Charles Hettesheimer. 
 
 $1541.01 was paid Jan. 1, 1905. Find the amount due 
 Aug. 15, 1905. 
 
 51. How much does it cost annually to insure the "Celtic" 
 for $1,525,000, if 2J% is paid for the insurance ? 
 
 52. $ 150 is paid an agent for purchasing 1200 barrels of 
 ■flour on a commission of 2£%. How much was paid per 
 barrel for the flour ? 
 
Review. 407 
 
 53. An agent received $2562.50 for purchasing land at 
 $62.50 per acre, and his commission of 2|%. How many 
 acres did he buy ? 
 
 54. Reduce the fraction -3 — 4S + « — I* 5 • 
 
 55. Divide J by 2.5, to the quotient add the divisor, and 
 from that sum subtract the dividend. Give the fractional 
 part of the answer in a decimal. 
 
 56. If the interest on $300 for 1 yr. 8 mo. is $36, find 
 what would be the interest on $ 212.50 for 3 yr. 4 mo. 24 da* 
 at the same rate. 
 
 57. Reduce .0468 T. to a compound number. 
 
 58. Find the prime factors of 20,930. 
 
 59. A man paid $ 999 for the rent of a house from 
 June 29, 1903, to May 5, 1905. What was the rent per 
 year? 
 
 60. What per cent of 3 lb. 7 oz. is 7 lb. 9 oz. ? 
 
 61. At 50 cents per running yard, what will be the cost 
 of fencing a square field containing 10 acres ? 
 
 62. At the rate of 20 problems an hour for A, and 15 in 
 55 minutes for B, in what time can both together solve 100* 
 problems ? 
 
 63. Find the entire surface of a cube whose edge meas- 
 ures 15 inches. 
 
 64. A dealer buys books at $ 1.50 each, less 33J and 
 10 per cent. At what price per copy must he sell them 
 to gain 43£ per cent? 
 
 65. Abraham Lincoln died at the age of 56 yr. 2 mo. 3 da.,, 
 after serving as President 4 yr. 1 mo. 11 da. Give the date 
 of his birth, the date of his inauguration being March 4, 
 1861. 
 
408 Chapter Six. 
 
 66. A dealer buys 150 barrels of flour. He sells one- 
 third of it at $ 4.50 per barrel, losing 10 per cent. The 
 remainder he sells at a profit of 6 per cent. What is his net 
 gain or loss ? 
 
 67. Sixty per cent of 66% per cent of a number equals 
 810. What is the number ? 
 
 68. A ladder 40 feet long is so placed in a street, that 
 without being moved at the foot, it will reach a window 
 on one side 33 feet, and on the other side 21 feet from the 
 ground. What is the breadth of the street ? 
 
 69. Four men hired a coach for $ 13, to convey them to 
 their respective homes, which were at distances from the 
 place of starting as follows: A's 16 miles, B's 24 miles, 
 C's 28 miles, and D's 36 miles. What ought each to pay ? 
 
 70. What is a pile of wood 8 feet long, 7 feet wide, and 
 5 feet high worth, at $ 4.50 per cord ? 
 
 71. When bank stock sells at a discount of 1\ per cent, 
 what amount of stock, at par value, will % 3700 purchase ? 
 
 72. The pound sterling is worth $4.8665. How much 
 U. S. coin would it require to pay a debt of £ 780 18s. lid. ? 
 
 73. A merchant imported 120 tons of English iron, cost- 
 ing 1£ pence per pound, on which he paid a duty of 20 per 
 cent. The freight was 5 shillings sterling per ton. What 
 was the total cost in U. S. currency ? (1 ton =2240 pounds. 
 £1 = $4.8665.) 
 
 74. How many rods of fence are required to enclose a 
 square lot whose area is 5184 square feet ? 
 
 75. Property worth $6000 is insured for { of its value, 
 at f of one per cent. What will be the loss, including pre- 
 mium, in case of total destruction by fire ? 
 
 76. How many acres of land, in the form of a square, 
 may be enclosed by 160 rods of fence ? 
 
Review. 
 
 409 
 
 77. Find the square root of .441 correct to two decimal 
 places. 
 
 78. Reduce 17 lb. 10 oz. Avoirdupois weight to pounds, 
 ounces, pennyweights, and grains, troy weight. (1 pound 
 Avoirdupois = 7000 Troy grains.) 
 
 79. Reduce |4if to ^s lowest terms. 
 
 80. Find the solid contents of a cube, the area of one face 
 of which is 256 square feet. 
 
 81. A car contains 21,643 pounds of wheat. Find the 
 value of the load at 92 ^ per bushel of 60 pounds. 
 
 82. Find the area of a triangle whose base is 22 ft. 8 in., 
 and altitude 19 ft. 9 in. 
 
 83. The list price of a certain stove is $ 38, and the 
 retail dealer is allowed commercial discounts of 20 per cent, 
 5 per cent, and 3 per cent. What price does he pay for the 
 stove ? 
 
 84. If a ton of coal lasts a family 21 days, what will be 
 the cost of coal used by it from Oct. 17, 1904, to April 25, 
 1905, exclusive of either day named, at $ 4.50 per ton ? 
 
 85. Find the cost of a pile of 4-foot wood, 27 feet long 
 and 6 feet high, at $ 5.50 per cord. 
 
 86. How many rods of fence will be required to enclose 
 a field in the form of a right-angled triangle, whose area is 
 13i acres and whose base measures 48 rods ? 
 
 87. What is the balance of a bill of $ 64.50, after two 
 discounts have been made; the first of 20% on the $ 64.50, 
 the other of 5 % on what then remained ? 
 
 88. There was shipped to Liverpool from New York in 
 one week $ 6,870,205 in specie. What amount of English 
 currency could be bought with it ? (£ 1 = $ 4.8665.) 
 
 89. What is the freight on 9860 pounds iron at $ 1.75 per 
 ton? 
 
41 o Chapter Six. 
 
 90. What is the value of 10 lb. 7 oz. 16 pwt. of gold at 
 9 -75 a pennyweight ? 
 
 91. The dividend is 6171, the quotient 17, the remainder 
 102. What is the divisor ? 
 
 92. Divide the L. C. M. of 132 and 156 by their G. C. D. 
 
 93. The product of three numbers is .0728 j one of them 
 is 1.3, another .07. Find the third. 
 
 94. If 5 men can make 38 rd. 5 yd. of fence in a day, 
 how much can they build in 30 days ? 
 
 95. The distance from New York to New Haven being 
 73 mi. 8 rd., at what rate does a train run per hour to cover 
 the distance in 2 hr. 10 min. ? 
 
 96. Eeduce 4 da. 4 hr. 48 min. to the decimal of a week. 
 
 97. After 4 per cent of a flock of sheep had been killed 
 by dogs, and 68 had been sold to a butcher, four-sevenths 
 of the original flock were left. Required the number of 
 sheep in the flock at first. 
 
 98. Six men bought a ship worth $ 45,268, for which A 
 paid \ of the whole, B J, and the others paid the remainder 
 equally. How much did each of the latter pay ? 
 
 99. A man agrees to dig a cellar 30 feet long, 24 feet 
 wide, and 6 feet deep. What per cent of the work has he 
 done when he has removed 16 cubic yards ? 
 
 100. How many boards 16 feet long, and 4 inches wide, 
 are required to floor a room 48 feet long, and 32 feet wide ? 
 
 101. How much walking does a man save by crossing 
 diagonally a field 28 rods long, and 21 rods wide, instead of 
 going along the end and the side ? 
 
 102. In order to have an annual income of $2500, what 
 sum must be invested at 5% ? 
 
 103. At $2 a rod, what is the difference in the cost of 
 fencing a lot of land 20 rods square, and another lot contain- 
 ing the same area which is 40 rods long ? 
 
Review. 41 1 
 
 104. If a man owning 45% of a steamboat sells J of his 
 share for $ 5860, what is the value of the whole boat ? 
 
 105. A farmer having 6 bu. 8 qt. of cranberries lost by- 
 decay 7 pk. 7 qt. What % had he left ? 
 
 106. Sold tea for 114% of its cost, and made a profit of 
 7^ a pound. Find selling price. 
 
 107. In § of an acre of land how many building lots, each 
 60 feet by 121 feet? 
 
 108. I bought a store for a certain sum, and after paying 
 a tax of 2\°/o on the cost and \<fo more for insurance I sold 
 it for $ 7828, which exactly covered the cost, tax, and insur- 
 ance. What was the cost ? 
 
 109. Parker P. Simmons, of Vermont, sent to Nostrand 
 Bros, of Boston, to be sold on commission, the following 
 goods : 25 tons of hay, 2 tons of butter, 1500 pounds of 
 maple sugar, 75 gallons maple syrup. Nostrand Bros, sell 
 the hay at $ 18 a ton, the butter at 20 tf a pound, the sugar 
 at 7P a pound, the syrup at 90^ a gallon. 
 
 Nostrand Bros, charge 2% commission. How. much da 
 they send to Parker P. Simmons ? 
 
 110. How much will a granite block weigh which is 
 7 feet long, 2 ft. 6 in. wide, 3 ft. 4 in. high ? (12 cubic- 
 feet of granite weigh a ton.) 
 
 111. A coal dealer bought 350 tons of coal, weighing- 
 2240 pounds each, at $3.50 a ton. He sold the coal at 
 $4.25 a ton, each ton weighing 2000 pounds. What was- 
 his profit ? 
 
 112. Mrs. Burns buys 40 yards of carpet f of a yard 
 wide. She uses 10% of it for a rug, and the remainder to 
 carpet a floor. How many square yards does she use for 
 the floor ? 
 
 113. Mr. Burns sold his carriage for $224, which was f 
 of its cost. What per cent would he have gained if he had 
 sold it for $210? 
 
412 Chapter Six. 
 
 114. What is the difference between four thousand nine 
 and seven hundred eighty-six ten-thousandths, and four hun- 
 dred thousand nine and seven hundred eighty-six millionths ? 
 
 115. Discover a fraction which, multiplied by %, equals -J. 
 
 116. What % of i of | of | is %? 
 
 117. How many inches in y 1 ^ of a mile ? 
 
 118. Bought a horse for $90, and sold him for $95. 
 What per cent of gain? Bought another horse for $95, 
 and sold him for $90. What per cent was lost ? 
 
 119. Bought land at $62.50 per acre, and sold it again at 
 $75 per acre, thereby making $8468.75. How many acres 
 were bought ? 
 
 120. Two ships sail from the same port; one goes due 
 north 128 miles, and the other due east 72 miles. How far 
 are the ships from each other ? Illustrate. 
 
 121. B and C, trading together, find their stock to be 
 worth $3500, of which C owns $2100. They have gained 
 40% on their first capital. What did each put in ? 
 
 122. A general wished to remove 80,000 pounds of provi- 
 sions from a fortress in 9 days. It was found that in 6 days 
 18 men had carried away but 18 tons. How many men 
 would be required to carry away the remainder in 3 days ? 
 
 123. A schoolroom is 40 feet long, 30 feet wide, and 14 feet 
 high. Find the difference between the length of a diagonal 
 drawn on the floor and one drawn from the floor to the ceiling. 
 
 124. Find the solid contents and the surface of a sphere 
 12 inches in diameter. 
 
 125. The number of copies in the first edition of the 
 " Lady of the Lake " was 2050, and was to the number in 
 the second edition as 41 to 69. Find the number in the 
 second edition. 
 
Review. 413 
 
 126. Find the proceeds of the following note : 
 
 $ 1050 AV Chicago, Feb. 13, 1905. 
 
 Six months after date I promise to pay to the order of 
 John G-. Agar One Thousand Fifty Dollars, with interest at 
 6 per cent. Henry R. M. Cook. 
 
 Discounted at 8 per cent, May 13. 
 
 127. A can do ^ of a piece of work in 4 days, and B can 
 do \ of it in 5 days. In what time can they do the whole 
 work together ? 
 
 128. A square is inscribed in a circle whose diameter is 
 84 inches. Find the area of the four segments of the circle 
 outside of the square. 
 
 129. Find the difference between the volume of a cylinder 
 whose diameter and height are 12 inches, and the volume of 
 a sphere whose diameter is the same. 
 
 130. A certain cistern can be filled by one pipe in 10 
 hours, by another in 6 hours, and can be emptied by a third 
 in 5 hours. In how many hours can it be filled if all three 
 pipes are opened at once ? 
 
 131. Two men start from two towns 105 miles apart and 
 walk toward each other. They meet at the end of 15 hours. 
 The first has travelled 3 miles per hour. At what rate has 
 the second travelled ? 
 
 132. If a cipher is added at the right of the decimal, 
 what effect has this on the value of the decimal ? Explain 
 the reason. 
 
 133. What is the easiest method of dividing a decimal 
 by 10? 
 
 134. If the numerator of a common fraction is divided 
 by 3, what is the effect upon the value of the fraction ? 
 
 135. If the denominator is divided by 3, what is the 
 effect upon the value of the fraction? 
 
414 Chapter Six. 
 
 136. What is the effect on the value of a decimal of 
 moving the decimal point two places to the right? Explain 
 the reason. 
 
 137. What is the exact interest on $400, from March 1 
 to December 17, at 5 per cent ? (365 days to the year.) 
 
 138. In what time will a principal amount to 2\ times 
 itself, at 10 per cent ? 
 
 139. A and B in partnership have together a capital of 
 $7500, and gain $1200. A's share of the gain is $250. 
 What is B's share of the gain ? What is B's share of the 
 capital ? 
 
 140. The circumference of a circle is 15.708 feet. What 
 is the radius of it ? 
 
 141. The radius of a circle is 42. What is the circum- 
 ference of it ? 
 
 142. Find the entire surface of a cylinder 10 inches long, 
 and 8 inches in diameter. Find the number of cubic inches 
 in the same cylinder. 
 
 143. Explain the reason for multiplying the second and 
 third terms together and dividing by the first term in solv- 
 ing an example in simple proportion. 
 
 144. Divide thirty-two hundred-millionths by sixty -four 
 ten-thou sandths. 
 
 145. A, B, and C gained by speculation $11,480, of which 
 A's share was twice as much as C's, and B's five times as 
 much as C's. How much did each gain ? 
 
 146. A pole was broken 52 feet from the bottom, and it 
 fell so that the top struck 39 feet from the foot, while the 
 other end of the broken portion remained attached. Re- 
 quired the length of the pole. 
 
 147. Sold a horse so that £ of the gain equalled ^ of the 
 cost. What was the gain per cent ? 
 
 148. In what time will the interest on £57 Is. 8c?. amount 
 to £ 2 lis. 4£& at 1\ per cent per annum ? 
 
CHAPTER VII. 
 
 ALGEBEAIO EQUATIONS. 
 ONE UNKNOWN QUANTITY. 
 
 516. A number increased by 12 equals 16. 
 This may be written, x + 12 = 16. 
 
 The second way is shorter. Here x stands for the 
 number. 
 
 517. PreHminary Exercises. 
 
 Tell what x may stand for, and write in a short way each 
 of the following : 
 
 1. A number increased by 5 equals 7. 
 
 2. 6 is added to a number. The sum is 9. 
 
 3. 4 subtracted from a number leaves 1. 
 
 4. 12 diminished by a number has 8 for a remainder. 
 
 5. A number is subtracted from 10. The remainder is 3. 
 
 6. 10 is subtracted from a number. The remainder is 3. 
 
 7. The number of years of John's age added to 3 years 
 equals 15 years. 
 
 8. In two years Mary will be 11 years old. 
 
 9. 5 years ago Thomas was 8 years old. 
 
 10. If William should add 5 marbles to the number he 
 now has, he would have 15 marbles. 
 
 11. If Kate spends 10 cents, she will have 15 cents left. 
 
 12. When paying for a top, Henry received 7 cents 
 change from 10 cents. 
 
 415 
 
41 6 Chapter Seven. 
 
 13. A ball and a bat together cost 40 cents. The bat 
 cost 15 cents. 
 
 14. A watch cost Mr. Smith $ 60. He bought the case 
 and the works separately. The works cost $ 20. 
 
 15. The weight of a loaded wagon is 3200 pounds. The 
 load weighs 2000 pounds. 
 
 518. Sight Exercises. 
 
 
 
 
 If 
 
 
 x + 7 
 
 = 9, 
 
 
 then 
 
 
 X 
 
 = 2, 
 
 
 because 
 
 
 2 + 7 
 
 = 9. 
 
 
 Find the value of x : 
 
 
 
 
 1. 
 
 x + 6 = 9. 
 
 
 9. 
 
 x + 5 = 15. 
 
 2. 
 
 x- 4= 1. 
 
 
 10. 
 
 x+ 5= 8. 
 
 3. 
 
 x+ 5 = 7. 
 
 
 11. 
 
 x- 7 = 10. 
 
 4. 
 
 10- x= 3. 
 
 
 12. 
 
 x - 10 = 15. 
 
 5. 
 
 x -10= 3. 
 
 
 13. 
 
 x + 20 = 60. 
 
 6. 
 
 12- x= 8. 
 
 
 14. 
 
 x + 15 = 40. 
 
 7. 
 
 x+ 2 = 11. 
 
 
 15. 
 
 a; + 2000 = 3200. 
 
 8. 
 
 x+ 3 = 15. 
 
 
 16. 
 
 x+ J= 1. 
 
 
 COEFFICIENTS. 
 
 
 519. (1) 3 x means 3 times x. 
 
 (2) 2£ a means 2 J times a. 
 
 (3) ax means a times a; . 
 
 In (1), 3 is the coefficient of x. 
 In (2), 2£ is the coefficient of a. 
 In (3), a is the coefficient of x. 
 
 Notice that the coefficient and its letter are written side 
 by side. Is there any sign between them ? What sign is 
 understood ? What is a coefficient ? 
 
Algebraic Equations. 417 
 
 520. "Written Exercises. 
 
 Write in a short way, and tell what x stands for. 
 
 1. 8 times a number = 64. 
 
 2. A butcher receives 63 cents for a piece of meat at 
 9 cents a pound. 
 
 3. 2$ yards of muslin cost 25 cents. 
 
 4. A lady paid 40 cents for 3 spools of black silk and 
 2 spools of white silk at the same price per spool. 
 
 5. A man worked by the day 10 days on my barn and 
 8 days on my house. For all this work he received $ 36. 
 
 6. A man spent \ of his week's wages for a pair of 
 boots. The boots cost him $ 3. 
 
 7. 11 times a number less 2 times the number is 27. 
 
 8. John's money is in pennies and nickels. He has the 
 same number of each. He has 42 cents. 
 
 521. Sight Exercises. 
 
 If 
 
 10a;- 
 
 ■7X: 
 
 = 18, 
 
 
 then 
 
 
 Sx 
 
 = 18, 
 
 
 and • 
 
 
 X 
 
 = 6. 
 
 
 Proof : 60 - 42 = 18. 
 
 
 
 
 
 Give value of x at sight : 
 
 
 
 • 
 
 1. 8a; = 64. 
 
 
 
 8. 
 
 lla;-2a; = 27. 
 
 2. 9x = 63. 
 
 
 
 9. 
 
 3a; — 2a; + 5x = 54. 
 
 3. 3 a; + 2 a; = 40. 
 
 
 
 10. 
 
 10a; -f 8a; -4a; = 42. 
 
 4. 2| a; = 25. 
 
 
 
 11. 
 
 2x + 4a; = 52 -16. 
 
 5. ix = S. 
 
 
 
 12. 
 
 3a; + 4a; = 30 -9. 
 
 6. 10a; -f 8a; = 36. 
 
 
 
 13. 
 
 12a;-5a; = 25 4-10. 
 
 7. 4 a; — 3 x = 72. 
 
 
 
 14. 
 
 6a; + 6a; = 16 + 8. 
 
41 8 Chapter Seven. 
 
 522. Oral Exercises. 
 
 If a = 2, and if 6 = 3, and if c = 7. 
 
 1. a + b = ? 8. ac — & = ? 
 
 2. b — a=? 9. abc=? 
 
 3. a + b + c = ? 10. ? = 9. 
 
 4. c-a + 6 = ? 11. ? = 21. 
 
 5. c-a-& = ? 12. ? = 23. 
 
 6. a& = ? 13. ? = 19. 
 
 7. ab + c = ? 14. ?=4. 
 
 523. Written Problems. 
 
 1. A horse and a wagon cost together $600. What is 
 the price of each, if the wagon costs twice as much as the 
 horse ? 
 
 Let x = cost of horse ; 
 
 then 2 x = cost of wagon. 
 
 Cost of both = 2 x + x = 600. 
 
 8aj = 600. 
 x = 200. 
 2 x = 400. 
 Ans. Cost of horse, $ 200 ; of wagon, $ 400. 
 
 2. Divide 100 into two parts, one of which shall be four 
 times as large as the other. 
 
 Let x = one part ; 
 
 then 4 x = the other. 
 
 x + 4 x = 100. 
 
 3. $ 18,000 is divided among three children, the second 
 of whom receives twice as much as the first, and the third 
 of whom receives six times as much as the first. Kequired 
 the share of each. x % x q x 
 
Algebraic Equations. 419 
 
 4. In a class of 54 pupils, there are twice as many boys 
 as girls. How many are there of each ? 
 
 5. The sum of two numbers is 78. One is five times as 
 large as the other. What are the numbers ? 
 
 6. 156 is equal to seven times a number added to five 
 times the same number. Find the number. 
 
 7. The difference between three times a certain number 
 and nine times the same number is 66. What is the number ? 
 
 8. $ 27,000 is divided among three children, the second 
 of whom receives twice as much as the first, and the third 
 of whom receives three times as much as the second. What 
 is the share of each ? 
 
 9. The sum of two numbers is 72, and the greater is 5 
 times the other. What are the numbers ? 
 
 10. John, Henry, and James have 54 marbles. Henry 
 has twice as many as John, and James has as many as the 
 other two. How many has each ? 
 
 11. The sum of the ages of mother and daughter is 42 
 years. What is the age of each, if the mother's age is six 
 times that of her daughter ? 
 
 12. A man paid $96 for an equal number of hats and 
 ■coats, paying $ 2 apiece for the former and $ 10 apiece for 
 the latter. How many of each did he buy ? 
 
 Let x = number of each, 
 
 then 2 x = cost of hats, 
 
 10 x = cost of coats. 
 
 13. Divide 41 into four parts, the first being twice the 
 second, the second three times the third, and the third four 
 times the fourth. (Let x _ the fourth>) 
 
 14. The sum of three numbers is 180. The first is double 
 the second, and the third is three times as large as the sum 
 of the other two. What are the numbers ? 
 
420 Chapter Seven. 
 
 15. Mr. Smith paid 81 cents for sugar and flour, the 
 same quantity of each. For the sugar he gave 5 ^ per pound, 
 and for the flour 4/ per pound. How many pounds of each 
 did he buy ? 
 
 16. The length of a rectangular field is 24 rods, its breadth 
 is x rods, its area is 456 square rods. Find the value of x. 
 
 17. It takes 340 feet of fence to enclose a square lot. 
 What are the dimensions of the lot ? 
 
 18. Mrs. B. divides $ 120 between her son and her daugh- 
 ter. She gives the latter twice as much as she gives the 
 former. What is the share of each ? 
 
 19. The earnings of a man and his son during January 
 amounted to $ 175, both having worked the same number of 
 days. The father's wages were $ 4 per day, and the son's 
 wages were $ 3 per day. How many days did they work ? 
 
 20. The sum of $240 is divided among four children, two 
 boys and two girls. Find the share of each, if each girl's 
 share is double that of each boy. 
 
 21. A man worked twice as many days as his son. Their 
 combined earnings amounted to $ 165. Find the number of 
 days each worked, if the father earned $4 per day and the 
 son three-fourths as much per day. 
 
 22. A boy's bank contains 78 tf in dimes, nickels, and 
 cents. There are twice as many nickels as there are dimes, 
 and three times as many cents as there are nickels. How 
 many are there of each ? 
 
 23. I paid 75^ more for a roll of 15-cent ribbon than I did 
 for a roll of 12-cent ribbon of the same length. How many 
 yards did each roll contain ? 
 
 24. A rectangular field whose length is four times its 
 breadth requires 250 rods of fence to enclose it. What are 
 the dimensions of the field ? (Make diagram.) 
 
Algebraic Equations. 421 
 
 25. A girl paid 60 cents for a speller and a reader, the 
 cost of the former being one-third that of the latter. Find 
 the cost of each. 
 
 26. The sum of two numbers is 72, and the smaller is 
 one-fifth of the other. What are the numbers ? 
 
 Let x = smaller. 
 
 27. Mary, Susan, and Jane have 54 hickory nuts. Susan 
 has one-half as many as Mary, and Jane has as many as the 
 other two. How many has each ? 
 
 Let x = number Susan has. 
 
 EQUATIONS. 
 
 524. An expression like 3 x + 16 = 28 is an equation. 
 3 x -f- 16 is the first member of the equation. 
 
 28 is the second member of the equation. 
 
 What is the part of the equation to the left of the equal- 
 ity sign called ? What is the second member of an equa- 
 tion? What sign is between the two members? What 
 does this sign show about the value of the two members ? 
 What is an equation ? 
 
 525. Written Exercises. 
 Suppose a = 2 and 6 = 3. 
 
 Complete the equations : 
 
 1. ab + ? = 7. 6. ab + b = 5 + ?a. 
 
 2. a + 6-? = 4. 7. 17 - ab = 5a + ? 
 
 3. ?a + 6 = ll. 8. 17-a6-? = 5a. 
 
 4. ab = a + b + ? 9. 12 + a6 = 26 + ?a. 
 6. a& = 12-?6. 10. 12 + a6-?a = 26. 
 
422 Chapter Seven. 
 
 CLEARING OF FRACTIONS. 
 
 526. Oral Exercises. 
 
 1. One-fifth of a number is 4. What is the number ? 
 
 2. £ of a number is 8. What is £ of the number ? 
 
 3. % of a number is 12. What is the number ? 
 
 4. ^ of a number is 10. What is J of the number ? 
 
 5. If | of a number is 30, what is the number ? 
 
 6. One-half a number added to \ of the same number 
 equals what fraction of the number ? 
 
 7. One-half a number added to \ of the same number 
 equals 30. What is the number ? 
 
 8. One-third of a number -f- one-sixth of the number = 
 what fraction of the number ? 
 
 9. One-third of a number added to \ of the number = 
 what fraction of the number ? 
 
 10. \x + \ x = what fraction of x? |+|=? 
 
 527. When x = 32, find the value of three-fourths of x ; 
 
 When — (3 x divided by 4) = 24, what is the value of 
 
 3z? 0f»? 
 
 Find the value of y, when | = 12. Of 2y, when ^ = 24. 
 
 4z ^ 
 
 Given the equation —- = 20 ; by what whole number can 
 o 
 
 we multiply the first member to get rid of the fraction ? If 
 we multiply one member of an equation by any number^ 
 what must we do to the second member in order to preserve 
 the equality ? 
 
 If equals are multiplied by equals the products are equal. 
 
Algebraic Equations. 423 
 
 528. Sight Exercises. 
 
 Give values of x, y, z, etc. : 
 
 1. |=4. , g+j-u 9 . ?+|= 8 . 
 
 9 . | = 8. 6 . | + |=5. 10. f + ^=32. 
 
 3. |=7. 7. | + |=10. 11. f + |=9. 
 4 Jo 4 o 
 
 4. ^ = 21. 8. | + | = 7. 12. | + ^ = 9. 
 
 13. |-|=2. 15. |-?=6. 
 
 14. §-£ = 8. 16. 2_? =5 . 
 3 12 2 7 
 
 529. Written Exercises. 
 
 Find the value of the unknown quantity (x). 
 
 Multiplying by 12, we have 6x + 4x + 3x = 312. 
 
 2 . .+1+5-44. • . 
 
 Multiply by 6. 6x + 3x + 2x = 264. 
 
 To clear an equation of fractions multiply each term of both 
 members by the least common denominator of the fractions. 
 
 3. ^ + ^=35. 6 * t* + f*«fl2. 
 
 23 7. ^ + ^=102. 
 
 4. * + *=49. 3 4 
 
 3 4 8. 21 x =115. 
 
 5. * + ^ = 28. 9. 1^-^=48. 
 2 3 5 3 
 
424 Chapter Seven. 
 
 10. 
 
 a?- — = 156. 
 
 19. 
 
 75 a; 33 x _ gl 
 
 
 40 
 
 
 100 50 
 
 11. 
 
 3fz = 116. 
 
 20. 
 
 8 a; 2 a; ^ oa 
 — - — = ldb. 
 
 
 ¥= 27 - 
 
 
 3 5 
 
 12. 
 
 21. 
 
 2^3^4 
 
 13. 
 
 Ux=x27. 
 
 
 
 
 2 
 
 22. 
 
 a;_^_? = 37. 
 
 14. 
 
 il^=22. 
 
 
 2 3 
 
 
 4 
 
 
 4 a; 2 a; . 3 a; , 
 
 
 
 23. 
 
 — 2< 
 
 15. 
 
 2fa; = 44. 
 
 
 5 9 4 
 
 16. 
 
 2 a; -f — = 33. 
 4 
 
 24. 
 
 a; . £ x x Krt 
 
 17. 
 
 3£a;-2fa; = 45. 
 
 25. 
 
 aj_^ = 80. 
 4 
 
 18. 
 
 a;+f=24. 
 
 26. 
 
 a; -f 2a; + — = 24. 
 
 
 5 
 
 
 7 
 
 530. Written Problems. 
 
 1. Divide 100 into two parts, one of which shall be 1J 
 times the other. 
 
 2. After losing ^ of his money, a man has $714. How 
 many dollars had he at first ? 
 
 (* -|=714) 
 
 3. A horse was sold for $240, the seller thereby gaining 
 one-third of what he originally paid for it. How much did 
 he pay for it ? 
 
 K) 
 
 4. One-half of a number added to one-fourth of the same 
 number equals 66f . What is the number ? 
 
 5. The difference between f of a number and £ of the 
 same number is 15. Find the number. 
 
 6. One number is £ of another. Their sum is 55. What 
 are the numbers ? 
 
Algebraic Equations. 425 
 
 7. Find a fraction equivalent to J, the sum of its numer- 
 ator and its denominator being 60. 
 
 (Let 7 x = numerator and 8 x = denominator.) 
 
 8. Find a fraction equivalent to j-, the difference between 
 its numerator and its denominator being 24. 
 
 9. The sum of two numbers is 480, and the quotient ob- 
 tained by dividing the greater by the less is 7. What are 
 the numbers ? 
 
 10. Find two numbers whose difference is 522 and whose 
 quotient is 30. 
 
 11. A boy buys apples at 2^, pears at 3^, and oranges at 
 4^, the same number of each. How many of each does he 
 buy, if he pays 81 4 for all ? 
 
 12. A girl bought 70 cents' worth of peaches and plums. 
 She paid 3^ each for the peaches and 2^ each for the plums, 
 buying four times as many of the former as of the latter. 
 How many of each did she buy ? 
 
 13. $ 1500 is divided among three persons, the second of 
 whom receives three times as much as the first, and the third 
 three and one-half times as much as the first. Find the 
 share of each. 
 
 14. A farmer paid for a cow three-sevenths as much as 
 he paid for a horse. How much did he pay for each, if the 
 latter cost $80 more tl- n the former ? j 
 
 15. Three times a man's money increased by two-thirds 
 of his money is equal to $ 1100. How much money has he ? 
 
 16. After giving away f of his marbles 'and losing \ of 
 them, Joseph has 24 left. How many had he at first ? 
 
 17. Bought a coat, a hat, and an umbrella for $15, pay- 
 ing for the hat 1^- times as much as for the umbrella, and 
 for the coat 3^ times as much as for the hat. Find the price 
 of each. 
 
426 Chapter Seven. 
 
 18. A merchant purchased two pieces of cloth for $240, 
 paying for one piece twice as much per yard as for the other. 
 The former contains 36 yards and the latter 48 yards. How 
 much does he pay per yard for each ? 
 
 19. A farmer sold 4 times as many cows as horses, receiv- 
 ing for all $840, at the rate of $40 for a cow and $120 for 
 a horse. How many of each did he sell ? 
 
 20. The weight of a team with a loaded wagon is 5500 
 pounds. The wagon weighs f as much as the load. The team 
 weighs twice as much as the wagon. How many pounds 
 does the load weigh ? 
 
 53i. Oral Exercises. 
 
 
 
 
 Give values of x, y, z, 
 
 etc.: 
 
 
 
 1. a + 15 = 21. 
 
 
 7. 
 
 Sy + 6 = 15. 
 
 2. 22/ + 15 = 21. 
 
 
 8. 
 
 7 y - 13 = 15. 
 
 3. 0-7 = 21. 
 
 
 9. 
 
 9 y + 13 = 58. 
 
 4. 4w-7 = 21. 
 
 
 10. 
 
 3y-10 = 56. 
 
 5. | + 3 = 8. 
 
 
 11, 
 
 5^+1 = 7. 
 4 
 
 6. --3 = 12. 
 
 
 12. 
 
 i^-i=n. 
 
 532. If x + 15 = 21, x = 21 - what ? 
 
 When x- 7 = 21, x = 21 + what ? 
 
 If in the equation 2 x -f 15 = 21, we take away 15 from 
 the first member, what must we do to the second member to 
 preserve the equality ? 
 
 If equals are subtracted from equals, the remainders are 
 equal. 
 
 ' By transposing we mean bringing the unknown quantities 
 (jr, /, z, etc.) to one side of the equation and the known quan- 
 tities to the other. 
 
 Note. — In bringing a quantity from one side of the equation to the 
 other, the sign of the quantity is changed. 
 
Algebraic Equations. 427 
 
 533. Written Exercises. 
 
 Find values of the unknown quantities. 
 
 Note. — Clear of fractions when necessary ; then transpose. 
 
 1. x + 37 = 56. 
 
 5. 
 
 x + 3 x = 25 + 11. 
 
 2. 4a;-5 = 83. 
 
 6. 
 
 5 x = x + 40. 
 
 3. 3 a; -43 = 98. 
 
 7. 
 
 3 x - 20 = x - 8. 
 
 4. 7x + 13 = lll. 
 
 8. 
 
 12 - 3 x = 45 - 4 x. 
 
 9. 3z-6 = 48 + z. 
 
 15. 
 
 7rc-5a; = 20 + a;+4 
 
 0. 3z + 6 = 9-2a + 12. 
 
 16. 
 
 6x -14 = 16 + x 
 
 1. 2z-2-16 = a; + 10. 
 
 17. 
 
 2z-ll+6a;-60=5a; 
 
 2. --8 = 24. 
 
 18. 
 
 l+|- 6 = 10 - 
 
 3. ^ + 4-7 = 21. 
 6 
 
 19. 
 
 2*-6 = 16 + |-|. 
 
 14. | + ^ = 10 + 5. 20. 2z + ^-| = ^ + 27. 
 
 534. Written Problems. 
 
 1. The sum of three numbers is 51. The second is 5 
 less than the first, and the third is 10 less than the first. 
 What are the numbers ? 
 
 Let x = first number, 
 
 x — 5 = second number, 
 x — 10 = third number ; 
 x + x - 5 + x - 10 = 51. 
 Transposing, x + x + x = 51 -f 5 + 10, 
 
 3 x = 66, 
 x = 22, first number, 
 a; — 5 = 17, second number, 
 X — 10 = 12, third number. 
 
428 Chapter Seven. 
 
 2. Add 45 to four times a number, and you will have 
 seven times that number. What is the number ? 
 
 (7 x = 45 -f 4 x.) 
 
 3. Nine times a number less 27 equals six times the 
 number. Find the number. 
 
 4. Two boys have together 48 marbles. One has 18 
 more than the other. How many has each? 
 
 (x, x+ 18.) 
 
 5. The length of a rectangular lot is 75 feet more than 
 the breadth. The distance around it is 250 feet. What are 
 its dimensions ? 
 
 6. A piece of land containing 86 acres is to be divided 
 into two fields, one of which shall be 8 acres larger than the 
 other. How many acres in each field ? 
 
 7. At a certain election 2436 votes were cast for two 
 candidates, the successful one receiving 318 more votes than 
 his opponent. How many votes did each receive ? 
 
 8. A man, being asked his age, replied that if he were 
 half as old again and 7 years more he would be 100. What 
 was his age ? 
 
 9. The sum of two numbers is 96, and their difference is 
 72. Find the numbers. 
 
 (Let x = less, x + 72 = greater.) 
 
 10. After paying £ and J of my debts, I still owe $ 45. 
 How much did I owe originally ? 
 
 s_?_? = 45. 
 3 4 
 
 11. Divide 45 into two parts, one of which shall be 6 less 
 than twice the other. 
 
 12. William has $5 more than John, and three times 
 William's money added to five times John's would be $ 103. 
 How many dollars has each ? 
 
Algebraic Equations. 429 
 
 13. I bought 3 cows and 4 horses for $635, paying $ 80 
 apiece less for the cows than for the horses. How many 
 dollars apiece did I pay for each ? 
 
 14. Mary has a dollar in dimes and five-cent pieces. 
 She has 11 more of the latter than of the former. Find 
 the number of pieces of each denomination. 
 
 15. Divide 100 into two parts whose difference shall 
 be 48. 
 
 16. In a class of 54 pupils, the girls outnumber the 
 boys by 12. How many are there of each? 
 
 17. $ 18,000 is divided among three persons, the second 
 of whom receives 9 2400 more than the first, and the third 
 of whom receives $2400 more than the second. Find the 
 share of each. 
 
 18. The greater of two numbers is 11 more than 3 times 
 the less. Their difference is 33. What are the numbers ? 
 
 19. A boy spent a dollar for postal cards, 2-cent stamps, 
 and 5-cent stamps. He bought 15 more 2-cent stamps than 
 5-cent stamps, and 15 more postal cards than 2-cent stamps. 
 How many of each did he buy ? 
 
 Let x = number of 5-cent stamps, 
 
 then x + 15 = number of 2-cent stamps, 
 
 x -f- 30 = number of postal cards. 
 
 5 x = value of 5-cent stamps, 
 2 x + 30 = value of 2-cent stamps, 
 x + 30 = value of postal cards. 
 5£-f-2s + 30 + a + 30 = 100 
 
 20. A farmer has 88 head of stock — horses, cows, and 
 sheep. He has 17 more cows than horses, and the number 
 of sheep is 22 greater than that of the cows and horses 
 together. How many are there of each ? 
 
430 Chapter Seven. 
 
 ADDITION OF ALGEBRAIC QUANTITIES. 
 535. Oral Exercises. 
 Add: 
 
 1. 2 fours 
 
 2. 6 hundredths i 
 
 3. $4 4. 
 
 3f 
 
 5. 7 a? 
 
 3 fours 
 
 8 hundredths 
 
 $5 
 
 5? 
 
 4a? 
 
 4 fours 
 
 10 hundredths 
 
 $7 
 
 *f 
 
 2x 
 
 5 fours 
 
 12 hundredths 
 
 $3 
 
 9^ 
 
 5x 
 
 ? fours 
 
 ? hundredths 
 
 i? 
 
 U 
 
 ~x 
 
 When no 
 
 coefficient is expressed, 
 
 1 is understood. 
 
 Thus, 
 
 abc = 1 abc. 
 
 
 
 
 
 Where nc 
 
 > sign is expressed, + is 
 
 understood 
 
 . 
 
 
 6. — 2a 
 
 7. + 3x 8. -5xy 
 
 9. 9 abc 
 
 10. 
 
 — 24:xyz 
 
 - 4a 
 
 + 4a; — 4:xy 
 
 15 abc 
 
 
 — 5xyz 
 
 - 6a 
 
 + 5a? — xy 
 
 6 abc 
 
 
 - xyz 
 
 - la 
 
 +10 a; —2xy 
 
 abc 
 
 
 —15xyz 
 
 -19 a 
 
 + ? x -?xy 
 
 ? abc 
 
 
 — ? xyz 
 
 NEGATIVE QUANTITIES. 
 
 536. Suppose three men as follows : 
 
 The first man has $5 and owes nothing. 
 The second man has $5 and owes $5. 
 The third man has nothing and owes $5. 
 The first man is worth $5. 
 The second man is worth nothing. 
 
 The third man is worth 5 less than nothing. So we may 
 say he is worth — $ 5. 
 
 537. Quantities like — $5, —17, and —2a are called negative 
 quantities. 
 
 What sign precedes a negative quantity ? 
 
 Quantities with a plus sign expressed or understood are called 
 positive quantities. 
 
Algebraic Equations. 433 
 
 SUBTRACTION OF ALGEBRAIC QUANTITIES. 
 
 541. Preliminary Exercises. * 
 
 1. A man sold a horse for $ 100 at a gain of $ 25. Find 
 the cost. (Cost = selling price — gain.) 
 
 $ 100 = selling price $ 100 
 
 subtract 25 = gain or add — 25 
 
 remainder $ 75 = cost $ 75 
 
 2. A man sold a horse for $ 100 at a gain of — $ 25. 
 Find the cost. 
 
 $ 100 = selling price $ 100 
 
 subtract - 25 = gain or add -f 25 
 
 $ 125 = cost $ 125 
 
 In the first of the above examples, subtracting 4- $ 25 is the same 
 as adding — $ 25. 
 
 In the second of the above examples, subtracting — $ 25 is the same 
 as adding + $ 25. 
 
 We changed the first example from subtraction to addition by 
 changing the sign of the subtrahend from + to — . 
 
 We changed the second example from subtraction to addition by 
 changing the sign of the subtrahend from — to +. 
 
 To subtract in algebra, change the sign of the subtrahend 
 and proceed as in addition. 
 
 3. Add 7 and - 3. 6. Add - 7 and 3. 
 
 4. From 7 subtract - 3. 7. Add - 7 and - 3. 
 
 5. From — 7 subtract 3. 8. From — 3 subtract — 7. 
 9. Subtracting — 7 is the same as adding what ? 
 
 10. Is a positive quantity increased or decreased by sub- 
 tracting a negative quantity ? 
 
 Note. — When you become familiar with the process of subtraction 
 it will not be necessary to write the subtrahend with a changed sign. 
 You can conceive the sign changed and add. 
 
43 2 Chapter Seven. 
 
 We may put down the above statements thus : 
 
 $ 6 $-3 
 
 -3 2 
 
 4 -2 
 
 8 4 
 
 -2 -1 
 
 _I -2 
 
 $20 $-2 
 
 In the first example we add all the positive quantities, $ 6 -f $ 4 
 + $8 + $7 = $25. Then we add all the negative quantities, — $3 
 
 — $ 2 = — $ 5. Adding $ 25 and — 1 5 the result is $ 20. 
 
 In the second example we add all the positive quantities, and get 
 $ 6. The sum of the negative quantities is — $ 8. Adding $ 6 and 
 
 — $ 8 the result is — $ 2. 
 
 Can you give the rule for addition where the quantities 
 have different signs ? Which sign does the sum take ? 
 
 540. Written Exercises. 
 Add: 
 
 1. —2a 2. 7x 3. — 5xy 4. — 9abc 5. — 24an/z 
 
 — 4:xy 15 abc 5 xyz 
 
 xy 6 abc xyz 
 
 2xy — abc 15 xyz 
 
 6. 3a; + 14, —7a; + 9, -23, 4a; -5, -2a;, and 3a; + 11. 
 
 3a; + 14 
 Write like quantities in the same column. __ j x , g 
 
 Find the sum of the positive terms, also —23 
 
 the sum of the negative terms; subtract 4#— 5 
 
 the less from the greater, and prefix the __2a; 
 
 sign of the greater. 3 a; + 11 
 
 7. 4a + 3a;, —2a, —7a; — 3a, —5a;, — 9a + a;. 
 
 8. -36 + c, 4a + 6 6, 56-9 c, -3a, -2a-3& + 4& 
 
 9. ix-S, -a; + 4, — \x — 3, 7a; + 16, -5a;-10. 
 10. 4a; + 23, -%x + 2\, -|a; + ll, -a; + 5, 9a;-3. 
 
 -2a 
 
 2. 
 
 7x 
 
 -4a 
 
 
 — 4a; 
 
 — 6a 
 
 
 -2a; 
 
 la 
 
 
 5x 
 
Algebraic Equations. 433 
 
 SUBTRACTION OF ALGEBRAIC QUANTITIES. 
 541. Preliminary Exercises. • 
 
 1 . A man sold a horse for $ 100 at a gain of $ 25. Find 
 the cost. (Cost = selling price — gain.) 
 
 $ 100 = selling price § 100 
 
 subtract 25 = gain or add — 25 
 
 remainder $ 75 = cost $ 75 
 
 2. A man sold a horse for $ 100 at a gain of — $ 25. 
 Find the cost. 
 
 1 100 = selling price $ 100 
 
 subtract — 25 = gain or add + 25 
 
 $ 125 = cost $ 125 
 
 In the first of the above examples, subtracting + $ 25 is the same 
 as adding — $ 25. 
 
 In the second of the above examples, subtracting — $ 25 is the same 
 as adding + $ 25. 
 
 We changed the first example from subtraction to addition by 
 changing the sign of the subtrahend from + to — . 
 
 We changed the second example from subtraction to addition by 
 changing the sign of the subtrahend from — to + . 
 
 To subtract in algebra, change the sign of the subtrahend 
 and proceed as in addition. 
 
 3. Add 7 and - 3. 6. Add - 7 and 3. 
 
 4. From 7 subtract —3. 7. Add —7 and —3. 
 
 5. From — 7 subtract 3. 8. From — 3 subtract — 7. 
 9. Subtracting — 7 is the same as adding what ? 
 
 10. Is a positive quantity increased or decreased by sub- 
 tracting a negative quantity ? 
 
 Note. — When you become familiar with the process of subtraction 
 it will not be necessary to write the subtrahend with a changed sign. 
 You can conceive the sign changed and add. 
 
434 
 
 Chapter Seven. 
 
 542. Sight Exercises. 
 
 1. /What is the difference between + 52° and +33°? 
 
 2. Between + 90° and - 10°? 
 Show by a diagram. 
 
 3. A has $600, B owes $400. What are they worth 
 together ? ^ + g 600 ^ + (_ $ 40 0) = ? 
 
 4. How much better off is A than B ? 
 
 (+$ 600) -(-$400) = ? 
 
 5. From — 8 a take —2a. 
 
 6. From —2a take 8 a. 
 
 7. From — 2 a take — 8 a. 
 
 8. From 2 a take — 8 a. 
 
 9. From 3 x +14 take as +10. 
 
 3a; + 14 
 - a? -10 
 
 10. From 5a;— 8 take— 3a;— 9. 
 
 11. From a;— 28 take 5 a,*— 37. 
 
 12. From 7a;+16 take 9a;— 4. 
 
 13. From 6 x take 2 x — 5. 
 
 14. From 8 x take 9 x + 3. 
 
 15. From 3 a; + 2 a — 5 take a; — a — 9. 
 
 16. From ly — 2z + b take — 8y + 66 — 2. 
 
 17. From c — d + e take c + d —f. 
 
 543. Written Exercises. 
 
 1. 
 
 From 8 
 
 a take 2 a. 
 
 8a 
 - 2a 
 
 
 Ans. 
 
 6a 
 
 2. 
 
 From 2 
 
 a take 8 a. 
 
 2a 
 - 8a 
 
 
 -4ns. 
 
 - 6a 
 
 3. 
 
 From - 
 
 - 8 a take 2 a. 
 
 - 8a 
 
 - 2a 
 
 
 Ans. 
 
 -10a 
 
 4. 
 
 From 8 
 
 a take — 2 a. 
 
 8a 
 + 2a 
 
 
 Ans. 
 
 10 a 
 
Algebraic Equations. 435 
 
 REMOVING PARENTHESES. 
 
 544. Written Exercises. 
 
 1. From 6 x + 15 y take 4 x + 10 y. 
 
 We may write the above in a shorter way, thus : 
 
 6x + 15y-(4x + 10 y). 
 
 The minus sign before the parenthesis shows that the 
 quantity within the parenthesis is to be subtracted. What 
 sign is before 10 y ? What sign is understood within the 
 parenthesis before 4 x ? In subtraction, what is done with 
 the signs of the subtrahend? If the whole expression is 
 written without using the parenthesis, what must be done 
 with the signs of the quantities within the parenthesis ? 
 
 a — (b — c) may be written a — b + c. Why ? 
 
 a + (6 — c) may be written a + b — c. Why ? 
 
 When removing a parenthesis preceded by a minus sign, 
 change the signs of all quantities within the parenthesis. 
 
 545. Written Exercises. 
 
 Write the following without parentheses : 
 
 1. 57 + (33 - 16) = 74. 4. (17 - 8) - (16 - 14) = 7. 
 
 2. 92 -(63 + 25) =4. 5. 75 + 4 x (15 - 10) = 95. 
 
 3. (43 -10) + (24 -5) = 52. 6. 75 -4 x (15- 10) = 55 
 
 7. 4 x + 5 y + (2 x — 6 y) = 6 x — y. 
 
 8. 4z + 5?/— (2ic + 6?/) = 2a; — y. 
 
 9. 4:X — 5y — (x — 6y) = Sx-{-y. 
 
 10. 4# — 5?/ — (— ic+ 6?/) = 5<c — 11 y. 
 
 11. 4X+5?/ — (— 2a — 6y)=t?. 
 
 12. -4 -5 y + (2 -6 ?/) = ?. 
 
436 Chapter Seven. 
 
 546. Solve the following equations. Prove the correct- 
 ness of your answers. 
 
 1. 6(2 x -5) = 5 x + 12. 
 
 Note. 6(2 x — 5) means 6 times (2 x — 6), or 12 x — 30. 
 
 2. 7(x + 2) = 3x + 50. 4. 3(16 - a?) = 4(13 - x). 
 
 3. 5(3 + a?) + 16 = 61. 5. 15(x - 3) = 2(189 - 16 x) 
 
 6. 38 -(11 -9 a) = 10 a. 
 Removing the parenthesis, we have 
 
 38 - 11 + 9 x = 10 x. 
 
 Transposing, 9 x - 10 x = - 88 + 11, 
 
 or, - x = - 27. 
 
 Bringing — x to the right side of the equation, and — 27 to the left 
 side, we have (+)27 = (+)x 
 
 In practice, however, when the result is such as the above, — x = 
 — 27, the signs of both members are changed, and the result becomes 
 
 x=27. 
 
 7. 2(x-l)-2(2x-19) = 3(x-3). 
 
 8. 6(2 a? -6) -5* = 12. 
 
 9. 5 x - 6(2 x - 5) = - 12. 
 
 10. 11 - 3a? + 5x = 19. 
 
 547. !^_24-4 = 2 . 
 
 Z o 
 
 Clear of fractions by multiplying both members of the equation by 
 10, and observe which sign must be changed to preserve the equality. 
 When x = 6, the above may be written 
 
 3s-6 4s-4 _ 2 
 2 6 
 
 Clearing of fractions, 16 x - 30 - (8 x - 8) = 20. 
 
Algebraic Equations. 437 
 
 Removing the parenthesis, 
 
 15x_30-8z + 8 = 20. 
 Transposing, 15 x - 8 x = 20 + 30 - 8, 
 
 or, 7x = 42, 
 
 x = 6. 
 
 Note. — The horizontal line between the numerator and the de- 
 nominator of the foregoing fractions has the effect of a parenthesis, 
 the entire quantity above the line being divided by the number below. 
 
 Hence when an equation is cleared of fractions, what must be done 
 with the signs of the terms obtained from a fraction with a minus sign ? 
 
 l 8 -^= (18-6) -2, 24^4 = ^ of ( 24-4). 
 
 1 iof(3x-6), 4x ~ 4 =(43-4)-*-5. 
 
 2 
 548. Solve 
 
 11. ^l + ^2 = 8. 
 
 2 3 
 
 12. ^1_E^2 = 2 . 
 
 2 3 
 
 13. ^zl_^zl_^zl + 2 = 0. 
 
 2 3 4 
 
 ' 2x — 5 , x — 7 5x — 3 
 "• — 2- + — = — 6" 
 
 15. l£^_(« + 2 ) = **±«_!I±2. 
 
 .. 40-5a: 52 + 9* 
 16- — — = — — ' 
 
 17. 9|-/|,_|\- f |» + 3|* 
 
 18. 2as = 3 + 2Ja!-(5+|a!)+2|. 
 
 19. fa; + 9 = 2* + (|x-|a!). 
 
 20 - I+H+I+ 81 -* 
 
438 Chapter Seven. 
 
 21. |a>-120 = | + 10. 
 
 22. a? -20 = (1 + 15^4 
 
 23. a + | + | = 19. 
 
 24. 9(8a + l)-4 = 4(9a; + 5)+3. 
 
 25. 2 a; + 3 = — 
 
 549. Written Problems. 
 
 1. A certain number is multiplied by 3f; 7 is subtracted 
 from the product ; the remainder is divided by 16, giving a 
 quotient of 3. What is the number ? 
 
 2. Three-eighths of what number is 60 less than the 
 number itself ? 
 
 3. Four persons are of the same age. If the first were 
 \ of his age older, the second \ of his age older, the third \ 
 of his age older, and the fourth \ of his age older, the sum 
 of their ages would be 99 years. What is the age of each ? 
 
 4. A man spends \ of his earnings on board and lodging, 
 ■§- on clothing and repairs, and \ on sundries. At the end of 
 the year he has $ 280 left. What are his yearly earnings ? 
 
 3 = ? + | + | + 280. 
 
 2i o 5 
 
 5. A boy gave \ of his marbles to one companion, and \ 
 of them to another. He then bought \ as many as he origi- 
 nally had, and had 4 marbles more than he had at first. 
 How many did he have at first ? 
 
 6. A father's age and a son's age added together amount 
 to 138 years. Twelve years ago the father was twice as old 
 as the son. How old is each now ? 
 
 Let x = son's age 12 years ago. 2 x = father's age then. 
 
Algebraic Equations. 439 
 
 7. John has 80 cents, and William has 60 cents. How 
 many cents should William give John so that the latter 
 might have 2\ times as much money as the former ? 
 
 After William gives John x cents, the former has (60 — x) cents, 
 and the latter has (80 -f x) cents. 
 
 8. In how many years will a man, now 25, be double the 
 age of his 11-year-old brother ? 
 
 Let x = number of years. 25 + x and 11 + x = ages after x years. 
 
 9. A man has a cask of 60 gallons' capacity. He draws 
 off one-fourth of its contents, and then fills it. If it takes 
 24 gallons to fill it, how many gallons did the cask originally 
 contain ? 
 
 10. A number is divided by 3, and 40 is subtracted from 
 the quotient, leaving a remainder of 104. What is the 
 number ? 
 
 11. The difference between two numbers is 430. When 
 the greater is divided by the less, the quotient is 4, and the 
 remainder is 76. What are the numbers ? 
 
 Let x = less. EH*L r = 4 + It 
 less less 
 
 12. A person pays $103 with 29 $2 and $5 bills. How 
 many are there of each denomination ? 
 
 13. A father is 30 years older than his daughter. In 
 4 years, his age will be four times her age. What are their 
 present ages ? 
 
 x and x + 30 = present ages, x + 4 and x + 34 = ages 4 years later. 
 
 14. The product of two numbers is 180. If the smaller 
 number be increased by 3, the product of the two numbers 
 will be 225. What are the numbers ? 
 
 smaller = x ; — = greater. 
 
 x 
 
 15. A man's wages are $ 1 per day more than his son's. 
 For 33 days' work, the father receives $ 12 more than the 
 son earns in 40 days. Find the wages of each. 
 
440 Chapter Seven. 
 
 TWO UNKNOWN QUANTITIES. 
 
 550. Preliminary Problems. 
 
 1. I paid a dollar for two 25^ balls and five bats. How 
 much did I pay apiece for the latter ? 
 
 2. When three times one number is added to five times 
 another, the sum is 84. If the second number is 12, what 
 is the first number ? 
 
 3. A girl paid 75^ for £ pound of tea and 2\ pounds of 
 coffee. The coffee cost 20 ^ per pound. What was the price 
 of the tea per pound ? 
 
 4. A man sold pigs at $5 each and lambs at $8 each, 
 receiving $42. He sold 4 lambs. How many pigs did he 
 sell? 
 
 5 . Four times a father's age added to twice his daughter's 
 age amounts to 180 years. The girl is 10 years old. What 
 is the father's age ? 
 
 6. Eight peaches and seven pears cost 44^. The peaches 
 cost 2^ each. What is the cost of a pear? 
 
 7. Two pieces of cloth and eleven pieces of silk contain 
 152 yards. There are 10 yards in each piece of cloth. 
 How many yards in each piece of silk ? 
 
 8. Two-thirds of a yard of linen and three-fourths of a 
 yard of lace cost 40^. The price of the lace is 32^ a yard. 
 Find the price of the linen. 
 
 9. Three and one-half times one number added to four 
 and one-third times a second number equals 60. The second 
 number is 9. What is the first number ? 
 
 551. Written Exercises. 
 
 Find the value of the unknown quantity : 
 
 1. 8 x + ly = 44. When a = 2, find the value of y, 
 
 2. 3 y + 5 z m 34. Find the value of z ; y = 3. 
 
 3. 2a + ll3 = 152. a; = 10;2 = ?. 
 
 4. 14a + 7y = 98. x = 3%-,y=?. 
 
Algebraic Equations. 441 
 
 5. !'*+ f*x=40. 2 = 32. 
 
 6. 9x- 25y = 8. x = 12. 
 
 7 . 3Jy + 4i*==eO. z = 9. 
 
 8. 16a>-19z = 49. z = 5. 
 
 9. 7y- 3* = 18. 2/ = 6f 
 10. 32x + 50?/ =2600. 2/ =20. 
 
 552. Written Problems. 
 
 1. The cost of 3 apples and 2 peaches is 7 cents. The 
 cost of 2 apples and 2 peaches is 6 cents. 
 
 Subtracting the second lot of fruit from the first lot we have 1 apple. 
 Subtracting the price of the second lot from the price of the first 
 lot we have 1 cent. 1 apple costs 1 cent. 
 If equals are subtracted from equals, the remainders are equal. 
 
 2. A boy gave 2S$ for 3 lemons and 8 oranges, another 
 boy paid 11$ for 3 lemons and 4 oranges. How much did 
 the lemons cost apiece ? 
 
 x = cost of lemons, 3 x + 8 y = 25 (1) 
 
 y = cost of oranges, 3 x + 4 y = 17 (2) 
 
 Subtracting (2) from (1), 4y= 8 
 
 The oranges cost 2f each, y = 2. 
 
 How much apiece was paid for the lemons ? 
 
 3. If 3 coats and 14 vests cost $ 78, and 2 coats and 14 
 vests, at the same rate, cost $66 } how much does 1 coat 
 cost ? What is the price of a vest ? 
 
 4. Given 4z + ly = 53, (1) 
 
 2x + 32/ = 25, ' (2) 
 
 to find the value of y. 
 
 First multiply (2) by 2, making it 4 re + 6 x = 50. Why ? 
 
 5. What is the value of x in equation (1) in above ex- 
 ample, when the value found for y is substituted therein ? 
 Substitute the same value for y in equation (2) and find the 
 value of x. 
 
442 Chapter Seven. 
 
 553. Written Exercises. 
 
 Find the values of x and y in the following equations : 
 
 1. x + 2/ = 15, 3. 2x + 3y = 18, 
 2x + 3 y = 38. 4# + 3y == 24. 
 
 2. 2z + 2y = 30, 4. 2a; + 3?/ = 40, 
 
 x + 32/ = 27. 3z + 2?/ = 35. 
 
 Multiply first equation by 3, 6 x + 9 y = 120. 
 Multiply second equation by 2, 6 a; + 4 y = 70. 
 
 5. 7 a? + 5 ?/ = 82, 6. 5 a -f- 9 y = 14, 
 
 2 * + 3 ?/ = 36. 9*4 5 ?/ =14. 
 
 7. 3a> + 53/ = 17, 8. 2x-3y = l$, 
 
 8 8. + 2 2/ = 17. 3z + 52/ = 65. 
 
 Given ; ft ( _ . * ' \ To find values of x and y. 
 (2) 7 # — 4 2/ = 22. J * 
 
 Multiply (1) by 7, 7 x + 21 y = 322 
 
 (2) Ix- 4y = 22 Subtract. 
 25 y = 300 
 y= 12 
 
 Substituting this value of y in (1), we have 
 x + 36 = 46, 
 
 a; = 46 - 36 = 10. Arts, x = 10, y = 12. 
 
 9. a; -J- y = 18, Add or subtract. 
 a-y- 4. 
 
 10. 4 a; + 3 # = 17, (1) Multiply (2) by 2 and subtract. 
 2a- y= I- (2) 
 
 11. 3 a; + 4 y = 48, Add. 
 
 a? — 4 y = 0. 
 
 12. 3 a; + 5 y = 13, (1) Multiply (1) by 7 and (2) by 3. 
 7x + 3y = 13. (2) Subtract. 
 
Algebraic Equations. 443 
 
 13. 4 x + 5y = 32, Add. 
 6a; — 5y = — 2. 
 
 14. 3 4 + 4 2/ = 3, (1) Multiply (2) by 2. Add. 
 12 x -2 ?/ = 3. (2) 
 
 15. 5 a; = 6 2/ + 5, Transpose. 
 3 x = 5 y — 4. 
 
 16. 3z + 5y+ 8 = 0, 17 
 
 y — 2x=8x — l, 
 
 2 as - 2/ - 12 = 0. 
 
 2y — 4=x = y + x-+ 9. 
 
 18. 5x+ 7y = 55, 
 
 (1) 
 
 9 x + 18 y = 126. 
 
 (2) 
 
 Divide (2) by 9 getting x + 2 y = 14. 
 
 C3) 
 
 Then multiply (3) by 5. 
 
 
 19. S + ?J? = 17. Clear of fractions. 
 4 3 
 
 5a; 5j/ = 20 
 
 4 8 
 
 
 20. £a?-f£y = 42, 24. 
 
 4^a; + 3|2/ = 67, 
 
 i*-Hy=F 17* 
 
 7^a;-5i2/ = 12. 
 
 21. 23x-7y= 3a; + 51, 25. 
 
 3 (a> + 7) =9(2/ -9), 
 
 ll2/ = 15a; + 2. 
 
 4(3 a> -8) = 17 y -155. 
 
 22. x -\-y = 100,000, 26. 
 
 2(* - 11) -2(2/-9)= 6, 
 
 fo + S=^°- 
 
 a? + 9 32 
 y-S 15 
 
 23 - f^I= 5 > 
 
 »-4 y-l_ 6 
 3 4 
 
 7a;-6_ 2 
 5y + 3 
 
 a;_4 2/-l_i 
 3 4 
 
 28. 2a>4-5*/ + 3 = 6 
 3a;-42/-2 ' 
 
 4a;-7?/4-5_^ 
 
 a;-22/ + 2 
 
 
444 Chapter Seven. 
 
 554. Written Problems. 
 
 1. The sum of two numbers is 37. Twice the first added 
 to three times the second is 96. What are the numbers ? 
 
 Let x = first number ; y = second number. 
 
 2. The difference between two numbers is 28. Five 
 times the first less twice the second is 197. What are the 
 numbers ? x-y = 28; hx-2y- 197. 
 
 3. The product of the first of two numbers by 5, added to 
 the product of the second by 3, gives 37. The product of 
 the first by 6, diminished by five times the second, equals 
 10. Find the numbers. 
 
 4. Divide 65 into two parts whose difference shall be 19. 
 Let x and y = parts. Solve also by one unknown quantity. 
 
 5. A person pays $103 with 32 bills, some of them $2 
 bills, the others $ 5 bills. How many of each does he use ? 
 
 6. For 25 head of pigs and sheep, a farmer received 
 $ 145. How many of each did he sell, if he sold the former 
 at $7 each, the latter at $5 each ? 
 
 7. 10 oranges and 4 peaches cost 38^; 6 oranges and 7 
 peaches cost 32 ^. Find the cost of an orange. Of a peach. 
 
 8. 5 pounds of tea and 3 pounds of coffee cost $3.75; 
 8 pounds of tea and 1 pound of coffee cost $ 5.05. What is 
 each worth per pound ? 
 
 9. A farmer buys a certain number of horses at $125 
 each and a certain number of cows at $ 40 each. They cost 
 together $ 740. If he had bought half as many horses and 
 twice as many cows they would have cost $730. How many 
 of each did he buy ? 
 
 10. A man paid 75^ for 2 pounds of raisins and 3 
 pounds of cheese. 5 pounds of raisins and 2 pounds of 
 cheese at the same prices would have cost 94^. What did 
 each cost per pound ? 
 
Algebraic Equations. 445 
 
 11. The sum of two numbers is 19. The sum of the 
 second number and ten times the first, minus the sum of 
 the first and ten times the second, equals 45. What are the 
 numbers ? 
 
 12. Reduce ^ to an equivalent fraction, the sum of whose 
 numerator and denominator shall be 126. 
 
 x = numerator ; y = denominator. 
 
 * = ±.;x + y = 126. 
 
 y 13 
 
 13. What fraction equivalent to -^ has 147 for the dif- 
 ference between its numerator and denominator ? 
 
 x - y = - 147. Why ? 
 
 14. 10 pounds of coffee at 30^ per pound are mixed with 
 x pounds of coffee at 25^ per pound. What is x equal to, 
 when the mixture is worth 26^ per pound ? 
 
 25x + (10 x 30) = 26 (10 + x). 
 
 15. A grocer mixes green tea costing 60^ per pound with 
 black tea costing 40^ per pound. He uses 100 pounds in 
 all, and the mixed tea costs him 48 ^ per pound. How many 
 pounds of each does he use ? 
 
 Let x = number of pounds of black tea ; y = number of green. 
 Then x + y — number of pounds of mixed tea. 
 
 x + y = 100 ; 40 x + 60 y = 48 (x + y). 
 
 THREE UNKNOWN QUANTITIES. 
 555. 1. Given the following : 
 
 3x + 2y- 2 = 12, (a) 
 5 x _ 4 y + 3 z = 16, (6) 
 2 x + 3 y + 2 z = 35, (c) 
 to find the values of x, y, and z. 
 
446 Chapter Seven. 
 
 (a) multiplied by 5, 15 x + 10 y - 5 * = 60 
 
 (6) multiplied by 3, 15 x - 12 y + 9 z - 48 
 
 Subtract, 22 y - 14 z = 12 (d) 
 
 an equation containing only two unknown quantities. 
 
 (6) multiplied by 2, 10 x - 8 y + 6 z = 32 
 
 (c) multiplied by 5, 10 a + 15 y + 10 z = 175 
 
 Subtract, - 23 y - 4 3 = - 143 («) 
 
 an equation containing only two unknown quantities. 
 
 Compare the two equations (d) and (e), which contain the same 
 two unknown quantities. 
 
 (d) multiplied by 2, 44 y - 28 z — 24 
 
 (e) multiplied by 7, — 161 y - 28 z = - 1001 
 
 Subtract, 205 y m 1025 
 
 y m 5 
 
 Substituting this value of y in (<Z), we have 
 
 110 -Uz = 12, - 14 z = - 98, z = 7. 
 
 Substituting values of y and z in (a), we have 
 
 3 x + 10 - 7 = 12, 3 x = 9, a; = 3. 
 
 ^ns. a; = 3, 
 
 y = 5, 
 
 3 = 7. 
 
 2. Find the values of the unknown quantities in the 
 following equations: 
 
 x-Sy + 2z= 3, (a) 
 
 2x+ y + 3z = 22, (6) 
 
 5x + 2y + 7z = 5l. (c) 
 
 Multiply (a) by 2, and subtract from (6). Multiply (a) by 5, and 
 subtract from (c). This gives two equations, each of which contains 
 two unknown quantities. 
 
 Compare these two resulting equations, and eliminate y. 
 
Algebraic Equations. 447 
 
 3. 5x-2y+ z = 10, (a) 
 
 3x + 8y-5z = 120, (b) 
 
 7x-Sy-2z = 8. (c) 
 
 Eliminate z by comparing (a) and (6), multiplying the former by 5 
 Compare (a) and (c), multiplying the former by 2. 
 
 4. 13ic- ±y + 15z = 317, 
 
 7z + 2y- 3^= 89, 
 21a-17# + 9z = -104. 
 
 5. - 8x + y — 12z = -259, 
 
 7x- ±y + 25z= 418, 
 13<c + 2y-4lz = -500. 
 
 "e. § + =±*-H 
 
 x + y x-y .„ 
 2 6 
 
 7. 3a;~5y g== 2a? + y 
 
 2 5 
 
 8 s — 2y _a? . y 
 4 2"V 
 
 8 . 2 + 5 -^^ = 4^3a ; , 
 
 12 + 5a ? -6y ==2 3^-2^ 
 6 4 
 
 2x-\-y 9a; — 7 ^ 3 y 4-9 4a? + 5y 
 2 8 ~ 4 16 
 
448 Chapter Seven. 
 
 556. Written Problems. 
 
 1. A man placed § of his capital at 5% and the other 
 third at 6%. At the end of a year, capital and interest 
 amounted to $ 31,600. What was his capital ? 
 
 — x and - x — = interest. 
 
 3 100 3 100 
 
 2. A has 18 chestnuts more than B. If each finds 4 
 more, A will have four times as many as B. How many 
 chestnuts has each? 
 
 3. Two mechanics earn together $ 8 per day. One 
 works 23 days and tke other 17 days, for which they receive 
 together $ 166. What does each earn per day ? 
 
 4. The sum of the first and the second of three numbers 
 is 55, of the first and the third 62, of the second and the 
 third 83. What are the numbers ? 
 
 Suggestion. — Add together the three equations. 
 
 5. The sum of two numbers is 53. Four times the first 
 is 20 more than twice the second. Find the numbers. 
 
 6. A certain sum of money is divided among four per- 
 sons. The first takes £ of it, the second takes f of the 
 remainder, the third takes £ of what then remains, the 
 fourth receives the balance, $24. What is the share of 
 each of the other three? 
 
 7. A merchant sold a lot of goods for $510, thereby 
 losing ^ of their cost. What did the goods cost ? 
 
 8. A man collected a bill for a physician and deducted 
 ^ of the amount for his services. If he gave the physician 
 $ 147, what was the amount collected ? 
 
 9. Divide 130J acres of land among three persons, giving 
 the first 27£ acres more than the second, and the second 13f 
 acres more than the third. 
 
 10. A merchant has sold ^ of a piece of cloth, and has 
 remaining 16 yards more than £ of the piece. How many 
 yards did the piece contain originally ? 
 
Algebraic Equations. 449 
 
 11. A servant is engaged for a year for $280 and a suit 
 of clothes 5 he leaves at the end of six months, and receives 
 $ 130 and the suit. What is the value of the clothes ? 
 
 Yearly wages = 280 + x. Wages for six months = 140 + ^ = 130 + x. 
 
 EXPONENTS. 
 
 557. (1) x 2 means x times x, or xx. 
 
 (2) y 3 means y times y times y, or yyy. 
 
 (3) a 4 means aaaa. 
 
 In (1) 2 is an exponent. 
 
 In (2) 3 is an exponent. 
 
 In (3) 4 is an exponent. 
 
 Notice that the exponent is written above the quantity to 
 which it belongs. 
 
 On which side of the quantity is it written ? 
 
 How many times is the quantity used as a factor if the 
 exponent is 2 ? If the exponent is 3 ? If the exponent 
 is 5? 
 
 Tell two facts concerning the location of the exponent. 
 Tell one fact concerning the meaning of the exponent. 
 
 What is an exponent ? 
 
 Note. — When no exponent is written, 1 is understood. 
 
 558. Oral Exercises. 
 
 If a = 2, and if b = 3, and if c = 5, 
 
 1. a 2 = ? 7. a 3 + a 2 -fa = ? 
 
 2. a 2 + a = ? 8. a 3 + 6 2 + c = ? 
 
 3. a 2 + b + c = ? 9. a 4 + 6 3 + c 2 = ? 
 
 4. a 2 + 6 2 =? 10. a 2 + ?c = 19. 
 
 5. 6 2 -a 2 = ? 11. a 3 + & 3 = a 2 -f& 2 +c 2 ~? 
 
 6. a 2 + 2ab + b 2 =? 12. b 2 + a 4 = c\ 
 
45° Chapter Seven. 
 
 559. 3 6 2 means 3 6 times 6 and not 3 6 times 36. The 
 exponent 2 belongs to the letter 6 and not to the expression 
 3 6. 
 
 ab 3 means a666 and not ab ab ab. 
 
 In 5 ab 3 to what part of the expression does the exponent 
 belong ? 
 
 If we use a parenthesis, the effect is different. (36) 2 
 means 3 6 times 3 6. (ab) 3 means ab ab ab. 
 
 What does (2 a) 4 mean ? 
 
 560. Oral Exercises. 
 
 If a = 2, and if 6 = 3, and if c = 5, 
 
 1. 5a 2 =? 7. 7ab 2 = ? 
 
 2. (5a) 2 = ? 8. 7a 2 6 = ? 
 
 3. (26) 2 -26 2 =? 9. a6c 2 =? 
 
 4. 4 6 2 -c 2 = ? 10. a 2 6 2 c = ? 
 
 5. (3a) 3 -3a 3 =? 11. c 2 -a 2 = ? (c-a). 
 
 6. a6c 2 = ? 12. 4 6 2 -2ac = a ? . 
 
 MULTIPLICATION. 
 
 561 . What coefficient is understood when none is written ? 
 What exponent is understood when none is written ? 
 
 2 a times 3 = 6 a. 
 
 2 a times 3 a = 6 a 2 . 
 
 3 a times 4 a = 12 a 2 . 
 5 a 2 times 2 a = 10 a 8 . 
 5 a times 2 6 = 10 ab. 
 
 Multiply together the numerical coefficients and affix all the 
 letters, giving each letter the sum of the exponents of that letter 
 in both factors. 
 
Algebraic Equations. 4ji 
 
 562. Oral Exercises. 
 Multiply ; 
 
 1. 3a by 5a. .7. 5a 2 6by3c. 
 
 2. 2ab by 3a. 8. 5ab by 3ac. 
 
 3. 2a6by36. 9. Sab by 3 a&V. 
 
 4. 2 a 2 6 by 3 a. 10. ab 2 <? by a6c. 
 
 5. 2a6 2 by3a. 11. £a&by6a 2 . 
 
 6. 2a 2 6 2 by36. 12. 10a 2 6byja. 
 
 SICNS IN MULTIPLICATION. 
 
 563. Preliminary Exercises. 
 
 1. -f a times + b = -h ab. 
 
 Here the two factors, + a and + b, have like signs (both 
 are +). The product has the -}- sign. 
 
 2. — a times — b = -f a&. 
 
 Here the two factors have like signs (both are — ). The 
 product has the + sign. 
 
 3. — a times -f- b = — ab. 
 
 Here the two factors have unlike signs (one + and one — ). 
 The product has the — sign. 
 
 4. -f a times — b = — ab. 
 
 Here the two factors have unlike signs. The product has 
 the — sign. 
 
 From the above we may see the law for signs in multipli- 
 cation. 
 
 Like signs give +• 
 
 
 
 Unlike signs give — . 
 
 
 
 Multiply : 
 
 
 
 5. —3 a by 5 a. 
 
 9. 
 
 — 5 a 2 b by — 3 c. 
 
 6. 2 a by 4 a. 
 
 10. 
 
 - 5 a 2 b by 3c. 
 
 7. — 2 a by — 4 a. 
 
 11. 
 
 5a 2 6by -3c. 
 
 8. 2a6 2 by -3a. 
 
 12. 
 
 -5a 2 b by -1. 
 
45 2 Chapter Seven. 
 
 564. Written Exercises. 
 Multiply : 
 
 1. a 2 + 3aby2a. * Ans. 2a 3 + 6a 2 . 
 
 2. 4a-6by2a&. Ans. 8 a 2 6 - 2 ab\ 
 
 3. 5 a + 3 b by 4 a. 8. a 2 -6 2 bya. 
 
 4. a« — c» by — 3 a. 9. a 2 — 6 2 by — a. 
 
 5. a 2 + a + 1 by a. 10. aW — a by a&. 
 
 6. a 2 + 2a + lby -2a. 11. a 2 + 2a& + b 2 by 2a&. 
 
 7. a 2 -M + lbya&. 12. a 2 -2a& + 6 2 by -2ab. 
 
 565. Written Exercises. 
 
 1. Multiply x 4- 2 by x + 3. 
 
 x + 2 
 
 3 + 3 
 
 Multiplying x -f 2 by x, x 2 4- 2 x 
 
 Multiplying x + 2 by 3, 3a;4-6 
 
 Adding the two parts of the product, x 2 + 5 x 4 6 
 
 Multiply each term of the multiplicand by each term of the 
 multiplier and combine the products. 
 
 2. Multiply x -f- 3 by x — 4. 
 
 
 x + 3 
 
 
 
 x-4 
 
 
 
 x 2 + 3 a 
 
 
 • 
 
 -4a 
 
 ;-12 
 
 
 x 2 -x- 
 
 -12 
 
 Multiply : 
 
 
 
 3. x 4- 3 by x + 4. 
 
 
 7. 2 a; -8 by a? + 9. 
 
 4. a; + 5 by a? — 2. 
 
 
 8. 3z + l by x +7. 
 
 5. x + 8bya; — 9. 
 
 
 9. 2a; + lby2a: + l 
 
 6. 2 x + 5 by a; 4- 2. 10. a; —5 by a; 4- 4. 
 
Algebraic Equations. 
 
 453 
 
 566. Written Exercises. 
 
 Find products : 
 
 Note : (x — 3) (x + 9) means x — 3 multiplied by x + 9. 
 
 1. 
 
 (x -3)(<c + 9). 
 
 13. 
 
 (a; -5) (a; -9). 
 
 2. 
 
 (x-6) (x + 7). 
 
 14. 
 
 (a; -f 5) (a; + 5). 
 
 3. 
 
 (a; - 5) (a + 5). 
 
 15. 
 
 (x -3) (x + 8). 
 
 4. 
 
 (x + 5)(x-5). 
 
 16. 
 
 + 7) (a; -6). 
 
 5. 
 
 (2x-6)(x + l). 
 
 17. 
 
 (a._4)(a;-7). 
 
 6. 
 
 (x-6)(2x + l). 
 
 18. 
 
 (2a; -4) (3a; -6). 
 
 7. 
 
 (2 s -6) (3* + 3). 
 
 19. 
 
 (2x + 6) (3a; -7). 
 
 8. 
 
 (3a? + 6) (2 a? -3). 
 
 20. 
 
 (2a; + 7) (3 a; + 3). 
 
 9. 
 
 (2 a; + 3) (2 a; -3). 
 
 21. 
 
 (2 a; -3) (3 a; -2). 
 
 10. 
 
 (a; -5) (a; -4). 
 
 22. 
 
 (2 a; -3) (2 a; + 3). 
 
 11. 
 
 ( x _ 7) (a; -9). 
 
 23. 
 
 (2 a; + 9) (4a; -6). 
 
 12. 
 
 (a- 7) (a -7). 
 
 24. 
 
 (3a;-4)(3a; + 4) 
 
 567 
 
 . Written Exercises. 
 
 
 
 
 1 . Multiply x + y by x + y. 
 
 s + y 
 x + y 
 
 x* + xy 
 
 xy + y* 
 
 x 2 + 2xy + y* 
 
 Multiply : 
 
 2. a + 6 by a + 6. 
 
 3. a — i/ by a — y. 
 
 6. a; + 2/ by x — y. 
 
 7. a + lby a-1. 
 
 8. m + n by m — w. 
 
 11. a 2 + a+l by a-1. 
 
 12. a 2 — a + 1 by a + 1. 
 
 4. a + 3 x by a + 3 x. 
 
 5. 2a + x by a + 2x. 
 
 Ans. aP — y 2 . 
 9. a 2 + 6 by a 2 + 6. 
 10. a 2 + 6 2 bya-6. 
 
 -4ws. a 3 — 1. 
 13. ar* + a;+l by a;-l. 
 
454 Chapter Seven. 
 
 TERMS. 
 
 568. Preliminary Exercises. 
 
 1. 2abc. 3. a 2 + 2a& + 6*. 
 
 2. 2 + a + 6 + c. 4. lab 2 — !. 
 
 The first of the above expressions has one term 
 
 The second has 4 terms. 
 
 The third has 3 terms. 
 
 The fourth has 2 terms. 
 
 An expression containing one term is called a monomial ; 
 one containing two terms, a binomial ; one containing three 
 terms, a trinomial; one containing four or more, a poly- 
 nomial. 
 
 How many terms has each of the following ? 
 6. 2 acx 2 y. 8. lla 2 + a. 
 
 6. a 4- 1. 9. a 2 4- a + abc. 
 
 7. a 2 + a + 11. 10. a 2 + a + b + c. 
 
 LIKE TERMS. 
 
 569. Like terms are those containing the same letters and 
 the same exponents for each letter. 
 
 570. Oral Exercises. 
 
 ' Which of the following expressions contain like terms ? 
 
 1. a and b. 6. 2 a 2 and 2 a 3 . 
 
 2. 2 a and a. 7. 2a 2 6 and 3a?b. 
 
 3. a 2 and a. 8. 2a 3 b and 3a 2 6. 
 
 4. a 2 and 2 a. 9. xy 2 and x*y. 
 
 5. 2 a 2 and 3 a 2 . 10. barfy and ax*y. 
 
Algebraic Equations. 455 
 
 COMBINING LIKE TERMS. 
 
 571. Two or more like terms may be combined into a single 
 term. Unlike terms cannot be so combined. 
 
 572. "Written Exercises. 
 Combine when, possible : 
 
 1. 3 abc 2 — abc + abc 2 = 4 abc 2 — abc. 
 The first and third terms are combined. 
 
 2. 3 abc 2 — ab 2 c + a 2 bc. 
 
 3. 3 abc — 2 abc + abc. 
 
 4. 2 abx — ab -f 3 bx. 
 
 5. a 3_ a 2 + 2 a-4 + 3a 2 -a. 
 
 6. ±a 2 -2ab + b 2 + a 2 -b 2 . 
 
 7. 8 arfy — xy -f- a? — xy — 3 #. 
 
 8. 4 ra 2 + 4 ran + n 2 — mw 2 . 
 
 573. Written Exercises. 
 1. (a 2 + &)(a + 6) = ? 
 
 Multiplying by a, we have a 3 + ab. Multi- a 2 + 5 
 plying a 2 by 6, gives a 2 6. As there is no like a + b 
 
 term, this product is placed after ab. The a z + ^& -f a 2 6 + b 2 
 product of & by 6 is placed last. 
 
 Rearrange the terms in the order of the size of the exponents of a. 
 Ans. a s + a?b + ab -f 6 2 . 
 
 2. (2a&- &)(&-l) = ? 5. (a 2 -6 2 )(a + &) = ? 
 
 3. (2a&-&)(2a-l) = ? 6. (a + b)(c+d) = ? 
 
 4. (2a&-&)(a&-&) = ? 7. (3a + b) (a- a&) = ? 
 
 8. (2j92-a)(2pg + a;) = ? 
 
 9. (x* + ±y 2 )(±x-y) = ? 
 10. (^ + «-2)'(a;-2)=? 
 
456 Chapter Seven. 
 
 DIVISION. 
 
 574. Preliminary Exercises. 
 
 1. 6 a -r- 3 = 2 a. 4. 5 a 2 a; -f- 5 a 2 = jc. 
 
 2. a 2 -r- a = a. 5. 10 a 2 a; -7- 5 a; = 2 a 2 . 
 
 3. 6a 2 -j-3a=2a. 6. 10 a 2 x ■+■ 2 ax = 5 a. 
 7. (12a 3 a;?/ 2 -27aVi/)^3a 2 a;2/ = 4a?/-9a;. 
 
 In the above examples what is done with the coefficients ? 
 
 What is done with the exponents of the same letter ? 
 
 When the divisor is a monomial, divide the numerical coeffi- 
 cient of each term of the dividend by the numerical coefficient 
 of the divisor. Then write the letters of the dividend, giving 
 each an exponent equal to the exponent in the dividend dimin- 
 ished by that in the divisor. 
 
 575. Like signs give +. Unlike signs give — . 
 
 576. Written Exercises. 
 Divide : 
 
 1. 5 a 3 by a 2 . 4. — xPy 3 by — xy 2 . 
 
 2. — xPy 3 by xy 2 . 5. x 2 + 2 xy by x. 
 
 3. xPy 3 by — xy 2 . 6. X s — 5 x 2 + 3 x by x. 
 
 7. 15 rf 4 - 10 a 8 + 20 a?" by -5a 2 . 
 
 8. 4 a 3 ?/ 3 — 3 x 2 y + #?/ 2 by xy. 
 
 9. -12afy 2 + 33.'c 2 ?/ 3 -24<c3/ 4 by -3^. 
 
 10. — 7 a 5 by — 7 a 4 . 
 
 11. 7 a 5 by — 7 a 5 . 
 
 12. a 2 + a by J a. -4ns. 2 a + 2, 
 
 13. oj 8 + 3 05* by — -J a?. 
 
 14. 6 a 3 - 3 a 2 by - 9 a 2 . 
 
 15. —x + xy — xzby —x. 
 
 16. £afy-2ayby -*afy. 
 
Algebraic Equations. 457 
 
 577. Written Exercises. 
 1. Divide x 2 + 7 x + 12 by x + 3. 
 We write a division like the above as follows : 
 
 x 2 + 7 x + 12 
 
 3 + 3 
 
 The first term in the divisor is x. 
 
 The first term in the dividend is 3 2 . 
 
 Dividing x 2 by x we get x for the first term of the quotient, which 
 
 3 + 3 
 x 
 
 we put down thus, 3 2 + 7 3 + 12 
 
 then we multiply the divisor by the first term of the quotient and write 
 the result under the dividend. 
 
 x 2 + 7 x + 12 I 3 + 3 
 (re + 3) x times x 2 + 3 x \x 
 
 Next, we subtract x 2 + 3 x from the dividend, 
 
 x 2 + 7 x + 12 | a; + 3 
 subtract a 2 + 3 a; | x 
 
 remainder 4 x + 12 
 
 Next, we divide the first term of the remainder by the first term of 
 the divisor and get + 4, which we write in the quotient thus, 
 
 x 2 + 7 x + 12 
 
 x 2 + Zx 
 
 3 + 3 
 3 + 4 
 
 43 + 12 
 Next, we multiply the divisor by 4 and write the product thus, 
 
 3 + 3 
 
 X 2 + 7 x + 12 
 3 2 + 3 3 
 
 43+12 
 (3 + 3) 4 times 43 + 12 
 
 3 + 4 
 
 Subtracting, there is no remainder. The whole quotient is x + 4. 
 
 3 2 + 7 3 + 12 I 3 + 3 
 
 3 2 + 33 I 3 + 4 An8. 
 
 43 + 12 
 4 3 + 12 
 
45 8 Chapter Seven. 
 
 2. Divide a^ + lSx + 56 by m 4-4. 
 x 2 + 18 x + 56 I x + 4 
 
 x 2 + 4x 
 
 14 x + 56 
 
 14 a; + 56 
 
 
 
 | se + 14 u4/w. 
 
 3. Divide a 2 -2ab~ 
 
 246 2 by a-f 46. 
 
 a 2 - 2 ab - 24 6 2 1 
 a 2 + 4 a6 
 
 - 6 a6 - 24 6 2 
 
 - 6 a6 - 24 6 2 
 
 a + 46 
 
 a — 6 6 Ans. 
 
 
 Prove that the answers in the above examples are correct. 
 
 578. Written Exercises. 
 Divide : 
 
 1. x* + 5x + 6 by x 4-3. 
 
 2. aj 8 4-5o 2 + 6a; by a? 4- 3. 
 
 3. ar* 4- 5 a? 4- 6 a? by x 2 + 3x. Ans. x + 2. 
 
 4. a^+7a;2/4-102/ 2 by x + 2y. 
 
 5. a 2 — 7a# + 10y*by a?-2y. 
 
 6. 3aj 2 + 14ajy + 82/ 8 byoj + 4y. 
 
 7. 3a^ + 10a;2/-8^ 2 by » + 4y. 
 
 8. 3a^-10^-8?/ 2 by aj-4?/. 
 
 9. 305 2 — 14 an/ 4- 8 ?/ 2 by a; — 4y. 
 
 10. 3^ + 10^ — 8?/ 2 by 3a — 2y. 
 
 11. 8^ + 22a;?/ + 152/ 2 by 2» + 3y. 
 
 12. 8a? — 2ajy-15y*by 4x + 5y. 
 
 13. 8a; 2 -22a;?/ + 15?/ 2 by4a;-5y. 
 
 14. n¥ + 3 ana? + 2 a 2 by wz 4. a. ^4ws. no; 4- 2 a. 
 
 15. n¥ 4- anx — 2a* by nx— a. 
 
 16. 6a 2 b 2 — 13dbw + 6w 2 by 3ab-2iv. 
 
 17. 4^4-2a*/z-1322/ 2 * 2 by 2x-llyz. 
 
Algebraic Equations. 
 
 459 
 
 18. ±ax* + 2axyz — 132ai/Vby 2x — 11 yz. 
 Id. 8 ax 2 — 26 axyz -f 15 ay 2 z 2 by 4 cue — 3 ayz. 
 
 579. 
 
 Written Exercises. 
 
 
 
 l. 
 
 Divide X s   
 
 -1 
 
 by x 
 
 — 
 
 1. 
 
 
 
 X s 
 
 -l | 
 
 X 
 
 -1 
 
 
 
 
 
 -3 2 
 
 X 2 
 
 + X+1 
 
 
 
 X* 
 
 - 1 
 
 
 
 
 
 X 2 
 
 — X 
 
 
 
 
 
 X — 
 
 1 
 
 
 
 
 
 X — 
 
 1 
 
 
 Ans. 
 
 2. Divide X s - 13 x - 10 - 2 or 2 by a; - 5. 
 
 The terms of the dividend should be arranged according to the size 
 of the exponents of £, thus, x z — 2 x 2 — 13 x — 10. 
 
 a 8 - 2 z 2 - 13 re - 
 
 9 
 
 x*-bx 2 
 
 3 a 2 - 13 x - 
 3z 2 -15a; 
 
 9 
 
 x 2 + 3 a; + 2 + 
 
 x-5 
 
 S«- 9 
 
 2s-10 
 
 + 1 
 
 Divide : 
 
 3. a 3 — 6 3 by a — 6. 
 
 4. cs-lby c-1. 
 7. c 3 + 2 by c -f- 1. 
 
 5. c 8 + 1 by c 4- 1. 
 
 6. 9a 4 -166 2 by 3a 2 +46. 
 
 ^4ns. c 2 + c + 1 + -^— • 
 c + 1 
 
 8. ar 5 + 6afy + :12a;2/ 2 + 8? / 3by x + 2y. 
 
 9. a 3 + 3 a — 3 a 2 — 1 by a — 1. Rearrange dividend. 
 
 10. or 3 + 27 ^ by x + 3 y. 
 
 11. a 2 + 2a& + & 2 -l by a + & + l. ^Ins. a + &-l. 
 
 12. 12a 3 -20a 2 + 33a-5by 6o-l. 
 
 13. 8 a& 3 - 125 aafy 3 by 2ab-5axy. 
 
 14. l + 5a 3 -6a 4 by l-a + 3a 2 . * 
 
 15. 6a 5 + a;-12^ + 9ic 3 -3-lla; 4 by2» 3 -3^-l. 
 
460 Chapter Seven. 
 
 FACTORING. 
 
 580. The expression x + xy may be divided by x ; the 
 quotient is 1 + y. The factors of x + xy are x and 1 + y ; 
 that is, (# + xy) = se(l + ?/). 
 
 In a similar manner we find that the factors of 2 a 2 b + 4 a& 2 
 are 2 a& and a -f 2 6 ; that is, 2 a 2 6 + 4 a& 2 = 2 ab (a + 2 6). 
 
 581. Factor: 
 
 1. m + mn. 4. a? + a\ 7. c 3 + c 2 + c. 
 
 2. m 2 /i + 2m. 5. 6 6 3 -96 2 . 8. 7 a 2 6 + 21 a 2 c. 
 
 3. 6afy-3a?/ 2 - 6. a 3 6 + 6a& 3 . 9. 4x 4 +6a 3 . 
 
 10. 4x 4 + 6ar J + 8x?. 18. 20 ax 2 - 15 bx 3 + 16 ex 4 . 
 
 11. 4x 4 + 6ar J + 8x 2 + 10a>. 19. 20ax 2 + 15a 2 ar } -20a 3 a; 4 . 
 
 12. 4z 4 +6ar 5 +8z 2 +10x+12. 2 0. a¥ + aV. 
 
 13. ^a^-gacr'+eac 3 . 21. 6 a? + 1 a* + 2 a«. 
 
 14. 3a6 2 + 2a 2 6 + 2a 3 . 22. a 2 x A + a 3 ^ 3 + a 4 s 4 . 
 
 15. 3a 2 6 + 6a& 2 -15a& 8 . 23. 12 m 3 n -f 5 m 2 n 2 + 15 ny. 
 
 16. #yz + xyz 2 . 24. a 7 — a 5 6 2 -f a 4 ^. 
 
 17. 9 m% - 27 m s n 2 y. 25. 70 a 7 + 60 x 6 - 50 a 5 . 
 
 582. The square of x -f- y is x 2 -f- 2 xy -f y\ 
 
 Note that x 2 and y 2 are the squares of x and y, respectively, 
 and that 2 scy is twice the product of x and y. 
 
 The square of the sum of two quantities is equal to the square 
 of the first, plus twice the product of the first and the second, 
 plus the square of the second. 
 
 Any expression in the form of x 2 -f 2 xy -f- y 2 is composed 
 of two equal factors. 
 
 a 2 + 2a& + & 2 = (a + 6)(a + b) or (a + &)*. 
 
Algebraic Equations. 461 
 
 583 
 
 1 Factor: 
 
 
 
 1. 
 
 c 2 + 2cd + d?. 
 
 14. 
 
 c 2 d 2 46cdm49m ! . 
 
 2. 
 
 m 2 4 2 m + 1. 
 
 15. 
 
 4a 2 + 12a + 9. 
 
 3. 
 
 4 + 4w + m; 2 . 
 
 16. 
 
 9a 2 4- 12 ab 4 4 6*. 
 
 4. 
 
 a¥ 4 2 axy 4 y 2 . 
 
 17. 
 
 4 a 2 4- 4 ac 4- c 2 . 
 
 5. 
 
 r 2 + 2 rs£ 4 s 2 * 2 . 
 
 18 
 
 a*4-2a 2 &4& 2 . 
 
 6. 
 
 e 2 + 6e+9. 
 
 19. 
 
 a 2 4 2 abx 4 &W 
 
 7. 
 
 x 2 + 8 a; 4 16. 
 
 20. 
 
 a 2 6 2 4 2 afcca" 4 c¥. 
 
 8. 
 
 4 6 2 4 46d4d 2 . 
 
 21. 
 
 m 2 n 2 4l0mn + 25. 
 
 9. 
 
 a 2 4 4 ay 4 4 y 2 . 
 
 22. 
 
 96 2 + 30 6c4 25c 2 . 
 
 10. 
 
 a 2 4 4 ayz 4 4 ?/ 2 2 2 . 
 
 23. 
 
 16 4l6a*/4-4afy 2 . 
 
 11. 
 
 4a 2 z 2 + 4a6a;4& 2 . 
 
 24. 
 
 x 4 4 2 afy 2 z 4 2/V. 
 
 12. 
 
 M *4-6ttv + 9v*. 
 
 25. 
 
 a 2 6 2 4-12a6cH-36c 2 . 
 
 13. 
 
 a 4 4-2a 2 6 2 4& 4 . 
 
 26. 
 
 4 ra 2 + 8 mw -|- 4 w 2 . 
 
 584. The square of a — b is a 2 — 2 a& 4 6 2 . 
 
 Compare this form with that of Article 582, and note the 
 difference in signs. 
 
 Give a general statement for the square of the difference 
 of two quantities. 
 
 585. Factor: 
 
 1. a 2 - 2 ay + if. 14. x 2 -2x + l. 
 
 2. l-2x + x*. 15. 4z 4 -4ar J 4l. 
 
 3. m 2 -2mnr + w¥. 16. 4 x 2 — 12 xy 4 9 y 2 . 
 
 4. a 2 -4a4 4. 17. 9 y 2 - 12 xy 4 4ar>. 
 
 5. 9-6&4& 2 . 18. a 4 - 2 a 2 a 4 ar>. 
 
 6. rV - 2 rs£ 4 Z 2 . 19. 9 - 12 b 4 4 6 2 . 
 
 7. c*-4 cd4 4d 2 . 20. 4^44 a; 41. 
 
 8. 16 b 2 - 8 a; + 1. 21. x 2 + 4 y 2 - A xy. 
 
 9. 4 b 2 - 4 6c 4 c 2 . 22. 16 x 2 - 40 a;z 4 25 z*. 
 
 10. a 2 ?/ 2 - 4 ayz 4 4 z 2 . 23. b 4 - 2 b 2 cd 2 + c 2 d\ 
 
 11. 4 a 2 ?/ 2 -4 ayz + z 2 . 24. 25 z 4 - 30 x 2 4 9. 
 
 12. 9a 2 -6ah + h 2 . 25. 25 ar* - 30 ar* 4 9 a. 
 
 13. 6 2 c 2 -66cd49d 2 . 26. 503^-60^4183?. 
 
462 Chapter Seven. 
 
 586. The product of a + b and a — b is a 2 — 6 2 . 
 
 Give a general statement for the product of the sum and 
 the difference of two quantities. 
 
 An expression consists of the difference of the squares of 
 two quantities. What are the factors of the expression ? 
 
 In factoring an expression first examine it to see if it contains a 
 monomial factor. o 2 — be 2 = b (b 2 — c 2 ). The factors of b 2 - c 2 are 
 6 + cand6-c. 6 3 - be 2 = b (b + c)(b - c). 
 
 12. m 2 -9n 2 . 
 
 13. b 5 -9b. 
 
 14. 9 m 2 — n 2 p 2 . 
 
 15. 9 6 2 -4. 
 
 16. 4 c 2 -9^. 
 
 17. 6c 2 -4 6d 2 . 
 
 18. a A -b 2 . 
 
 19. x*-y 2 . 
 
 20. xif — a 2 b 2 x. 
 
 21. afy 2 -a 2 6 2 . 
 
 22. -m 2 + aV. 
 
 588. The product of x + 2 and a? + 3 is ar 2 4- 5 a; + 6. 
 
 Note that the coefficient of the second term of the product 
 is the sum of the second terms of the factors ; 5 = 2 4- 3. 
 
 Note that the last term of the product is the product of 
 the second terms of the factors ; 6 = 2 times 3. 
 
 Factor x 2 + 7 x + 12. 
 
 We must find two numbers whose sum is 7 and whose 
 product is 12. These numbers are 3 and 4. 
 a? + 7 x + 12 = (x + 3) (x + 4). 
 
 In a similar manner, we find 
 
 x 2 + 6 x + 5 = (x + 1) (a? + 5). 
 
 587. Factor: 
 
 1. 
 
 m 2 — n 2 . 
 
 2. 
 
 1-m 2 . 
 
 3. 
 
 a^-4. 
 
 4. 
 
 c 2 - ay. 
 
 5. 
 
 9-v 2 . 
 
 6. 
 
 h 2 - 16. 
 
 7. 
 
 4 ar* — y 2 . 
 
 8. 
 
 oV — # 2 . 
 
 9. 
 
 a 2 — 4 ?/ 2 . 
 
 10. 
 
 4 a 2 6 2 — w 2 . 
 
 11. 
 
 aw 2 — 4 a. 
 
Algebraic Equations. 463 
 
 589. Factor: 
 
 1. ^ + 43 + 3. 14. 7* + 10r + 9. 
 
 2. m 2 + 6m + 8. 15. s^lls + lS. 
 
 3. 6 2 + 7& + 10. \ 16. 24 + 10a + a 2 . 
 
 4. c 2 + 7c + 6. 17. ^ + 11^ + 24. 
 
 5. a 2 + 8a + 7. 18. 20 + 12^ + ^. 
 
 6. a 2 + 9a + 14. 19. x 2 + 12x + 32. 
 
 7. y* + 8y + 12. 20. 27 + 12x + x*. 
 
 8. d 2 + 10d + 16. 21. a 2 + 14a + 24. 
 
 9. 7t 2 + 87i + 15. 22. c* + 20c + 19. 
 
 10. a 2 + 9 a; + 18. 23. 2/ 2 + 12?/ + 35. 
 
 11. a; 2 + 9a; + 20. 24. ra 2 + 13m + 30. 
 
 12. z 2 + 10z + 21. 25. ar' + llaj + SO. 
 
 13. w 2 + 12w + 20. 26. / 2 + 10/+9. 
 
 590. The product of x — 2 and a; — 3 is ar* — 5 a; + 6. 
 Compare with Article 588 and note the difference in 
 
 signs. a 2 - 11 a + 10 = (a - 10) (a - 1). 
 
 591. Factor: 
 
 1. c 2 -7c+12. 9. w 2 -14w + 45. 
 
 2. <z 2 -5d + 4. 10. n 2 -18n + 45. 
 
 3. a 2 -13a; + 22. 11. ^-16^ + 28. 
 
 4. 2/2-142/ + 33. 12. 36-15z + z 2 . 
 
 5. 2 2 -12z + ll. 13. & 2 -17& + 30. 
 
 6. a 2 -13a + 40. 14. 30-316 + 6 2 . 
 
 7. m 2 -18m + 32. 15. s?-16s + 55. 
 
 8. w 2 -14n + 13. 16. y 2 -16y + 63. 
 
464 Chapter Seven. 
 
 17. 48-14c + c*. 22. 80 + 18?/ + 2/ 2 - 
 
 18. z 2 -27z+50. 23. d 2 + 17d + 72. 
 
 19. tv 2 + 20w + 51. 24. a^-20a; + 99. 
 
 20. 50-fl5* + * 2 . 2*. 99-100a + a 2 . 
 
 21. h 2 + 17h + 4:2. 
 
 592. 1. The product of x + 5 and x - 3 is a 2 + 2 a; - 15. 
 2. The product of ic — 5 and x + 3 is or 2 — 2 re — 15. 
 
 What gives a minus sign in a product ? 
 
 Why is the sign of the last term of the product minus in 
 each of the above statements ? 
 
 Note that in each of the products the coefficient of the 
 second term is the difference between 5 and 3. 
 
 In the factors in the first statement, has the larger num- 
 ber a plus or a minus sign ? What is the sign of the second 
 term of the product ? 
 
 In the factors in the second statement, has the larger 
 number a plus or a minus sign ? What is the sign of the 
 second term of the product ? 
 
 Factor a^-4«- 45. 
 
 We must find two numbers whose product is 45 and whose 
 difference is 4. We see that 9 and 5 are such numbers. 
 
 The sign of the second term in the given expression is 
 minus, so we must give the minus sign to the larger of the 
 two numbers which we have found. 
 
 rf _ 4 x - 45 = (x - 9) (x + 5). 
 
 In a similar manner, we would get 
 
 x 2 + 4 x - 45 = (x + 9)(x — 5). 
 x 2 + x-56 = (x + S)(x-7). 
 x 2 -x-56 = (x-$)(x + 7). 
 
Algebraic Equations. 465 
 
 593. Factor: 
 
 1. a^-x-12. 14. ar* -5 a; -36. 
 
 2 . a* + x -12. 15. a 2 -5a-14. 
 
 3. a 2 + 3 a _io. 16. b 2 -7b-S0. 
 
 4. a 2_ 3a _io. 17. z 2 -2z-35. 
 
 5. 2/ 2_4 2/ _i2. 18. ra 2 + 4m-60. 
 
 6 . y* + 4y-12. 19. & 2 + 3&-54. 
 
 7. c2 + 5 c _6. 20. x 2 + 2x-4,S. 
 
 8. d 2 -7d-18. 21. * 2 -10*-ll. 
 
 9. d 2 + 7d-18. 22. ?/ 2 + 7 ?/ - 60. 
 
 10. k 2 -7k-$. 23. /i 2 -9/i-36. 
 
 11. 02_ 2 0_24. 24. a 2 -2a-120. 
 
 12. a 2 -5 a; -24. 25. d 2 + 5 d - 150. 
 
 13. ^ + 5^-36. 26. d 2 -5d-150. 
 
 594. If we divide 2? — y 3 by x — y, the quotient is ar* + xy 
 + 2/ 2 - a 3_ 53 = ( a _ 6 )( a 2 + ab + j^ 
 
 The difference of the cubes of two quantities may be 
 divided by the difference of the quantities. The quotient 
 consists of three terms, namely : the square of the first, plus 
 the product of the first and second, plus the square of the 
 second. ( a s + 6 sj + ( a + fy = a 2_ a0 + 6 2 # 
 
 Give a general statement for the quotient obtained by 
 dividing the sum of the cubes of two quantities by the sum 
 of the quantities. 
 
 Factor a 3 4- 8. 8 = 2 3 
 
 hence a 3 + 8 = a 3 + 2 3 , 
 
 a 3 + 2 3 = (a + 2) (a 2 - 2 a + 2 2 ), 
 
 since 2 2 = 4, 
 
 a 3 + 8 = (a + 2) (a 2 - 2 a + 4). 
 
56 
 
 Chapter Seven. 
 
 
 
 595. Factor: 
 
 
 
 
 1. ra 3 — x 5 . 
 
 10. m 3 -f-8n 3 . 
 
 18. 
 
 a*-tf. 
 
 2. m 3 + ar*. 
 
 11. a 3 6 3 + 8. 
 
 19. 
 
 7^ + 27. 
 
 3. a 3 + l. 
 
 12. 8yV + l. 
 
 20. 
 
 a 3 -64. 
 
 4. a 3 -l. 
 
 13. (3x) s -f. 
 
 21. 
 
 a 3 + 64. 
 
 5. 1-a 3 . 
 
 14. ar>+(3) 3 . 
 
 22. 
 
 a? - 64 y 5 . 
 
 6 . &-<*&. 
 
 15. l + 27d 3 . 
 
 23. 
 
 a 3 6 3 + <W. 
 
 7. fcV+r 3 . 
 
 16. 27 a 3 -1. 
 
 24. 
 
 64 -z 3 . 
 
 8. (2a) 3 -6 3 . 
 
 17. (a 2 ) 8 -6 3 . 
 
 25. 
 
 a 3 - 125. 
 
 9. S^-c 8 . 
 
 
 
 
 596. The following supplementary exercises are applica- 
 tions of the preceding cases. Sometimes it is possible to 
 apply more than one case to an exercise. 
 
 1. Factor 2 (fix — 2 a¥- 24 a 5 . 
 First divide by 2 x. 
 
 2a 4 x -2a 2 x*- 24 a 5 = 2a(a 4 - aV - 12a 4 ), 
 but a 4 -aV- 12a 4 = (a 2 + 3ar 2 )(a 2 - 4^); 
 
 hence 2 a 4 * - 2 aV - 24 a^> = 2 x(a 2 + 3 ar 2 ) (a 2 - 4 ar 2 ), 
 but a 2 - 4 ar> = (a + 2 a?) (a - 2 ar) ; 
 
 hence 2a 4 a;-2cW- 24rf = 2aj(a 2 + 3« 8 ) (a + 2 a?) (a -2 a). 
 
 2. Factor ai 6 — # 6 . 
 
 Since x e =(x i ) 2 , and ^=(jf) 2 9 
 
 x*-tf=(x* + tf)(7*-tf). 
 Factoring the second member of the above equation, 
 ** — f = 0* + V) («* - ^ + 2Z 2 ) (* — 2/) 0* 2 + xy + t/ 2 ) . 
 When in doubt prove your work by either multiplication 
 or division. 
 
Algebraic Equations. 467 
 
 597. Factor: 
 
 1. 7 s?y + 21 xyz 2 . 27. Sb 5 -5b\ 
 
 2. ab 2 -2abc + ac 2 . 28. (z + ?/) 2 -l. 
 
 3. a 2 t> + 3a&c + 2 6c 2 . 29. a 2 -2 ab + 6 2 -l. 
 
 4. a 3 + 2a 2 + a. 30. z 2 -* 5 . 
 
 5. 2^-216. 31. 125 x + x\ 
 
 6. a?b 2 c? + 6abc + 9. 32. <e 4 -5a: 2 + 4. 
 
 7. 3n 4 + 81tt. 33. a 4 -2a 2 -8. 
 
 8. n 4 -81w. 34. 100 + 25 ?/ 2 . 
 
 9. a 4 -a. 35. 100- 25 y 2 . 
 
 10. 2a 2 -2a&-246*. 36. ab 2 - 144 ac 2 . 
 
 11. a 4 + 2cc 2 2/ 2 + y 4 . 37. 12 a*/ 2 - 144 xz\ 
 
 12. a^ + 5x. 38. 3z 2 -9a-84. 
 
 13. tf — 25tf. 39. a^c 2 — a + 2 a&x + ab\ 
 
 14. m 2 + 25 mn + 100 n 2 . 40. a 4 +2 ar' + l -a 2 . 
 
 15. a 2 - 20 a&c + 100 W. 41. z 4 + 2 ar> + l -s 2 . 
 
 16. ^-16^. 42. a 4 + a 2 + l. 
 
 17. a 4 — 16 a 2 6 + 55 b 2 . (Change 42 to same form as 
 
 18. o^ + aY- 41 ° 
 
 19. z 4 + *y. 43 ' 2/ 2 - 13 2/ + 36. 
 
 20. * 4 -*y. 44 ' ^"13 6 2 + 36. 
 
 21. fl/-^f-'6« 45 ' *+'*+* 
 
 22. ^-5^ 2 + 6^. 46 ' ™ 4 -16. 
 
 23. a 3 6 2 + 5a 2 6 3 + 10a6 5 . 47. 16-wiW. 
 
 24. (3m) 2 + 2(3m) + l. 48. 16m + 2m 4 . 
 
 25. (x + y) 2 + 2(x + y) + l. 49. 16 a -a 4 . 
 
 26. (x + y) 2 -2(x + y)z + z 2 . 50. a 6 -64. 
 
468 Chapter Seven. 
 
 FRACTIONS. 
 598. Preliminary Exercises. 
 
 1. Keduce 9! to its lowest terms. Ans. ^-> 
 
 xz z 
 
 Divide the numerator and the denominator by x. 
 
 2. Change ^ to an equivalent fraction whose denominator 
 
 is be. Ans. ^ 
 
 be 
 Multiply the numerator and the denominator by c. 
 
 The value of a fraction is not changed when both numerator 
 and denominator are either multiplied or divided by the same 
 quantity. 
 
 599. Oral Exercises 
 
 
 
 
 
 Eeduce : 
 
 
 
 
 
 1. $?*. 2. 
 
 3x 
 
 2a& 
 
 o 
 
 7 b h? 7 a 3 
 21 &W a 3 
 
 3a 2 ' 
 
 
 
 ax-\- ay 
 
 7. 
 
 2ab 2 
 
 
 9 nxy + a; 
 
 Sab 
 
 
 a-\- ab 
 
 
 ax-f-x 2 
 
 6 12a * b \ 
 
 8. 
 
 a 2 + 2ab 
 
 10 2px + 3xy 
 
 6 ab 
 
 
 3ab 
 
 
 px + xy 
 
 600. Sight Exercises. 
 
 
 
 
 Give answers at sight : 
 
 
 
 
 , b ? 
 
 
 
 
 a 16 a 3 
 
 1. - = — . 
 
 
 
 6. 
 
 4a = — - — 
 
 3 3x 
 
 
 
 
 ? 
 
 2 2 V- ? . 
 ' 3x 3a 2 
 
 
 
 7. 
 
 2/ + *_ ? . 
 2 a 2a& 
 
 2a 2ac 
 
 
 
 8. 
 
 2^- 2 f- 
 
 ' 3 ~ ? " 
 
 
 
 
 ? 
 
 4 5a*/_ ? . 
 
 
 
 9. 
 
 3 7^ = _J 
 
 7 ra 14 ran 
 
 
 
 
 3 rat 
 
 5. 4a = -. 
 
 a 
 
 
 
 10. 
 
 . . » 2a 2 & + 2a# 
 a + = - 
 
Algebraic Equations. 469 
 
 601. Written Exercises. 
 
 , .p , a? + lire + 30 
 1. Keduce — -*- - 1 
 
 x + 5 
 
 Divide the numerator by the denominator. Ans. x + 6. 
 
 a . Keduoe *+i2»+35. 
 
 x + 5 
 
 3. Keduce 
 
 ^ + 10a; + 20 . Jjm. a + 6 
 
 a;-f4 # + 4 
 
 1 2c + 3 
 
 6. Change x+2 to a fraction whose denominator is x+3. 
 • _j. 2 = ?-i — Multiply the numerator and the denominator by x + 3. 
 
 7. a -3 = —^—. 8. a^-ic + l = 
 
 x + 5 a+1 
 
 a f racti 
 
 3^ + 23; 
 
 x — 3 
 
 9. Change a; H to a fraction. 
 
 x-\- 2 
 
 x — 
 
 x + 2 
 
 x* + 2x x-3 
 
 x + 2 x + 2 x + 2 
 What may be done with the numerators when the denominator is 
 
 common 
 
 10 . Eednce **-}. 13. -^+ 3aJ 
 
 2x-l x+2 x+2 
 
 11. x — 1 - = -• 14. - = • 
 
 z + 2 x + 2 x 2x* + x 
 
 3 5? 2 9 
 
 12. 7^- + —^ = ' - — 15. 
 
 2x x + 2 2x? + 4:X y y* — y 
 
 16 . §+•=« + *-« 
 
 a; 2a; + l 2^ + a; 2^ + a; 
 
470 Chapter Seven. 
 
 "• [x-l)[x-2j ■' e -' (x-l)(x-2) C 
 | multiplied by f = ? 
 
 18 ^±1^^L? = 9. u ( a; + 1 )^- 2 ) ==? 
 * a_i * a;_2 *' . ' («-l)(a + 2) ' 
 
 19. By what quantity must a; — 5 be multiplied to give a 
 product of a 2 + z -30? 
 
 By what number must 7 be multiplied to give a product of 63 ? 
 1 ? 
 
 20. 
 21. 
 22. 
 23. 
 
 x-5 ^ + 05-30 
 
 a ? 
 
 a;_6 a^_a;-30* 
 
 a aac-f»q 
 
 a-l~~? 
 
 x-2\fx + 5\ = l> 
 x-5j\x + 2j ' 
 
 24. 
 
 05 + 1 a? + 2x + l 
 x-1 ? 
 
 
 25. 
 
 x-5 ? 
 
 
 s + 1 a^-l 
 
 
 26. 
 
 a; — 5 , x — 1 9 
 
 a; + 3 ' a;-h4 - 
 
 
 27. 
 
 Add*-*and*-?. 
 a; -}- 1 a— 1 
 
 Ans. 2^-7» + 8 
 as»-l 
 
 28. 
 
 a; — 2 x — 5 _ 9 
 a; — 1 a;-f 1 
 
 i 
 
 29. 
 
 z + 5_ ? 
 
 
 a;_3 3^-^-6 
 
 
 30. 
 
 27a 8 a?-2_ ? 
 
 
 3CUB-1 
 
Algebraic Equations. 471 
 
 PURE QUADRATICS. 
 
 602. Given ^±-^ = 3a?2 ~ 66 , to find the value of x. 
 
 5 9 
 
 Clearing of fractions, 9 z 2 + 54 = 16 z 2 - 330. 
 Transposing and combining, — 6 z 2 = — 384. 
 
 Dividing by - 6, x 2 = 64. 
 
 Extracting square root, z = ± 8. 
 
 Since (—8) x (— 8) = 64, the square root of 64 may be 
 either + 8 or — 8. It is written ± 8, and is read "positive 
 or negative 8." (It is sometimes less correctly called plus or 
 minus 8.) 
 
 603. Written Exercises. 
 Find value of x, y, z, etc. : 
 
 1. ^-13 = 36. 
 
 2. 3/ +25 = 100. 
 
 3. 5z 2 -13 = 3z 2 + 37. 
 
 4. 5(ar } -fl7)-3ar J +63 = 198. 
 
 5. 5(^ + 17) -3(^-21) =198. 
 
 6. 2/ 2 + 2 2 / + l- 2 / 2 = 49. 
 
 7. (a;-f-l) 2 -ar J =49. 
 
 8 i/ 2 + 5 2y 2 -18 _o 
 
 8. — 4— 2 ' 
 
 9 z + 7 = z-5 _ 
 z _3 z _9* 
 
 10 20a; = 30a; 
 x — 1 a; + 1 
 
47 2 Chapter Seven. 
 
 11. (a; -3) (a? + 3) = 40. 
 
 12. (x + 5)(x + 5)=10x + 26. 
 
 13. (x + 4:) 2 =8x + 80. 
 
 14. z 2 + 64 = 5z 2 . 
 
 15. 3^ + 18 = 21^ + 36. 
 
 16. (x-3) 2 -(x-5y = 12. 
 
 17. (aj + 7)(aj-9) = (aj-3)(a?-6). 
 
 
 18. 
 
 4 a; 
 
 X 
 
 5 
 
 +5- 
 
 a; 
 
 
 19. 
 
 « + 7 
 a; — 5 
 
 a; 
 
 a; 
 
 -3 
 -9 
 
 
 20. 
 
 2/-9_ 
 y — 5 
 
 2/ 
 
 -3 
 
 + 7 
 
 604. 
 
 "Written Problems. 
 
 
 
 1. Find the dimensions of a field, the length of which is 
 twice its breadth, its area being 1800 square rods. 
 
 2. The surface of the six equal faces of a cube contains 
 96 square inches. Find the length of one edge. 
 
 3. One number is four-fifths of another, and their product 
 is 80. What are the numbers ? 
 
 4. One-third of a number multiplied by two-fifths of the 
 same number gives a product of 270. Find the number. 
 
 5. Thirty per cent of a number multiplied by forty per 
 cent of the same number gives a product of 300. What is 
 the number? 
 
 6. Thirty per cent of twenty per cent of a number is 
 300. What is the number ? 
 
Algebraic Equations. 473 
 
 7. The base of a right-angled triangle is f as long as 
 the perpendicular, and the area of the triangle is 96 square 
 rods. Find the length of the base. What is the length of 
 the hypotenuse ? 
 
 8. The base of a right-angled triangle measures x yards, 
 
 3 x 
 
 the perpendicular measures — yards. What is the length 
 
 of the hypotenuse ? If the hypotenuse measures 15 yards, 
 find the length of the base. 
 
 9. The base of a right-angled triangle measures x feet, 
 the hypotenuse measures (x-\- 9) feet, the perpendicular 
 measures 15 feet. What is the length of the base ? 
 
 10. The difference between the squares of two consecutive 
 numbers is 49. What are the numbers ? 
 
 11. The difference between two numbers is 6. The sum 
 of their squares is 146. What are the numbers ? 
 
 Let x — 3 = smaller number, 
 
 and x + 3 = greater number. 
 
 AFFECTED QUADRATICS. 
 605. Preliminary Exercises. 
 
 (x + 1 ) (x + 1 ) = z? + 2 x + 1 . 
 Compare with Article 582. 
 
 0-l)(a;-l)=a; 2 -2a; + l. 
 Compare with Article 584. 
 
 (a + 6) 2 =a 2 + 2a& + 6 2 . 
 (m — n) 2 =m 2 — 2 mn -f- n 2 . 
 (10 + 5) 2 = 10 2 + 2 x 10 x 5 + 5 2 . 
 (10-3) 2 = 10 2 -2xl0x3 + 3 2 . 
 
474 Chapter Seven. 
 
 606. Oral Exercises. 
 Square : 
 
 1. 3J + 3. 4. 3J + 10. 7. 30 — 1. 10. x-y. 
 
 2. *— 7, 5. a-6. 8. 40-1. 11. 80 + 5 
 
 3. x — 9. 6. # + y. 9. m + n. 12. 60 — 5. 
 
 607. Oral Exercises. 
 Extract the square root of 
 
 1. a^ + 6a; + 9. 6. x 2 + 2xy + y 2 . 
 
 2. a 2 -14a; + 49. 7. a? - 2 an/ + ?/ 2 . 
 
 3. ar>-18a; + 81. 8. a 2 -2ab + b 2 . 
 
 4. a^ + 20a; + 100. 9. a 2 - 24 x + 144. 
 
 5. a 2 + 2a6 + 6 2 . 10. z 2 + 22 a; + 121. 
 
 The square of (x + 3) consists of how many terms ? Of 
 how many terms does (x + 4) 2 consist ? (a; + 5) 2 ? 
 
 608. Supply term necessary to make a complete square : 
 
 1. x* + 6x+? 6. x* + 2x + ? 
 
 2. a^-12a; + ? 7. 3^-43;+? 
 
 3. 3^-83; + ? 8. « 2 -10a; + ? 
 
 4. 3^-163?+? 9. 3^+143; + ? 
 
 5. 3^ + 183; + ? 10. x 2 -22a; + ? 
 
 609. "Written Exercises. 
 
 Given x 2 + 6x = 27. 
 
 What number must be added to the first member of the 
 equation to make it a " complete " square ? 
 
 If a number is added to one member of an equation, what 
 must be done to the other member to preserve the equality ? 
 
Algebraic Equations. 475 
 
 610. Extract the square root of both members of the fol- 
 lowing equations, adding to both, where necessary, such a 
 number as will make the first member a complete square. 
 
 1. x* + 6x + 9 = 40 + 9. 2. a?—12x + 36 = 28 + 36. 
 Remember that (+ 7) x (+ 7) = 49, and that (- 7) x (- 7) = 49. 
 .-. V49 = + 7 or - 7, written ± 7. 
 
 3. ^-8^ + 16 = 20 + 16. 7. »*- 14 a; =15. 
 
 4. ar J -16a + 64 = -39 + 64. 8. ar»-22a; = 23. 
 
 5. <c 2 + 18a; + ? = 19 + ? 9. a" + 14 a = 51. 
 
 6. jb 2 + 2oj+? = 24 + ? 10. x 2 -22x = AS. 
 
 611. Given a 2 -10* = 24. 
 
 Completing the square, we have x 2 — 10 x + 25 = 24 + 25 = 49. 
 Extracting the square root of both sides, we have 
 
 x - 5 = ± 7, 
 
 x = 7 + 5 = 12, orx = -7+5 = -2. 
 Ans. 12 or - 2. 
 
 612. Written Exercises. 
 Find values of x : 
 
 1. x*-6x = 7. 9. ar* - 24a = 0. 
 
 2. ^-12^ = 108. 10. ar 2 -8a; = 384. 
 
 3. or* + 2a; = 48. 11. a? 2 — 4a; = — 3. 
 
 4. a 2 + 18 a; = 115. 12. ar> + 30a; = 175. 
 
 5. a^-14a; = -13. 13. ar> + 28a; = 29. 
 
 6. ^-10a; = 0. 14. x 2 + 22 x = 104. 
 
 7. x> + 20x = 125. 15. x 2 -16x = -6±. 
 
 8. ar J + 26a; = 56. 16. ar 5 + 36a; = 76. 
 
 To make the first member a complete square, you added 
 the square of what part of the coefficient of x ? 
 
476 Chapter Seven. 
 
 613. Written Exercises. 
 Find values of x : 
 
 1. a? + a; = 12, 5. x 2 + 9x = -20. 
 x> + x + $y = 12 + (by. 6. ar J -lla;=-28. 
 
 2. a?-3a; = 10, 7. a? + 13x = -42. 
 a?-3a;+(f) 2 =10+(f) 2 . 8. a 2 -15 a; =76. 
 
 3. a? + 5a; = -4. 9. a?-17a; = 18. 
 
 4. a?-7a; = 8. 10. x 2 + 19 x = - 18. 
 
 614. When a? has a coefficient, divide both members by 
 the coefficient. 3 a? + 9 a; = 84. 
 
 Dividing by 3, x 2 + Sx = 28. 
 
 Completing the square, 
 
 x* + 3x + (f)2 = 28 + f = 112 + 9 = 121 t 
 
 4 4 
 
 Extracting square root, x + f = ± ty. 
 
 ...ar=¥-f = * = 4; or ~y -* = -¥ = - *• 
 
 ^4ns. 4 or — 7. 
 
 615. Written Exercises. 
 
 1. 6a 2 - 6a; = 36. 6. 3a? + 9a; = 54. 
 
 2. 9a? + 9a; = 180. 7. 8 a 2 - 72 x = - 160. 
 
 3. 7a? + 28a;=147. 8. 7a? + 49a; = 56. 
 
 4. 4a?-40a; = -64. 9. 3 x 2 + 21 a; = 54. 
 
 5. 83? -16a; =504. 10. 5a? -25a; = -20. 
 
 616. Five times nothing = ? 
 Zero multiplied by one million = ? 
 
 If a;»=5, 
 
 a;-5 = ? 
 
 lQ(as - 5) = ? 
 
 (. T + 5)(a; _ 5) = ? 
 
Algebraic Equations. 477 
 
 If one of two factors is zero, the product is zero. 
 The converse is also true. 
 
 If the product of two factors is zero, one of the factors is zero. 
 Given (x - 2)(x - 3) = 0. 
 
 One of the factors in the above equation is equal to zero. 
 If x - 2 = 0, 
 
 by transposing we get x = 2. 
 
 If x - 3 = 0, 
 
 a; = 3. 
 
 617. A quadratic equation may sometimes be readily 
 solved by factoring. 
 
 1. x>-5x = -6. 2. x*- 5x = 14. 
 
 ^-5^ + 6 = 0. x 2 -5x-U = 0. 
 
 (x - S)(x - 2) = 0. (a; - 7)(x + 2) = 0. 
 
 x = 3 or 2. x = 7 or — 2. 
 
 Solve by factoring: 
 
 3. a 2 + a- 6 = 0. 8. z 2 - 4z + 7 = 19. 
 
 4. a? + 2x-3 = 0. 9. ?/ 2 + 10 = 28 + 3y. 
 
 5. 3^-3^ + 2 = 12. 10. ^-2^-24 = 0. 
 
 6. y 2 + 7y + 15 = 3. 11. ^-15^ = 16. 
 
 7. ar>- 7 a; +20 = 8. 12. 2/ 2 + 19?/ = 20. 
 
 618. Written Problems. 
 
 1. The sum of two numbers is 12; their product is 32. 
 What are the numbers ? 
 
 x and 12 — x = numbers. (12 — x) x = product. 
 
 2. The base of a rectangle is 
 60 feet longer than its altitude, x 
 Its area is 2400 square feet. How 
 long is the base ? 
 
 Area x 2 + 50 X 
 2400 sq. ft. 
 
 x+ 50 
 
478 
 
 Chapter Seven. 
 
 3. The perpendicular of a right-angled tri- 
 angle measures 15 yards more than the base. 
 The hypotenuse is 75 yards. Find the length 
 of the perpendicular. 
 
 x 2 + (15 + xy = 752. 
 
 4. The hypotenuse of a right-angled tri- 
 angle is 1J times as long as the base. The 
 area of the triangle is 150 square yards. How- 
 long is the hypotenuse ? 
 
 Perpendicular = V(£x) 2 - x 2 ; area = £ base x per- 
 pendicular. 
 
 5. The entire surface of a square prism is 170 square 
 feet. Its altitude is 6 feet, and one side of its base is x 
 feet. Find the value of x. 
 
 50+2Z 
 
 6. A garden 50 feet long, 40 feet 
 wide, has a walk just outside it x feet 
 wide. Find the area of the walk. 
 
 If the area of the walk is 784 square 
 feet, what is its width ? 
 
 7. A field, ABCD, contains 
 12 acres. Its length is 1J times 
 its breadth. How many rods 
 long is the diagonal BG ? 
 
 8. A flag-staff, AB, 50 feet high, was 
 broken off at the point C. The broken 
 part, resting on G, reached the ground 
 D, 30 feet from the base of the staff. 
 Find the length of the part broken off. 
 
Algebraic Equations. 
 
 479 
 
 9. A ladder, CE or DE, placed at 
 a point E, in a street 58 feet wide be- 
 tween the opposite houses, just touches 
 the top of a house, DB, 60 feet high on 
 one side of the street, or the top of a 
 house, CA, 56 feet high on the other 
 side. Find the length of the ladder. 
 
 DE 2 = 60 2 + (58 - x)' 2 = CE 2 = 56 2 + a?. 
 
 10. ABC is a triangle. The side 
 AB measures 13 feet ; the side BC, 4 
 feet ; AC, 15 feet. Find the altitude 
 BD. 
 
 BD* 
 
 AD 2 = BC 2 -CD 2 . 
 
 DxQ 
 
 11. ABCD is a trapezium. AB = 
 34 feet ; BC = 20 feet ; CD = 40 feet ; 
 DA = 26 feet. The perpendicular BF 
 measures 16 feet. Find the length of 
 the diagonal AC and of the perpen- 
 dicular ED. 
 
CHAPTER VIII. 
 
 GEOMETBY. 
 
 619. Vertical Lines. 
 
 Hang a weight from a fixed point by a string. When the 
 weight stops swinging the string is in a vertical line. What 
 way does the lower end of the string point ? the higher end ? 
 Hold a sheet of ruled paper so that the lines are vertical. 
 
 620. Oblique and Horizontal Lines. 
 
 Hold a pointer so that it points upward but not straight 
 up. It is in an oblique line. 
 
 Hold a pointer so that it does not point or slant either up 
 or down. It is in a horizontal line. 
 
 Note. — In representing vertical, horizontal, or oblique lines on the 
 page of a book or a sheet of paper it is assumed that the book or paper 
 is held in an upright position. 
 
 621. Oral Exercises. 
 
 1. What kind of line is represented by the course of a 
 drop of water running down a roof ? 
 
 2. By the course of a falling raindrop when there is no 
 wind? 
 
 3. By the course of a falling raindrop when there is a 
 wind? 
 
 4. By straws floating on the surface of still water ? 
 Use the object for the four following exercises. 
 
 6. How many lines are there in the edges of a cube or 
 a rectangular box ? 
 
 480 
 
Exercises in Geometry. 481 
 
 6. When the cube is placed on a level table, how many- 
 edges are vertical ? How many are horizontal ? How many 
 are oblique ? 
 
 7. Hold the cube so that four edges are horizontal. 
 How many are vertical ? How many are oblique ? 
 
 8. Hold the cube so that no edges are horizontal. How 
 many are oblique ? How many are vertical ? 
 
 9. A straight line is 3 feet long. What kind of line is 
 it if one end is 4 feet from the floor and the other end is 
 1 foot from the floor ? 
 
 10. If one end of a 3-foot straight line is 4 feet from the 
 floor and the other end is 2 feet from the floor, what kind of 
 line is it ? 
 
 11. If each end of a straight line is 5 feet from the floor, 
 what kind of line is it ? 
 
 12. If one end of a straight line is 4 feet from the floor 
 and the middle is 4 feet from the floor, what kind of line 
 is it? 
 
 13.- A vertical straight line is 5 feet long. The middle is 
 5 feet from the floor. How far is each end from the floor ? 
 
 14. A vertical line is 4 feet long. One end is 5 feet from 
 the floor. How far from the floor is the other end ? Why 
 are there two answers ? 
 
 622. Angles. 
 
 When the ends of two straight lines meet they form an 
 angle. A \ 
 
 What two lines form the angle ABC in 
 the above figure ? 
 
 At what point do they meet ? *2? 
 
 The point B is the vertex of the angle ABC. 
 
 What is the vertex of an angle ? 
 
482 Chapter Eight. 
 
 623. Designation of Angles. 
 
 The angle formed by the lines ST arid. TU may be called 
 the angle T. It is frequently better to 
 call it the angle STU or UTS, the letter 
 at the vertex being placed between the 
 two others. 
 
 The use of the three letters is necessary 
 where two or more angles have vertices /^ 
 
 at the same point, as in the accompanying / ^^ 
 
 figure, where UX, VX, and WX meet at x ^^__ _ w 
 the point X. 
 
 624. Exercises. 
 
 Draw a horizontal line 3 inches long. Mark a point in 
 this line one inch from the left end. From this point draw 
 a line upward slanting towards the right. Mark each end 
 of each line by a letter. How many angles have you formed ? 
 Designate each of these angles by three letters. 
 
 Note. — The above exercise may be varied for blackboard drill — 
 draw a vertical line 11 inches long ; mark a point 4 inches from the 
 top ; draw a line to the left slanting downward, etc. 
 
 How many angles are formed when two lines meet at 
 their ends ? When two lines pass through the same point ? 
 When from a point in one line another line is drawn ? 
 
 625. Circular Measure. 
 
 60 seconds (") 1 minute. 
 60 minutes (') 1 degree. 
 360 degrees (°) 1 circle, or circumference. 
 
 626. Exercises. 
 
 1. What part of a circumference is 180°? 90°? 60°? 
 30°? 45°? 36°? 72°? 
 
 2. 1° on the circumference of a circle is 5 inches. What 
 is the length of the circumference ? 
 
Exercises in Geometry. 
 
 483 
 
 3. The circumference of a circle is 9000 feet. 1° = ? 
 1'=? 
 
 4. How many degrees are there between the XII and the 
 I on the face of a clock ? between the XII and VI ? between 
 the XII and III ? between the III and VII ? 
 
 5. If one degree of the earth's circumference is 69^- miles, 
 find the circumference. 
 
 6. Through how many degrees does the minute hand of a 
 clock pass in 1 hour? in \ hour? in 15 minutes? in 5 
 minutes ? in 10 minutes ? in 1 minute ? in 3 minutes ? 
 
 627. Angular Measure. 
 
 The angle at the centre of a circle has the same number 
 of degrees as the arc between the sides of the angle. 
 
 Thus, in the following figure the angle AOC has the same 
 number of degrees as the arc ABC 
 
 The circumference of this circle is divided into 36 equal 
 parts. How many degrees are there in each part ? How 
 many degrees are there in each of the following angles ? 
 
 AOB, BOC, COD, DOE, EOF, FOG, GO A, AOE, DOF. 
 
484 Chapter Eight. 
 
 628. The Protractor. 
 
 The number of degrees in an angle may be measured by a 
 protractor. 
 
 D 
 
 unnnm 
 
 SEMICIRCULAR PROTRACTOR 
 
 To measure an angle, XYZ, for instance, produce the 
 lines YX and YZ. Place the point A of 
 the protractor on the vertex (Y) of the 
 angle, and the edge AC on the line YZ 
 produced. Using the lower line of figures, 
 read off from the protractor the number of 
 degrees at the point where the line YX produced cuts the 
 semicircle. 
 
 In measuring the angle DEF, the line AB D* 
 is placed on EF, the point A on the vertex 
 
 E. The number of degrees in this case is p ^ 
 
 read from the upper row of figures. 
 
 Note. — There is only one point on the protractor where the num- 
 bers of the upper and lower lines of figures are equal. What is the 
 number of degrees at that point ? What kind of angle is measured 
 at that point ? If an angle is acute, would you read its measure by 
 the larger or by the smaller number ? 
 
Exercises in Geometry. 485 
 
 EXERCISES IN CONSTRUCTION. 
 
 629. Note. — In the following exercises, the ruler, the compasses, 
 and the protractor may be used. 
 
 The drawing should be carefully done with a sharp, hard pencil. 
 
 1. Draw an obtuse angle formed by two lines, each one 
 inch long. Draw an acute angle formed 
 by two lines, each six inches long. 
 Which is the larger ? 
 
 2. The lines OH and IJ intersect at 
 K, making four right angles. Which 
 arc is longer, 7 8 or cd ? Which con- 
 tains the greater number of degrees? 
 
 3. Draw two lines meeting at an 
 angle of 45°. Two lines meeting at an angle of 90°. Two 
 meeting at an angle of 135°. 
 
 4. Draw two lines making two angles, one 
 of which measures 60°. How many degrees 
 does the other angle contain ? 
 
 5. To a horizontal line draw a line making two equal ad- 
 jacent angles. How many degrees does each angle contain ? 
 
 Two angles are said to be adjacent when they have one side in 
 common. 
 
 To a vertical line draw a line making two equal adjacent 
 angles. How many degrees does each angle contain ? 
 
 To an oblique line draw a line making two equal adjacent 
 angles. How many degrees does each angle contain ? 
 
 6. How many degrees are there in a right angle ? 
 
 7. To an oblique line draw a line making two unequal 
 adjacent angles. How many degrees are there in the sum 
 of the two angles ? 
 
 Two angles are said to be supplementary when they are together 
 equal to two right angles. 
 
486 
 
 Chapter Eight. 
 
 •fc 
 
 8. How many degrees in the angle T y if 
 8 contains 75° ? 
 
 V measures 110°. How many degrees does 
 U measure ? 
 
 If one of two supplementary angles meas- 
 ures 63£°, how many degrees are there in the 
 other angle ? 
 
 How many degrees are there in an angle supplementary 
 to one of 47° 45'? 
 
 9. Construct angle 5, 60°; angle 4, 50°. 
 Measure angle 3. 
 
 How many degrees and minutes will there 
 be in angle 5 when 3 contains 49£° and 4 contains 83J° ? 
 
 When angle 3 contains 36° 30' and angle 5 contains 
 79° 45', how many degrees and minutes will angle 4 contain ? 
 
 10. Erect a perpendicular at each extremity of a hori- 
 zontal line. At each extremity of a vertical line. At each 
 extremity of an oblique line. 
 
 Note. — A line making a right angle with another line is said to be 
 perpendicular to it. 
 
 11. Construct a square upon a horizontal line, 
 oblique line. 
 
 12. Draw two lines intersecting at an 
 angle of 100°. Mark in each of the other 
 three angles the number of degrees it con- 
 tains. 
 
 13. Draw two lines making an angle (6) 
 of 150°. Construct an adjacent angle (7) 
 containing 80°. How many degrees will 
 angle 8 contain? 
 
 14. How many degrees will there be in the 
 sum of five angles having the same vertex ? 
 
 Upon an 
 
Exercises in Geometry. 487 
 
 15. Draw five equal angles having a common vertex. 
 
 16. Draw six equal angles having a common vertex. Is 
 any angle supplementary to the angle next it ? Why ? 
 
 Are any of the angles vertical ? Why ? 
 
 17. Draw two angles,, one of 65° and the other of 25°. 
 Draw a third angle equal to the sum of both. 
 
 Draw an angle equal to their difference. 
 
 18. Draw an angle equal to the sum of three angles 
 measuring, respectively, 40°, 50°, and 60°. 
 
 630. Parallels. 
 
 Lines which lie in the same plane and which cannot meet, 
 no matter how far produced, are said to be parallel. 
 
 19. Using the protractor, draw two or more lines that 
 shall be perpendicular to a horizontal line. Where will they 
 meet ? 
 
 Draw two or more that shall be perpendicular to a verti- 
 cal line. Where will they meet ? 
 
 Draw two or more that shall be perpendicular to an 
 oblique line. Where will they meet ? 
 
 20. To a horizontal line draw two or more lines running 
 in the same direction, and each making an angle of 35° with 
 the first line. Will the oblique lines meet ? 
 
 Draw two or more lines running in. the same direction, 
 and each making an angle of 125° with a vertical line. Will 
 the oblique lines meet if produced very far ? 
 
 Draw two or more lines running in the same direction, 
 and each making an angle of 74° with an oblique line. Will 
 the former lines meet ? 
 
 21. Draw two lines making angles of 30° and 60°, respec- 
 tively, with a third line. Will the two former lines meet if 
 produced in either direction ? 
 
488 
 
 Chapter Eight. 
 
 22. Draw a line, AB, meeting a horizontal line, BO, at 
 an angle of 58°. Draw a third line, DE, 
 parallel to the horizontal line, and cut- 
 ting the oblique line. What angles does 
 it make with the oblique line ? 
 
 Draw a fourth line, EG, parallel to 
 the oblique line, and cutting both hori- 
 zontal lines. 
 
 Mark in each of the twelve angles the number of degrees 
 it contains. 
 
 23. QR and UV are parallel lines, 
 cut by a line ST. If the angle b 
 measures 50°, how many degrees does 
 a measure ? 
 
 Find the number of degrees in each 
 of the other six angles. 
 
 631. Triangles. 
 
 24. From the extremities of the line AB, draw lines that 
 shall make angles of 60° and 40°, respec- 
 tively, with AB. Prolong the lines until 
 they meet at C, forming a triangle. 
 
 Measure the angle at C. How many 
 degrees does it contain? How many 
 degrees are there in the sum of the three angles of the tri- 
 angle ? 
 
 25. Construct a triangle having one angle of 90° and one 
 of 30°. Measure the third angle. 
 
 How many degrees are there in the sum 
 of the three angles ? 
 
 26. Construct a triangle, KLM, 
 making the angles at the base 28° 
 and 120°, respectively. Draw 
 NO, parallel to LM. 
 
Exercises in Geometry. 489 
 
 Is the angle e equal to any angle of the triangle ? How 
 many degrees does it contain ? Is the angle / equal to any 
 angle of the triangle ? How many degrees does it contain ? 
 
 How many degrees are there in the sum of the angles e, g, 
 and /? How many degrees are there in the angle g ? 
 
 27. How many degrees are there in the three angles of 
 any triangle ? 
 
 28. Two angles of a triangle measure 36° and 65°, respec- 
 tively. How many degrees does the third angle contain ? 
 
 29. Draw a triangle containing two angles of 50° and 70°, 
 respectively. How many degrees are there in the third 
 angle ? 
 
 Measure each side, and mark on the side its length. 
 Opposite which angle is found the longest side ? Opposite 
 which, the shortest side ? 
 
 30. Draw a triangle having two angles of 75° each. Are 
 any two of its sides equal ? 
 
 Draw a triangle having two angles of 50° each. Are any 
 of its sides equal ? 
 
 31. Draw a triangle having two angles of 60° each. How 
 many degrees does the third angle contain ? 
 
 Are any of its sides equal ? 
 
 32. If a triangle has two of its sides equal, what is true 
 of its angles? 
 
 33. If a triangle has three of its sides equal, what is true 
 of its angles? 
 
 632. A triangle having all its sides equal, is called an 
 equilateral triangle. 
 
 A triangle having two equal sides, is called an isosceles 
 triangle. 
 
 A triangle having all its sides unequal, is called a scalene 
 triangle. 
 
49° 
 
 Chapter Eight. 
 
 34. How does a perpendicular let fall upon the 
 of an isosceles triangle from the opposite 
 angle divide the angle ? How does it divide 
 the base ? How do the angles at the base 
 of an isosceles triangle compare with each 
 other as to size ? 
 
 The unequal side of an isosceles triangle is called 
 the base. 
 
 35. Draw an isosceles triangle having the base a vertical 
 line. 
 
 An isosceles triangle having the vertex below the base. 
 One having an oblique line for the base. 
 
 36. Draw a right-angled isosceles triangle. How many- 
 degrees will there be in each of the other angles ? 
 
 Draw an obtuse-angled isosceles triangle. 
 
 37. How many degrees will there be in each angle of an 
 equilateral triangle ? 
 
 Draw an equilateral triangle having one side vertical. 
 Draw an equilateral triangle having jy q ^ 
 
 its vertex below the base. 
 
 38. DEF is an isosceles triangle, 
 DF and EF being the equal sides. If 
 the angle 1 measures 50°, how many 
 degrees are there in each of the other 
 five angles, when the line FO bisects 
 the base.? 
 
 39. ABC is a right-angled A 
 triangle, the angle at B measur- 
 ing 90°, and the angle at C 
 measuring 30°. If the line AX 
 is so drawn as to make the 
 angle AXB equal to 60°, find 
 the number of degrees in the 
 angles m, n, and p, respectively. 
 
Exercises in Geometry. 491 
 
 633. Quadrilaterals. 
 
 A plane figure of four sides is called a quadrilateral. 
 
 When the opposite sides are parallel, the quadrilateral is 
 called a parallelogram. (Figs. 1 to 8.) 
 
 A rectangle is a parallelogram all of whose angles are right 
 angles. (Figs. 1 to 4.) 
 
 When the four sides of a rectangle are equal to each other, 
 it is called a square. (Figs. 1 and 2.) 
 
 The term oblong is frequently applied to rectangles whose 
 adjacent sides are unequal. (Figs. 3 and 4.) 
 
 Fig. 1. 
 
 Fig. 2. 
 
 Fig. 3. 
 
 Fig. 4. 
 
 A rhombus is a parallelogram all of whose sides are equal, 
 but whose angles are oblique. (Figs. 5 and 6.) 
 
 When the adjacent sides of a parallelogram are unequal 
 and the angles are oblique, it is called a rhomboid. (Figs. 7 
 and 8.) 
 
 VA 
 
 Fig. 5. 
 
 Fig. 6. 
 
 Fig. 7. 
 
 Fig. 8. 
 
 A trapezoid is a quadrilateral having only two of its sides 
 parallel. (Figs. 9 and 10.) 
 
 A trapezium is a quadrilateral having no two sides paral- 
 lel. (Figs. 11 and 12.) 
 
 Fig. 9. 
 
 Fig. 10. 
 
 Fig. 11. 
 
 Fig. 12. 
 
492 Chapter Eight. 
 
 634. The altitude of a parallelogram is the perpendicular 
 distance between its base and the side opposite. 
 
 M 
 
 G 
 
 x j; b- IT 
 
 The altitude of a triangle is the perpendicular distance 
 between the vertex and the base, or between the vertex and 
 base produced. 
 
 AB is the altitude of MANT; DX is the altitude of 
 DBE-, OYoi OHI. 
 
 40. Draw a parallelogram. How many angles does it con- 
 tain? Into how few triangles can you divide a parallelo- 
 gram? How many degrees are there in the sum of the 
 angles of each triangle ? How many degrees are there in 
 the sum of the angles of a parallelogram ? 
 
 41. Construct a parallelogram, the adjacent sides of which 
 shall measure 2 inches and 3 inches, respectively, and the 
 angle between them 60°. How long will each of the other 
 two sides be? Measure each of the other angles. How 
 many degrees are there in the sum of the four angles ? 
 
 42. Construct a trapezoid having a base of 5 inches, alti- 
 tude 3 inches, the angles at the base measuring 90° and 60°, 
 respectively. Measure the remaining angles, and find the 
 sum of the four angles. How long is each of the remaining 
 sides ? 
 
 43. Fold a piece of paper twice at right angles, and cut 
 off the folded corner, making a rhombus when the part cut 
 off is opened out. 
 
 Can you cut out a rhombus having two angles of 60° each ? 
 A rhombus having two angles of 80° each ? 
 
Exercises in Geometry. 493 
 
 44. Can you so cut a piece of paper, folded twice at right 
 angles, that the part cut off will be a square ? 
 
 45. Draw a rectangle, base 2\ inches, altitude 2 inches. 
 A rhomboid, base 2\ inches, altitude 2 inches. 
 
 46. Make, out of paper, a rectangle and a rhomboid, each 
 having the above dimensions, and endeavor to ascertain, by 
 cutting, whether or not they are equal to each other in area. 
 
 635. The Circle. 
 
 47. Draw a circle. Between two points on the circum- 
 ference draw a line that does not pass through the centre. 
 
 This line is called a chord. 
 
 48. Draw a circle. In it draw .two diameters, a radius, 
 and three chords. Write on each line its name. 
 
 49. Draw a part of the circumference of a circle greater 
 than one-half of it. Draw the chord. 
 
 A part of the circumference is called an arc. 
 
 50. Draw an arc less than a semi-circumferenee. Draw a 
 chord. Write the name on each. 
 
 Can you make a chord that will be longer than the 
 diameter ? 
 
 51. Draw two equal circles. In the first draw the chord 
 of an arc of 120°. In the second, the chord of an arc of 
 240°. What is the ratio between the two chords you have 
 drawn ? 
 
 52. In a circle draw a chord equal in length to the radius. 
 How many degrees are there in the arc whose chord has 
 been drawn ? 
 
 53. Draw an arc of 72°. To its extremities draw two 
 radii. 
 
 The part of the surface of a circle enclosed by two radii and the 
 intercepted arc is called a sector. 
 
494 * Chapter Eight. 
 
 54. Draw a sector of 60° (a sextant). A sector of 90° (a 
 quadrant). 
 
 55. Draw an arc of 120°. Draw the chord. 
 
 The part of the surface of a circle bounded by an arc and 
 its chord is called a segment. 
 
 56. Draw several circles having the same centre, but of 
 unequal radii (concentric circles). 
 
 57. Draw two equal circles just touching each other (tan- 
 gent). Draw two unequal circles tangent to each other. 
 
 Within a large circle draw a smaller one tangent to it. 
 
 58. Draw circles of equal radii cutting each other. Draw 
 intersecting circles of unequal radii. 
 
 636. Pentagons, Hexagons, Octagons. 
 
 59. Divide the circumference of a circle into four equal 
 arcs. Draw the chords, forming an inscribed square. 
 
 60. If you wish to inscribe in a circle a 
 figure of five equal sides, into how many equal 
 arcs must the circumference be divided ? How 
 many degrees will each arc contain ? 
 
 637. A plane figure bounded by straight lines is called a polygon. 
 
 A five-sided polygon is called a pentagon ; one of six sides, a hexa- 
 gon; of seven, a heptagon; of eight, an octagon; of nine, a nonagon; 
 of ten, a decagon; etc. 
 
 A regular polygon is one that is both equilateral and equi- 
 angular. 
 
 61. Inscribe a regular pentagon in a circle. Use the 
 protractor. 
 
 62. Inscribe in a circle a regular hexagon. A regular 
 octagon. An equilateral triangle. 
 
Exercises in Geometry. 495 
 
 63. Inscribe in a circle a regular hexagon. Connect the 
 opposite corners by lines passing through the 
 centre of the circle, forming six triangles. 
 
 How many degrees are there in each of the 
 six angles about the centre of the circle ? In 
 each of the twelve angles at the circumference ? 
 How many degrees are there in the sum of angles 1 and 2 ? 
 
 Is each of the six triangles scalene, equilateral, or 
 isosceles ? 
 
 64. Divide a regular inscribed pentagon into five equal 
 triangles by lines drawn from the centre of the circle. 
 
 What kind of triangles are formed; isosceles, scalene, or 
 equilateral ? 
 
 How many degrees are there in each angle at the centre ? 
 In each angle at the circumference? How many degrees 
 are there in the sum of two adjoining angles at the circum- 
 ference ? In each angle of the pentagon ? 
 
 65. About a circle circumscribe a square. An equilateral 
 triangle. A regular pentagon. A regular hexagon. A 
 regular octagon. 
 
 PROBLEMS IN CONSTRUCTION. 
 
 638. In drawing the following exercises, only the ruler and the 
 compasses are to be used. Use neither the protractor nor the triangle. 
 
 66. Draw a circle, radius an inch and a half. Outside of 
 it, and tangent to it, draw a second circle of an inch radius. 
 
 How far apart are the centres ? 
 
 67. Draw two tangent circles having radii of an inch and 
 a half and an inch, respectively, one within the other. 
 
 How long is the line joining the centres ? 
 
 68. With centres 3 inches apart draw two equal circles 
 tangent to each other. How long is the radius of each ? 
 
49^ Chapter Eight. 
 
 69. With centres three inches apart draw two equal 
 circles of 2 inches' radius. Connect the centres. 
 
 Draw a line joining the two points in which the circles 
 intersect. How does this line divide the line connecting 
 the centres ? 
 
 Draw radii from each centre to each point of intersection. 
 
 70. Construct an isosceles triangle, base 3 inches, equal 
 sides 2 inches. 
 
 Note. — Use circles or arcs where necessary. 
 
 71. Construct an isosceles triangle, base 3£ inches, equal 
 sides 4 inches. 
 
 Divide it into two equal parts. Do not locate the centre 
 of the base by measurements. 
 
 72. On a vertical line construct an isosceles triangle. 
 Without measuring the length of the base draw a perpen- 
 dicular to the centre of the base. 
 
 73. Bisect a vertical line. An oblique line. 
 Do not measure the length of the line. 
 
 74. Construct an equilateral triangle on a two-inch line. 
 
 75. Construct an equilateral triangle on a vertical line. 
 On an oblique line. 
 
 76. Cut out two equal right-angled triangles. Put them 
 together in different ways so as to form two different 
 isosceles triangles. 
 
 
 
 77. Construct a scalene triangle. 
 
 A triangle having sides measuring 1, 1J, 2 inches, respec- 
 tively. 
 
 One whose sides measure 2, 2|, and 3 inches, respectively. 
 
 78. Can you construct an isosceles triangle whose base 
 measures 4 inches, equal sides 2 inches ? 
 
 Try to construct a scalene triangle with sides measuring 
 1, 2, and 3 inches, respectively. 
 
Exercises in Geometry. 497 
 
 79. Draw a circle. In it draw a chord. 
 
 Bisect the chord, using as few lines and as short ones as 
 you can. 
 
 Note. — Do not use the ruler to ascertain the length of the chord 
 before bisecting it. 
 
 80. Divide a sector into two equal parts. 
 
 81. Draw a circle. Draw a chord. Draw a radius through 
 the centre of the chord. 
 
 Is the radius perpendicular to the chord ? Why ? 
 
 82. Bisect the arc of a circle and its chord. 
 Bisect the arc of a circle without drawing the chord. 
 
 83. Draw a perpendicular to the middle point of a hori- 
 zontal line. To the middle point of a vertical line. To the 
 middle point of an oblique line. 
 
 84. Draw in a circle two diameters perpendicular to each 
 other. 
 
 85. Divide the circumference of a circle into four equal 
 parts. Into eight equal parts. 
 
 Inscribe a square in a circle. 
 
 86. Inscribe a regular octagon in a circle. 
 
 87. Connect the opposite vertices of a regular octagon 
 inscribed in a circle by lines passing through the centre of 
 the circle. 
 
 Lines connecting the opposite vertices of a polygon are called 
 diagonals. 
 
 88. Inscribe a square in a circle. Circumscribe a square 
 whose sides shall be perpendicular to the diagonals of the 
 inscribed square. 
 
 89. Cut out the circumscribed square and show by folding 
 that it is twice the area of the inscribed square. 
 
498 Chapter Eight. 
 
 90. Construct an equilateral triangle on a horizontal line 
 1 inch long. On the right side as a base, construct a second 
 equilateral triangle. On the left side of the first triangle, 
 construct a third Construct three more, completing the 
 hexagon. 
 
 91. Can you circumscribe a circle about the above hexa- 
 gon ? What is the radius of the circle ? 
 
 92. Inscribe a regular hexagon in a circle whose radius is 
 1 \ inch. What is the length of each side of the hexagon ? 
 
 93. Inscribe in a circle an equilateral triangle. On each 
 of its three sides construct an equilateral triangle. 
 
 94. Construct an arc of 60°. Draw two lines meeting at 
 an angle of 60°. 
 
 95. Bisect an arc of 60°. Draw two lines meeting at an 
 angle of 30°. 
 
 96. Construct an angle of 60° and an angle of 30°. Draw 
 two lines making an angle equal to the sum of the two angles 
 first constructed. 
 
 97. Erect a perpendicular at the end of a horizontal line. 
 At the end of a vertical line. At the end of an oblique line. 
 
 98. Construct an angle of 45°. An angle of 22£°. An 
 angle of 135°. An angle of 15°. An angle of 75°. 
 
 99. Draw a circle, radius 1 inch. Draw a diameter, and 
 produce it an inch beyond the circumference. At the centre 
 of the circle erect a perpendicular to the diameter. 
 
 100. An inch from one end of a 3-inch line, erect a perpen- 
 dicular, using as few and as short lines as possible. 
 
 101. Draw a horizontal line. Take a point above the line 
 as a center. Draw an arc that cuts the line in two places. 
 
 102. Draw a line. From a point above the line, let fall 
 a perpendicular to the line. 
 
Exercises in Geometry. 499 
 
 EQUAL TRIANGLES. EQUIVALENT TRIANGLES. 
 
 639. Note. — The protractor and the triangle may be used in the 
 following exercises. 
 
 1. Draw a rectangle, base 2\ inches, altitude 2 inches. 
 Draw a rhomboid, base 2£ inches, altitude 2 inches. Find 
 the area of each. 
 
 2. With a base 2 \ inches, altitude 2 inches, draw 
 
 (a) A right-angled triangle. 
 
 (b) An isosceles triangle. 
 
 (c) One or more acute-angled scalene triangles. 
 
 (d) One or more obtuse-angled triangles. 
 Calculate the area of each. 
 
 3. Can you show, by cutting from paper, that a right- 
 angled triangle having its base and perpendicular 4 inches 
 and 3 inches, respectively, has the same surface as an acute- 
 angled triangle whose base and altitude are 4 inches and 
 3 inches respectively, and an obtuse-angled triangle whose 
 base and altitude are 4 inches and 3 inches, respectively ? 
 
 Two triangles that have the same area are called equivalent tri- 
 angles ; those having their corresponding sides and angles equal, each 
 to each, are called equal triangles. 
 
 4. Construct a triangle whose sides measure 1J, 2, and 
 2\ inches, respectively. Construct another triangle having 
 its sides of the same lengths. Are the angles of the second 
 equal to the angles of the first ? Are the triangles equal ? 
 
 5. Draw two triangles each of which has two sides 
 measuring 1^ and 3 inches, respectively, and the included 
 angle 60 degrees. Is the third side of one triangle equal to 
 the third side of the other? Are the remaining angles 
 of the first triangle equal to the remaining angles of the 
 second ? 
 
500 
 
 Chapter Eight. 
 
 6. Construct two triangles with equal bases, and angles at 
 the bases respectively equal. Are the triangles equal ? 
 
 7. A person wishing to ascertain the length, AB, of a 
 pond, places a pole at a convenient 
 point, G, visible from A and B. The 
 distance BO is measured, and a pole is 
 set up, on a line with B and G, at D, 
 the distance CD being made equal to 
 BG. A pole is also placed at E, on a 
 line with A and C, the distance CE 
 being made equal to AG. 
 
 Can you show that the length, AB, of the pond can be 
 ascertained by measuring the distance DE ? 
 
 CALCULATING HEIGHTS AND DISTANCES. 
 
 640. To verify the results obtained by calculation, the pupil should 
 make diagrams, drawing the figures to a convenient scale. (7 
 
 1. If AB in a right- 
 angled triangle measures ^ 
 120 feet, and a perpendicu- ^S 
 lar, vw, erected 10 feet from 
 A measures 5 feet, calcu- 
 late the length of BG. 
 
 Aw:AB::wv:BC: 
 
 — w 
 i.e. 10 : 120 : 
 
 5: BG. 
 
 2. A post 6 feet above ground throws a shadow of 7£ feet 
 How high is a tree whose shadow measures 60 feet ? 
 
 t^ 
 
Exercises in Geometry. 
 
 5<» 
 
 3. Wishing to ascertain the distance between two houses, 
 R and S, on opposite sides 
 of a stream, I measure a 
 line, SV, at right angles to 
 SR, 200 feet. At T, 90 feet 
 from V, the perpendicular 
 TW measures 60 feet. Re- 
 quired the distance SR. 
 
 VT:TW::VS:SR 
 
 4. Beginning at B, 100 feet from the bank of a rive^ a 
 line, BC, is measured 1200 feet 
 long. At D, distant from C 
 50 feet, the perpendicular DE 
 is found to measure 90 feet. 
 What is the distance from B to 
 A, a tree on the opposite bank ? 
 How wide is the river ? 
 
 5. A boy, whose eye (E) is 4 feet from the ground, can 
 just see the top (A) of a steeple when he stands back 3 feet 
 from a fence (CG) 6 feet high. The distance from the 
 foot of the fence to the centre of the base of the steeple is 
 177 feet. Find the height of the steeple, AB. 
 
 CD = ? EH=? ED: CD:: EH: AH. 
 
 When AH is found, how may you get AB ? 
 
502 
 
 Chapter Eight. 
 
 6. Wishing to ascertain the distance AB, I measure a 
 line, AD, at right angles to AB, 
 12 chains ; DE, at right angles to 
 AD, 5 chains ; and find that a line 
 sighted from E to B intersects 
 AD at C, distant from D 3.25 
 chains. What is the distance from 
 AtoB? 
 
 Note. —The triangles DCE and ACB 
 are similar. Why ? 
 
 7. Wishing to find the height of a tower, //, I set up 
 a pole, cd, 12 feet long above the ground. Another pole, ab, 
 4J feet above ground, is set up at such a distance that the 
 tops of the two poles 
 and of the tower are 
 in a line. The distance 
 between the poles (ae 
 or db) is 10J feet. The 
 distance from d to the 
 foot of the tower is 195 
 feet. The width of the 
 tower Qcj) is 30 feet. 
 
 The similar triangles aec and ahf give us the proportion 
 
 kij 
 
 ae :ah : : ec: hf. 
 
 What is the distance ec? ah = bi=bd+dk+ki 
 fh is found, what must be added 
 to get the height of the tower ? 
 
 8. To determine the height 
 of a building, MN, a person 
 attached a straight strip of 
 wood, ab, to a post, OP, in 
 such a manner that sighting 
 from a, he could just see M> 
 
 ki=\kj. When 
 
Exercises in Geometry. 
 
 503 
 
 the top of the building. He then sighted down from b, and 
 marked on the ground the point R, on a line with ab. 
 
 PQ was found by measurement to be 4 feet, MP 6 feet, 
 PN 120 feet. Eequired, MN. 
 
 9. Wood-choppers, desiring 
 to know the height of a tree 
 before cutting it, sometimes 
 make an isosceles right-angled 
 triangle of wood or paper, and 
 " step off " the distance on level 
 ground from the point at which 
 they find they can just see the 
 top of the tree looking along 
 the hypotenuse of the triangle, 
 the base being parallel to the ground. 
 
 How high is the tree AB, if ^4(7 is 36 paces of 
 3 feet each, and the angle AGB is 45° ? 
 
 10. B is a point on the bank of a stream 
 due east of A on the other bank. A boy 
 walks due south of A until he reaches a 
 point at which he finds, from his pocket 
 compass, that he is directly southwest of 
 B. If the distance AC measures 119 yards, 
 how wide is the stream ? 
 
TABLES 
 
 LINEAR MEASURE 
 
 12 inches (in.) . . 
 
 3 feet 
 
 5£ yards, or 16| feet 
 40 rods .... 
 320 rods .... 
 
 = 1 foot ft. 
 
 — 1 yard yd. 
 
 = 1 rod rd. 
 
 = 1 furlong fur. 
 
 = 1 mile mi. 
 
 1 mi. 
 
 320 rd. = 1760 yd. = 5280 ft. = 63,360 in. 
 
 A hand, used in measuring the height of horses, = 4 in. A knot, used 
 in measuring distances at sea, = 1.15 mi. A fathom, used in measuring 
 the depth of the sea, 
 
 Oft. 
 
 SQUARE MEASURE 
 
 144 square inches (sq. in.) . . = 1 square foot . 
 
 9 square feet = 1 square yard . 
 
 30£ sq. yd., or 272^ sq. ft. . . = 1 square rod . 
 
 160 square rods = 1 acre . . . 
 
 640 acres . . . . < . . . = 1 square mile . 
 
 sq. ft. 
 sq. yd. 
 sq. rd. 
 A. 
 sq. mi. 
 
 1 A. = 160 sq. rd. = 4840 sq. yd. = 43,560 sq. ft. 
 A Section of land is a square mile. 
 
 Koofing, flooring, and slating are often estimated by the square, 
 -which contains 100 square feet. 
 
 SURVEYORS' MEASURE 
 
 In measuring land, surveyors use a chain (ch.) which contains 100 
 links (1.) and is 4 rods long. Since the chain isf4 rods long, a square 
 chain contains 16 sq. rd., and 10 sq. ch. = 160 sq. rd., or 1 acre. 
 
 CUBIC MEASURE 
 
 1728 cubic inches (cu. in.) . . = 1 cubic foot 
 
 27 cubic feet = 1 cubic yard 
 
 128 cubic feet = 1 cord . . 
 
 16 cubic feet = 1 cord ft. . 
 
 8 cord feet = 1 cord . . 
 
 cu. ft. 
 cu. yd. 
 cd. 
 
 cd. ft. 
 cd. 
 
 Note. — In computing the contents of an enclosing wall, masons and 
 brick-layers regard it as one straight wall whose length is the distance 
 around it on the outside. Corners are thus measured twice. 
 
 A perch of stone or masonry is 16£ ft. long, 1J ft. thick, and 1 ft. 
 high, and contains 24 1 cu. ft. 
 
YB 35889 
 
 MEASURES OF CAPACITY 
 
 Liquid Measure Dry Measure 
 
 4 gills = 1 pint . . . pt. 2 pints = 1 quart . . qt. 
 
 2 pints sb 1 quart . . . qt. 8 quarts s= 1 peck . . pk. 
 
 4 quarts = 1 gallon . . gal. 4 pecks = 1 bushel . . bu. 
 
 The standard gallon contains 231 cubic inches. 
 The standard bushel contains 2150.42 cubic inches. 
 
 The capacity of cisterns, reservoirs, etc., is often expressed in barrels 
 (bbl.) of 315 gallons each, or in hogsheads (hhd.) of 63 gallons each. In 
 commerce, these vary in size. 
 
 AVOIRDUPOIS WEIGHT 
 
 16 ounces (oz.) 
 100 y 
 2000] 
 
 1 pound lb. 
 
 The long 
 and in weigl 
 
 1 bushel of 
 1 bushel of 1 
 1 bushel of 
 1 bushel of 
 
 24 gr: 
 20 pe 
 12 oul 
 
 54884 
 
 UNIVERSITY OF CALIFORNIA LIBRARY 
 
 APOTHECARIES' WEIGHT 
 
 60 grains (gr. ) . . . . = 1 dram . . . dr., or 3. 
 
 8 drams =1 ounce . . . oz., or %. 
 
 12 ounces = 1 pound . . . lb., or lb. 
 
 One pound Apothecaries' weight = 576C grains. 
 
 BRITISH OR STERLING MONEY 
 
 4 farthings =1 penny . .... A 
 
 . . . s. 
 
 . . . £. 
 
 12 pence = 1 shilling 
 
 20 shillings =1 pound 
 
 5 shillings =1 crown. 
 
 The value of £1 is $4.8665 in United States gold coin. 
 
 The unit of French money is 1 franc, which is 19.3 cents. The unit of 
 German money is 1 mark, which is 23.85 cents. 
 
Ililittil 
 
 1 
 
 
 
 
 1 
 
 ■i 11 
 
 i in! 
 
 1 
 
 Jar 
 
 '111!