i 
 
 m. 
 
IN MEMORIAM 
 FLORIAN CAJORI 
 
*{.. 
 
WENTWORTH'S 
 SERIES OF MATHEMATICS. 
 
 First Steps in Number. 
 
 Primary Arithmetic. 
 
 Grammar School Arithmetic. 
 
 High School Arithmetic. 
 
 Exercises in Arithmetic. 
 
 Shorter Course in Algebra. 
 
 Elements of Algebra. Complete Algebra. 
 
 College Algebra. Exercises in Algebra. 
 
 Plane Geometry. 
 
 Plane and Solid Geometry. 
 
 Exercises in Geometry. 
 
 PI. and Sol. Geometry and PI. Trigonometry. 
 
 Plane Trigonometry and Tables. 
 
 Plane and Spherical Trigonometry. 
 
 Surveying. 
 
 PI. and Sph. Trigonometry, Surveying, and Tables. 
 
 Trigonometry, Surveying, and Navigation. 
 
 Trigonometry Formulas. 
 
 Logarithmic and Trigonometric Tables (Seven}, 
 
 Log. and Trig. Tables (Complete Edition). 
 
 Analytic Geometry. 
 
 Special Terms and Circular on Application, 
 
GRAMMAR SCHOOL 
 
 ARITHMETIC 
 
 BY 
 
 G. A. WENTWORTH, A.M., 
 
 i * 
 1'KnKK^soii OK MAYHKMATK > IN I'll I 1. 1. 1 PS EXETER ACADEMY. 
 
 REVISED EDITION. 
 
 BOSTON, U.S.A.: 
 
 PUBLISHED BY GINN & COMPANY. 
 1892. 
 
Entered, according to Act of Congress, in the year 1889, by 
 
 Q. A. WENT WORTH, 
 in the Office of the Librarian of Congress, at Washington. 
 
 ALL RIGHTS RESERVED. 
 
 TYPOGRAPHY BY J. S. CUSHING & Co., BOSTON, U.S.A. 
 
 PRESSWORK BY GINN & Co., BOSTON, U.S.A. 
 
f <?ca 
 
 PREFACE. 
 
 ^F^IIIS Arithmetic is designed to give pupils of the grammar-school 
 age an intelligent knowledge of the subject and a moderate 
 power of independent thought. 
 
 Whether Arithmetic is studied for mental discipline or for practical 
 uiustury over the every-day problems of common life, mechanical pro- 
 cesses and routine methods are of no value. Pupils can be trained 
 to logical habits of mind and stimulated to a high degree of intel- 
 lectual energy by solving problems adapted to their capacities. 
 They become practical arithmeticians, not by learning special 
 business forms, but by founding their knowledge on reasoning 
 which they fully comprehend, and by being so thoroughly exer- 
 cised in logical analysis that they are independent of arbitrary 
 rules. 
 
 The book contains a great number of well-graded and progres- 
 sive problems, made up for youths from ten to fourteen years of 
 age. Definitions and explanations are made as brief and simple 
 as possible. It is not intended that definitions shall be committed 
 to memory, but that they shall be simply discussed by teacher 
 and pupils. Every teacher, of course, will be at liberty to give 
 better definitions, and to make a better presentation of methods, 
 than those given in the book. In short, the chief object in view 
 will be gained if pupils are trained to solve the problems by neat 
 and intelligent methods, and are kept free from set rules and 
 formulas. 
 
 A great many number- problems are given in the first pages of 
 the book, so that the necessary facility and accuracy in computing 
 
 1306072 
 
IV PREFACE. 
 
 under the four fundamental rules may be acquired ; as want of 
 accuracy and rapidity in mere calculations distracts the attention 
 which should be given to the investigation and correct statement 
 of clothed exercises. The pupil should be required to do only so 
 many of these number-problems as are found to be necessary to 
 give him facility and accuracy in the four fundamental operations , 
 and he should be allowed to omit some of the harder-clothed prob- 
 lems until he reviews the book. 
 
 The chapter on the Metric System is put at the end of the lok 
 because many grammar-school pupils have no time for it, while 
 those who have time can as well learn the system at this stage of 
 their progress as earlier. 
 
 The chapter on Miscellaneous Problems is intended as a review 
 of the subject-matter of Arithmetic and as a test of the learner's 
 knowledge. 
 
 The author is under obligations to many teachers who have 
 given valuable suggestions and assistance in the preparation ot 
 this work. 
 
 G. A. WENTWORTH, 
 PHILLIPS EXETER ACADEMY, April, 1889. 
 
CONTENTS. 
 
 PAGE 
 
 CHAPTER I. NOTATION AND NUMERATION .... 1 
 
 II. ADDITION 12 
 
 III. SUBTRACTION 31 
 
 IV. MULTIPLICATION ...... 46 
 
 V. DIVISION 59 
 
 VI. DECIMALS 76 
 
 VII. MULTIPLES AND MEASURES .... 98 
 
 VIII. COMMON FRACTIONS Ill 
 
 IX. COMPOUND QUANTITIES 155 
 
 X. PKRCENTACIK 203 
 
 XI. INTEREST AND DISCOUNT 225 
 
 XII. PROPORTION 254 
 
 XIII. POWERS AND ROOTS 266 
 
 XIV. MENSURATION 278 
 
 XV. MISCELLANEOUS PROBLEMS . . . 293 
 
 XVI. METRIC SYSTEM . 319 
 
VOCABULARY. 
 
 Abstract number. This phrase is employed to designate numbers 
 used without reference to any particular unit, as 8, 10, 21. But 
 all numbers are in themselves abstract whether the kind of thing 
 numbered is or is not mentioned. 
 
 Addition. The process of combining two or more numbers so as to 
 form a single number. 
 
 Aliquot part. A number which is contained an integral number of 
 times in a given number. Thus, 5, 6}, 12J, 16J, are aliquot parts 
 of 100. 
 
 Amount. The sum of two or more numbers. In Interest, the sum 
 of principal and interest. 
 
 Analysis. The separation of a question into parts, to be examined 
 each by itself. 
 
 Antecedent. The first of the two terms named in a ratio. 
 
 Area of a surface. The area of a surface is the number of units of 
 surface it contains ; the unit of surface being a square whose side 
 is a unit of length. 
 
 Arithmetic. The science that treats of numbers and the methods 
 of using them. 
 
 Assets. All the property belonging to an estate, individual, or cor- 
 poration. 
 
 Average. The mean of several unequal numbers, so that, if substi- 
 tuted for each, the aggregate would be the same. 
 
 Bank. An establishment for the custody, loaning, and exchange of 
 money ; and often for the issue of money. 
 
 Bank discount. An allowance received by a bank for the loan of 
 money, paid at the time of lending as interest. 
 
 Bonds. Written contracts under seal to pay specified sums of money 
 at specified times, issued by national governments, states, cities, 
 and other corporations. 
 
 Cancellation. The striking out of a common factor from the divi- 
 dend and divisor. 
 
v iii VOCABULARY. 
 
 Commission. Compensation for the transaction of business, reck- 
 oned at some per cent of the money employed in the transaction. 
 
 Common denominator. A denominator common to two or more 
 fractions. 
 
 Common factor. A factor common to two or more numbers. 
 
 Common multiple. A multiple common to two or more numbers. 
 
 Complex fraction. A fraction that has a fraction in one or both of 
 its terms. 
 
 Composite number. The product of two or more integral factors, 
 each factor being greater than unity. 
 
 Compound denominations. Several denominations used to express 
 parts of one quantity. 
 
 Compound interest. When the interest due is left unpaid, and con- 
 sidered as an increase made to the principal, the whole interest, 
 accruing in any time, is called compound interest. 
 
 Compound fraction. A fraction of another fraction. 
 
 Concrete number. A phrase used to denote numbers applied to 
 specified things ; as 6 horses, 8 desks. 
 
 Consequent. The second of the two terms named in a ratio. 
 
 Consignee. The person or firm to whom goods are sent. 
 
 Consignor. The person or firm who sends goods to another. 
 
 Corporation. An association of individuals authorized by law to 
 transact business as a single person. 
 
 Couplet. The two terms of a ratio taken together. 
 
 Coupon. A certificate of interest attached to a bond, to be cut off 
 when due and presented for payment. 
 
 Creditor. A person or firm to whom money is due. 
 
 Cube root. One of the three equal factors of a number. 
 
 Customs. Duties or taxes imposed by law on merchandise imported, 
 and sometimes on merchandise exported. 
 
 Debtor. A person who owes money to another. 
 
 Decimal fractions. Fractions of which only the numerators are 
 written, and the denominators are ten or some power of ten. 
 
 Decimal point. A dot placed after the units' figure to mark its place. 
 
 Decimal system. The common system of numbers founded on their 
 relations to ten, ten tens, etc. 
 
 Denominator. The number which shows into how many equal parts 
 a unit is divided. 
 
 Difference. The number which, added to a given number, makes a 
 sum equal to another given number. 
 
VOCABULARY. ix 
 
 Discount. Allowance made for the payment of money before it be- 
 comes due. Also, the difference between the market value and 
 the face value when the market value is below the face value. 
 
 Dividend. In division, the given number which is equal to the 
 product of a given factor (called divisor) and required factor 
 (called quotient). In business, the share of profits which belongs 
 to each owner of stock, on his proportion of the capital. 
 
 Division. The operation by which, when a product and one of its 
 factors are given, the other factor is found. 
 
 Divisor. The number by which a given dividend is to be divided. 
 
 Draft. A written order directing one person to pay a specified sum 
 of money to another. 
 
 Drawee of a draft. The person to whose order the sum of money 
 named in a draft is to be paid. 
 
 Drawer of a draft. The person who signs the draft. 
 
 Duties. Taxes required by the government to be paid on goods 
 imported, exported, or put on the market for consumption. 
 
 Equation. A statement that two expressions of number are equal. 
 
 Equation of payments. The finding of an average time at which 
 several payments may be justly made. 
 
 Exchange. A system of paying debts, due to persons living at a 
 distance, by transmitting drafts instead of money. 
 
 Exponent. A small figure placed at the right of a number to show 
 how many times the number is taken as a factor. 
 
 Extremes. The first and last terms of a proportion. 
 
 Evolution. The process of finding the root of a number. 
 
 Factors. The factors of a number are a set of numbers whose prod- 
 uct is the given number ; they are assumed to be integral, except 
 in the extraction of roots. In commerce, agents employed by 
 merchants to transact business. 
 
 Figures. Symbols used to represent numbers in the common system 
 of notation. Also diagrams used to represent geometrical forms. 
 
 Firm. The name under which a company transact business. 
 
 Fractions. One or more of the equal parts into which the unit is 
 divided. 
 
 Grace. An allowance of three days, after the date a note becomes 
 due, within which to pay the note. 
 
 Gram. The unit of weight in the metric system. 
 
 Greatest common measure. The greatest number which is a com- 
 mon factor of two or more given numbers. 
 
x VOCABULARY. 
 
 Improper fraction. A fraction whose numerator equals or exceeds 
 '.ne denominator. 
 
 Index. A figure written at the left and above the radical sign to 
 show what root of the number under the radical sign is required. 
 A fraction written at the right of a number, of which the nume- 
 rator shows the required power of that number, and the denomi- 
 nator the required root of that power. 
 
 Instalment. A payment in part. 
 
 Insurance. A guarantee of a specified sum of money in the event 
 of loss of property by fire, storm at sea, or other disaster; or of 
 loss of life. 
 
 Integral number. A number which denotes whole things. 
 
 Interest. Money paid for the use of inoiiey. 
 
 Involution. The process of finding a power of a number. 
 
 Latitude of a point. The angle made by ilia vertical line at that 
 point with the plane of the equator. 
 
 Least common multiple. The least number which is a common 
 multiple of several given numbers. 
 
 Liability. A debt, or obligation to pay. 
 
 Line. Length without breadth or thickness. The path of a moving 
 point. 
 
 Liter. The unit of capacity in the metric system equal in volume 
 to a cube each edge of which is one- tenth of a meter. 
 
 Long division. The method of dividing in which the processes are 
 written in full. 
 
 Longitude of a point. The angle between two planes supposed to 
 pass through the centre of the earth and to contain, the one the 
 meridian of that point, and the other the standard meridian. 
 
 Loss. The excess of the cost price above the selling price. 
 
 Maturity of a note. The date at which a note legally becomes due. 
 
 Mean proportional. A number which is both the second and third 
 terms of a proportion. 
 
 Means. The terms of a proportion between the extremes. 
 
 Meter. The unit of length in the metric system. 
 
 Minuend. The given number in subtraction which is equal to the 
 sum of another given number called the subtrahend, and a 
 required number called the difference or remainder. 
 
 Mixed number. A number that expresses both entire things and 
 parts of things taken together. 
 
VOCABULARY. xi 
 
 Multiple of a number. The product obtained by taking the given 
 
 number an integral number of times. 
 Multiplicand, The number to be multiplied by another. 
 Multiplication. The operation of finding a number bearing the 
 
 same ratio to the multiplicand which the multiplier bears to unity 
 Multiplier. The number by which the multiplicand is multiplied. 
 Net proceeds. The money that remains of the money received for 
 
 property after all expenses and discounts are paid. 
 Notation. A system of expressing numbers by symbols. 
 Note. A written agreement to pay a specified sum of money at a 
 
 specified time. 
 
 Number. The answer to the question, How many ? 
 Numeration. A system of naming numbers. 
 Obligation. A debt, or liability to pay. 
 Order of units. A name used to designate the number of things 
 
 in a group, as tens, hundreds, thousands, etc. 
 Partial payment. Part payment on a note. 
 Partnership, An association of two or more persons to carry on 
 
 business. 
 
 Par value. Face or nominal value. 
 Pendulum. A body suspended by a straight line from a fixed point, 
 
 and moving freely about that point as a centre. 
 Percentage, A part of any given number reckoned at some rate 
 
 per cent. 
 
 Period. A group of three figures. 
 Policy A written contract of insurance. 
 Poll tax. A tax levied by the head or poll. 
 Power. The product of two or more equal factors. 
 Premium. Money paid for insurance computed at some rate per 
 
 cent of the value insured Also the excess of market value 
 
 above par value. 
 
 Present worth. The present value of a debt due at some future day. 
 Prime number. A number which has no integral factors except 
 
 itself and one. 
 
 Principal. Money drawing interest 
 Problem. A question to be solved, 
 Product. The result obtained by multiplying the multiplicand by 
 
 the multiplier 
 Profit. The excess of selling price aoove cost 
 
xii VOCABULARY. 
 
 Proof The evidence by which the accuracy of any result is estab- 
 lished. 
 
 Proper fractioi . A fraction, the numerator of which is less than 
 the denominator. 
 
 Proportion. A statement that two ratios are equal. 
 
 Quantity. The answer to the question, How much ? 
 
 Quotient. The number sought in division. 
 
 Rate per cent. Rate by the hundred. 
 
 Ratio. The relative magnitude of two numbers or of two quantities. 
 
 Reciprocal of a number. One divided by that number. 
 
 Reduction. The process of changing the unit in which a quantity 
 is expressed without changing the value of the quantity. 
 
 Remainder. The number which, added to the iubtrahend, gives a 
 sum equal to the minuend. 
 
 Root of a number. One of the equal factors of the number. 
 
 Rule. The statement of a prescribed method. 
 
 Security. Property used to guarantee the payment of any debt. 
 
 Share. One of a certain number of equal parts into which the capi- 
 tal of a company is divided. 
 
 Short division. The method of dividing in which the operations of 
 multiplying and subtracting are performed mentally. 
 
 Solid. A magnitude which has length, breadth, and thickness. 
 
 Solution. The process by which the answer to a question is obtained. 
 
 Specific gravity of a substance. The ratio of the weight of a 
 given volume of it to that of an equal volume of water. 
 
 Square root. One of two equal factors. 
 
 Stock. Capital invested in business. 
 
 Subtraction. The process of finding a number which added to one 
 of two given numbers will produce the other. 
 
 Sum. The number which results from combining two or more num- 
 bers by addition. 
 
 Surd. An indicated root the value of which cannot be exactly ex- 
 pressed in figures. 
 
 Surface. That which has only length and breadth. 
 
 Thermometer. An instrument for measuring heat. 
 
 Units. The standards by whicK we count separate objects or measure 
 magnitudes. 
 
 Verify. To establish, by trial, the truth of any statement. 
 
 Volume of a solid. The volume of a solid is the number of units of 
 volume it contains ; the unit of volume being a, cube whose edge 
 is a unit of length. 
 
GRAMMAR SCHOOL ARITHMETIC. 
 
 
 
 CHAPTER I. 
 PRELIMINARY DEFINITIONS. 
 
 1, Number, A fundamental idea, like that of number, 
 cannot be defined. A simple, direct answer to the ques- 
 tion " How many? " is a number. 
 
 2, A collection of several similar objects (as a collection 
 of apples) gives the idea of number. 
 
 3, Units, In counting separate objects the standards by 
 which we count are called units. Thus : 
 
 In counting the eggs in a nest, the unit is an egg. 
 In selling eggs by the dozen, the unit is a dozen eggs. 
 In selling bricks by the thousand, the unit is a thousand 
 bricks. 
 
 4, Measurement, Continuous magnitudes, such as length, 
 surface, space, time, heat, cannot be counted ; they are 
 measured. Magnitudes, whether continuous or separate, are 
 generally measured, and the standards by which they are 
 measured are fixed by law, or by common consent. 
 
 5, Units of Measure. The standards by which we meas- 
 ure magnitudes are called units of measure. Thus : 
 
 An inch, a foot, a yard, a rod, a mile, are units of length. 
 A square inch, a square foot, are units of surface. 
 A cubic inch, a cubic foot, are units of volume. 
 
NOTATION AND NUMERATION. 
 
 6, Abstract Numbers, Numbers standing alone, as 4, 7, 
 13, which mean 4 units, 7 units, 13 units, but do not specify 
 the kind of objects counted or the kind of units of measure 
 taken, are called abstract numbers. They signify simply 
 the number of repetitions of some unit. 
 
 7, Concrete Numbers, Expressions that give the name of 
 the objects counted or of the unit of measure employed, 
 and the number of such objects, or of such units of measure, 
 are called concrete numbers. Thus, 5 horses, 7 feet, 6 
 pounds, 5 days, are called concrete numbers. Such expres- 
 sions consist of two parts, the number proper, and the kind 
 of units taken, and should, strictly speaking, be called 
 quantities. 
 
 8, Arithmetic treats of the simple properties of num- 
 bers, and the art of computing by numbers. 
 
 NOTATION AND NUMERATION. 
 
 9, The first numbers have special names, as follows : 
 one, two, three, four, five, six, seven, eight, nine, ten. 
 
 10, The first nine of these numbers are called Simple 
 Units, or units of the first order. 
 
 11, The group of ten units has received the name of a 
 Ten, or a unit of the second order ; and we count by tens as 
 by units ; thus : 
 
 one ten, two tens, three tens ... nine tens, ten tens. 
 
 12, The group of ten tens has received the name of a 
 Hundred, or a unit of the third order ; and we count by 
 hundreds, as by tens and units; thus: 
 
 one hundred, two hundreds ... ten hundreds. 
 
NOTATION AND NUMERATION. 
 
 13, A group of ten hundreds is called a Thousand, or a 
 unit of the fourth order. 
 
 14, From ten units of the fourth order is formed a ten 
 thousand, or a unit of the fifth order; and from ten units of 
 the fifth order is formed a hundred thousand, or a unit of 
 the sixth order. 
 
 15, Units of the seventh order are called Millions ; of the 
 eighth order, ten millions; of the ninth order, hundred 
 millions. Finally, units of the tenth order are called 
 Billions; units of the thirteenth order, Trillions; and so on. 
 
 16, The table of units of different orders is as follows : 
 
 First order, simple units, \ 
 
 Second order, tens of units, > first class. 
 
 Third order, hundreds of units, ) 
 
 Fourth order, thousands, \ 
 
 Fifth order, tens of thousands, > second class. 
 
 Sixth order, hundreds of thousands, J 
 
 Seventh order, millions, \ 
 
 Eighth order, tens of millions, > third class. 
 
 Ninth order, hundreds of millions, ) 
 
 Tenth order, billions, -\ 
 
 Eleventh order, tens of billions, > fourth class. 
 
 Twelfth order, hundreds of billions, J 
 
 Thirteenth order, trillions, \ 
 
 I fifth class. 
 
 17, The group of the first three orders is called the first 
 class of units, and the group of the three following orders, 
 the second class, and so on. 
 
 18, The unit of the second class is equal to a thousand 
 units of the first class, and a unit of the third class is equal 
 to a thousand units of the second class, and so on. 
 
NOTATION AND NUMERATION. 
 
 19, To read a number we decompose it into units of the 
 different orders, and state how many groups there are of 
 each kind, commencing with the highest order. Thus, for 
 example, two millions, three thousands, five hundreds, 
 seven tens, and four units. 
 
 20, It is clear that the names of all numbers up to a 
 billion are formed by combining the names of the first nine 
 numbers with the words ten, hundred, thousand, million. 
 
 21, Usage sanctions the following irregularities : 
 
 I. Instead of saying two tens, three tens, four tens, five 
 tens, six tens, seven tens, eight tens, nine tens, we say 
 twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety. 
 
 II. The names of the numbers between ten and twenty 
 are eleven, twelve, thirteen, fourteen, fifteen, sixteen, 
 seventeen, eighteen, nineteen. 
 
 22, The names of the numbers between twenty and a 
 hundred are : 
 
 twenty-one, twenty-two, twenty-three ... twenty-nine, 
 thirty-one, thirty-two, thirty-three ... thirty-nine, 
 
 ninety-one, ninety-two, ninety-three ... ninety-nine. 
 
 
 
 23, The names of the numbers between a hundred and 
 a thousand are : 
 
 hundred one, hundred two ... hundred ninety-nine, 
 two hundred one ... two hundred ninety-nine, 
 
 nine hundred one ... nine hundred ninety-nine. 
 
 24, The common system of notation employs ten figures 
 or digits : 
 
 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. 
 
NOTATION AND NUMERATION. 
 
 The first nine of these figures represent the first nine num- 
 bers ; the last, which is called Zero, Naught, or Cipher, is 
 used to denote the absence of units of the order in which 
 it stands. It is possible to express all numbers by these 
 ten digits by making the value of each figure increase ten- 
 fold for every place that it is moved to the left. 
 
 25, If we have given a number written in figures, the 
 position of each figure counting from the right indicates 
 the order of units that the figure represents. If we divide 
 the number into periods of three figures each, the first 
 period on the right will be the period of simple units, the 
 second period will be the period of thousands, the third 
 will be the period of millions, and so on. In each period 
 the first figure on the right expresses the units of that 
 class, the second figure the tens, and the third the 
 hundreds. Thus : 
 
 MILLIONS. THOUSANDS. UNITS. 
 
 Tens. Units. Hundreds. Tens, Units. Hundreds. Tens. Units. 
 
 21 334 334 
 
 Thus, the number 21,334,334 means and is read 21 
 millions, 334 thousands, 334 units. If the number is 
 applied to dollars, it means and is read 21 million, 334 
 thousand, 334 dollars. The next period is the billions' 
 period. 
 
 NOTE. The fundamental principle of forming and expressing 
 numbers should be illustrated by making little bundles of wooden 
 toothpicks, ten in each bundle, and then making bundles of hun- 
 dreds by taking for each hundred ten bundles of ten each. When 
 the pupil has become familiar with forming and expressing numbers 
 consisting of hundreds, tens, and units, he should be shown that the 
 method of forming and expressing numbers of hundreds, tens, and 
 units of thousands is precisely the same, the only difference being 
 that the unit of this period is not a single toothpick, but a pile of ten 
 bundles of a hundred each, which is a thousand. 
 
6 NOTATION AND NUMERATION. 
 
 26, To write a number in figures we write successively trie 
 number of units of each order from left to right, beginning 
 at the highest order and taking care to supply by zeros 
 orders of units that may be lacking. 
 
 27, To read a number written in figures we divide the 
 number into periods of three figures each from right to 
 left : this done, we begin to read at the left-hand period 
 and read as if the figures of that period stood alone, adding 
 the name of the period ; then the next period to the right 
 is read with the name of that period, and so on. 
 
 28, The number 1256 may be read one thousand two 
 hundred fifty-six, or it may be read twelve hundred fifty-six. 
 The number 5004 may be read five thousand four, or it 
 may be re&d fifty hundred four. The shortest method is the 
 best method of reading any number. Twelve hundred fifty- 
 six is shorter than one thousand two hundred fifty-six ; five 
 thousand four is shorter than fifty hundred four. 
 
 29, It will be seen that the value of each figure, in any 
 number expressed in figures, depends on two things : 
 
 First, the value attached to the figure without regard to 
 its position. 
 
 And, secondly, the value it acquires from the place it 
 holds in the number. 
 
 The value of a figure, without regard to its position, is 
 called its absolute value ; and the value it acquires by its 
 position is called its local value. 
 
 30, The art of expressing numbers by means of figures 
 is called Notation, and the art of expressing in words a 
 number written in figures is called Numeration, 
 
 31, The unit of money is the dollar. Instead of writing 
 the word dollars, this mark $ is used, which is called the 
 
NOTATION AND NUMERATION. 
 
 sign for dollars, or the " dollar mark." Thus, if we wish 
 to write five dollars, we write it $5. 
 
 It takes ten ten-cent pieces to make a dollar ; that is, a 
 ten-cent piece is one-tenth of a dollar. It takes ten single 
 cents to be equal in value to a ten-cent piece. If we have 
 one dollar and one ten-cent piece, we write it $1.10. If 
 we have one dollar, one ten-cent piece, and two cents, we 
 write it $1.12. 
 
 The dot which is placed after the one dollar is called 
 the Decimal Point. Figures to the left of the decimal point 
 denote whole units. Figures to the right of the decimal 
 point denote parts of a unit, and are called Decimal Frac- 
 tions, The expression $1.10 is read "one dollar and ten 
 cents" ; and the expression $ 1.12 is read "one dollar and 
 
 twelve cents." 
 
 Ex. . 
 Write in figures : 
 
 1. Two hundred thirty-six, one hundred forty, five hun- 
 
 dred two, seven hundred three. 
 
 2. Five hundred fourteen, three hundred seventy-six, 
 
 four hundred thirty, eight hundred two, nine hun- 
 dred twenty-seven. 
 
 3. One hundred ninety, four hundred six, eight hundred 
 
 ten, two hundred seven. 
 
 4. Three hundred ten, two hundred thirteen, six hun- 
 
 dred twenty-three, two hundred nineteen. 
 
 5. Five hundred fifty, four hundred four, four hundred 
 
 twenty-five, eight hundred sixty. 
 
 6. Eight hundred sixteen, seven hundred eight, nine 
 
 hundred, seven hundred three. 
 
 7. Nine hundred ninety-five, eight hundred eighty, seven 
 
 hundred, eignt hundred seven. 
 
 8. Two hundred seventeen, four hundred twelve, four 
 
 hundred eight, one hundred two. 
 
8 NOTATION AND NUMERATION. 
 
 9. Four hundred seventeen, six hundred nineteen, three 
 hundred six, one hundred eighteen. 
 
 Ex. 2. 
 Read (or write in words) : 
 
 1. 500, 700, 300, 200, 900, 100. 
 
 2. 830, 709, 506, 350, 819, 703. 
 
 3. 607, 312, 918, 810, 103, 560. 
 
 4. 752, 698, 405, 536, 121, 514. 
 
 5. 973, 356, 703, 409, 211, 713. 
 
 6. 225, 64, 970, 49, 83, 674. 
 
 7. 106, 170, 380, 759, 921, 538. 
 
 8. 481, 360, 593, 32, 296, 551. 
 
 9. 182, 802, 555, 705, '649, 630. 
 10. 314, 97, 613, 384, 992, 516. 
 
 Ex. 3. 
 
 Write in figures : 
 
 1. Eight thousand seven hundred three, four thousand 
 
 forty-five, six thousand three hundred eight, forty- 
 eight hundred. 
 
 2. Five thousand forty-eight, nineteen hundred ninety, 
 
 seven thousand eighty-two, eight thousand fifty. 
 
 3. Seven thousand two hundred forty, nine thousand 
 
 nine hundred nineteen, six thousand seven, eight 
 thousand seven hundred seventy-six. 
 
 4. Seven thousand one hundred seven, six thousand 
 
 eight hundred four, nine thousand one hundred ten, 
 five thousand five hundred fifty. 
 
 5. Six thousand eighty-six, four thousand forty, one 
 
 thousand ten, nine thousand ninety-nine. 
 
 6. Eight thousand eighty, seventeen hundred fifty-seven, 
 
 eleven hundred one, seven thousand seven, forty- 
 five hundred forty-five 
 
NOTATION AND NUMERATION. 
 
 7. Two thousand four hundred ninety-six, eighteen hun- 
 
 dred eighty-three, three thousand ninety-five, one 
 thousand eleven. 
 
 8. One thousand thirteen, one thousand one, fourteen 
 
 hundred, thirty-three thousand fourteen. 
 
 9. Seventeen hundred thirty-six, three thousand forty- 
 
 nine, eight thousand eighteen, nine thousand seventy. 
 10. Four thousand seven hundred nine, fifteen hundred ten, 
 one thousand sixty-nine, sixteen thousand sixteen. 
 
 Ex. 4. 
 
 Read (or write in words) : 
 
 1. 
 
 8,000, 
 
 5,000, 
 
 2,000, 
 
 6,000, 
 
 1,000, 
 
 9,000. 
 
 2. 
 
 9,210, 
 
 6,907, 
 
 7,402, 
 
 9,998, 
 
 4,060, 
 
 7,210. 
 
 3. 
 
 5,068, 
 
 4,020, 
 
 1,400, 
 
 7,031, 
 
 1,290, 
 
 1,010. 
 
 4. 
 
 8,808, 
 
 6,006, 
 
 8,482, 
 
 3,096, 
 
 4,720, 
 
 11,973. 
 
 5. 
 
 12,002; 
 
 11101 
 
 5,812, 
 
 1,739, 
 
 6760, 
 
 6,903. 
 
 6. 
 
 4,085, 
 
 1,169, 
 
 2,615 
 
 5,007, 
 
 1,110, 
 
 1,460. 
 
 7. 
 
 4,760, 
 
 4,190, 
 
 2,607, 
 
 5,180, 
 
 1,200, 
 
 3,746. 
 
 8. 
 
 9,008, 
 
 8,300, 
 
 6,804, 
 
 2,977, 
 
 6,202, 
 
 9,620. 
 
 9. 
 
 6,322, 
 
 7,450, 
 
 8,673, 
 
 2,603, 
 
 2,518, 
 
 1,508. 
 
 10. 
 
 7,080, 
 
 1,009, 
 
 8,070, 
 
 5,068, 
 
 1,397, 
 
 5,782. 
 
 Write in figures : Ex * 5 ' 
 
 1. Twelve and twelve hundredths, twenty-two and eight 
 
 tenths, three hundred twenty-five and six tenths, 
 one hundred one and one hundred one thousandths. 
 
 2. Seventy-five and seventy-five hundredths, eighty- 
 
 three and twenty-six thousandths, ninety-six and 
 seven hundred four thousandths, one thousand ten 
 and two tenths. 
 
 3. Five hundred seventy-three and five hundred seventy- 
 
 three thousandths, eleven thousand four and sixteen 
 hundredths, three hundred sixty-five and eighl 
 tenths, seventy-two and ninety-six hundredths. 
 
10 
 
 NOTATION AND NUMERATION. 
 
 4. Three and nineteen thousandths, six hundred fifty- 
 
 eight and two hundredths, eight hundred and eight 
 hundredths, thirty-seven and five thousandths. 
 
 5. Seventy-one and seven tenths, seven and seventeen 
 
 hundredths, seven hundred and seventeen thou- 
 sandths, eight hundred ten and one tenth. 
 
 6. Eighty-one and one hundredth, eight and one hundred 
 
 one thousandths, nine hundred sixty-three and two 
 tenths, ninety-six and thirty-two hundredths, nine 
 and six hundred thirty-two thousandths. 
 
 7. Six hundred and five tenths, sixty and five hundredths, 
 
 six and five thousandths. 
 
 8. Nine hundred eighty-three and three tenths, ninety- 
 
 eight and thirty-three hundredths, nine and eight 
 hundred thirty-three thousandths. 
 
 9. One hundred twelve and one tenth, eleven and 
 
 twenty-one hundredths, one and one hundred 
 twenty-one thousandths. 
 
 10. Eleven thousand and sixty-three thousandths, twenty- 
 three and eighty-six hundredths, one hundred ten 
 and eleven hundredths. 
 
 Ex. 
 Read (or write in words) : 
 
 6. 
 
 1 
 
 2. 
 3. 
 4. 
 5. 
 
 a 
 
 7. 
 8. 
 9. 
 10. 
 
 3010.3, 
 903.9, 
 234.5, 
 6187.8, 
 291.59, 
 360.4, 
 47.S28, 
 510.14, 
 65.002, 
 770.85, 
 
 477.12, 
 413.9, 
 3010.3, 
 785.33, 
 29.645, 
 3605.9, 
 59.184, 
 51.028, 
 69.949, 
 6994.9, 
 
 60.206, 
 17.918, 
 59.106, 
 90.849, 
 30.081, 
 361.16, 
 600.65, 
 580.35, 
 602.17, 
 712.06. 
 
 698.97, 
 113.94, 
 43.136, 
 92.294, 
 299.07, 
 39.041, 
 601.19, 
 5804.7, 
 6020.6, 
 719.66, 
 
 778.15, 
 14.613, 
 380.21, 
 27.989, 
 30.190, 
 468.64, 
 60.108, 
 641.97, 
 64.058, 
 833.87, 
 
 84.510. 
 204.12. 
 361.73. 
 28.012. 
 35.257. 
 463.59. 
 52.466. 
 6409.8. 
 76.343. 
 83.493. 
 
NOTATION AND NUMERATION 11 
 
 Ex. 7. 
 
 Write in figures : 
 
 1. Fifty thousand three dollars, eighty thousand nine 
 
 hundred ninety dollars. 
 
 2. Twenty-eight million seven hundred forty-four thou- 
 
 sand one hundred sixty-nine dollars. 
 
 3. Five hundred sixteen dollars and ten cents, twenty- 
 
 five hundred fifty dollars and sixty-nine cents. 
 
 4. Sixteen hundred million thirty thousand three hun- 
 
 dred eight dollars and fifty cents. 
 
 5. Twenty-seven hundred million one thousand one dol- 
 
 lars and eighty-seven cents. 
 
 6. Five hundred thousand two hundred one dollars and 
 
 seventy-five cents. 
 
 7. Eight million fourteen thousand three hundred 
 
 twenty-five dollars and twenty-five cents. 
 
 8. Ninety-seven million two hundred thousand one hun- 
 
 dred two dollars and five cents. 
 
 9. Ten million ten thousand ten dollars and ten cents. 
 10. Eleven hundred ten thousand dollars and eleven 
 
 cents. 
 
 Ex. 8. 
 Bead (or write in words) : 
 
 1. $259,132.10, $27,186.25. 
 
 2. $1,213,062.50, $2,763,001.75. 
 
 3. $3,675,321.12, $3,500,005.15. 
 
 4. $17,360,502.20, $27,132,857.33. 
 
 5. $55,333,263.36, ' $58,785,587.09. 
 
 6. $116,001,556.40, $275,363,750.11. 
 
 7. $ 660,878,640.69, $ 594,340,000.94. 
 
 8. $600,241,560.02, $124,271,000.01. , 
 
 9. $768,301,520.20, $802,631,516.73. 
 10. $505,631,880.04, $1,555,676,410.62. 
 
CHAPTER II. 
 
 ADDITION. 
 
 32. If you put 2 cents with 3 cents, how many cents 
 have you ? Answer, 5 cents. 
 
 How can you express this operation on your slate? 
 
 You can write the figure 2 ; then the figure 3 be- 
 neath it; draw a line underneath, and below the 
 line write the figure 5. The work is shown in the 5 
 margin. 
 
 Or, you can express it thus : 2 + 3 = 5. 
 
 The sign + is called plus, and means that the numbers 
 between which it is placed are to be counted together ; and 
 . the sign = means equals, So that 2 + 3 5 is read 2 plus 
 3 equals 5. 
 
 33. The operation of finding a number equal to two or 
 more numbers taken together is called addition ; and the 
 result is called their sum, The numbers to be added are 
 called addends, 
 
 Name the sums of the following numbers, and practise 
 naming them until you can name each sum the instant 
 your eye rests upon the numbers to be added. 
 
 Ex. 9. (Oral) 
 
 1 + 1- 3+1- 1+0- 1+7- 1+5= 
 2+1= 8+1- 1+4- 6+1- 9+1= 
 2+2- 2+0- 1+2- 8+2- 2+7- 
 

 
 ADDITION. 
 
 
 13 
 
 2 + 5 = 
 
 6 + 2- 
 
 3 + 2 = 
 
 2 + 4 = 
 
 2 + 9 = 
 
 3 + 4 = 
 
 2 + 3- 
 
 6 + 3 = 
 
 1 + 3 = 
 
 3 + 7 = 
 
 8 + 3 = 
 
 3 + = 
 
 3 + 6 = 
 
 3 + 3 = 
 
 9 + 3 = 
 
 + 4 = 
 
 5 + 4 = 
 
 4 + 7 = 
 
 9 + 4 = 
 
 4 + 1 = 
 
 8 + 4-= 
 
 4 + 3 = 
 
 2 + 4 = 
 
 4 + 4 = 
 
 4 + 6 = 
 
 5 + 5 = 
 
 5 + 7 = 
 
 3 + 5 = 
 
 + 5 = 
 
 5 + 9- 
 
 2 + 5 = 
 
 5 + 1 = 
 
 4 + 5 = 
 
 5 + 6 = 
 
 8 + 5- 
 
 6 + 3 = 
 
 1 + 6 = 
 
 5 + 6 = 
 
 6 + = 
 
 2 + 6- 
 
 6 + 6 = 
 
 4 + 6 = 
 
 6 + 9 = 
 
 7 + 6 = 
 
 6 + 8 --- 
 
 5 + 7 = 
 
 5 + 6 = 
 
 7-1 3 = 
 
 7 + 1 = 
 
 + 7 = 
 
 8 + 7 = 
 
 7 + 2 = 
 
 4+7 = 
 
 7 + 7 = 
 
 7 + 9=- 
 
 8 + 1 = 
 
 5 + 8 = 
 
 2 + 8 = 
 
 8 + = 
 
 7 + 8 - 
 
 8 + 3 = 
 
 8 + 8 = 
 
 4 + 8 = 
 
 8 + 6 = 
 
 9 + 8=- 
 
 9 + = 
 
 9 + 9 = 
 
 9 + 2 = 
 
 1+9 = 
 
 3 + 9 = 
 
 4 + 9 = 
 
 9 + 6 = 
 
 8 + 9 = 
 
 9 + 5 = 
 
 7 + 9 = 
 
 5 6 
 
 8 7 
 
 4 3 
 
 6 6 
 
 6 7 
 
 3 7 
 
 3 5 
 
 5 6 
 
 4 8 
 
 5 5 
 
 7 7 
 
 7 7 
 
 7 8 
 
 5 8 
 
 8 8 
 
 7 2 
 
 9 4 
 
 8 2 
 
 8 7 
 
 9 8 
 
 9 9 
 
 9 9 
 
 6 2 
 
 3 8 
 
 3 6 
 
 1 4 
 
 7 8 
 
 8 5 
 
 4 2 
 
 6 5 
 
 8 7 
 
 5 3 
 
 7 8 
 
 9 5 
 
 4 3 
 
 7 2 
 
 4 8 
 
 3 8 
 
 3 .3 
 
 3 3 
 
14 ADDITION. 
 
 Ex. 10. 
 Find the sums of the following numbers : 
 
 1. 28 9- 29 17 32 25. 47 33. 49 
 
 57985 
 
 2. 43 10. 56 18. 58 26. G5 34. G7 
 
 68999 
 
 3. 54 11. 78 19. 79 27. 27 35. 43 
 
 75487 
 
 4. 63 12. 78 20. 57 28. 27 36. 35 
 
 89869 
 
 5. 74 13. 32 21. 63 29. 42 37. 14 
 
 37896 
 
 6. 18 14. 27 22. 36 30. 12 38. 73 
 
 99898 
 
 7. 85 15. 37 23. 59 31. 50 39. 13 
 
 75459 
 
 8. 19 16. 37 24. 93 32. 89 40. 79 
 89976 
 
AUDITION 15 
 
 Copy the following, and fill the blanks : 
 
 42 + 9= 19 + 8= 17 + 9= 23 + 8 = 
 
 71 + 9= 26 + 7= 35 + 8= 47 + 6 = 
 
 85 + 8= 18+7= 17 + 3= 18 + 9 = 
 
 29 + 6= 38 + 5= 15 + 7= 14 + 8 = 
 
 34, Since 10 in any place is equal to 1 in the next place 
 to the left, if the sum of the digits of any column exceeds 9, 
 write the units' figure of the sum under the column added 
 and carry the number of tens to the next column. 
 
 Thus, in the following example : 872 
 
 The sum of the digits in the right-hand column is 3. The 99^ 
 
 sum of the digits in the second column is 16 ; the 6 is writ- 
 ten under this column and the 1 is carried to the third 1863 
 column. The sum of the digits of the third column, to- 
 gether with the 1 carried to it, is 18 ; the 8 is written under this col- 
 umn and the 1 is carried to the place of thousands. 
 
 Add: 
 
 1. 497 6. 689 11. 9535 16. 56902 
 735 297 9675 94876 
 
 2. 840 7. 477 12. 5557 17. 93689 
 869 335 5763 60086 
 
 3. 997 8. 449 13. 8284 18. 59857 
 289 483 7998 84556 
 
 4. 643 9. 857 14. 8956 19. 83897 
 937 816 7694 50799 
 
 5. 958 10. 842 15. 3448 20. 59988 
 294 863 4876 99939 
 
16 ADDITION. 
 
 35, Practise the following additions until you can name 
 the results as rapidly as you can count 1, 2, 3, 4, 5, etc. 
 
 Ex. 12. (Oral.) 
 
 Add by twos to 50, beginning 0, 2, 4, 6, 8. Add by 
 twos to 51, beginning 1, 3, 5, 7, 9. 
 
 Add by threes to 102, beginning 0, 3, 6. Add by threes 
 to 100, beginning 1, 4, 7. Add by threes to 101, beginning 
 2, 5, 8. 
 
 Add by fours to 100, beginning 0, 4, 8. Add by fours 
 to 101, beginning 1, 5, 9. Add by fours to 102, beginning 
 2, 6, 10. Add by fours to 103, beginning 3, 7, 11. 
 
 Add by fives to 100, beginning 0, 5, 10. Add by fives 
 to 101, beginning 1, 6, 11. Add by fives to 102, beginning 
 2, 7, 12. Add by fives to 103, beginning 3, 8, 13. Add 
 by fives to 104, beginning 4, 9, 14. 
 
 Add by sixes to 102, beginning 0, 6, 12. Add by sixes 
 to 103, beginning 1, 7, 13. Add by sixes to 104, beginning 
 2, 8, 14. Add by sixes to 105, beginning 3, 9, 15. Add 
 by sixes to 100, beginning 4, 10, 16. Add by sixes to 101, 
 beginning 5, 11, 17. 
 
 Add by sevens to 105, beginning 0, 7, 14. Add by sevens 
 to 106, beginning 1, 8, 15. Add by sevens to 100, begin- 
 ning 2, 9, 16. Add by sevens to 101, beginning 3, 10, 17. 
 Add by sevens to 102, beginning 4, 11, 18. Add by sevens 
 to 103, beginning 5, 12, 19. Add by sevens to 104, begin- 
 ning 6, 13, 20. 
 
 Add by eights to 104, beginning 0, 8, 16. Add by eights 
 to 105, beginning 1, 9, 17. Add by eights to 106, begin- 
 ning 2, 10, 18. Add by eights to 107, beginning 3, 11, 19. 
 Add by eights to 100, beginning 4, 12, 20. Add by eights 
 to 101, beginning 5, 13, 21. Add by eights to 102, begin- 
 ning 6, 14, 22. Add by eights to 103, beginning 7, 15, 23. 
 
ADDITION. 17 
 
 Add by nines to 108, beginning 0, 9, 18. Add by nines 
 to 100, beginning 1, 10, 19. Add by nines to 101, begin- 
 ning 2, 11, 20. Add by nines to 102, beginning 3, 12, 21. 
 Add by nines to 103, beginning 4, 13, 22. Add by nines 
 to 104, beginning 5, 14, 23. Add by nines to 105, begin- 
 ning 6, 15, 24. Add by nines to 106, beginning 7, 16, 25. 
 Add by nines to 107, beginning 8, 17, 26. 
 
 36, Practise adding columns of three digits until you 
 can name the sum of any three digits the instant you see 
 them. 
 
 Ex. 13. (Oral.) 
 Find the sums of : 
 
 1. 3245456743 
 5123323232 
 4334214327 
 
 2. 3456578345 
 4242421232 
 1323230423 
 
 3. 453543 8^6 7 8 
 2243151296 
 3132323549 
 
 4. 1876795796 
 3657286387 
 51799949 18 
 
 5. 7659765395 
 2450669898 
 3348468797 
 
18 ADDITION. 
 
 6. 
 
 3 
 
 5 
 
 7 
 
 6 
 
 8 
 
 9 
 
 7 
 
 6 
 
 9 
 
 8 
 
 
 9 
 
 6 
 
 2 
 
 3 
 
 5 
 
 6 
 
 5 
 
 3 
 
 8 
 
 8 
 
 
 7 
 
 8 
 
 5 
 
 4 
 
 4 
 
 4 
 
 7 
 
 5 
 
 8 
 
 8 
 
 7. 
 
 6 
 
 6 
 
 7 
 
 7 
 
 4 
 
 4 
 
 3 
 
 5 
 
 3 
 
 6 
 
 
 2 
 
 5 
 
 8 
 
 7 
 
 9 
 
 4 
 
 8 
 
 5 
 
 7 
 
 8 
 
 
 5 
 
 3 
 
 9 
 
 7 
 
 4 
 
 4 
 
 7 
 
 5 
 
 7 
 
 7 
 
 8. 
 
 8 
 
 3 
 
 2 
 
 3 
 
 5 
 
 5 
 
 9 
 
 4 
 
 2 
 
 2 
 
 
 4 
 
 3 
 
 2 
 
 4 
 
 4 
 
 6 
 
 3 
 
 7 
 
 9 
 
 2 
 
 
 4 
 
 3 
 
 9 
 
 8 
 
 8 
 
 6 
 
 6 
 
 5 
 
 8 
 
 8 
 
 9. 
 
 9 
 
 5 
 
 7 
 
 8 
 
 9 
 
 6 
 
 7 
 
 4 
 
 5 
 
 8 
 
 
 9 
 
 6 
 
 7 
 
 4 
 
 5 
 
 7 
 
 4 
 
 3 
 
 4 
 
 2 
 
 
 7 
 
 6 
 
 5 
 
 7 
 
 5 
 
 6 
 
 8 
 
 9 
 
 7 
 
 9 
 
 10. 
 
 3 
 
 2 
 
 2 
 
 3 
 
 6 
 
 8 
 
 7 
 
 8 
 
 9 
 
 6 
 
 
 8 
 
 9 
 
 2 
 
 3 
 
 7 
 
 5 
 
 2 
 
 2 
 
 8 
 
 9 
 
 
 7 
 
 8 
 
 9 
 
 7 
 
 4 
 
 4 
 
 3 
 
 2 
 
 7 
 
 7 
 
 37. The quickest way to add columns of four or more 
 digits is to train the eye to see at a glance sums 
 
 of 20, and simply add these sums. If you add the 8 
 
 column given in the margin by single digits, you say 9 
 
 to yourself, ten, thirteen, seventeen, twenty-two, twenty- 6 
 
 eight, thirty-seven, forty-five; if you add by taking 5 
 
 two digits at a time, you say ten, seventeen, twenty- 4 
 
 eight, forty-five; if you add by taking three digits at 3 
 
 a time, you say thirteen, twenty-eight, fo?*ty-five; if 2 
 
 you add by 20's, you say twenty (separating 5 into 3 8 
 and 2), forty-five. 
 
ADDITION. 19 
 
 Ex. 14. 
 
 Find 
 
 the sums of: 
 
 1- 5 
 
 9 
 
 6 
 
 4 
 
 5 
 
 7 
 
 1 
 
 7 
 
 8 
 
 2 
 
 4 
 
 5 
 
 4 
 
 6 
 
 8 
 
 3 
 
 4 
 
 3 
 
 4 
 
 6 
 
 8 
 
 7 
 
 9 
 
 5 
 
 7 
 
 8 
 
 5 
 
 6 
 
 3 
 
 9 
 
 6 
 
 3 
 
 7 
 
 2 
 
 3 
 
 6 
 
 3 
 
 8 
 
 2 
 
 hr 
 < 
 
 4 
 
 8 
 
 5 
 
 8 
 
 4 
 
 9 
 
 8 
 
 5 
 
 7 
 
 4 
 
 7 
 
 4 
 
 8 
 
 6 
 
 8 
 
 2 
 
 6 
 
 4 
 
 8 
 
 5 
 
 3 
 
 9 
 
 2 
 
 9 
 
 5 
 
 7 
 
 7 
 
 3 
 
 2 
 
 3 
 
 5 
 
 7 
 
 6 
 
 4 
 
 9 
 
 3 
 
 9 
 
 6 
 
 8 
 
 5 
 
 2. 8 
 
 5 
 
 4 
 
 4 
 
 5 
 
 2 
 
 3 
 
 5 
 
 3 
 
 6 
 
 2 
 
 3 
 
 2 
 
 2 
 
 3 
 
 6 
 
 8 
 
 3 
 
 8 
 
 4 
 
 4 
 
 6 
 
 5 
 
 5 
 
 2 
 
 4 
 
 6 
 
 8 
 
 4 
 
 7 
 
 9 
 
 8 
 
 9 
 
 7 
 
 8 
 
 5 
 
 7 
 
 9 
 
 9 
 
 3 
 
 3 
 
 2 
 
 7 
 
 6 
 
 4 
 
 3 
 
 9 
 
 5 
 
 7 
 
 5 
 
 7 
 
 8 
 
 6 
 
 9 
 
 8 
 
 5 
 
 5 
 
 7 
 
 3 
 
 8 
 
 6 
 
 5 
 
 9 
 
 4 
 
 6 
 
 3 
 
 4 
 
 6 
 
 8 
 
 6 
 
 5 
 
 3 
 
 4 
 
 3 
 
 2 
 
 7 
 
 8 
 
 9 
 
 6 
 
 7 
 
 2 
 
 4 
 
 1 
 
 8 
 
 1 
 
 9 
 
 2 
 
 4 
 
 2 
 
 9 
 
 3. 7 
 
 5 
 
 4 
 
 2 
 
 4 
 
 8 
 
 4 
 
 6 
 
 9 
 
 7 
 
 3 
 
 6 
 
 4 
 
 6 
 
 8 
 
 7 
 
 9 
 
 8 
 
 3 
 
 9 
 
 5 
 
 7 
 
 9 
 
 8 
 
 6 
 
 3 
 
 6 
 
 
 
 8 
 
 6 
 
 8 
 
 3 
 
 6 
 
 4 
 
 7 
 
 5 
 
 7 
 
 4 
 
 2 
 
 3 
 
 2 
 
 9 
 
 7 
 
 7 
 
 4 
 
 5 
 
 4 
 
 5 
 
 8 
 
 4 
 
 8 
 
 4 
 
 9 
 
 9 
 
 5 
 
 4 
 
 3 
 
 3 
 
 7 
 
 5 
 
 6 
 
 2 
 
 3 
 
 8 
 
 4 
 
 7 
 
 7 
 
 6 
 
 3 
 
 8 
 
 3 
 
 8 
 
 7 
 
 2 
 
 6 
 
 9 
 
 6 
 
 9 
 
 4 
 
 6 
 
 4 
 
 1 
 
 7 
 
 3 
 
 2 
 
 9 
 
 2 
 
 7 
 
 6 
 
 8 
 
20 ADDITION. 
 
 4. 6 
 
 5 
 
 9 
 
 8 
 
 5 
 
 9 
 
 8 
 
 7 
 
 9 
 
 6 
 
 8 
 
 8 
 
 5 
 
 6 
 
 9 
 
 6 
 
 4 
 
 9 
 
 3 
 
 8 
 
 5 
 
 3 
 
 3 
 
 7 
 
 4 
 
 7 
 
 5 
 
 6 
 
 8 
 
 3 
 
 3 
 
 7 
 
 4 
 
 9 
 
 3 
 
 5 
 
 9 
 
 3 
 
 9 
 
 5 
 
 7 
 
 9 
 
 9 
 
 5 
 
 7 
 
 9 
 
 7 
 
 4 
 
 8 
 
 3 
 
 4 
 
 4 
 
 6 
 
 8 
 
 2 
 
 8 
 
 6 
 
 5 
 
 7 
 
 6 
 
 6 
 
 8 
 
 8 
 
 4 
 
 9 
 
 3 
 
 8 
 
 8 
 
 3 
 
 9 
 
 8 
 
 3 
 
 2 
 
 3 
 
 6 
 
 9 
 
 3 
 
 6 
 
 4 
 
 5 
 
 2 
 
 6 
 
 1 
 
 2 
 
 8 
 
 4 
 
 2 
 
 1 
 
 
 
 2 
 
 5. 8 
 
 5 
 
 3 
 
 8 
 
 6 
 
 3 
 
 5 
 
 1 
 
 3 
 
 4 
 
 5 
 
 9 
 
 9 
 
 6 
 
 5 
 
 4 
 
 5 
 
 3 
 
 3 
 
 3 
 
 5 
 
 6 
 
 1 
 
 4 
 
 7 
 
 7 
 
 6 
 
 5 
 
 3 
 
 2 
 
 1 
 
 1 
 
 1 
 
 3 
 
 4 
 
 3 
 
 1 
 
 8 
 
 2 
 
 6 
 
 1 
 
 1 
 
 8 
 
 4 
 
 9 
 
 1 
 
 5 
 
 5 
 
 9 
 
 1 
 
 9 
 
 7 
 
 1 
 
 
 
 3 
 
 4 
 
 1 
 
 7 
 
 6 
 
 6 
 
 7 
 
 7 
 
 9 
 
 4 
 
 9 
 
 3 
 
 1 
 
 4 
 
 2 
 
 7 
 
 1 
 
 1 
 
 3 
 
 3 
 
 1 
 
 1 
 
 1 
 
 
 
 9 
 
 8 
 
 6 
 
 4 
 
 7 
 
 9 
 
 2 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 Ex. 15. 
 
 Find the sums of : 
 
 1. 
 
 50 
 
 2. 40 
 
 3. 60 
 
 4. 30 
 
 5. 10 
 
 6. 80 
 
 
 20 
 
 80 
 
 50 
 
 10 
 
 70 
 
 90 
 
 
 70 
 
 20 
 
 80 
 
 90 
 
 10 
 
 30 
 
 
 60 
 
 30 
 
 20 
 
 40 
 
 90 
 
 80 
 
 
 30 
 
 70 
 
 50 
 
 20 
 
 40 
 
 60 
 
 
 90 
 
 80 
 
 30 
 
 50 
 
 70 
 
 30 
 
 
 80 
 
 60 
 
 40 
 
 70 
 
 30 
 
 40 
 
 
 10 
 
 50 
 
 70 
 
 80 
 
 20 
 
 50 
 
ADDITION. 21 
 
 7. 40 
 
 8. 80 
 
 9. 52 
 
 10. 30 
 
 11. 42 
 
 12. 60 
 
 21 
 
 70 
 
 60 
 
 23 
 
 40 
 
 40 
 
 90 
 
 31 
 
 70 
 
 52 
 
 50 
 
 32 
 
 50 
 
 42 
 
 30 
 
 91 
 
 80 
 
 54 
 
 83 
 
 60 
 
 81 
 
 70 
 
 70 
 
 90 
 
 70 
 
 51 
 
 42 
 
 33 
 
 34 
 
 82 
 
 62 
 
 90 
 
 50 
 
 80 
 
 90 
 
 91 
 
 13. 51 
 
 14. 56 
 
 15. 48 
 
 16. 36 
 
 17. 25 
 
 18. 17 
 
 46 
 
 63 
 
 31 
 
 42 
 
 52 
 
 82 
 
 30 
 
 72 
 
 45 
 
 50 
 
 49 
 
 25 
 
 25 
 
 81 
 
 82 
 
 81 
 
 38 
 
 13 
 
 32 
 
 17 
 
 19 
 
 14 
 
 41 
 
 80 
 
 47 
 
 26 
 
 21 
 
 35 
 
 57 
 
 45 
 
 19. 18 
 
 20. 57 
 
 21. 15 
 
 22. 44 
 
 23. 19 
 
 24. 91 
 
 24 
 
 31 
 
 8 
 
 21 
 
 27 
 
 42 
 
 91 
 
 28 
 
 23 
 
 36 
 
 48 
 
 36 
 
 33 
 
 63 
 
 70 
 
 8 
 
 39 
 
 82 
 
 64 
 
 90 
 
 61 
 
 14 
 
 7 
 
 71 
 
 75 
 
 9 
 
 55 
 
 27 
 
 9 
 
 54 
 
 37 
 
 81 
 
 83 
 
 59 
 
 87 
 
 65 
 
 25. 48 
 
 26. 52 
 
 27. 8 
 
 28. 16 
 
 29. 33 
 
 30. 54 
 
 9 
 
 61 
 
 43 
 
 48 
 
 52 
 
 46 
 
 17 
 
 26 
 
 52 
 
 85 
 
 27 
 
 8 
 
 29 
 
 28 
 
 67 
 
 7 
 
 38 
 
 19 
 
 83 
 
 83 
 
 9 
 
 26 
 
 41 
 
 92 
 
 75 
 
 94 
 
 17 
 
 35 
 
 9 
 
 57 
 
 21 
 
 77 
 
 84 
 
 , 54 
 
 94 
 
 83 
 
22 ADDITION. 
 
 31. 55 
 
 32. 68 33. 
 
 9 34. 
 
 13 35. 48 
 
 36. 35 
 
 67 
 
 5 
 
 23 
 
 99 6 
 
 42 
 
 78 
 
 43 
 
 25 
 
 7 51 
 
 57 
 
 9 
 
 67 
 
 68 
 
 85 9 
 
 64 
 
 4 
 
 25 
 
 79 
 
 64 23 
 
 49 
 
 18 
 
 14 
 
 7 
 
 39 88 
 
 87 
 
 
 
 Ex. 16. 
 
 
 
 Add: 
 
 
 
 
 
 1. 123 
 
 2. 516 
 
 3. 321 
 
 4. 225 
 
 5. 871 
 
 205 
 
 341 
 
 75 
 
 716 
 
 215 
 
 310 
 
 236 
 
 184 
 
 348 
 
 64 
 
 79 
 
 110 
 
 769 
 
 519 
 
 371 
 
 118 
 
 196 
 
 815 
 
 96 
 
 296 
 
 6. 123 
 
 7. 205 
 
 8. 310 
 
 9. 79 
 
 10. 118 
 
 516 
 
 341 
 
 236 
 
 110 
 
 196 
 
 321 
 
 75 
 
 184 
 
 769 
 
 815 
 
 225 
 
 716 
 
 348 
 
 519 
 
 96 
 
 871 
 
 215 
 
 64 
 
 371 
 
 296 
 
 11. 213 
 
 12. 421 
 
 13. 85 
 
 14. 231 
 
 15. 526 
 
 327 
 
 87 
 
 222 
 
 624 
 
 448 
 
 98 
 
 116 
 
 376 
 
 785 
 
 379 
 
 716 
 
 615 
 
 584 
 
 923 
 
 87 
 
 825 
 
 399 
 
 972 
 
 84 
 
 999 
 
 ia 213 
 
 17. 327 
 
 18. 98 
 
 19. 716 
 
 20. 825 
 
 421 
 
 87 
 
 116 
 
 615 
 
 379 
 
 85 
 
 222 
 
 376 
 
 584 
 
 972 
 
 231 
 
 624 
 
 785 
 
 923 
 
 84 
 
 526 
 
 448 
 
 379 
 
 87 
 
 999 
 
ADDITION. 
 
 23 
 
 
 Ex. 
 
 17. 
 
 
 Add: 
 
 
 
 
 1. 1234 
 
 2. 4321 
 
 3. 2345 
 
 4. 345 
 
 368 
 
 6450 
 
 3456 
 
 2783 
 
 5721 
 
 378 
 
 4567 
 
 1497 
 
 1050 
 
 4291 
 
 5678 
 
 5840 
 
 4862 
 
 5782 
 
 689 
 
 9010 
 
 9215 
 
 6431 
 
 7890 
 
 2709 
 
 5. 5207 
 
 6. 3426 
 
 7. 2358 
 
 8. 9210 
 
 3584 
 
 783 
 
 7291 
 
 1029 
 
 2671 
 
 5279 
 
 5946 
 
 291 
 
 987 
 
 1085 
 
 7368 
 
 3587 
 
 3512 
 
 9270 
 
 5492 
 
 2785 
 
 6705 
 
 876 
 
 876 
 
 8899" 
 
 
 Ex. 
 
 18. 
 
 
 Add: 
 
 
 
 
 1. 12345 
 
 2. 23456 
 
 3. 5 
 
 4. 92583 
 
 3275 
 
 72564 
 
 23 
 
 4620 
 
 4721 
 
 3785 
 
 936 
 
 973 
 
 371 
 
 23584 
 
 6543 
 
 25 
 
 51028 
 
 987 
 
 92840 
 
 9 
 
 61234 
 
 96 
 
 72104 
 
 17 
 
 5. 23504 
 
 6. 358 
 
 7. 56789 
 
 8. 123456 
 
 4368 
 
 9246 
 
 3587 
 
 258071 
 
 25 
 
 14376 
 
 296 
 
 589347 
 
 9 
 
 845 
 
 89 
 
 258923 
 
 36 
 
 29 
 
 7 
 
 720145 
 
 378 
 
 7 
 
 12345 
 
 396012 
 
24 
 
 ADDITION. 
 
 9. 580921 
 
 10. 654321 
 
 11. ' 5 
 
 12. 345 
 
 13. 584321 
 
 42364 
 
 41058 
 
 24 
 
 6197 
 
 92047 
 
 527913 
 
 3792 
 
 358 
 
 52718 
 
 3681 
 
 80235 
 
 589 
 
 1497 
 
 6904 
 
 927 
 
 726048 
 
 75 
 
 36725 
 
 871 
 
 1078 
 
 4386 
 
 9 
 
 187348 
 
 89 
 
 92569 
 
 Ex. 19. 
 
 Add: 
 
 
 
 1. 5203461 
 
 2. 2587609 
 
 3. 1357924 
 
 9350472 
 
 3582764 
 
 6804281 
 
 1456849 
 
 1357908 
 
 5975325 
 
 2604030 
 
 4670253 
 
 7101584 
 
 5876543 
 
 8492056 
 
 9276432 
 
 1234567 
 
 4759841 
 
 6789009 
 
 4. 8274108 
 3509270 
 4680259 
 3584672 
 9876543 
 5279614 
 
 5. 5791350 
 246801 
 1384650 
 2794589 
 6532108 
 7999888 
 
 38, It is obvious that numbers can be added only when 
 they refer to the same things. Five oranges and three 
 books when "put together" are still 5 oranges and 3 books, 
 and not 8 oranges or 8 books. 
 
 It is also obvious that digits can be added only when 
 they refer to the same order of units. Nine hundreds and 
 eight tens when put together are still 9 hundreds and 8 
 tens, and not 17 hundreds or 17 tens. 
 
ADDITION. 
 
 25 
 
 Care must be taken, therefore, in writing numbers to be 
 added, that all the units digits shall fall in one column, all 
 the tens' digits in the next column (to the left), and all the 
 hundreds digits in the next column, and so on. 
 
 39. To add columns of digits with absolute accuracy and 
 great rapidity is a real accomplishment, and the operation 
 of addition should be continued until both these results are 
 secured. The beginner, however, will need some test of the 
 accuracy of his work. One test is to begin at the bottom 
 of the right-hand column in adding, and write on a piece of 
 waste-paper the entire sum of each column ; then to begin 
 at the top of the left-hand column and write also the entire 
 sum of each column ; finally, to add the sums obtained in 
 the first addition, and the sums obtained in the second 
 addition, and compare the results. 
 
 The study of an example will make the process under- 
 stood. 
 
 Beginning at the top 
 
 
 Beginning at the bot- 
 
 of the left-hand column 
 
 
 tom of the right-hand col- 
 
 in adding, and writing 
 
 871254 
 
 umn in adding, and writ- 
 
 the entire sum of each 
 
 123456 
 
 ing the entire sum of each 
 
 column, we have : 
 
 789098 
 
 column, we have : 
 
 28 
 
 357912 
 
 26 
 
 31 
 
 993286 
 
 28 
 
 23 
 
 
 17 
 
 
 17 
 
 3135006 
 
 23 
 
 28 
 
 
 31 
 
 26 
 
 
 28 
 
 3135006 
 
 3135006 
 
 By comparing the results we find each sum to t>e 
 3,135,006, and so infer that the operation is correct. 
 
26 ADDITION. 
 
 Find the sums of: 
 
 1. 427, 342, 856, 728. 
 
 2. 483, 1000, 8000, 648, 3750, 9840. 
 
 3. 15, 603, 1145, 6342. 
 
 4 41, 725, 60, 425, 7000, 4900, 398. 
 
 5. 39, 876, 5742, 3000, 478, 9873. 
 
 6. 327, 4960, 5000, 749, 3000, 7849. 
 
 7. 4284, 32, 679, 43, 5006, 7897. 
 
 8. 325, 6007, 983, 4050, 678, 9874. 
 
 9. 856, 9193, 8765, 4287, 6696, 9185, 979. 
 
 10. 7964, 5000, 303, 9784, 5673, 9004. 
 
 11. 9007, 34, 6876, 400, 9344, 7879. 
 
 12. 45,678, 96, 375, 4784, 9673, 11,980. 
 
 13. 7865, 3586, 4321, 8576. 
 
 14. 900,542 + 308,970 + 555,674 + 498,785. 
 
 15. 456,789 + 304,590 + 600,792 + 480,893 + 514,763. 
 
 16. 357,963 + 478,497 + 323,484 + 596,372 + 300,409. 
 
 17. 706,963 + 78,405 + 907,342 + 503,476. 
 
 18. A man bought a sleigh for $142, a carriage for $325, 
 
 and a pair of horses for $476. What was the cost of 
 all? 
 
 19. A man collected on Monday, $ 1290 ; on Tuesday, $ 340 ; 
 
 on Wednesday, $1008. How much was collected in 
 all? 
 
 20. A lady paid $912 for a piano, $342 for furniture, $ 187 
 
 for linen, $46 for silver. What did she pay for all? 
 
 21. A farmer had in one flock of sheep, 407 ; in another, 
 
 96 ; and in a third, 2584. How many had he in all? 
 
ADDITION. 27 
 
 22. A man owns four houses; the first is worth $47,050; 
 
 the second, $9106; the third, $1492; the fourth, 
 $ 512. What is the value of them all ? 
 
 23. Five loads of flour weighed as follows : 3500 pounds, 
 
 4967 pounds, 3974 pounds, 7982 pounds, 7963 
 pounds. What was the weight of the whole? 
 
 24. A house was bought for $ 7895 ; repairs amounted to 
 
 $1500; new fences, $97; repairs on stable, $463 ; 
 furniture, $1285. What was the cost of the whole ? 
 
 25. The population of six towns is : 1674, 9008, 3769, 
 
 4000, 7096, 3784. Find the whole population. 
 
 26. A house-lot cost $675 ; for building the house and fur- 
 
 nishing materials the carpenters were paid $2245, 
 the masons $540, the painters $320. What was ex- 
 pended on house and lot? 
 
 27. A merchant bought carpets to the amount of $4670 ; 
 
 c.urtains, $300; paper-hangings, $1275; matting, 
 $9765. What was the cost of the whole ? 
 
 28. Find the sum of three hundred thousand six hundred 
 
 fifty, seven thousand eight hundred thirty-two, eleven 
 thousand five hundred sixty-seven, ten thousand fifty- 
 six, four hundred seventy-two. 
 
 29. Find the sum of one hundred sixty-seven thousand, 
 
 three hundred sixty-seven thousand, nine hundred 
 six thousand, two hundred forty-seven thousand, ten 
 thousand, seven hundred thousand, nine hundred 
 seventy-six thousand, one hundred ninety-five thou- 
 sand, ninety-seven thousand. 
 
 30. Find the sum of two hundred seven, three hundred 
 
 sixty-two, nine hundred forty-five, two thousand 
 three hundred forty-three, fifteen thousand six hun- 
 dred twenty- two, forty-five thousand eight. 
 
28 ADDITION. 
 
 31. Add 3 thousand 4 hundred 92, one thousand four, 6 
 
 thousand 5 hundred seventy, 42 hundred eleven. 
 
 32. Add 386 million 591, 546 million 311 thousand 122, 
 
 796 thousand 351, 84 hundred 1, 9 thousand, 86 
 thousand 521, 3 hundred fifty-eight thousand 6 hun- 
 dred, 8 million 888 thousand eight hundred eighty- 
 eight, 1 hundred million. 
 
 33. Find the sum of six million sixty thousand six, seven 
 
 million nine hundred fifty thousand ninety-nine, ten 
 million nine thousand eight hundred seven, three 
 hundred sixty-seven thousand forty-five. 
 
 34. Find the sum of 200 million 302 thousand, 200 thou- 
 
 sand two hundred, 50 million 50 thousand 50, 25 
 million 860 thousand, 47 million 467 thousand, 202 
 million 6367. 
 
 35. What is the sum of eighteen thousand three hundred 
 
 twenty, seventy-four thousand five hundred six, ten 
 hundred seventeen thousand nine hundred twenty- 
 one, fifty-three thousand seven hundred eleven, five 
 hundred seventy-six thousand three hundred four, 
 six hundred fifty thousand forty-four ? 
 
 36. A man drew five loads of bricks ; in the first load there 
 
 were 4068 ; in the second, 1342 ; in the third, 3927 ; 
 in the fourth, 1694 ; in the fifth, 2009. How many 
 in all the loads? 
 
 37. What is the united population of the following cities : 
 
 Utica, 28,804 ; Lowell, 40,928 ; Lynn, 28,236 ; Sa- 
 lem, 24,100; Erie, 19,500; Auburn, 17,225? 
 
 38. A fruit-grower sent to market the produce of six peach 
 
 orchards ; from the first, 7000 baskets ; from the 
 second, 6973 ; from the third, 1004 ; from the fourth, 
 3276; from the fifth, 1594; from the sixth, 3976. 
 How many baskets in all ? 
 
ADDITION. 29 
 
 39. The distance from Boston to Springfield is 98 miles, 
 
 from Springfield to New Haven 62 miles, from New 
 Haven to New York 76 miles. How many miles is 
 it from Boston to New York ? 
 
 40. An army officer paid at one time $7038 for horses, at 
 
 another time $7776, at another time $9948. How 
 many dollars did he pay in all ? 
 
 41. A farmer sold his wheat for $8742, his corn for 
 
 $13,569, and his oats for $9528. How much did 
 he receive for the whole ? 
 
 42. A bank has $40,317 in specie, $91,256 in bills, $18,317 
 
 in cash items. Find the whole amount. 
 
 43. The army of Napoleon at Waterloo consisted of 48,950 
 
 infantry, 15,765 cavalry, 7732 artillery. What was 
 the whole number? 
 
 44. The Duke of Wellington's army at Waterloo consisted 
 
 of 20,661 infantry, 8735 cavalry, 6877 artillery. 
 There were also 33,413 allies. What was the whole 
 number of his army ? 
 
 45. The area of England is 50,535 square miles, of Scot- 
 
 land 29,167 square miles, and of Wales 8125 square 
 miles. How many square miles in England, Scot- 
 land, and Wales together ? 
 
 46. New Hampshire furnished 12,497 soldiers for the 
 
 Revolution, Massachusetts 67,907, Rhode Island 
 5908, Connecticut 31,939. How many did these 
 four states furnish? 
 
 47. A country merchant has in his store flour worth $656, 
 
 sugar worth $480, molasses worth $325, cotton 
 cloth worth $125, tea worth $56, canned goods 
 worth $78. What is the whole value of his goods? 
 
30 ADDITION. 
 
 48. A farmer sold four loads of hay. The first weighed 
 
 2007 pounds, the second 1963 pounds, the third 
 2585 pounds, the fourth 2614 pounds. How many 
 pounds did the whole weigh ? 
 
 49. If Abraham was born at the beginning of the year 
 
 B.C. 1996, how many years from the date of his birth 
 to the end of the year 1889? 
 
 50. An orchard contains 112 apple trees, and an equal 
 
 number of pear trees ; 56 peach trees, and an equal' 
 number of plum trees ; and 19 cherry trees. How 
 many trees are there in the orchard? 
 
 51. How many times does a clock strike from half past 
 
 twelve o'clock at night to half past twelve o'clock at 
 noon? 
 
 52. The area of Maine in square miles is 29,895, of New 
 
 Hampshire 9005, of Vermont 9135, of Massachu- 
 setts 8040, of Rhode Island 1085, of Connecticut 
 4845. What is the area of New England in square 
 miles? 
 
 53. The area of New York in square miles is 47,620, of 
 
 Pennsylvania 44,985, of Virginia 40,125, of North 
 Carolina 48,580, of Ohio 40,760. What is the area 
 of these five states in square miles ? 
 
 54. The area of Illinois in square miles is 56,000, of Mich- 
 
 igan 57,430, of Wisconsin 54,450, of Iowa 55,475, 
 of Missouri 68,735. What is the area of these five 
 states in square miles? 
 
 55. The area of Texas in square miles is 262,290, of 
 
 California 155,980, of Dakota 147,700, of Montana 
 145,310, of New Mexico 122,460, of Arizona 112,920. 
 Find their total area in square miles. 
 
CHAPTEE III. 
 SUBTRACTION. 
 
 40. What number must be added to four to make seven? 
 What, then, will be left if 4 is taken from 7 ? 
 
 What number must be added to seven to make ten? 
 What, then, will be left if 7 is taken from 10 ? 
 
 Copy the following set of numbers, and find what num- 
 ber must be added to each one in the upper row to make 
 the number below the line. Write the required numbers 
 in the empty places above the lines : 
 
 76 12 429056 
 17 14 20 7 12 10 5 7 12 
 
 13 19 24 7 13 8 9 4 10 
 
 15 25 28 12 20 10 11 16 21 
 
 When you have done this, you will see that, since 7 and 
 10 make 17, 7 taken from 17 leaves 10; since 6 and 8 
 make 14, 6 taken from 14 leaves 8 ; so with each set of 
 numbers. 
 
32 SUBTRACTION. 
 
 41. In the following set, under each number in the lower 
 row, write the number that must be added to it to make 
 the upper number : 
 
 9 12 7 12 15 10 6 9 7 
 342865054 
 
 11 18 17 5 10 9 16 8 3 
 52 16 132520 
 
 To 3 we have to add 6 to make 9, so we write 6 under 
 the 3. To 4 we must add 8 to make 12, so we write 8 
 under the 4. 
 
 Now in finding what number must be added to 3 to 
 make 9, we have really found what number will be left if 
 3 is taken from 9. In finding what number must be added 
 to 4 to make 12, we have really found what number will 
 remain if 4 is taken from 12. 
 
 42. The operation of finding the number that remains, 
 when a smaller number is taken from a larger, is called 
 subtraction. The result is called the remainder or difference, 
 
 43. The number which is to be subtracted is called the 
 subtrahend; and the number which is to be diminished (that 
 is, the number from which the subtraction is made), is called 
 the minuend. 
 
 44. A dash is the sign of subtraction, and when 
 placed between two numbers means that the first number is 
 to be diminished by the second. It is called the minus sign. 
 
 The expression 4 1 = 3 is read four minus one equals 
 three. 
 
SUBTRACTION. 33 
 
 45, Three dots .*. are often used for the word therefore. 
 The expression 6 + 2 = 8, .'. 8 6 = 2, is read six plus 
 two equals eight, therefore eight minus six equals two. 
 
 Ex. 21. (Oral.) 
 
 1. What number with 5 makes 10 ? 
 What number with 3 makes 10? 
 What number with 2 makes 10 ? 
 What number with 4 makes 10? 
 
 2. What number taken from 10 leaves 2? 
 What number taken from 10 leaves 4 ? 
 What number taken from 10 leaves 3 ? 
 What number taken from 10 leaves 5 ? 
 
 3. 5 is one part of 12, what is the other? 
 
 8 is one part of 12, what is the other? 
 3 is one part of 12, what is the other? 
 
 7 is one part of 12, what is the other ? 
 
 9 is one part of 12, what is the other? 
 6 is one part of 12, what is the other? 
 
 10 is one part of 12, what is the other? 
 
 4. What number taken from 12 leaves 11 ? 
 What number taken from 12 leaves 9 ? 
 What number taken from 12 leaves 5 ? 
 What number taken from 12 leaves 8 ? 
 What number taken from 12 leaves 2 ? 
 What number taken from 12 leaves 6 ? 
 What number taken from 12 leaves 7 ? 
 What number taken from 12 leaves 1 ? 
 
 5. 9 + 2= .'.11 2= and 11 9 = 
 
 8 + 3= .'.11-3= and 11- 8 = 
 6 + 5= .-.11 5= and 11- 6 = 
 
 10 + 1 = /. 11 - 1 = and 11 - 10 = 
 
34 SUBTRACTION 
 
 6. 
 
 8 + 5 = 
 
 .-.13 
 
 -5 = 
 
 and 
 
 13- 
 
 8 
 
 a 
 
 
 6 + 7 = 
 
 .\13 
 
 -7 = 
 
 and 
 
 13- 
 
 6 
 
 = 
 
 
 9 + 4 = 
 
 .-.13 
 
 -4 = 
 
 and 
 
 13- 
 
 9 
 
 = 
 
 7. 
 
 6 + 8^ 
 
 .-.14 
 
 -6 = 
 
 and 
 
 14- 
 
 8 
 
 
 
 
 5 + 9 = 
 
 .\14 
 
 -9 = 
 
 and 
 
 14- 
 
 5 
 
 = 
 
 
 7 + 7 = 
 
 .-.14 
 
 f-T 
 
 
 
 
 
 8. 
 
 7 + 8 = 
 
 .-.15 
 
 7 = 
 
 and 
 
 15- 
 
 8 
 
 =r 
 
 
 9 + 6 = 
 
 .-.15 
 
 -6 = 
 
 and 
 
 15- 
 
 9 
 
 tss 
 
 
 9 + 3 = 
 
 .-.12 
 
 9 = 
 
 and 
 
 12- 
 
 3 
 
 sat 
 
 9. 
 
 8 + 8 = 
 
 .-.16 
 
 -8 = 
 
 
 
 
 
 
 7 + 9 = 
 
 .-.16 
 
 Y 
 
 and 
 
 16- 
 
 9 
 
 = 
 
 
 9 + 8 = 
 
 .-.17 
 
 -9 = 
 
 and 
 
 17- 
 
 8 
 
 = 
 
 10. Subtract by threes, from 100 to 1 ; from 102 to 0; by 
 
 fours, from 101 to 1 ; from 102 to 2 ; from 103 to 3. 
 
 11. Subtract by fives, from 102 to 2 ; from 103 to 3 ; from 
 
 104 to 4 ; from 100 to 5. 
 
 12. Subtract by sixes, from 103 to 1 ; from 104 to 2 ; from 
 
 105 to 3 ; from 100 to 4 ; from 102 to 6. 
 
 13. Subtract by sevens, from 106 to 1 ; from 100 to 2 ; 
 
 from 101 to 3 ; from 102 to 4 ; from 103 to 5 ; from 
 
 104 to 6 ; from 105 to 7. 
 
 14. Subtract by eights, from 105 to 1 ; from 106 to 2; from 
 
 107 to 3 ; from 100 to 4 ; from 101 to 5 ; from 102 
 to 6 ; from 103 to 7 ; from 104 to 8. 
 
 15. Subtract by nines, from 100 to 1 ; from 101 to 2 ; 
 
 from 102 to 3 ; from 103 to 4 ; from 104 to 5 ; from 
 
 105 to 6 ; from 106 to 7. 
 
SUBTRACTION. 35 
 
 Ex. 22. (Oral.) 
 
 5 + 4 = 
 
 .-. 9 
 
 -5 = 
 
 9- 
 
 4 = 
 
 9 + 3 = 
 
 .-.12 
 
 9 _ 
 
 12- 
 
 
 
 6 + 5 = 
 
 .-.11 
 
 -6 = 
 
 11- 
 
 5 = 
 
 7 + 6 = 
 
 .-.13 
 
 -7 = 
 
 13- 
 
 6 = 
 
 9 + 6 = 
 
 .-.15 
 
 -6 = 
 
 15- 
 
 9 = 
 
 7 + 9 = 
 
 .-.16 
 
 -9 = 
 
 16- 
 
 7 
 
 14-8= 16-9= 18-6= 17-8= 25-9 = 
 
 11-3= 33-8= 45-6= 76-8= 32-9 = 
 
 16-7= 24-9= 37-8= 48 6= 53-9 = 
 
 17-8= 35-8= 43-7= 50-4= 63-6 = 
 
 12-4= 44 7= 24-8= 31-3= 26-9 = 
 
 15-7= 68-9= 56-7= 43-5= 29-7 = 
 
 13_6= 27-8= 34-9= 40-9= 50-7 = 
 
 11-8= 13-8= 15-8- 13-9- 31 3 = 
 
 27-9= 86-8= 85-9= 87-6= 84-5 = 
 
 32-8= 73-5= 62-7= 26-9= 23-7 = 
 
 25-4= 75-9= 73-7= 72-6= 83-8 = 
 
 17_9=: 31 8= 42 9= 50 3- 39 8 = 
 
 42-3= 30-6= 38-9= 40-4- 93-7 = 
 
 37-9= 58-9= 52-6= 63-8= 41-3 = 
 
 24-7= 70-8= 21-9= 22-7= 38-9 = 
 
 45-8= 42-3= 54-7= 71-8= 65-7 = 
 
 19_8 = 60-3= 65-9= 64-6= 17-9 = 
 
 34-6= 95-6= 82-8= 79-9= 76-8 = 
 
 28-9= 72-7= 90-9= 65-6= 81-7 = 
 
 54-5= 77-8= 85-7= 69-9= 71-4- 
 
36 SUBTRACTION. 
 
 From 876 take 631. 
 
 Write units under units, tens under tens, and so on. Then 1 unit 
 
 from 6 units leaves 5 units, and we write 5 
 
 Operation. under the units' column ; 3 tens from 7 tens 
 
 Minuend 876 leave 4 tens, and we write 4 under the tens' 
 
 a U4. k A AQ1 column; 6 hundreds from 8 hundreds leave 
 bubtranend, ool 
 
 2. hundreds, and we write 2 under the nun- 
 Remainder, 245 dreds' column. The remainder, therefore, is 
 2 hundreds 4 tens 5 units ; that is, 245. 
 
 46, The minuend is the sum of the subtrahend and the 
 remainder. Hence, to test the accuracy of the work, add 
 the subtrahend and remainder together, and if the work is 
 correct, their sum will be equal to the minuend. 
 
 47, It is obvious that one number can be subtracted from 
 another only when both numbers refer to the same things. 
 Thus, we can subtract 3 oranges from 5 oranges, but we 
 cannot subtract 3 apples from 5 oranges. 
 
 Ex. 23. 
 Find the results of: 
 
 1. 
 
 59- 
 
 23. 
 
 13. 
 
 89- 
 
 41. 
 
 25. 
 
 786- 
 
 45. 
 
 2. 
 
 54- 
 
 23. 
 
 14. 
 
 67- 
 
 23. 
 
 26. 
 
 674- 
 
 52. 
 
 3. 
 
 67- 
 
 14. 
 
 15. 
 
 58- 
 
 17. 
 
 27. 
 
 569- 
 
 38. 
 
 4. 
 
 65- 
 
 32. 
 
 16. 
 
 75- 
 
 34. 
 
 28. 
 
 857- 
 
 43. 
 
 5. 
 
 78- 
 
 25. 
 
 17. 
 
 96- 
 
 53. 
 
 29. 
 
 294- 
 
 82. 
 
 6. 
 
 75- 
 
 41. 
 
 18. 
 
 87- 
 
 42. 
 
 30. 
 
 348- 
 
 37. 
 
 7. 
 
 85- 
 
 33. 
 
 19. 
 
 69- 
 
 37. 
 
 31. 
 
 489- 
 
 76. 
 
 8. 
 
 78- 
 
 25. 
 
 20. 
 
 78- 
 
 26. 
 
 32. 
 
 768- 
 
 47. 
 
 9. 
 
 96- 
 
 42. 
 
 21. 
 
 64- 
 
 43. 
 
 33. 
 
 976- 
 
 53. 
 
 10, 
 
 97- 
 
 54. 
 
 22. 
 
 98- 
 
 35. 
 
 34. 
 
 897- 
 
 75. 
 
 11. 
 
 87- 
 
 54. 
 
 23. 
 
 89- 
 
 53. 
 
 35. 
 
 588- 
 
 64. 
 
 12. 
 
 86 
 
 31. 
 
 24. 
 
 77- 
 
 46. 
 
 36. 
 
 467- 
 
 45. 
 
SUBTRACTION. 
 
 37 
 
 37. 874 - 632. 42. 6982 - 5431. 47. 725,419 - 613,208. 
 
 38. 792 - 261. 43. 7629 - 4518. 48. 965,420 - 342,100. 
 
 39. 798 - 627. 44. 7824 - 6821. 49. 854,267 - 723,150. 
 
 40. 764 - 532. 45. 8542 - 6131. 50. 549,830 - 438,820. 
 
 41. 862 - 741. 46. 8792 - 6281. 51. 628,300 - 517,200. 
 
 48, If the number of units of any order in the minuend 
 is less than the number of units of the corresponding order 
 in the subtrahend, one of the next higher order of units in 
 the minuend must be added to the units of the order we 
 are considering. The process will be understood by an 
 example. 
 
 From 783 take 469. 
 
 Since we cannot take 9 units from 3 units, we add 1 of the 8 tens 
 to the 3 units, making 13 units; then 9 units 
 from 13 units leave 4 units. Now as we have 
 added 1 of the 8 tens to the 3 units of the min- 
 uend, we have only 7 tens remaining, and 6 
 tens from 7 tens leave 1 ten ; 4 hundreds from 
 7 hundreds leave 3 hundreds. The remainder, 
 therefore, is 3 hundreds 1 ten 4 units ; that is, 314. 
 
 Operation. 
 Minuend, 783 
 Subtrahend, 469 
 Remainder, 314 
 
 From 359 take 186. 
 
 Here 6 units from 9 units leave 3 units. Since we cannot take 8 
 . tens from 5 tens we add 1 of the 3 hundreds to 
 
 on ' the 5 tens, making 15 tens; then 8 tens from 
 
 Minuend, 359 15 tens leave 7 tens> N ow as we have added 
 
 1 of the 3 hundreds to the 5 tens of the minu- 
 end, we have only 2 hundreds remaining ; and 
 1 hundred from 2 hundreds leaves 1 hundred. 
 
 Subtrahend, 186 
 Remainder, 173 
 
 The remainder, therefore, is 1 hundred 7 tens 3 units ; that is, 173. 
 
38 SUBTRACTION. 
 
 Ex. 24. 
 
 1. 
 
 867 
 
 -325. 
 
 13. 
 
 90- 
 
 35. 
 
 25. 
 
 70 - 28. 
 
 2. 
 
 985 
 
 -312. 
 
 14. 
 
 40- 
 
 13. 
 
 26. 
 
 50 - 13. 
 
 3. 
 
 746 
 
 -213. 
 
 15. 
 
 70- 
 
 26. 
 
 27. 
 
 80 - 37. 
 
 4. 
 
 384 
 
 - 132. 
 
 16. 
 
 50- 
 
 24. 
 
 28. 
 
 60 - 48. 
 
 5. 
 
 479 
 
 -235. 
 
 17. 
 
 80- 
 
 32. 
 
 29. 
 
 90- 
 
 -25. 
 
 6. 
 
 679 
 
 -215. 
 
 18. 
 
 60- 
 
 33. 
 
 30. 
 
 50 - 27. 
 
 7. 
 
 857 
 
 -324. 
 
 19. 
 
 60- 
 
 47. 
 
 31. 
 
 80 - 43. 
 
 8. 
 
 956 
 
 -532. 
 
 20. 
 
 70- 
 
 45. 
 
 32. 
 
 70-36. 
 
 9. 
 
 795 
 
 -362. 
 
 21. 
 
 70- 
 
 52. 
 
 33. 
 
 90 - 32. 
 
 10. 
 
 687 
 
 -321. 
 
 22. 
 
 80- 
 
 36. 
 
 34. 
 
 60 - 27. 
 
 11. 
 
 978 
 
 -333. 
 
 23. 
 
 90- 
 
 28. 
 
 35. 
 
 80-49. 
 
 12 
 
 835 
 
 -214. 
 
 24. 
 
 90- 
 
 27. 
 
 36. 
 
 90 - 36. 
 
 Ex. 25. 
 
 1. 
 
 5.2-26. 
 
 13. 
 
 63- 
 
 29. 
 
 25. 
 
 680 
 
 -247. 
 
 2. 
 
 73 - 38. 
 
 14. 
 
 74- 
 
 37. 
 
 26. 
 
 570 
 
 -236. 
 
 3. 
 
 81 - 49. 
 
 15. 
 
 92- 
 
 68. 
 
 27. 
 
 860 
 
 -218. 
 
 4. 
 
 94-57. 
 
 16. 
 
 81- 
 
 56. 
 
 28. 
 
 690 
 
 -254. 
 
 5. 
 
 72 - 48. 
 
 17. 
 
 75- 
 
 38. 
 
 29. 
 
 750 
 
 -419. 
 
 6. 
 
 91-64. 
 
 18. 
 
 96- 
 
 48. 
 
 30. 
 
 830 
 
 -214. 
 
 7. 
 
 75- 
 
 -48. 
 
 19. 
 
 85- 
 
 57. 
 
 31. 
 
 690 
 
 -275. 
 
 8. 
 
 92 - 48. 
 
 20. 
 
 93- 
 
 75. 
 
 32. 
 
 750 
 
 -326. 
 
 9. 
 
 83- 
 
 -26. 
 
 21. 
 
 ^4 
 j i 
 
 18. 
 
 33. 
 
 860 
 
 -247. 
 
 10. 
 
 95 - 47. 
 
 22. 
 
 81- 
 
 27. 
 
 34. 
 
 970 
 
 -358. 
 
 11. 
 
 86 - 57. 
 
 23. 
 
 75- 
 
 29. 
 
 35. 
 
 580 
 
 -149. 
 
 12. 
 
 95 66. 
 
 24. 
 
 94- 
 
 58. 
 
 36. 
 
 870 
 
 -146. 
 
 Ex. 26. 
 
 1. 
 
 407 
 
 -84. 
 
 7. 
 
 462-38. 
 
 13. 
 
 608 
 
 247. 
 
 2. 
 
 308 
 
 -75. 
 
 8. 
 
 374-57. 
 
 14. 
 
 706 
 
 -253. 
 
 3. 
 
 609 
 
 -58. 
 
 9. 
 
 281 65. 
 
 15. 
 
 805 
 
 -364. 
 
 4. 
 
 205 
 
 -81. 
 
 10. 
 
 592 - 83. 
 
 16. 
 
 904 
 
 -472. 
 
 5. 
 
 506 
 
 -63. 
 
 11. 
 
 476 - 68. 
 
 17. 
 
 809 
 
 -581. 
 
 6. 
 
 807 
 
 -42. 
 
 12. 
 
 852 - 39. 
 
 18. 
 
 705 
 
 -r694. 
 
SUBTRACTION. 39 
 
 19. 
 
 508 
 
 -294. 
 
 25. 
 
 781 
 
 - 246. 
 
 31. 
 
 461- 
 
 239. 
 
 20. 
 
 609 
 
 -385. 
 
 26. 
 
 892 
 
 -387. 
 
 32. 
 
 572 
 
 238. 
 
 21. 
 
 707 
 
 -246. 
 
 27. 
 
 643 
 
 -418. 
 
 83. 
 
 693- 
 
 447, 
 
 22. 
 
 806 
 
 -324. 
 
 28. 
 
 954 
 
 - 216. 
 
 34. 
 
 754- 
 
 536. 
 
 23. 
 
 405 
 
 -132. 
 
 29. 
 
 763 
 
 -419. 
 
 35. 
 
 835- 
 
 226. 
 
 24. 
 
 709 
 
 -328. 
 
 30. 
 
 655 
 
 -247. 
 
 36. 
 
 973- 
 
 237. 
 
 
 
 
 
 Ex. 
 
 27. 
 
 
 
 
 1. 
 
 612 
 
 -78. 
 
 13. 
 
 732 
 
 -458. 
 
 25. 
 
 531 
 
 352. 
 
 2. 
 
 523 
 
 -64. 
 
 14. 
 
 816 
 
 -237. 
 
 26. 
 
 642- 
 
 263. 
 
 3. 
 
 845 
 
 -87. 
 
 15. 
 
 624 
 
 -158. 
 
 27. 
 
 763- 
 
 174. 
 
 4. 
 
 417 
 
 -58. 
 
 16. 
 
 936 
 
 -489. 
 
 28. 
 
 824- 
 
 296. 
 
 5. 
 
 731 
 
 -94. 
 
 17. 
 
 567 
 
 -298. 
 
 29. 
 
 915- 
 
 468. 
 
 6. 
 
 324 
 
 -65. 
 
 18. 
 
 715 
 
 -348. 
 
 30. 
 
 812- 
 
 357. 
 
 7. 
 
 942 
 
 -74. 
 
 19. 
 
 623 
 
 -417. 
 
 31. 
 
 514- 
 
 136. 
 
 8. 
 
 635 
 
 -89. 
 
 20. 
 
 861 
 
 -375. 
 
 32. 
 
 972- 
 
 489. 
 
 9. 
 
 522 
 
 -56. 
 
 21. 
 
 453 
 
 - 286. 
 
 33. 
 
 624- 
 
 248. 
 
 10. 
 
 417 
 
 -68. 
 
 22. 
 
 817 
 
 -329. 
 
 34. 
 
 512- 
 
 136 
 
 11. 
 
 325 
 
 -86. 
 
 23. 
 
 643 
 
 -457. 
 
 35. 
 
 713- 
 
 364. 
 
 12. 
 
 712 
 
 -94. 
 
 24. 
 
 415 
 
 -186. 
 
 36. 
 
 817- 
 
 259 
 
 
 
 
 
 Ex. 
 
 28. 
 
 
 
 
 1. 
 
 500 
 
 -78. 
 
 13. 
 
 600 
 
 235. 
 
 25. 
 
 902 
 
 146. 
 
 2. 
 
 600 
 
 -83. 
 
 14. 
 
 800 
 
 -217. 
 
 26. 
 
 805- 
 
 347. 
 
 3. 
 
 700 
 
 -92. 
 
 15. 
 
 900 
 
 -386. 
 
 27. 
 
 704- 
 
 215. 
 
 4. 
 
 800 
 
 fi4 
 
 \J Jt. 
 
 16. 
 
 700 
 
 - 427. 
 
 28. 
 
 607- 
 
 238. 
 
 5. 
 
 600 
 
 -57. 
 
 17. 
 
 400 
 
 -128. 
 
 29. 
 
 503 
 
 267. 
 
 6. 
 
 400 
 
 -76. 
 
 18. 
 
 800 
 
 -372. 
 
 30. 
 
 906- 
 
 387. 
 
 7. 
 
 802 
 
 -68. 
 
 19. 
 
 600 
 
 -345. 
 
 31. 
 
 904- 
 
 328. 
 
 8. 
 
 304 
 
 -95. 
 
 20. 
 
 700 
 
 -562. 
 
 32. 
 
 802- 
 
 467. 
 
 9. 
 
 506 
 
 -87. 
 
 21. 
 
 800 
 
 -427. 
 
 33. 
 
 705- 
 
 258. 
 
 10. 
 
 403 
 
 -75. 
 
 22. 
 
 900 
 
 -368. 
 
 34. 
 
 603- 
 
 318. 
 
 11. 
 
 902-94. 
 
 23. 
 
 500 
 
 -321. 
 
 35. 
 
 701- 
 
 427. 
 
 12. 
 
 504 
 
 -69. 
 
 24. 
 
 600 
 
 -487. 
 
 36. 
 
 705- 
 
 348. 
 
40 SUBTRACTION. 
 
 Ex. 29. 
 
 1. 7689-2345. 9. 9580 5136. 17. 8300-2746. 
 
 2. 6837-4216. 10. 7480-2367. 18. 7400-2843. 
 
 3. 9876 - 1234. 11. 9560 - 1423. 19. 8020 - 3647. 
 
 4. 8697-3274. 12. 8670-4324. 20. 7050-6873. 
 
 5. 7586-2145. 13. 8700-3218. 21. 6040-2895. 
 
 6. 6789-4321. 14. 9600-2745. 22. 8030-2746. 
 
 7. 8470-2138. 15. 9600-4347. 23. 7050-4873. 
 
 8. 6790-3245. 16. 7200-3647. 24. 6020-2748. 
 
 Ex. 30. 
 
 1. 6005-2347. 9. 8021-3472. 17. 9000-3725. 
 
 2. 8002-2636. 10. 8064-2397. 18. 9000-2745. 
 
 3. 8003-2746. 11. 9012-3684. 19. 6324-2538. 
 
 4. 6005 2748. 12. 7054-2768. 20. 6245-3789. 
 
 5. 9004-2615. 13. 7000-2546. 21. 4517-1638. 
 
 6. 6003-2846. 14. 7000-3748. 22. 7253-4867. 
 7.7035-2648. 15. 8000-5318. 23. 9215-4757. 
 8. 7023-2896. 16. 8000-3526. 24. 7214-4869. 
 
 Ex. 31. 
 
 1. 56,739-24,316. 13. 59,001-16,739. 
 
 2. 68,507-47,623. 14. 89,076- 569. 
 
 3. 47,865-12,341. 15. 60,020-24,156. 
 
 4. 72,006-48,315. 16. 57,490- 598. 
 
 5. 65,043-17,872. 17. 70,000-25,487. 
 
 6. 81,000-25,143. 18. 70,000- 4,139. 
 
 7. 90,000-30,906. 19. 60,300-36,428. 
 
 8. 90,503-47,628. 20. 70,302- 5,648. 
 
 9. 41,009-31,214. 21. 80,040-23,619. 
 
 10. 43,020-36,748. 22. 63,008-47,236. 
 
 11. 26,735- 9,856. 23. 50,004-47,825. 
 
 12. 75,986-43,264. 24. 80,047-26,578. 
 
SUBTRACTION. 41 
 
 Ex. 32. 
 
 1. 431,250 153,697. 8. 842,003-459,687. 
 
 2. 920,503-476,829. 9. 715,324-369,857. 
 
 3. 523,146-286,759. 10. 900,500-465,783. 
 
 4. 647,352-268,574. 11. 512,435-126,867. 
 
 5. 502,304186,475. 12. 600,000-285,436. 
 
 6. 625,030-274,384. 13. 723,514-536,945. 
 
 7. 720,301-368,596. 14. 801,050-469,872. 
 
 Ex. 33. 
 
 1. What number must be added to 7428 to make 8047? 
 
 2. What number must be taken from 3015 to leave 2405? 
 
 3. If the minuend is 78,206, and the subtrahend 35,264, 
 
 what is the remainder ? 
 
 4. A man owed $4689. He paid at one time $3894. 
 
 How much did he still owe ? 
 
 5. A flour merchant had on hand 2038 barrels of flour, 
 
 He sold 1299 barrels. How many barrels had 
 he left? 
 
 6. Mr. Brown's yearly income is $5067. His expenses 
 
 are $4093. How much does he save? 
 
 7. The population of New England in 1870 was 3,487,924, 
 
 in 1880, 4,010,529. Find the increase. 
 
 8. A house cost $9468. If payments to the amount of 
 
 $5889 have been made to the builder, how much 
 still remains due? 
 
 9. The sum of two numbers is 890,375, and one of them is 
 
 309,007. What is the other ? 
 
42 SUBTRACTION. 
 
 10. A is worth $98,760 ; B is worth $4586 less than A. 
 
 How much is B worth? 
 
 11. In 1880 the population of Boston was 369,832, and 
 
 the population of Baltimore was 332,313. How 
 much greater was the population of Boston than 
 that of Baltimore ? 
 
 12. A tank holding 370 gallons of water was filled by 
 
 pouring 77 gallons into it. How many gallons 
 were there already in the tank ? 
 
 13. What number increased by 15,639 will be 28,984 ? 
 
 14. What number subtracted from nine hundred eighty- 
 
 seven thousand three hundred fifty-nine will leave 
 three hundred thousand two hundred eight? 
 
 15. A cotton planter raised 9675 pounds of cotton. He 
 
 sold 7876 pounds. How many pounds had he left ? 
 
 16. There were 322 apples on a tree, of which 198 were 
 
 gathered, and 87 were blown off by the wind. How 
 many were left on the tree ? 
 
 17. There are 60 minutes in an hour; how many minutes 
 
 between 4 minutes after 10 o'clock and 3 minutes 
 before 11 o'clock ? Between 9 minutes after 1 
 o'clock and 3 minutes before 2 o'clock? 
 
 18. A man purchases a farm for $24,669, and pays down 
 
 $13,708. How much remains unpaid? 
 
 19. Eight hundred seventy-six thousand four hundred 
 
 twenty-five added to a certain number makes 
 eleven million seven hundred nine thousand three 
 hundred four. What is the number? 
 
SUBTRACTION. 43 
 
 20. Two men, A and B, start together from the same place 
 
 and travel in the same direction. A walks the 
 first day 29 miles, B rides the first day 67 miles. 
 How many miles apart are they at the end of the 
 first day ? How many miles would they have been 
 apart if they had travelled in opposite directions ? 
 
 21. A merchant deposited in a bank $10,040; and after- 
 
 wards drew a check for $3780. How much had he 
 in the bank after the check was paid ? 
 
 22. What is the difference between 106,074 and 28,999 ? 
 
 23. A man went to market with $10.25. He paid for 
 
 steak $2; for sugar, $1 ; for coffee, $1; for fruit, 
 $2 ; for flour, $2. How much money had he left? 
 
 24. A horse cost $397 and was sold for $563. How much 
 
 was gained? 
 
 25. A horse and carriage were bought for $458, and were 
 
 sold for $539. What was the gain ? 
 
 26. A cow was sold for $171.25. The cow cost $152. 
 
 What was the gain ? 
 
 27. A man bought a house lot for $1290 and sold it for 
 
 $ 1196. How much did he lose ? 
 
 28. A horse, harness, and saddle were bought for $378, 
 
 and were sold for $423.50. How much was gained? 
 
 29. A man owing $7862.50 has paid $5678. How much 
 
 is still due ? 
 
 30. From a $50 bank-note a bill of $38.50 was paid. 
 
 What change was given back ? 
 
 31. In the siege of Gibraltar (1779-1783) the English 
 
 fired 57,163 round shot, and the French, 175,741. 
 How many more did the French fire than the 
 English ? 
 
44 SUBTRACTION. 
 
 32. The length of the Missouri River from its source to the 
 
 Mississippi is three thoasand ninety-six miles, and 
 from its source to the Gulf of Mexico four thousand 
 five hundred six miles. How many miles is it from 
 the junction of the two rivers to the Gulf of Mexico ? 
 
 33. A lady bought articles in a store amounting to nine 
 
 dollars and seventy-five cents. She gave in pay- 
 ment a ten-dollar bill. How much change should 
 she receive? 
 
 34. A gentleman received from his father $65,784. He 
 
 paid for a house $28,598. How much had he left? 
 
 35. If the area of the Mississippi Valley is 1,237,111 
 
 square miles, and the area of the Atlantic slope is 
 967,576 square miles, find the excess of the Missis- 
 sippi Valley over the Atlantic slope in square miles. 
 
 36. Lake Erie covers 9600 square miles, and Massachusetts 
 
 contains 8040 square miles. How many more square 
 miles in Lake Erie than in Massachusetts? 
 
 37. The foreign immigration into the United States was, 
 
 in 1883, 603,322, and in 1885, 395,346. How 
 much greater was the number in 1883 than in 1885 ? 
 
 38. The consumption of imported sugar in the United 
 
 States was, in 1882, 866,517 tons, and in 1880, 
 730,519 tons. How many more tons were con- 
 sumed in 1882 than in 1880? 
 
 39. A lady bought goods amounting to two dollars and 
 
 thirty-four cents. She gave a five-dollar bill in pay- 
 ment, What change should she receive ? 
 
 40. The polar diameter of the earth is 41,707,620 feet, and 
 
 the equatorial diameter is 41,847,426 feet. Find 
 the difference in feet. 
 
SUBTRACTION. 45 
 
 41, The Secretary of the Treasury of the United States 
 estimated the revenue for 1885 to be $330,000,000. 
 The actual revenue was $323,690,706. How much 
 did the actual fall short of the estimated revenue ? 
 
 42 The population of Chicago in 1860 was 109,260, in 
 1880, 503,185. Find the increase. 
 
 43. The gross earnings of the Eastern Railroad for 1883 
 
 were $3,584,506, and the expenses were $2,310,830. 
 Find the net earnings for the year. 
 
 44. The population of New York City in 1880 was 
 
 1,206,299, in 1860 it was 805,651. Find the in- 
 crease for twenty years 
 
 45. In 1880 Kentucky raised 149,017,855 pounds of tobacco, 
 
 and Virginia raised 78,421,860 pounds. How many 
 more pounds did Kentucky raise than Virginia ? 
 
 46. The value of the tobacco raised in Kentucky in 1880 
 
 was $10,431,250, the value of that raised in Virginia 
 was $6,273,749. Find the difference. 
 
 47. The population of Massachusetts in 1880 was 1,783,085, 
 
 and of Virginia 1 .512 565. Find the difference. 
 
 48. The population of New York in 1880 was 5,082,871, 
 
 and of Ohio 3,198,062. Find the difference. 
 
 49. The population of the United States in 1880 was 
 
 50,155,783, in 1870, 38,558,371. Find the increase. 
 
 50. In 1885 the railroads of the United States earned 
 
 from freight $519,690,992, and from passengers 
 $200.883,911. How much more was earned from 
 freight than from passengers? 
 
 51. The imports of raw cotton into England in 1871 were 
 
 1,778,139,776 pounds, the exports were 362,075,616 
 pounds. How many more pounds were imported 
 than exported ? 
 
CHAPTER IV. 
 
 MULTIPLICATION. 
 
 49, If the cost of G tons of coal at $ 7 a ton is required, 
 the amount can be found by writing $7 six times in 
 
 a column convenient for adding, as in the margin, ^L 
 and finding the sum of the column. 7 
 
 7 
 
 50, If the cost of a whole cargo of coal was re- 7 
 
 quired, this operation would be long and tedious, 7 
 
 and therefore a shorter process has been devised, 
 called Multiplication, 
 
 By this process $7 is written only once, 6 is written be- 
 neath the $7 to show the number of times $7 must be 
 taken in order to obtain the required amount, and this 
 amount is found by saying 6 times $ 7 are $42. Thus : 
 
 o>7 
 $< 
 
 6 
 $42 
 
 51, In this operation $7 is called the multiplicand, 6 the 
 multiplier, and $42 the product, The multiplier 6 is the 
 sum of six 1's, and the product 42 is the sum of six 7's. 
 Hence, it will be seen that : 
 
 Multiplication is an operation by which, when two num- 
 bers are given, called multiplicand and multiplier, a third 
 number is found called product, which is formed from the 
 multiplicand as the multiplier is formed from unity. 
 
MULTIPLICATION. 47 
 
 52, The multiplicand is the number to be multiplied. 
 The multiplier is the number by which we multiply. The 
 product is the result obtained. The multiplicand and mul- 
 tiplier are called factors of the product. The 
 product of two or more factors is the same in 
 whatever order they are taken. Thus, 3x4 
 4 X 3. The dots in the margin, read hori- 
 zontally, make 3 fours ; read vertically, make 4 threes. 
 
 53, The sign of multiplication is X. When the multi- 
 plier precedes the multiplicand, the sign X is read times. 
 Thus, 6 X $7 $42 is read 6 times $7 equal $42. 
 
 54, When the multiplier follows the multiplicand the 
 sign X is read multiplied ly. Thus, $7 X 6 =$42 is read 
 $7 multiplied by 6 equal $42; and means $7 taken 6 
 times equal $42. In all cases the product refers to the 
 same kind of units as the multiplicand. 
 
 55, Products of two factors, which are each less than 
 ten, must be learned by heart. 
 
 They can all be readily found by addition. Thus, if the 
 product of 4 times 6 is required, we see thaj the multiplier 
 4 is the sum of four 1's, and the multiplicand is 6; hence, 
 the product is the sum of four 6's, and we write 
 
 6 
 6 
 
 6 Thus, 4 X 6 = 24. 
 J> 
 
 24 
 
 In the same way every product is found when each of 
 its two factors is less than ten; and the results are all 
 written in the following multiplication table : 
 
48 
 
 MULTIPLICATION. 
 
 MULTIPLICATION TABLE. 
 
 2 
 
 3 
 
 4 
 
 5 
 
 TIMES 
 
 TIMES 
 
 TIMES 
 
 TIMES 
 
 1 ARE 2 
 
 1 ARE 3 
 
 1 ARE 4 
 
 1 ARE 5 
 
 2 ARE 4 
 
 2 ARE 6 
 
 2 ARE 8 
 
 2 ARE 10 
 
 3 ARE 6 
 
 3 ARE 9 
 
 3 ARE 12 
 
 3 ARE 15 
 
 4 ARE 8 
 
 4 ARE 12 
 
 4 ARE 16 
 
 4 ARE 20 
 
 5 ARE 10 
 
 5 ARE 15 
 
 5 ARE 20 
 
 5 ARE 25 
 
 6 ARE 12 
 
 6 ARE 18 
 
 6 ARE 24 
 
 6 ARE 30 
 
 7 ARE 14 
 
 7 ARE 21 
 
 7 ARE 28 
 
 7 ARE 35 
 
 8 ARE 16 
 
 8 ARE 24 
 
 8 ARE 32 
 
 8 ARE 40 
 
 9 ARE 18 
 
 9 ARE 27 
 
 9 ARE 36 
 
 9 ARE 45 
 
 6 
 
 7 
 
 8 
 
 9 
 
 TIMES 
 
 TIMES 
 
 TIMES 
 
 TIMES 
 
 1 ARE 6 
 
 1 ARE 7 
 
 1 ARE 8 
 
 1 ARE 9 
 
 2 ARE 12 
 
 2 ARE 14 
 
 2 ARE 16 
 
 2 ARE 18 
 
 3 ARE 18 
 
 3 ARE 21 
 
 3 ARE 24 
 
 3 ARE 27 
 
 4 ARE 24 
 
 4 ARE 28 
 
 4 ARE 32 
 
 4 ARE 36 
 
 5 ARE 30 
 
 5 ARE 35 
 
 5 ARE 40 
 
 5 ARE 45 
 
 6 ARE 36 
 
 6 ARE 42 
 
 6 ARE 48 
 
 6 ARE 54 
 
 7 ARE 42 
 
 7 ARE 49 
 
 7 ARE 56 
 
 7 ARE 63 
 
 8 ARE 48 
 
 8 ARE 56 
 
 8 ARE 64 
 
 8 ARE 72 
 
 9 ARE 54 
 
 9 ARE 63 
 
 9 ARE 72 
 
 9 ARE 81 
 
MULTIPLICATION. 49 
 
 Ex. 34. (Oral.) 
 
 1. Of what number are 2 and 4 the factors? 3 and 3? 
 
 5 and 3? 2 and 5? 3 and 6? 
 
 2. What are the factors of 14? of 9? of 8? of 18? of 6? 
 
 of 21? of 10? 
 
 3. 4 is one factor of 8 ; what is the other ? 
 
 3 is one factor of 12 ; what is the other ? 
 9 is one factor of 18 ; what is the other ? 
 
 4. 3 times what number make 15? 
 
 6 times what number make 18? 
 
 5 times what number make 25 ? 
 
 4 times what number make 28 ? 
 3 times what number make 21 ? 
 
 7 times what number make 14 ? 
 
 5. 8 times what number make 24 ? 
 
 6 times what number make 24 ? 
 
 6 times what number make 12? 42? 30? 18? 
 
 6. 7 times what number make 21? 63? 35? 49? 56? 14? 
 
 7. 8 times what number make 32? 64? 16? 40? 24? 56? 
 
 8. 9 times what number make 27? 72? 45? 63? 36? 18? 
 
 9. 6x2 10. 7x3^- 11. 8x4= 12. X 5 = 
 
 8x2 = 
 
 9x3 = 
 
 1x4 = 
 
 3x5 = 
 
 3x2 = 
 
 0x3 = 
 
 0x4 = 
 
 2x5- 
 
 9x2 = 
 
 4x3 = 
 
 3x4 = 
 
 7x5- 
 
 7x2 = 
 
 3x8 = 
 
 9x4 = 
 
 1x5 = 
 
 1x2 = 
 
 3x3 = 
 
 7x4 = 
 
 8x5 = 
 
 0x2 = 
 
 3x0 = 
 
 4x4 = 
 
 6x5 = 
 
 5x2 = 
 
 3x5 = 
 
 6x4 = 
 
 4x5 = 
 
 4x2 = 
 
 3x1 = 
 
 4x4- 
 
 9x5 = 
 
50 MULTIPLICATION. 
 
 13. 1x6 = 
 
 14. 3x7 = 
 
 15. 4x8 = 
 
 16. 1x9 = 
 
 9x6 = 
 
 0x7 = 
 
 0x8 = 
 
 7x9 = 
 
 8x6 = 
 
 1x7 = 
 
 3x8 = 
 
 0x9 = 
 
 3x6 = 
 
 2x7 = 
 
 9x8 = 
 
 3x9 = 
 
 5x6 = 
 
 4x7 = 
 
 7x8 = 
 
 8x9 = 
 
 0x6 = 
 
 8x7 = 
 
 1x8 = 
 
 6x9 = 
 
 7x6 = 
 
 5x7 = 
 
 5x8 = 
 
 9X9 = 
 
 2x6 = 
 
 9x7 = 
 
 8x8 = 
 
 5x9 = 
 
 4x6 = 
 
 6x7 = 
 
 8x6 = 
 
 9x9 = 
 
 17. 4x6- 
 
 18. 3x2 = 
 
 19. 5x2 = 
 
 20. 7x4 = 
 
 7x3 = 
 
 7x9 = 
 
 8x2 = 
 
 8x8 = 
 
 9x2 = 
 
 8x3 = 
 
 6x4 = 
 
 0x2 = 
 
 5x3 = 
 
 4x5 = 
 
 7x3 = 
 
 1x9 = 
 
 7x4 = 
 
 9x6 = 
 
 0x9 = 
 
 6x5 = 
 
 8x2 = 
 
 7x4 = 
 
 7x6 = 
 
 7x7 = 
 
 5x7 = 
 
 8x9 = 
 
 8x5 = 
 
 9x9 = 
 
 9x3 = 
 
 6x5 = 
 
 9x5 = 
 
 4x8 = 
 
 4x9 = 
 
 7x8 = 
 
 3x6 = 
 
 7x2 = 
 
 56, When the multiplicand consists of two or more 
 digits, and the multiplier is a single digit, it is necessary to 
 multiply each digit of the multiplicand by the multiplier. 
 Thus, the product of 6 X 4587 is the sum of six numbers, 
 each the same as the multiplicand. 
 
 The sum of the six 7's is 6 times 7 = 42, and we write the 2 units 
 in the column of units, and reserve the 4 tens to 
 be added to the product of the tens ; then 6 times 4587 
 8 tens = 48 tens, which, with the 4 tens, make 4587 
 52 tens, or 5 hundreds and 2 tens, and we write 4587 . 
 
 the 2 tens in the column of tens ; then 6 times 5 4587 
 
 hundreds = 30 hundreds, which, with the 5 hun- 4587 27522 
 dreds, make 35 hundreds, or 3 thousands and 5 4587 
 hundreds, and we write the 5 hundreds in the 
 column of hundreds : then 6 times 4 thousands = 24 thousands, which, 
 with the 3 thousands, make 27 thousands, and we write 27 to the left 
 of the 5 hundreds. 
 
MULTIPLICATION. 51 
 
 57, When the multiplier is 10, 100, 1000, etc., the pro- 
 duct is obtained by simply annexing as many zeros to the 
 multiplicand as are found in the multiplier. Thus : 
 
 10 x 4587 = 45,870. 
 
 Likewise, when the multiplier is any one cf the nine 
 significant* digits followed by zeros, the product is obtained 
 by multiplying the multiplicand by the significant digit 
 and annexing to the result as many zeros as are found in 
 the multiplier. Thus, if the multiplicand is 4587, and the 
 multiplier is 600, we multiply by 6 and obtain 27,522, and 
 annex to this result 2 zeros, and have for the required pro- 
 duct 2,752,200 : 458? 
 
 600 
 2,752,200 
 
 Ex. 35. 
 Find the products of: 
 
 1. 
 
 4x 
 
 80. 
 
 13. 
 
 6x 
 
 32. 
 
 25. 
 
 3x 
 
 97. 
 
 37. 
 
 9x 
 
 96. 
 
 2. 
 
 8X 
 
 40. 
 
 14. 
 
 2x 
 
 62. 
 
 26. 
 
 8x 
 
 57. 
 
 38. 
 
 6x 
 
 59. 
 
 3. 
 
 9x 
 
 70. 
 
 15. 
 
 7 x 
 
 47. 
 
 27. 
 
 8x 
 
 75. 
 
 39. 
 
 4x 
 
 83. 
 
 4. 
 
 7x 
 
 60. 
 
 16. 
 
 3x 
 
 53. 
 
 28. 
 
 9x 
 
 74. 
 
 40. 
 
 7x 
 
 84. 
 
 5. 
 
 5x 
 
 60. 
 
 17. 
 
 8x 
 
 54. 
 
 29. 
 
 9x 
 
 28. 
 
 41. 
 
 5x 
 
 94. 
 
 6. 
 
 9x 
 
 80. 
 
 18. 
 
 4x 
 
 87. 
 
 30. 
 
 2x 
 
 86. 
 
 42. 
 
 8x 
 
 96. 
 
 7. 
 
 6x 
 
 90. 
 
 19. 
 
 9x 
 
 63. 
 
 31. 
 
 2x 
 
 67. 
 
 43. 
 
 8x 
 
 86. 
 
 8. 
 
 9x 
 
 40. 
 
 20. 
 
 5x 
 
 96. 
 
 32. 
 
 3x 
 
 95. 
 
 44. 
 
 9x 
 
 78. 
 
 9. 
 
 7 x 
 
 40. 
 
 21. 
 
 5x 
 
 78. 
 
 33. 
 
 7x 
 
 85. 
 
 45. 
 
 7x 
 
 53. 
 
 10. 
 
 5x 
 
 90. 
 
 22. 
 
 6x 
 
 58. 
 
 34. 
 
 4x 
 
 79. 
 
 46. 
 
 8x 
 
 83. 
 
 11. 
 
 8x 
 
 50. 
 
 23. 
 
 4x 
 
 86. 
 
 35. 
 
 Sx 
 
 74. 
 
 47. 
 
 9x 
 
 68. 
 
 12. 
 
 5x 
 
 70. 
 
 24. 
 
 7x 
 
 89. 
 
 36. 
 
 5x 
 
 68. 
 
 48. 
 
 7x 
 
 94. 
 
 The digits 1, 2, 3, 4, 5, 6, 7, 8, 9 are called significant digits. 
 
52 MULTIPLICATION. 
 
 1. 
 
 2. 
 
 Find the 
 7 X 800. 
 4 x 200. 
 
 Ex. 
 products off 
 13. 6 X 703. 
 14. 9 x 507. 
 
 36. 
 
 25. 5 x 974. 
 26. 4 x 789. 
 
 37 
 
 38 
 
 
 8 x 948. 
 9 x 827. 
 
 3. 
 
 9 
 
 X 
 
 700. 
 
 15. 5 
 
 X809. 
 
 
 27. 4 x 947. 
 
 39 
 
 
 7 x 825. 
 
 4. 
 
 5 
 
 X 
 
 300. 
 
 16. 7 
 
 X604. 
 
 
 28. 5x987. 
 
 40 
 
 . 
 
 8 x 493. 
 
 5. 
 
 8 
 
 X 
 
 600. 
 
 17. 4 
 
 X906. 
 
 
 29. 6 x 896. 
 
 41 
 
 
 9 x 672. 
 
 6. 
 
 7 
 
 X 
 
 400. 
 
 18. 6 
 
 X803. 
 
 
 30. 6x456. 
 
 42 
 
 . 
 
 7 x 756. 
 
 7. 
 
 6 
 
 X 
 
 750. 
 
 19. 2 
 
 X986. 
 
 
 31. 7x627. 
 
 43 
 
 . 
 
 8 x 359. 
 
 8. 
 
 4 
 
 X 
 
 340. 
 
 20. 2 
 
 X593. 
 
 
 32. 7 X 645. 
 
 44 
 
 
 6 x 387. 
 
 9. 
 
 M 
 
 I 
 
 X 
 
 960. 
 
 21. 3 
 
 X593. 
 
 
 33. 5x865. 
 
 45 
 
 
 9 X 865. 
 
 10. 
 
 6 
 
 X 
 
 580. 
 
 22. 3 
 
 X486. 
 
 
 34. 8x329. 
 
 46 
 
 
 5 x 739. 
 
 11. 
 
 8 
 
 X 
 
 680. 
 
 23. 4 
 
 X867. 
 
 
 35. 6x496. 
 
 47 
 
 
 9 x 648. 
 
 12. 
 
 8 
 
 X 
 
 630. 
 
 24. 3 
 
 X837. 
 
 
 36. 9 x 584. 
 
 48 
 
 
 
 4 X 867. 
 
 Ex. 
 
 37. 
 
 
 
 
 Find the 
 
 products of: 
 
 1. 
 
 9 
 
 X 
 
 6000. 
 
 
 13. 8 
 
 X 
 
 6070. 
 
 25. 
 
 2 X 6007. 
 
 2. 
 
 4 
 
 X 
 
 8000. 
 
 
 14. 4 
 
 x 
 
 9080. 
 
 26. 
 
 9 
 
 X 7008. 
 
 3. 
 
 7 
 
 X 
 
 8000. 
 
 
 15. 6 
 
 X 
 
 5080. 
 
 27. 
 
 3 
 
 X 8005. 
 
 4. 
 
 7 
 
 X 
 
 9000. 
 
 
 16. 7 
 
 X 
 
 4070. 
 
 28. 
 
 8 
 
 X 4007. 
 
 5. 
 
 8 
 
 X 
 
 6000. 
 
 
 17. 3 
 
 X 
 
 7040. 
 
 29. 
 
 4 
 
 X 6009. 
 
 6. 
 
 6 
 
 X 
 
 7000. 
 
 
 18. 9 
 
 X 
 
 3050. 
 
 30. 
 
 7 
 
 X 5006. 
 
 7. 
 
 6 
 
 X 
 
 7300. 
 
 
 19. 9 
 
 X 
 
 6320. 
 
 31. 
 
 7 
 
 X 8026. 
 
 8. 
 
 6 
 
 X 
 
 7400. 
 
 
 20. 7 
 
 X 
 
 3980. 
 
 32. 
 
 6 
 
 X 7054. 
 
 9. 
 
 7 
 
 X 
 
 8500. 
 
 
 21. 6 
 
 X 
 
 8570. 
 
 33. 
 
 5 
 
 X 9045. 
 
 10. 
 
 6 
 
 X 
 
 8600. 
 
 
 22. 5 
 
 X 
 
 7390. 
 
 34. 
 
 4 
 
 X 6072. 
 
 11. 
 
 5 
 
 X 
 
 3900. 
 
 
 23. 6 
 
 X 
 
 8570. 
 
 35. 
 
 
 
 X 6038. 
 
 12. 
 
 7 
 
 X 
 
 7500. 
 
 
 4. 8 
 
 X 
 
 6780. 
 
 36. 
 
 5 
 
 X 5076. 
 
MULTIPLICATION. 53 
 
 Ex. 38. 
 
 Find the products of: 
 
 1.7x7204. 13.2x4716. 25.6x3725. 
 
 2. 3x6305. 14. 3x3825. 26. 7x5273. 
 
 3. 8 X 9308. 15. 4 x 6918. 27. 8 x 6531. 
 
 4. 6 x 4706. 16. 5 x 5724. 28. 9 x 1365. 
 
 5. 4 x 6407. 17. 6 x 6375. 29. 2 x 8417. 
 
 6. 9 x 3809. 18. 7 x 8413. 30. 3 x 7148. 
 
 7. 7 X 3628. 19. 8 X 5823. 31. 4 x 6528. 
 
 8. 8 x 6984. 20. 9 X 3285. 32. 5 X 8256. 
 
 9. 8 X 5746. 21. 2 x 7619. 33. 6 x 3748. 
 
 10. 4 x 4968. 22. 3 x 9167. 34. 7 X 4873. 
 
 11. 9x9786. 23. 4x4682. 35. 8x5329. 
 
 12. 7x3715. 24. 5x2864. 36. 9x9235. 
 
 Ex. 39. 
 
 Multiply by 2 ; by 3 ; and so on to 9 : 
 
 1. 2739. 4. 7658. 7. 7463. 10. 6483. 
 
 2. 4519. 5. 5396. 8. 8367. 11. 3526. 
 
 3. 8526. 6. 5783. 9. 8562. 12. 5417. 
 
 Multiply by 20 ; by 30 ; and so on to 90 : 
 
 13. 5732. 14. 6749. 15. 8345. 16. 7952. 
 
 Multiply by 200 ; by 300 ; and so on to 900 : 
 
 17. 6738. 13. 3579. 19. 5742. 20. 5793. 
 
 Multiply by 2000; by 3000; and so on to 9000: 
 
 21. 4827. 22. 9357. 23. 6519. 24. 7953. 
 
54 MULTIPLICATION. 
 
 58, Suppose the product of 649 X 4587 is required. The 
 multiplier 649 is 600 + 40 + 9, and the product is found 
 by multiplying by 9, then by 40, and then by 600, and 
 adding the partial products. Thus, 
 
 4587 
 649 
 
 9 times the multiplicand = 41283 ) 
 40 times the multiplicand = 183480 I Fartl; 
 600 times the multiplicand = 2752200 j P roducts - 
 
 649 times the multiplicand = 2976963 
 
 59, The zeros at the right of the partial products do not 
 affect the result of the addition, and may be omitted if care 
 is taken to put the right-hand digit of each partial product 
 directly under the multiplier used. Thus, 
 
 4587 
 649 
 41283 
 18348 
 
 27522 
 2976963 
 
 60, If the multiplier contains zeros, the products that 
 correspond to them will be zero, and need not be written. 
 
 Find the product of 2007 X 4587. 
 
 4587 
 
 2007 
 32109 
 
 9174 Proof: 
 
 9206109 
 
 2007 
 4587 
 14049 
 16056 
 10035 
 8028 
 
 I 9206109 
 
 61. To test the accuracy of the work in multiplication, 
 interchange the multiplicand and the multiplier. If the 
 numerical result is the same in both cases, as in the last 
 example, the work may be assumed to be correct. 
 
MULTIPLICATION. 55 
 
 Ex. 40. 
 
 Find the products of: 
 
 1. 27x8436. 13. 83x8495. 
 
 2. 26 X 7358. 14. 86 x 5283. 
 
 3. 36x3579. 15. 91x5246. 
 
 4. 37 X 5684. 16. 93 x 6475. 
 
 5. 45x5823. 17. 26x8167. 
 
 6. 43 x 4263. 18. 29 x 7384. 
 
 7. 53x4271. 19. 38x7496. 
 
 8. 54 x 7538. 20. 34 x 4976. 
 
 9. 64x9057. 21. 47x4982. 
 
 10. 65x8154. 22. 46x8217. 
 
 11. 78x6381. 23. 56x6284. 
 
 12. 74x9472. 24. 57x9582. 
 
 Ex. 41. 
 
 Find the products of : 
 
 1. 364x6492. 13. 843x6527. 
 
 2. 327x4756. 14. 935x5729. 
 
 3. 283 x 5718. 15. 297 X 7186. 
 
 4. 465x3862. 16. 487x8526. 
 
 5. 592x4718. 17. 752x3849. 
 
 6. 583x5926. 18. 594x6392. 
 
 7. 647x8529. 19. 265x6973. 
 
 8. 637 x 6548. 20. 378 x 7495. 
 
 9. 741 x 9438. 21. 374 x 8247. 
 
 10. 758x4857. 22. 648x9238. 
 
 11. 824x3741. 23. 864x9753. 
 
 12. 826x3297. 24. 798x5937. 
 
56 MULTIPLICATION. 
 
 Ex. 42. 
 
 1. What will 29 acres of land cost at $475 an acre? 
 
 2. What will 89 passenger cars cost at $3785 a car? 
 
 3. A square mile contains 640 acres. How many acres 
 
 in a county containing 936 square miles? 
 
 4. If a cotton factory makes 9660 yards of cloth daily, 
 
 how many yards will the factory make in a year 
 (313 days)? 
 
 5. The cost of building a certain road was, on the aver- 
 
 age, $ 1789 a mile. What was the cost of 327 miles 
 of this road ? 
 
 6. If a field contains 2340 hills of potatoes, and the 
 
 average number of potatoes in a hill is 12, how 
 many potatoes are there in the field ? 
 
 7. If a saw mill turns out 5708 feet of boards in a day, 
 
 how many feet will it turn out in 294 days? 
 
 8. A pound of platinum is worth $85. If 4730 pounds 
 
 are obtained yearly from South America and the 
 Ural Mountains, what is the value 'of the whole 
 amount? 
 
 9. Two cities 294 miles apart are to be connected by a 
 
 railroad, at a cost of $24,645 per mile. What will 
 be the cost of the road ? 
 
 10. If 125 tons of steel rails are required for one mile of 
 
 railroad, how many tons will be necessary for 389 
 miles ? 
 
 11. A mile contains 5280 feet. How many feet in 542 
 
 miles ? 
 
MULTIPLICATION. 57 
 
 12. The garrison of a fort consumes 785 pounds of bread 
 
 a day. How many pounds will be consumed in 3 
 years of 365 days ? 
 
 13. If a railway train runs 38 miles in an hour, how many 
 
 miles will it run in 84 trips of 3 hours each ? 
 
 14. A square mile contains 640 acres. How many acres 
 
 are there in 3481 square miles ? 
 
 15. At the rate of 1275 words in an hour, how many 
 
 words can be sent over a telegraph line in 108 
 hours? 
 
 16. A clock strikes 156 times a day. How many times 
 
 does it strike in a leap year (366 days) ? 
 
 17. If a swallow destroys daily 500 insects, how many will 
 
 it destroy in 92 days ? 
 
 18. At 27 bushels an acre, how many bushels of wheat 
 
 will be harvested from 640 acres ? 
 
 19. A good cow yields 168 pounds of butter a year. If it 
 
 takes 215,000 cows to supply London with butter, 
 how many pounds of butter are consumed in that 
 city annually ? 
 
 20. If sound travels at the rate of 1120 feet in a second, 
 
 how many feet distant is a cloud where the thunder 
 clap follows the flash of lightning in 9 seconds ? 
 
 21. Find the weight in pounds of 5792 iron bars, each 
 
 weighing 24 pounds. 
 
 22. From what number can 847 be subtracted 307 times, 
 
 and leave a remainder of 49 ? 
 
 23. If 19 men can do a piece of work in 31 days, how 
 
 many days will it take one man to do it ? 
 
58 MULTIPLICATION. 
 
 24. If an army consists of 24 regiments averaging 913 men 
 
 each, how many men are there in the whole army ? 
 
 25. If one acre produces 211 pounds of cotton, how many 
 
 pounds will 933,000 acres produce ? 
 
 26. If one acre produces 154 pounds of tobacco, how 
 
 many pounds will 10,070 acres produce ? 
 
 27. If one acre produces 28 bushels of oats, how many 
 
 bushels will 911,200 acres produce? 
 
 28. If one acre produces 42 bushels of corn, how many 
 
 bushels will 201,106 acres produce? 
 
 29. If one acre produces 17 bushels of wheat, how many 
 
 bushels will 613,263 acres produce? 
 
 30. If one acre produces 227 bushels of potatoes, how 
 
 many bushels will 19,121 acres produce? 
 
 31. If one acre produces 23 bushels of barley, how many 
 
 bushels will 237,769 acres produce ? 
 
 32. If one acre produces 19 bushels of winter rye, how 
 
 many bushels will 27,119 acres produce? 
 
 33. British India has a population of 150 to the square 
 
 mile, and contains 1,004,616 square miles. Find its 
 population. 
 
CHAPTER V. 
 
 DIVISION. 
 
 62, To divide $42 by 6 is to find the number of dollars 
 that must be taken 6 times to make $42. Again, to divide 
 $42 by $6 is to find the number of times that it is neces- 
 sary to take $6 to make $42. In either case, the product 
 and one factor are given and the other factor is required. 
 Hence, 
 
 63, Division is an operation by which when the product 
 and one factor are given the other factor is found. 
 
 64, The number to be divided is called the dividend, the 
 number by which the dividend is to be divided is called 
 the divisor, arid the result is called the quotient, 
 
 65, Division is indicated by the sign of division -s-, or by 
 writing the dividend over the divisor with a line between 
 them. Thus, each of the expressions 42-*- 6 = 7, and ^=7, 
 means and is read " forty- two divided by six equals seven." 
 
 Ex. 43. (Oral) 
 
 2x8- .-.16-5-2 = 2x6= .-. 12-*- 6 = 
 
 16-*- 8= 12-*- 2 = 
 
 2x2- .-. 4-*- 2= 2x3- ..6*1-8 = 
 
 6-s-2 = 
 
 2x5- .-.10-5-2=:; 2x7= .-.14-5-7 = 
 
 10-5-5 = 14-i-2 = 
 
60 DIVISION. 
 
 2x9 = 
 
 .-.18-4-2 = 
 
 5x9 = 
 
 .-.45 + 5 = 
 
 
 18 + 9 = 
 
 
 45 + 9 = 
 
 3x4 = 
 
 .-.12 + 4 = 
 
 5x8 = 
 
 .-.40 + 8 = 
 
 
 12 -4- 3 = 
 
 
 40 + 5 = 
 
 3x3 = 
 
 .*. 9 + 3- 
 
 5X3 = 
 
 .-.15 + 3 = 
 
 
 
 
 15 + 5 = 
 
 3x6 = 
 
 ..18-4-6 = 
 
 5x6 = 
 
 .-.30 + 6 = 
 
 
 18 + 8 = 
 
 
 30 + 5 = 
 
 3x9 = 
 
 ..27-9 = 
 
 5x4 = 
 
 .-.20 + 4 = 
 
 
 27 + 3 = 
 
 
 20 + 5 = 
 
 3x7 = 
 
 /.21 + 3 = 
 
 5x7 = 
 
 .-.35+7 = 
 
 
 21 + 7 = 
 
 
 35 + 5 = 
 
 3x8 = 
 
 .-.24 + 8 = 
 
 6x9 = 
 
 .-.54 + 6 = 
 
 
 24 + 3 = 
 
 
 54 + 9 = 
 
 3x5 = 
 
 .-.15 + 5 = 
 
 6x3 = 
 
 .-.18 + 6 = 
 
 
 15 + 3 = 
 
 
 18 + 3 = 
 
 4x5==. 
 
 .-.20-4 = 
 
 6x6 = 
 
 .-.36 + 6 = 
 
 
 20 + 5 = 
 
 
 
 4x3 = 
 
 /.12 + 4 = 
 
 6x7 = 
 
 .-.42 + 6 = 
 
 
 12 + 3 = 
 
 
 42 + 7 = 
 
 4x6 = 
 
 .-.24 + 6 = 
 
 6x8 = 
 
 .-.48 + 8 = 
 
 
 24 + 4 = 
 
 
 * 48 + 6 = 
 
 4x9 = 
 
 .-.36 + 9 = 
 
 6x5 = 
 
 .-.30 + 6 = 
 
 
 36 + 4 = 
 
 
 30 + 5 = 
 
 4x7 = 
 
 .-.28+7 = 
 
 6x4 = 
 
 .-.24 + 6 = 
 
 
 28 + 4 = 
 
 
 24 + 4 = 
 
 4x8 = 
 
 .-.32 + 4 = 
 
 7x3 = 
 
 .-.21 + 3 = 
 
 
 32 + 8 = 
 
 
 21 + 7 = 
 
 4x4 = 
 
 .-.16 + 4 = 
 
 7x9 = 
 
 .-.63 + 9 = 
 
 
 
 
 63 + 7 = 
 
 5x5 = 
 
 .-.25-5 = 
 
 7x7 = 
 
 .-.49 + 7 = 
 
DIVISION. 
 
 7x4 = 
 
 .-.28-4-4 = 
 
 8x9 = 
 
 .-.724-9 = 
 
 
 28-4-7 = 
 
 
 72-4-8 = 
 
 7x8 = 
 
 .-.56-4-7 = 
 
 8x4 = 
 
 .-.32-4-4 = 
 
 
 56-4-8 = 
 
 
 32-4-8 = 
 
 7x5 = 
 
 .-.35-f-7 = 
 
 9x3 = 
 
 .-.274-9 = 
 
 
 35-4-5 = 
 
 
 27 -*- 3 = 
 
 7x6 = 
 
 .-.42-^-6 = 
 
 9x5 = 
 
 .-.45-4-9 = 
 
 
 42-4-7 = 
 
 
 45-4-5 = 
 
 8x8 = 
 
 .-.64-5- 8 = 
 
 9x9 = 
 
 .-.81 4- 9 = 
 
 8x3 = 
 
 . -.24-5-8 = 
 
 9x6 = 
 
 .-.54^-9 = 
 
 
 24 -v- 3 = 
 
 
 54 -4- 6 = 
 
 8x7 = 
 
 .-.56-4-7 = 
 
 9x8 = 
 
 .-.72-8 = 
 
 
 56-^-8 = 
 
 
 72-4-9 = 
 
 8x5 = 
 
 ..40-4-5 = 
 
 9x7 = 
 
 .-.63-4-9 = 
 
 
 40-^-8 = 
 
 
 83-4-7 = 
 
 66. In the following exercises, the divisor for each line of 
 dividends is written at the left. The quotients should be 
 named without a moment's hesitation. 
 
 Ex. 44. (Om/.) 
 
 1. 6)54184212_62436603048 
 
 2. 4)281620324036_42412_8 
 
 3. 8)32486416402456_872_0 
 
 4. 9)_0277281186354_94536 
 
 5. 7) 21 _0 14 _7 28 42 35 56 63 49 
 
 6. 5)304520^405025153510 
 
 7. 3) 9 12 3 30 27 18 15 21 _6 24 
 
62 DIVISION. 
 
 Ex. 45. (Oral.) 
 
 Give the quotients and remainders in the following 
 examples : 
 
 1.2)11 _I_91^i31418191617 
 2. 3)j7111013'l6141715^519 
 3.4)19 _71317251522313329 
 
 4. 9)71831525341719622644 
 
 5. 7)15 192738405448601739 
 
 6. 8)23143117_92537687128 
 
 7. 6)20151927321013455740 
 
 8. 5)14_91321431249322938 
 9.7)11 182637345347591633 
 
 10. 5)13112216423248314937 
 11.6)21 163119112639465641 
 12. 8)33343918274169707563 
 13.9)73 168429354351648070 
 14.5)37 413427362328334448 
 15.6)37 4417105158253459 50 
 16. 8)26393042532036435157 
 17.9)21 372341471155506065 
 18.7)13 236146556925586218 
 19. 8)55677344615074655277 
 
DIVISION. 63 
 
 SHORT DIVISION. 
 
 67. When the divisor is so small that the work can be 
 performed mentally, the process is called Short Division, and 
 will be understood from the following examples : 
 
 (1) Divide 697,425 by 3. 
 
 The divisor is written at the left of the dividend, as in the margin. 
 Wording. 3 in 6, 2 ; in 9, 3 ; in 7, 2 ; in 14, 4 ; in 
 3)697425 22, 7 ; in 15, 5. 
 
 232475 Here the divisor is contained in 6 twice, in 9 three 
 times, and in 7 twice with remainder 1 ; this 1 is equal 
 to 10 of the next lower order, and with the 4, the next order of the 
 dividend, makes 14. Then 14 is divided by 3 ; the quotient is 4 with 
 remainder 2; this 2 is equal to 20 of the next lower order, and with 
 the 2 makes 22. Then 22 is divided by 3 ; the quotient is 7 with 
 remainder 1. Then 15 is divided by 3, and the quotient is 5. 
 
 (2) Divide 4,236,158 by 7. 
 
 7)4236158 
 
 605165 with remainder 3. 
 
 In this example, 7 is not contained in 3, BO is the second figure 
 of the quotient: then the next figure 6 of the dividend is joined to 
 the 3, making 36, and the division is continued. When the division 
 is finished, there is a remainder 3. 
 
 (3) Divide 54,123 by 9. 
 
 9)54123 
 
 6013 with remainder 6. 
 
 Each quotient figure is of the same order of units as the right-hand 
 figure of that part of the dividend used in obtaining it. Thus, 54 in 
 this example are 54 thousands, and the first figure of the quotient ia 
 6 thousands. 
 
 (4) Divide $23,087 by 5. 
 
 5) $23087 
 
 $4617 with $2 remaining. 
 
64 DIVISION. 
 
 In this example, we are required to divide 23087 dollars into five 
 equal parts, and find the number of dollars in each part. The answer 
 is 4617 dollars, with 2 dollars over. The complete quotient may be 
 written $4617$. 
 
 (5) Divide $23,087 by $5. 
 
 $5) $23087 
 
 4617 with $ 2 remaining. 
 
 In this example, we are required to find the number of times we 
 can take away $5 from $23,087, and the answer is 4617 times, with 
 $2 left over. The complete quotient may be written 4617$; and 
 the meaning is, that we can take $5 away 4617 times from $23,087, 
 and the next time have $ 2 to take away. 
 
 68, The last two examples illustrate the different mean- 
 ings of division. When the divisor corresponds to the 
 multiplier in multiplication the quotient corresponds to the 
 multiplicand, and denotes the same kind of units as the divi- 
 dend; when the divisor corresponds to the multiplicand 
 the quotient corresponds to the multiplier, and denotes the 
 number of times the divisor must be taken to obtain a 
 quantity equal to the dividend. 
 
 69. A number, when divided by 10, will have a quotient 
 consisting of the same series of figures, the last one being 
 cut off for the remainder. Thus, 35764-^10 = 3576 with 
 remainder 4. In this case, the value of each figure in the 
 result is diminished ten-fold, the tens becoming units, the 
 hundreds becoming tens, and so on. A number, when 
 divided by 100, 1000, etc., will have the same series of 
 figures in the quotient, the last two, three, etc., figures being 
 cut off for the remainder. Hence, 
 
 When a divisor ends in one. or more zeros, cut off the 
 zeros and an equal number of figures from the right of 
 the dividend, perform the division with the numbers left, 
 
DIVISION. 65 
 
 and for the total remainder annex the figures cut off from 
 the dividend to the remainder from the division. 
 
 Divide 5,786,342 by 200. 
 
 200)57863^2 
 
 28931 with remainder 142. 
 
 In this example, we cut off the two zeros at the right of the divisor 
 and two figures at the right of the dividend ; then we divide, putting 
 the first figure of the quotient under the figure 8, which is the right- 
 hand figure of the first partial dividend when the entire divisor 200 
 is used. 
 
 70, The product of the divisor and quotient increased by 
 the remainder is equal to the dividend. Hence, 
 
 To test the accuracy of the work of division, find the 
 product of the divisor and quotient, and to this product 
 add the remainder ; this result will be equal to the divi- 
 dend if the work is correct. 
 
 Thus, in the last example, 
 
 200 X 28,931 - 5,786,200, 
 and 5,786,200 + 142 = 5,786,342 (the dividend). 
 
 Ex. 46. 
 
 Find the quotients of : 
 
 1. 48-*- 2. 10. 75 +-5. 19. 91-*- 8. 28. 815-*- 5. 
 
 2. 
 
 72 
 
 -*-8. 
 
 11. 
 
 98+- 
 
 7. 
 
 20. 
 
 94+- 
 
 9. 
 
 29. 
 
 714 
 
 +-6. 
 
 3. 
 
 56 
 
 +-4. 
 
 12. 
 
 92+- 
 
 4. 
 
 21. 
 
 94+- 
 
 5. 
 
 30. 
 
 826 
 
 +-7. 
 
 4. 
 
 85 
 
 *-5, 
 
 13. 
 
 57+- 
 
 2. 
 
 22. 
 
 87+- 
 
 4. 
 
 31. 
 
 952 
 
 +-8. 
 
 5. 
 
 96 
 
 -f-6. 
 
 14. 
 
 83+- 
 
 3. 
 
 23. 
 
 95+- 
 
 6. 
 
 32. 
 
 972 
 
 +-9. 
 
 6. 
 
 84 
 
 +-7. 
 
 15. 
 
 75+- 
 
 4. 
 
 24. 
 
 77+- 
 
 3. 
 
 33. 
 
 912 
 
 +-8. 
 
 7. 
 
 96 
 
 f-8. 
 
 16. 
 
 48+- 
 
 5. 
 
 25. 
 
 734 H 
 
 -2. 
 
 34. 
 
 492 
 
 +-4. 
 
 8. 
 
 99 
 
 +-9. 
 
 17. 
 
 77+- 
 
 6. 
 
 26. 
 
 768 H 
 
 h-a 
 
 35. 
 
 675 
 
 +-5. 
 
 9. 
 
 90 
 
 -1-8. 
 
 18. 
 
 82+- 
 
 7. 
 
 27. 
 
 956 H 
 
 n4. 
 
 36. 
 
 918 
 
 + 6. 
 
66 DIVISION. 
 
 37. 
 
 513-4- 
 
 2. 
 
 53. 
 
 9354-4- 
 
 6. 
 
 69. 
 
 4017 
 
 ._ 
 
 7. 
 
 38. 
 
 719-4- 
 
 3. 
 
 54. 
 
 8176 -4- 
 
 7. 
 
 70. 
 
 7139 
 
 -4- 
 
 8. 
 
 39. 
 
 623^- 
 
 4. 
 
 55. 
 
 9456-4- 
 
 8. 
 
 71. 
 
 9415 
 
 -4- 
 
 6. 
 
 40. 
 
 749^- 
 
 5. 
 
 56. 
 
 8568-4- 
 
 9. 
 
 72. 
 
 8793 
 
 -4- 
 
 5. 
 
 41. 
 
 875-r- 
 
 6. 
 
 57. 
 
 3712-4- 
 
 8. 
 
 73. 
 
 3794 
 
 4- 
 
 2. 
 
 42. 
 
 643-^ 
 
 7. 
 
 58. 
 
 2226-4- 
 
 7. 
 
 74. 
 
 7929 
 
 -4- 
 
 3. 
 
 43. 
 
 927-4- 
 
 8. 
 
 59. 
 
 2550-4- 
 
 6. 
 
 75. 
 
 6728 
 
 -4- 
 
 4. 
 
 44. 
 
 705-4- 
 
 9. 
 
 60. 
 
 2895-4- 
 
 5. 
 
 76. 
 
 6380 
 
 -4- 
 
 5. 
 
 45. 
 
 591-*- 
 
 8. 
 
 61. 
 
 5391^- 
 
 o. 
 
 77. 
 
 8322 
 
 -4- 
 
 6. 
 
 46. 
 
 853-4- 
 
 7. 
 
 62. 
 
 7418-4- 
 
 3. 
 
 78. 
 
 9219 
 
 ^ 
 
 7. 
 
 47. 
 
 735^- 
 
 6. 
 
 63. 
 
 5327-4- 
 
 4. 
 
 79. 
 
 7395 
 
 ^~ 
 
 2. 
 
 48. 
 
 923-4- 
 
 5. 
 
 64. 
 
 8236-4- 
 
 5. 
 
 80. 
 
 7684 
 
 -J- 
 
 3. 
 
 49. 
 
 7594-^-2. 
 
 65. 
 
 7129-4- 
 
 6. 
 
 81. 
 
 7315 
 
 -s- 
 
 4. 
 
 50. 
 
 7458 H 
 
 K& 
 
 66. 
 
 8513+- 
 
 7. 
 
 82. 
 
 8369 
 
 -4- 
 
 5. 
 
 51. 
 
 9656 -4- 4. 
 
 67. 
 
 9237^- 
 
 8. 
 
 83. 
 
 5869 -- 6. 
 
 52. 
 
 7985 H 
 
 h5. 
 
 68. 
 
 5682 -4- 
 
 9. 
 
 84. 
 
 4239 
 
 + 
 
 7. 
 
 Divide by 2 ; by 3 ; and so on to 9 : 
 85. 5794. 86. 4572. 87. 9785. 88. 7163. 
 
 Divide by 20 ; by 30 ; and so on to 90. 
 89. 8239. 90. 5197. 91. 3274. 92. 5834. 
 
 Divide by 200 ; by 300 ; and so on to 900 : 
 93. 4571. 94. 5768. 95. 9563. 96. 9876, 
 
 Ex. 47. 
 
 I. There were 72 children in a Sunday-school, and they 
 walked two and two to church. How many rows 
 would they make ? How many rows would there 
 have been if they had walked three and three ? 
 
DIVISION. 67 
 
 2. A boy had 97 filberts. He kept 34 for himself, and 
 
 divided the rest equally among his 9 class-mates. 
 How many did he give to each ? 
 
 3. How many times must we take the number 7 to make 
 
 819 ? How many times the number 9 ? 
 
 4. Divide a paper of 264 pins equally into 8 papers. 
 
 5. 2691 poles were used in a certain hop-yard, and 3 
 
 were required for each plant. How many plants 
 were there ? 
 
 6. A blacksmith uses 7 nails in putting on one shoe, and 
 
 in one day he used 336 nails. How many hoofs did 
 he shoe ? 
 
 7. A forest of 1995 trees is to be thinned by cutting down 
 
 1 tree in 7. How many will be taken out ? 
 
 8. A regiment consists of 1200 men and 60 officers. How 
 
 many men are there to each officer ? 
 
 9. When beef is $7 per hundred-weight, how many hun- 
 
 dred-weight can be bought for $9,700,327? 
 
 10. How many tons of coal, at $9, can be bought for 
 
 $3,596,301? 
 
 11. A wagon travels 58,068 feet. How many times will 
 
 a wheel 12 feet in circumference turn in going that 
 distance ? 
 
 12. A square yard contains 9 square feet. How many square 
 
 yards in 3,917,502 square feet? 
 
 13. Aaron Reed left $325,645 for his wife and four chil- 
 
 dren. How much had each, if the property was 
 divided equally among them ? 
 
68 DIVISION. 
 
 14. A grocer sells brown sugar at $9 per hundred- weight. 
 
 If he receives $976,482, how many hundred-weight 
 does he sell ? 
 
 15. John Brown paid $375,008 for a tract of wild land, at 
 
 $8 per acre. How many acres did he buy? 
 
 16. How many tons of coal, at $7 per ton, can be pur- 
 
 chased for $3,785,908? 
 
 17. A merchant received $397,640 in selling a quantity 
 
 of flour, at $8 per barrel. How many barrels did 
 he sell ? 
 
 18. What must be paid for 12 yards of cloth, if 5 yards 
 
 cost $25? 
 
 SOLUTION. If 5 yards cost $25, to find the cost of 1 yard 
 $25 must be divided by 5; $25-*- 5 = $5, cost of 1 yard. 12 
 yards will cost 12 x $5 = $60. Ans. 
 
 19. A drover paid $20 for 5 sheep. What will be the cost 
 
 of 125 sheep? 
 
 20. Three cows cost $156. What must be paid for 27 
 
 cows? 
 
 21. If 7 tons of hay cost $105, what will be the cost of 63 
 
 tons? 
 
 22. If 9 barrels of flour are worth $63, how many barrels 
 
 of apples, at $3 a barrel, will pay for 72 barrels of 
 flour? 
 
 23. If 7 cords of birch wood are worth $28, how many 
 
 cords of birch wood will pay for 6 barrels of sugar 
 worth $16 a barrel? 
 
 24. If 12 men do a piece of work in 12 hours, how many 
 
 hours would it take 8 men to do the same work ? 
 
DIVISION 
 
 69 
 
 LONG DIVISION. 
 
 71, The process of Long Division is the same as that of 
 Short Division, except that the work is written in full, and 
 the quotient is written over the dividend. 
 
 Divide 41,668 by 78. 
 
 The beginner will find it convenient to form a table of 
 products of the divisor by the numbers 1, 2, 3, , as follows : 
 
 1 X 78 = 78 
 
 4x78-312 
 
 7 x 78 - 546 
 
 2 x 78 = 156 
 
 5 X 78 = 390 
 
 8 X 78 = 624 
 
 3 x 78 = 234 
 
 6 x 78 - 468 
 
 9 x 78 - 702 
 
 The third product is found by adding the first and sec- 
 ond products, the fourth by adding the first and third, and 
 so on. 
 
 As 78 is more than 41, it is necessary to take three figures of the 
 dividend for the first partial dividend. Of the products in the table 
 that do not exceed 419 the greatest is 390, 
 that is, 5 X 78. Hence the first quotient 
 figure is 5, and is written over the 9 in the 
 dividend ; then 390 is subtracted from 419. 
 To the remainder 29, the next figure 9 of 
 the dividend is annexed. Of the products 
 that do not exceed 299, the greatest is 234, 
 that is, 3 x 78. Hence 3 is the next figure 
 of the quotient, and the next remainder is 
 65, to which the 8 of the dividend is an- 
 nexed. Of the products that do not exceed 
 658, the greatest is 624, that is, 8 X 78. Hence the next figure of the 
 quotient is 8, and the remainder 34. 
 
 After a little practice the operation of division can be 
 performed without the aid of a table of products. Each 
 quotient figure is estimated by taking for a trial divisor the 
 left-hand figure of the divisor (or the left-hand figure in- 
 
 OPERATION. 
 538 
 
 78)41998 
 390 
 299 
 234 
 658 
 624 
 
 34 remainder. 
 
70 DIVISION. 
 
 creased by 1, when the next figure is greater than 5), and 
 by taking for a trial dividend one or two figures only of 
 each partial dividend. When the trial divisor is increased 
 by 1, the trial dividend should be increased by 1. 
 
 Divide 2,791,163 by 394. 
 
 The first partial dividend is 2791. As 9, the second figure of the 
 divisor, is greater than 5, we take 4 for a .trial divisor. As we have 
 increased the trial divisor, we increase 
 the trial dividend by 1, making it 28. 
 4 is contained in 28 7 times. We write 7084 
 
 the 7 over the 1, and multiply the divi- 394) 2791163 
 Bor 394 by 7. We subtract the product 2758 
 
 2758 from 2791 and have for a remain- 3316 
 
 der 33, to which we annex the 1 of the 3152 
 
 dividend. As 331 is less than 394, the 1643 
 
 next quotient figure is 0. To 331 we 1576 
 
 annex the next figure 6 of the dividend. 7 remainder. 
 
 4 is contained in 34 8 times. We there- 
 fore write 8 for the next quotient figure, and find the product of 
 8 X 394 to be 3152. The remainder obtained by subtracting 3152 is 
 164, to which the 3 of the dividend is annexed. 4 is contained 4 
 times in 17. The product of 4 x 394 is 1576, and this subtracted 
 from 1643 leaves 67 for the final remainder. 
 
 NOTE. If the product of the divisor by the quotient figure is 
 greater than the partial dividend, the quotient figure is too large, 
 and must be diminished ; and, if the difference between the partial 
 dividend and the product of the divisor by the quotient figure is 
 greater than the divisor, the quotient figure is too small and must be 
 increased. 
 
 Ex. 4a 
 
 Find the quotients of : 
 
 1. 4386-^21. 5. 9357 -f- 61. 9. 6985-^22. 
 
 2. 5271-^-31. 6. 5263 -f- 71. 10. 9876 -f- 32. 
 
 3. 8056-^-41. 7. 3046 -f- 82. 11. 2378-^42. 
 
 4. 7158-^51. 8. 7219 -J- 92. 12. 4068-^52. 
 
DIVISION. 
 
 71 
 
 13. 8359-5-63. 
 
 21. 6,543-5-68. 
 
 29. 79,853-4-63 
 
 14. 4573-73. 
 
 22. 8,319 -4- 78. 
 
 30. 82,569-4-73. 
 
 15. 7358-4-84. 
 
 23. 5,432-4-89. 
 
 31. 94,365-5-84. 
 
 16. 3985-94. 
 
 24. 9,753-4-99. 
 
 32. 98,765-94. 
 
 17. 6973-4-25. 
 
 25. 41,268-4-21. 
 
 33. 82,639-f-25. 
 
 18. 7413-36. 
 
 26. 74,306-4-31. 
 
 34. 64,372-5-35. 
 
 19. 8765-4-47. 
 
 27. 89,415 -i- 42. 
 
 35. 59,036-5-46. 
 
 20. 7654-4-57. 
 
 28. 67,834 -*- 52. 
 
 36. 42,837-4-56. 
 
 
 Ex. 49. 
 
 
 Find the quotients of : 
 
 1. 84,317 -f- 67. 
 
 13. 437,650-^-23. 
 
 25. 437,650-4-53. 
 
 2. 72,659-4-77. 
 
 14. 657,320-35. 
 
 26. 657,320 -*- 65. 
 
 3. 64,980-4-88. 
 
 15. 327,045-47. 
 
 27. 327,045 -*- 77. 
 
 4. 52,196 -f- 98. 
 
 16. 632,008-^-59. 
 
 28. 632,008-5-89. 
 
 5. 47,028 -f- 29. 
 
 17. 437,650-^-33. 
 
 29. 437,650-4-63. 
 
 6. 74,369-39. 
 
 18. 657,320-^-45. 
 
 30. 657,320-4-75. 
 
 7. 54,371-4-14. 
 
 19. 327,045-57. 
 
 31. 327,045-4-87. 
 
 8. 68,594-4-15. 
 
 20. 632,008 -i- 69. 
 
 32. 632,008-4-99. 
 
 9. 73,109-4-16. 
 
 21. 437,650-^-43. 
 
 33. 43*7,650-4-73. 
 
 10. 82,563-4-17. 
 
 22. 657,320-5-55. 
 
 34. 657,320-85. 
 
 11. 94,069 -f- 18. 
 
 23. 327,045 -i- 67. 
 
 35. 327,045-5-97. 
 
 12. 47,938-5-19. 
 
 24. 632,008-^-79. 
 
 36. 632,008-4-29. 
 
 
 Ex. 50. 
 
 
 Find the quotients of : 
 
 1. 50,576-5-101. 
 
 7. 76,593-^-415. 
 
 13. 96,432-4-781. 
 
 2. 50,576 -f- 102. 
 
 8. 76,593-4-516. 
 
 14. 96,432-4-592. 
 
 3. 50,576-4-203. 
 
 9. 76,593-^-621. 
 
 15. 96,432^-864. 
 
 4. 50,576-5-205. 
 
 10. 76,593-4-732. 
 
 16. 96,432-5-972. 
 
 5. 50,576-4-302. 
 
 11. 76,593-4-843. 
 
 17. 96,432 + 492. 
 
 6. 50,576-^-106. 
 
 12. 76,593-5-954. 
 
 18. 96.432-5-993. 
 
72 DIVISION. 
 
 Ex. 51. 
 
 Find the quotients of : 
 
 1. 861,345-^-4001. 7. 730,604-8403. 
 
 2. 861,345-2048. 8. 972,817-^-7184. 
 
 3. 861,345-^3507. 9. 854,235-8794. 
 
 4. 861,345-^-6409. 10. 730,604-^-5748. 
 
 5. 861,345-8157. 11. 972,817-4981. 
 
 6. 861,345-^-3965. 12. 730,604-1984. 
 
 Ex. 52. 
 
 1. How many stoves can be bought for $1120, if one 
 
 stove costs $35? 
 
 2. If a carriage is valued at $ 144, how many carriages 
 
 can be bought at the same rate for $54,000? 
 
 3. A horse dealer bought a horse for $125. How many 
 
 horses could he buy for $60,625, at the same rate? 
 
 4. How many barrels of sugar can be bought for $8352 
 
 when $36 is paid for one barrel? 
 
 5. A merchant sold 297 barrels of flour for $2673. How 
 
 much did he get a barrel ? 
 
 6. George Clifford paid $10,250 for oxen, paying on the 
 
 average $82 an ox. How many did he buy? 
 
 7. How many days will it take a man to dig a ditch 864 
 
 feet long, if he can dig 48 feet a day ? 
 
 8. One share of a certain bank stock is worth $98. How 
 
 many shares can be bought for $ 22,050 ? 
 
DIVISION. 73 
 
 9. A farmer sold 19 sheep for $152. For how much a 
 head did he sell them ? 
 
 10. John Jones paid $1752 for lambs at an average price 
 
 of $4. How many did he buy ? 
 
 11. A fruit grower received $1755 for 195 barrels of cran- 
 
 berries. What was the price per barrel? 
 
 12. In one square foot there are 144 square inches. How 
 
 many square feet in 1,375,920 square inches? 
 
 13. A public library has a yearly circulation of 56,966 
 
 books. How many books are taken daily, if the 
 library is open 313 days in a year ? 
 
 14. One mile contains 320 rods. How many miles in 
 
 348,160 rods ? 
 
 15. If the dividend is 514,478, the divisor 327, and the 
 
 remainder 107, what is the quotient ? 
 
 16. A railroad 478 miles in length cost $3,500,872. What 
 
 was the average cost per mile ? 
 
 17. How many house lots, at $321 for each, can be bought 
 
 for $772,326? 
 
 18. A company of 547 men took equal shares in a mine 
 
 valued at $705,083. How much money did each 
 man invest? 
 
 19. If 325 workmen are paid $583,700, what sum does 
 
 each receive ? 
 
 20. At $89 per acre, how many acres of land can be pur- 
 
 chased for $713,513? 
 
 21. Divide one million three hundred seventy-five thou- 
 
 sand eight hundred nine by two hundred eighty- 
 seven. 
 
74 DIVISION. 
 
 22. A ship averaging 215 miles per day has to sail 3678 
 
 miles. How many days will be required for the 
 trip? 
 
 23. A New Orleans merchant sends to New York 376,705 
 
 gallons of molasses. How many casks will there be 
 if each cask contains 235 gallons ? 
 
 24. The capital and surplus of a bank amounting to 
 
 $518,077 belonged to 679 stockholders. What is 
 the average amount belonging to each stockholder ? 
 
 25. If 34,823 tons of coal are required for 97 steamships, 
 
 what is the average number of tons for each ? 
 
 26. A carpet factory running 45 looms makes 17,820 yards 
 
 of carpet in a fortnight. What number of yards is 
 woven by each loom on the average ? 
 
 27. A cotton planter raises 428,243 pounds of cotton. If 
 
 the cotton is put into bales, weighing on the average 
 401 pounds, what will be the whole number of 
 bales? 
 
 28. In one cubic foot there are 1728 cubic inches. How 
 
 many cubic feet are there in a pile of wood contain- 
 ing 3,507,840 cubic inches? 
 
 29. In how many hours will a cistern holding 3330 gal- 
 
 lons be filled by a pipe that discharges into it 185 
 gallons an hour ? 
 
 30. An army officer paid $107 for a horse. At that rate 
 
 how many horses can he buy for $317,897 ? 
 
 31. A man having an income of $3874 a year (52 weeks) 
 
 spent $2314 and saved the rest. How much did he 
 save per week on the average? 
 
DIVISION. 75 
 
 32. What number subtracted 88 times from 80,005 will 
 
 leave 13 as a remainder ? 
 
 33. How many rolls of carpet at $75 a roll can be bought 
 
 for $1275? 
 
 34. A has 425 horses valued at $58,650 ; B has 382 acres 
 
 of land worth $48,514. What is the difference in 
 value between one of A's horses and an acre of B's 
 land? 
 
 35. If the dividend is 325,682, the divisor 284, and the 
 
 remainder 218, what is the quotient ? 
 
 36. What is the nearest number to 7196 that will contain 
 
 372 without a remainder ? 
 
 37. New York contains 47,620 square miles, Texas 262,290. 
 
 How many states as large as New York can be made 
 out of Texas, and how many square miles will be 
 left over ? 
 
 38. Dakota contains 147,700 square miles, Massachusetts 
 
 8040. How many states as large as Massachusetts 
 can be made out of Dakota, and how many square 
 miles will be left over ? 
 
 39. In 1880 Texas produced 550,872,000 pounds of cotton. 
 
 Allowing 400 pounds to a bale, how many bales of 
 cotton did Texas raise that year ? 
 
 40. If one pound of sugar is obtained from 18 sugar canes, 
 
 how many pounds will be obtained from 1,233,216 
 canes ? 
 
CHAPTER VL 
 
 DECIMALS. 
 
 72, Numbers which denote whole units are called Integral 
 numbers ; but it is often necessary to express parts of a 
 unit. 
 
 If a unit is divided into two equal parts, each part is 
 called one-half, and is expressed by . If a unit is divided 
 into three equal parts, each part is called one-third, and is 
 expressed by -J- ; two of the parts are called two-thirds, and 
 are expressed by -|. Again, if a unit is divided into four 
 equal parts, each part is called one-fourth, and is expressed 
 by \ ; into five equal parts, each part is called one-fifth, and 
 is expressed by ; into six equal parts, each part is called 
 one-sixth, and is expressed by % ; into seven equal parts, each 
 part is called one-seventh, and is expressed by ^ ; into eight 
 equal parts, each part is called one-eighth, and is expressed 
 by % ; into nine equal parts, each part is called one-ninth, 
 and is expressed by -J-; into ten equal parts, each part is 
 called one-tenth, and is expressed by -fa. 
 
 If AB (see page opposite) represent a unit of length, 
 each division of the line next below AB represents one- 
 half of a unit ; and each division of the second line below 
 AB represents one-third of a unit ; and so on. 
 
 How many halves of a unit make a whole unit ? 
 
 How many fourths make a half? how many make a 
 whole unit? 
 
 How many sixths make a third? how many make a half? 
 how many make a whole unit ? 
 
 How many eighths make a half? a fourth? a whole unit? 
 
 How many tenths make a fifth? a half? a whole unit? 
 
DECIMALS. 77 
 
 i 
 
 i 
 
 1 
 
 1 1 1 
 
 1 
 
 i 
 
 1 1 1 
 
 1 
 
 1 1 
 
 i 
 
 \ 
 
 i 
 
 i i 
 
 1 
 
 1 1 
 
 a i 
 i i 
 
 
 
 i 
 
 l 
 
 i 
 
 1 
 i 
 
 i 3 
 
 i * 
 
 1 1 
 
 $ f 
 
 i i i 
 
 I 
 
 \ 
 
 
 i 
 
 1 
 
 i 
 
 drAAdrAAAftfttt 
 
 73, When a unit is divided into ten equal parts, and we 
 wish to express in figures one or more of these parts, we do 
 not usually write them -fa, y 2 ^, etc., but we write 1, 2, 3, 
 etc., and separate the number which denotes parts of a unit 
 from the number which denotes whole units by a decimal 
 point. Thus, two units and three-tenths of a unit are 
 written, i3. 
 
 If each tenth of a unit is divided into ten equal parts, 
 that is, the entire unit into a hundred equal parts, each 
 part is called a hundredth of the unit ; and if each hun- 
 dredth is divided into ten equal parts, that is, the entire 
 unit into a thousand equal parts, each part is called a thou- 
 sandth of the unit ; and so on. 
 
 These tenth-parts are called Decimal parts, from the Latin 
 word decem, which means ten; and these parts are com- 
 monly called Decimal Fractions, 
 
78 
 
 DECIMALS. 
 
 Let A B, for example, represent the unit of length by which a certain 
 distance is to be measured. Suppose the given distance to contain 
 AS 137 times, and a remainder LM to be left, which 
 is less than AB. Take AC, a tenth of AB, and 
 suppose AC is contained in LM ^ times, with a re- 
 mainder OM less than AC. Again, suppose AD, a 
 tenth of A (that is, a hundredth of AB), to be con- 
 tained in OM 3 times, with a remainder less than 
 AD. And again, suppose a tenth of AD (that is, a 
 thousandth of AB), to be contained in this last re- 
 mainder 9 times. Then the whole distance expressed 
 in lengths of AB will be 137.439. 
 
 The series of figures 137.439 means 1 hun- 
 dred + 3 tens + 7 units + 4 tenths + 3 hun- 
 dredths + 9 thousandths ; as 1 hundred = 10 
 tens = 100 units, and 3 tens = 30 units, the 
 integral value is 137 units; so, 4 tenths =40 
 hundredths = 400 thousandths, and 3 hun- 
 dredths = 30 thousandths; the decimal value 
 therefore is 439 thousandths. 
 
 If the unit is the yard-stick, the whole is 
 read "one hundred thirty-seven and four 
 hundred thirty-nine thousandths yards"; if 
 the unit is the meter-stick, the whole is read 
 11 137 and 439 thousandths meters." 
 
 NOTE. The pupil will get the clearest notions of 
 decimals by taking a meter-stick (which is divided 
 in tenths, hundredths, and thousandths) and meas- 
 uring given lengths ; such as, the length of the side 
 of the room, of the platform, of the window-sill, etc., 
 etc., and writing down the result in each case. 
 Whenever the length measured is less than a meter, 
 he should write down 0, and after it the decimal 
 point, then the actual measure. Thus, if the length 
 is found to be 8 tenths 2 hundredths and 7 thou- 
 sandths, it is expressed by 0.827, and read " eight hundred twenty 
 seven thousandths of a meter." 
 
DECIMALS. 79 
 
 74, It will be seen that 1 tenth = 10 hundredths, 1 
 hundredth = 10 thousandths ; and, conversely, 10 thou- 
 sandths 1 hundredth, 10 hundredths 1 tenth, 10 tenths 
 = 1 unit ; so that in decimal numbers, as in integral num- 
 bers, 10 in any place is equal to 1 in the next place to the 
 left, and 1 in any place is equal to 10 in the next place to 
 the right. 
 
 Hence figures in the first decimal place denote tenths, in 
 the second place hundredths, in the third place thousandths, 
 in the fourth place ten-thousandths, in the fifth place hun- 
 dred-thousandths, in the sixth place millionths, and so on. 
 
 75, In reading decimals, read precisely as if the decimal 
 were an integral number, and add the name of the lowest 
 decimal place. It is best to pronounce the word " and " 
 at the decimal point, and omit it in all other places. Thus, 
 100.023 is read one hundred and twenty-three thousandths. 
 Ambiguity in reading, from having zeros at the end of a 
 decimal, is avoided by a pause ; thus, 0.300 is read three 
 hundred . . . thousandths, while 0.00003 is read three . . . 
 hundred-thousandths. 
 
 76, Read the following numbers : 
 
 0.3; 0.7; 0.65; 0.99; 37.5; 26.9; 425.312; 617.624; 
 94.57 ; 83.28 ; 0.9 ; 0.96 ; 57.09 ; 3.207 ; 2.03 ; 3.045 ; 
 40.7; 0.055; 0.074; 0.0215; 7.3945; 0.14875 ; 0.00005 ; 
 2.000375; 100.015625; 3.7525; 2.1136257. 
 
 77, Express in the decimal notation : 
 
 Seven tenths ; nine tenths ; eleven hundredths ; eight 
 hundredths ; one hundred thirty-four thousandths ; 
 twenty-five thousandths; two hundred and thirty-four 
 thousandths ; nineteen and forty-one hundred-thou- 
 sandths ; twenty-five and sixteen ten-thousandths ; 
 
80 DECIMALS. 
 
 thirteen and two hundred one hundred-thousandths ; 
 six hundred fifty-eight thousand three hundred forty-two 
 millionths ; eighty-six and eight hundred three thousand 
 three hundred four millionths ; three and twenty-nine 
 hundredths ; fifteen and six hundred seventy-one thou- 
 sandths ; fifty-three ten-thousandths ; twenty-two and 
 sixty-seven hundredths ; fourteen and two thousand 
 three hundred fifty-one ten-thousandths; two and two 
 hundred nineteen thousandths ; three and one hundred 
 fifty-seven thousandths. 
 
 78, Zeros occurring at the end of a decimal do not affect 
 its value. Thus, 3.50700 means 3 units + 5 tenths + 
 hundredths + 7 thousandths -f ten-thousandths + hun- 
 dred-thousandths, and is, therefore, 3 and 507 thousandths, 
 the same as 3.507. 
 
 78, The arrangement and method of working employed 
 in decimals is precisely like that employed in integral s 
 numbers, the decimal point being the only new considera- 
 tion. 
 
 ADDITION OF DECIMALS. 
 
 Add 17.5163, 236.3, 1.7162, 0.00132, 
 OPERATION. 
 
 17.5163 
 236.3 
 
 1.7162 
 
 0.00132 
 255.53382 
 
 Write the numbers in columns, units under units, tens 
 under tens, tenths under tenths, and so on, so that the 
 decimal -points will fall in a vertical line, and add as in 
 integral numbers. 
 
DECIMALS. 81 
 
 Ex. 53. 
 Find the value of: 
 
 1. 2.514 + 3.7 + 9.6304 + 0.24876. 
 
 2. 1.916 + 6.3 + 0.4782 + 9.35634. 
 
 3. 0.415 + 8.0 + 6.3746 + 8.29426. 
 
 4. 7.516 + 9.6 + 1.9238 + 7.21442. 
 
 5. 7.03 + 7.2456 + 0.483 + 9.23579 + 8.3. 
 
 6. 2.576 + 3.4203 + 1.5 + 6.27948 + 0.362357 
 
 7. 3.29+15.671 + 0.0053 + 22.67. 
 
 8. 14.2351 + 651 + 2.219 + 3.157. 
 
 9. 213.7 + 2.913 + 14.769 + 0.007871. 
 
 10. 1.4178 + 0.2 + 2.356709 + 1.14 + 2.0. 
 
 11. 4.96 + 3.2728 + 0.7 + 3.54219 + 4.7. 
 
 12. 1.198 + 3.5 + 7.635487 + 4.23 + 1.5724. 
 
 13. 4.372 + 9.5 + 7.369248 + 1.72 + 3.2948. 
 
 14. 0.4293 + 0.7 + 6.954326 + 3.14 + 7.005. 
 
 15. 3.87 + 2.6493 + 0.8 + 2.63495 + 9.3. 
 
 16. 6.9 + 5.71 + 0.0431 + 329.2 + 4.4. 
 
 17. 3.571 + 0.008+12.51 + 649 + 3.051. 
 
 18. 15.753 + 2.069+17.6143 + 3.2107. 
 
 19. 1.1 + 20.02+13 + 2.845 + 1.0001. 
 
 20. 31.826 + 3.471 + 0.004 + 45 + 0.6. 
 
 21. 82.537 + 2000 + 1.354 + 0.006 + 13. 
 
 22. 64.27 + 1.1 + 23 + 17.12 + 8.8. 
 
 23 . 72.5 + 140 + 340.03 + 21.5715 + 4.00087. 
 
 24. 0.96 + 7,3004 + 8010 + 0.00093 + 124650. 
 
82 
 
 DECIMALS. 
 
 SUBTRACTION OF DECIMALS. 
 
 80, Subtract 37.286 from 41.1325 ; and 1.00523 from 9.3. 
 
 OPERATION. OPERATION. 
 
 41.1325 9.30000 
 
 37.286 1.00523 
 
 3.8465 8.29477 
 
 Write the subtrahend under the minuend, so that the 
 decimal points may fall in a vertical line. If the number 
 of decimal places in the subtrahend exceed the number in 
 the minuend, zeros may be annexed to the minuend, as 
 such zeros have no effect on its value. 
 
 1. 0.58-0.39. 
 
 2. 0.67-0.59. 
 
 3. 3.927-1.836. 
 
 4. 4.825-1.763. 
 
 5. 4.325-1.672. 
 
 6. 6.283-3.576. 
 
 7. 9.025-6.387. 
 
 8. 6.275-3.829. 
 
 9. 7.57-6.385. 
 
 10. 9.26-2.375. 
 
 11. 8.4-3.228. 
 
 12. 9.5-2.732. 
 
 13. 14.3846-4.8003. 
 
 14. 3.4370-0.3045. 
 
 15. 0.3290-0.0089. 
 
 16. 136.0200-1.5423. 
 
 17. 1.9990-0.063. 
 
 18. 13.5298-10.0060. 
 
 Ex. 54. 
 
 19. 2.1808-0.0009. 
 
 20. 1.9870-1.0873. 
 
 21. 48.9370-30.3000. 
 
 22. 0.9990-0.9009. 
 
 23. 15.1409-3.8579. 
 
 24. 5.9009-0.0909. 
 
 25. 1.3993-0.9090. 
 
 26. 10.1010-0.0999. 
 
 27. 3.5-0.075. 
 
 28. 517-0.0076. 
 
 29. 1.325-0.4736. 
 
 30. 192.3-17.294. 
 
 31. 175.8-1.0024. 
 
 32. 186.257-13.794. 
 
 33. 0.715-0.70451. 
 
 34. 1111.116-22.22222. 
 
 35. 71.0047-9.0008167. 
 
 36. 9161.0098-7149.16776. 
 
DECIMALS. 83 
 
 MULTIPLICATION OF DECIMALS. 
 
 81. A change in position of the decimal point of a num- 
 ber will affect the local value of each figure of that number. 
 Thus, if in place of 79.213 we write 792.13, we increase 
 the value of each figure ten-fold, the 7 tens become 7 
 hundreds, the 9 units become 9 tens, the 2 tenths become 
 2 units, the 1 hundredth becomes 1 tenth, and the 3 thou- 
 sandths become 3 hundredths, and, as the value of every 
 figure is increased ten-fold, the entire number is increased 
 ten-fold. If the decimal point is moved one place to the 
 left, the local value of each figure is diminished ten-fold, 
 and consequently the value of the entire number is dimin- 
 ished ten-fold. Hence, 
 
 To multiply a decimal by 10, 100, 1000, etc., we have 
 only to move the decimal point in the multiplicand as many 
 places to the right, annexing zeros if necessary, as there 
 are zeros in the multiplier. 
 
 To divide a decimal by 10, 100, 1000, etc., we have only 
 to move the decimal point in the dividend as many places 
 to the left, prefixing zeros if necessary, as there are zeros 
 in the divisor. 
 
 Thus, 100 X 36.123 = 3612.3, and 1000 X 36.1 = 36100: 
 36.123-^10 = 3.6123, and 36.123 -*- 1000 = 0.036126. 
 
 82, To multiply a number by 0.1, 0.01, 0.001, etc., we 
 have, by the definition of multiplication, to divide the 
 multiplicand by 10, 100, 1000, etc. ; that is, to remove the 
 decimal point one place, two places, etc., to the left. 
 
 To divide by 0.1, 0.01, 0.001, etc., we have only to move 
 the decimal point in the dividend one place, two places, 
 etc., to the right. 
 
 Thus, 0.1 X 86.32 = 8.632, and 0.01 X 1.236 = 0.01236 ; 
 86.32 -- 0.1 = 863.2, and 1.236 -*- 0.01 = 123.6. 
 
84 DECIMALS. 
 
 83, Multiply 123.826 by 3. 
 
 Here 3x6 thousandths = 18 thousandths, or 1 hundredth and 8 
 
 OPERATION, thousandths ; the 8 therefore is written in the thou- 
 
 123 826 san dths' column ; then, 3x2 hundredths = 6 hundredths, 
 
 3 which, with the 1 hundredth, make 7 hundredths, and 
 
 S71 478 ^ e ^ * 8 wr ^ ten * n the bundredths' column ; then, 3x8 
 
 tenths = 24 tenths, or 2 units and 4 tenths, and the 4 
 
 is written in the tenths' column ; then, 3x3 units = 9 units, which, 
 
 with the 2 units, make 11 units, and so on. 
 
 Multiply 123.826 by 0.3. 
 
 OPERATION. 
 
 123.826 
 
 03 
 
 37.1478 
 
 The multiplier 0.3 = 3x0.1. We therefore multiply first by 3, 
 and the resulting product by 0.1. But multiplying by 0.1 simply 
 moves the decimal point in the product one place to the left. Hence, 
 the product will have three decimal places for the decimal in the 
 multiplicand, and one more place for the decimal in the multiplier. 
 
 Multiply 123.826 by 0.32. 
 
 OPERATION. 
 
 123.826 
 0.32 
 247652 
 371478 
 
 39.62432 
 
 The multiplier 0.32 = 32 X 0.01. We therefore multiply first by 
 32, and the resulting product by 0.01. But multiplying by 0.01 
 simply moves the decimal point in the product two places to the left. 
 Hence, the product has three decimal places for the decimal in the 
 multiplicand, and two more places for the decimal in the multiplier. 
 
 In the multiplication of decimals, therefore, point off in 
 the product as many decimal places as there are in the mul- 
 tiplicand and multiplier taken together. 
 
DECIMALS. 
 
 85 
 
 Find the products of : 
 
 1. 5x0.3. 26. 
 
 2. 8 X 0.27. 27. 
 
 3. 12 X 0.375. 28. 
 
 4. 15x0.256. 29. 
 
 5. 9 X 0.7. 30. 
 
 6. 6x0.75. 31. 
 
 7. 16 X 0.284. 32. 
 
 8. 11 X 0.386. 33. 
 
 9. 10x0.65. 34. 
 
 10. 100x0.721. 35. 
 
 11. 1000x3.736. 36. 
 
 12. 1000x0.074. 37. 
 
 13. 10x0.99. 38. 
 
 14. 100x0.615. 39. 
 
 15. 1000x2.409. 40. 
 
 16. 1000x0.055. 41. 
 
 17. 0.5x37. 42. 
 
 18. 0.9x99. 43. 
 
 19. 0.25x428. 44. 
 
 20. 0.36 X 7384. 45. 
 
 21. 0.9x26. 46. 
 
 22. 0.7 X 67. 47. 
 
 23. 0.48 X 237. 48. 
 
 24. 0.18x3692. 49. 
 
 25. 0.312x425. 50. 
 
 Ex. 55. 
 
 0.716 x 388. 
 0.725 x 96. 
 0.085 x 88. 
 0.624x617. 
 0.358 X 776. 
 0.145 X 48. 
 0.017 X 44. 
 57 X 9.4. 
 26 X 3.8. 
 3 x 972.3. 
 65 x 87.2. 
 2.8 x 83. 
 
 3.2 x 64. 
 7.8 x 369. 
 3.7 X 815. 
 1.44 x 9.6. 
 2.88 X 4.8. 
 3.21 x 72.5. 
 2.16 X 40.7. 
 3.26 x 4.37. 
 
 2.03 x 3.207. 
 2.472 X 9.525, 
 3.264 x 3.045 
 0.7 X 0.5. 
 
 0.9 X 0.57. 
 
 51. 0.45 x 0.57. 
 
 52. 0.72x0.324 
 
 53. 0.6 x 0.9. 
 
 54. 0.8 x 0.96. 
 
 55. 0.72 x 0.72. 
 
 56. 0.36 x 0.648. 
 
 57. 416 x 0.416. 
 
 58. 57 X 0.015. 
 
 59. 693 X 0.83. 
 
 60. 4.625 X 7.14, 
 
 61. 99.9 X 4.09. 
 
 62. 753 x 0.672. 
 
 63. 928 x 8.302. 
 
 64. 56.704x0.413. 
 
 65. 2.052x0.0037. 
 
 66. 0.00948 X 29. 
 
 67. 372 x 0.468. 
 
 68. 9.43 x 0.054. 
 
 69. 786 x 3.62. 
 
 70. 0.632 X 85. 
 
 71. 2.406 X 0.008. 
 
 72. 6824 X 3.7. 
 
 73. 42.53 X 0.685. 
 
 74. 0.832 X 59. 
 
 75. 763.24x4.078, 
 
86 DECIMALS. 
 
 DIVISION OF DECIMALS. 
 
 84, In Division, if the dividend and divisor are both 
 multiplied or both divided by the same number, the quo- 
 tient is not changed. Thus, 18 -s- 6 = 3, and (when both 
 dividend and divisor are multiplied by 2) 36 -s- 12 = 3. 
 Again (when both dividend and divisor are divided by 2), 
 9-^-3 = 3. 
 
 If, therefore, the divisor contains decimal places, we may 
 remove the decimal point from the divisor, provided we carry 
 the decimal point in the dividend as many places to the 
 right as there are dermal places in the divisor. 
 
 Divide 78.528 by 0.8. 
 
 Here the decimal point is removed from OPERATION. 
 the divisor, and the decimal point in the divi- 8) 785.28 
 dend is carried one place to the right ; that is, 98.16 
 
 both dividend and divisor are multiplied by 10. 
 
 When the divisor is a whole number, each quotient 
 figure is of the same order of units as the right-hand figure 
 of the partial dividend used in obtaining it. Hence, the 
 decimal point is put in the quotient as soon as the decimal 
 point in the dividend is reached. 
 
 Divide 28.3696 by 1.49. 
 
 OPERATION. 
 
 19.04 
 
 149)2836.96 
 149 
 1346 
 1341 
 
 596 
 596 
 
 Here the decimal point is removed from the divisor, and 
 is moved two places to the right in the dividend ; in other 
 words, both dividend and divisor are multiplied by 100. 
 
DECIMALS. 
 
 87 
 
 If the divisor is not contained in the dividend without a 
 remainder, ciphers may be mentally annexed to the divi- 
 dend, and the division continued. 
 
 Divide 0.39842 by 3.7164 to four decimal places. 
 OPERATION. 
 
 0.1072 
 
 37164)3984.2 
 37164 
 267800 
 260148 
 
 76520 
 
 74328 
 
 2192 
 
 If the divisor is a whole number, and ends in zeros, we 
 may cut off the zeros from the divisor, and move the deci- 
 mal point in the dividend as many places to the left as there 
 are zeros cut off. 
 
 Divide 42.08 by 8000. 
 
 OPERATION. 
 
 8)0.04208 
 
 0.00526 
 
 Here the three zeros are cut off from the divisor, and the 
 decimal point in the dividend is moved three places to the 
 left. In other words, both divisor and dividend are divided 
 by 1000. 
 
 Ex. 66. 
 Find the quotients of : 
 
 1. 34.24-^-4.28. 
 
 2. 24.56^-6.14. 
 
 3. 52.90-^-5.75. 
 
 4. 37.576 -H 
 
 5. 281.232- 
 
 6.832. 
 h 7.812. 
 
 6. 97.524-^5.16. 
 
 7. 738.0930-^-0.023. 
 
 8. 5.18466 -- 1.02. 
 
 9. 0.018 --9.6. 
 
 10. 34.96818^0.381. 
 
88 
 
 DECIMALS. 
 
 11. 
 
 0.003125 -f- 25. 
 
 36. 
 
 6 + 0.008. 
 
 12. 
 
 859.95 -i- 136.5. 
 
 37. 
 
 4.8 * 0.00016. 
 
 13. 
 
 5.468 -*- 0.08. 
 
 38. 
 
 1562.5 H- 0.00025. 
 
 14. 
 
 0.04922-*- 0.0023. 
 
 39. 
 
 64 + 0.016. 
 
 15. 
 
 0.00044408 -f- 0.01 12. 
 
 40. 
 
 5. 76 -H 0.0048. 
 
 16. 
 
 0.20412^-0.0084. 
 
 41. 
 
 3.012-^-0.0006. 
 
 17. 
 
 0.07504 -f- 23.45. 
 
 42. 
 
 91844152.5 * 1.1575. 
 
 18. 
 
 0.00025 -*- 2.5. 
 
 43. 
 
 7 -5- 0.0035. 
 
 19. 
 
 0.03217 - 1250. 
 
 44. 
 
 0.39237 -H 0.319. 
 
 20. 
 
 171.99 -f- 27.3. 
 
 45. 
 
 0.3230864 H- 0.5072. 
 
 21. 
 
 0.012305-^-1.07. 
 
 46. 
 
 3. I-H 0.0025. 
 
 22. 
 
 15.625 -f- 2.5. 
 
 47. 
 
 63.8406-^-0.18345. 
 
 23. 
 
 5.418-2.58. 
 
 48. 
 
 181.3 -f- 0.00037. 
 
 24. 
 
 0.59064 -- 0.0276. 
 
 49. 
 
 12.5 -H 2.56. 
 
 25. 
 
 0.73807 -!- 0.023. 
 
 50. 
 
 284.7432^-0.00004. 
 
 26. 
 
 15.4546 -*- 0.019. 
 
 51. 
 
 130.4 -^ 0.0004. 
 
 27. 
 
 6.7288 -H 64.7. 
 
 52. 
 
 1 13.4 -f- 0.0108. 
 
 28. 
 
 72.36 -* 144. 
 
 53. 
 
 68.97516 -*- 0.9246. 
 
 29. 
 
 0.01124-*- 11.24. 
 
 54. 
 
 0.022185 *- 0.0306. 
 
 30. 
 
 15.625 -*- 5. 
 
 55. 
 
 0.276766 -^ 0.371. 
 
 31. 
 
 8.192-^0.00128. 
 
 56. 
 
 286-^-0.013. 
 
 32. 
 
 0.00512 -i- 2.048. 
 
 57. 
 
 0.10724-^-0.003125, 
 
 33. 
 
 0.00972 -H 0.0004. 
 
 58. 
 
 0.03 -*- 0.001. 
 
 34. 
 
 0.07504 -*- 23.45. 
 
 59. 
 
 105-^-43.75. 
 
 35. 
 
 15.21-11.7. 
 
 60. 
 
 8.468 -f- 0,0292. 
 
DECIMALS. 89 
 
 Ex. 57. 
 
 1. To enclose a certain lot 225 yards of fence are needed. 
 
 What will be the cost of the fence at the rate of 
 $0.50 a yard? 
 
 2. A section of land costs $49,878; what must be paid 
 
 for 0.375 of a section ? 
 
 3. When 0.7 of a ton of coal is worth $6.30, what will be 
 
 the cost of 12.5 tons? 
 
 4. Coal being worth $7.00 per ton, what part of a ton 
 
 can be bought for $2.59? 
 
 5. If a man can build 0.45 of a rod of wall in one hour, 
 
 how many rods will 4 men build in 3.8 days, work- 
 ing 7.5 hours per day ? 
 
 6. Twelve dozen penknives cost .$90. If they are sold. 
 
 at $0.75 each, what will be the gain on each? 
 
 7. Divide $125. 15 by $25.03. 
 
 8. Twelve yards of velvet cost $150. At that rate, 
 
 what must be paid for 18 yards ? 
 
 9. What will be the cost of 9.75 cords of white oak wood 
 
 at the rate of $ 10 a cord ? 
 
 10. Twenty-five hundred ths of a farm cost $ 5000 ; what V 
 
 will nine-tenths of it cost ? 
 
 11. A merchant bought 575 pounds of sugar for $51.75; 
 
 he sold four-tenths of it at $0.11 per pound, and the 
 remainder at $0.125. What was his gain ? 
 
 12. A railroad train has 201 miles to run. If it averages 
 
 26.8 miles per hour, how many hours will be 
 required ? 
 
90 DECIMALS. 
 
 13. Hiram Jones owes to one man $1000, to another 
 
 $75.02, to another $198.75, and to a fourth 
 $325.50. How much money will he need to pay 
 these debts? 
 
 14. A man bought 80 yards of cotton cloth for $18.40. 
 
 Find the price of the cloth per yard. 
 
 15. How many yards of calico at $0.125 per yard can be 
 
 bought for $45.625? 
 
 16. A farmer spent $303.75 for corn at $ 1.35 a bag. How 
 
 many bags of corn did he buy ? 
 
 17. One pound of dry oak wood when burnt yields 0.022 
 
 of a pound of ashes. What part of the pound dis- 
 appears into the air ? 
 
 18. Before a storm the mercury in a barometer fell from 
 
 30.292 inches to 29.347 inches. What part of an 
 inch did it fall ? 
 
 19. John purchased goods to the amount of $7.48. He 
 
 gave the salesman a ten-dollar bill. How much 
 money should he receive ? 
 
 20. The distance from New York to Chicago by the New 
 
 York Central and Lake Shore route is 982.24 miles, 
 and the distance from New York to Buffalo is 
 441.75 miles. What is the distance from Buffalo 
 to Chicago ? 
 
 21. Excursion tickets from Boston to Fabyan's and return 
 
 cost $7.75, from Fabyan's to the summit of Mt. 
 Washington and return $3. If a person takes 
 this trip and pays $4.50 for supper, lodging, and 
 breakfast at the Summit House on Mt. Washington, 
 and $1.25 for dinner at Fabyan's the next day, how 
 much will the whole trip cost? 
 
DECIMALS. 91 
 
 22. From an ice-house containing 1000 tons of ice the 
 owner sold at different times 242.765 tons, 92.325 
 tons, 161.575 tons, and 479.312 tons. The rest 
 melted. How many tons melted ? 
 
 Ex. 58. 
 To find the cost of goods sold by the hundred, 
 
 Point off two places for decimals at the right of the 
 number denoting the quantity, and multiply the price of a 
 hundred by this number. 
 
 To find the cost of goods sold by the thousand, 
 
 Point off three places for decimals at the right of the 
 number denoting the quantity, and multiply the price of a 
 thousand by this number. 
 
 NOTE. M is often used for thousand, and C for hundred. 
 
 1. What will 9875 feet of boards cost at $9 per M ? 
 
 2. At $3.25 per C, what must be paid for 3784 feet of 
 
 Georgia pine ? 
 
 3. For the roof of a building 8000 tiles are to be used. 
 
 What will they cost at $9.875 per M ? 
 
 4. Required the cost of 98,762 laths at $0.35 per C. 
 
 5. An architect estimates that 1,500,784 bricks will be 
 
 needed for a schoolhouse. What will they cost at 
 $7.75 a thousand? 
 
 6. What must be paid for 4879 paving-stones at $9.375 
 
 a hundred ? 
 
 7. If the freight from New York to Boston is $0.12 per 
 
 hundred pounds, what must be paid on five boxes of 
 goods weighing respectively 348.25, 227.25, 429.25, 
 396.125, 419.125 pounds? 
 
92 DECIMALS. 
 
 8. A lumber dealer paid $4.50 per M for cedar shingles 
 
 and sold them for $5.75. What did he gain on 35 
 M? 
 
 9. A farmer contracted for boards for fencing at the rate 
 
 of $12.375 per M. His bill for lumber amounted 
 to $61.875. How many thousand feet did he buy? 
 
 To find the cost of goods sold by the ton, 
 Point off from the right of the number denoting the 
 quantity three decimal places, multiply the price of a ton 
 by this number, and divide this result by 2. The reason 
 for this operation follows from the fact that two thousand 
 pounds make a ton. 
 
 Find the retail price of 7846 pounds of coal at $8.75 a 
 
 ton. 
 
 OPERATION. 
 
 $8.75 
 7.846 
 
 5250 
 3500 
 7000 
 6125 
 
 2)68.65250 
 $34.32625 
 
 10. What must be paid for 9785 pounds of plaster at $6.75 
 
 per ton ? 
 
 11. If 25,000 pounds of plaster cost $131.25, what is that 
 
 per ton ? 
 
 12. A dealer in New York retails coal at $7.75 per ton. 
 
 If a ton costs $3.75 at the mine and $0.75 for 
 freight, what will he make on 8758 pounds of coal ? 
 
DECIMALS. - 93 
 
 13. At $10.50 per ton what is the cost of 25,000 pounds 
 
 of plaster? 
 
 14. What is the retail price of coal per ton when" 17,520 
 
 pounds are sold for $ 74.46 ? 
 
 15. How many pounds of plaster at $10.50 per ton can be 
 
 bought for $131.25? 
 
 16. An errand boy receives $2.75 a week. In how many 
 
 weeks will he earn $44? 
 
 17. How many cords of pine wood at $3.375 a cord must 
 
 be given for 12 yards of broadcloth at $2.25 a 
 yard? 
 
 18. The milk from a herd of 75 cows at 6 cents a quart 
 
 amounted in one summer to $2025. How many 
 quarts were sold ? 
 
 19. A merchant sold 3 pieces of matting, each containing 
 
 45.5 yards, at $0.375 per yard. How much money 
 did he receive ? 
 
 20. If a man earns $12 a week and spends on the average 
 
 $10.125, in how many weeks will he save $97.50? 
 
 21. A grocer bought 156 boxes of oranges at $5.625 each, 
 
 and sold the whole for $916.875. How much did 
 he gain ? 
 
 22. A Western farmer's wheat crop at $1.08 per bushel 
 
 brings $831.60. How many bushels did he raise? 
 
 23. Find the cost of 15,964 feet of boards at $39.25 a 
 
 thousand. 
 
 24. Find the cost of 19,500 laths at 35 cents a hundred. 
 
 25. What will be paid for shipping 1500 tons of wheat 
 
 from Buffalo to New York at the rate of 5 cents a 
 bushel ? (A bushel of wheat weighs 60 pounds.) 
 
94 DECIMALS. 
 
 Ex. 59. 
 
 1. Find the price of 30 Parian statuettes at $8.875 each. 
 
 2. In February, 1884, the number of days during which 
 
 rain fell in New England was 22, and the amount 
 which fell was 4.57 inches. Find the daily average 
 for the 22 days. 
 
 3. How many acres are in a park containing 0.08 of 
 
 115.1875 acres? 
 
 4. If 31.75 rods of fence are made for $10.90, what is 
 
 the cost of a rod ? 
 
 5. On a certain day in February, 1884, the thermometer 
 
 at the highest was 51.1, and at the lowest 29.4. 
 Find the difference. 
 
 6. Of 100 parts of matter in beans, sugar and gum form 
 
 61.10, other vegetable matter forms 31.55, and 
 moisture 5 parts. Of how many parts does the 
 remainder, which is mineral matter, consist? 
 
 7. If 0.1571 of the weight of superphosphate is organic 
 
 matter, find the weight in tons of organic matter in 
 80 tons of superphosphate. 
 
 8. In January, 1884, the barometer at the highest was 
 
 30.543 inches, and at the lowest 28.843 inches. 
 Find the difference. 
 
 9. A cubic inch of pure water weighs 252.458 grains. 
 
 Find the weight in grains of a cylindrical inch, 
 which is 0.7854 of a cubic inch. 
 
 10. Divide $31.40 among 6 men and 11 youths, giving a 
 youth 0.525 of a man's share. What is each man's 
 share ? 
 
DECIMALS. 95 
 
 11. .Four men together paid $20,000 for some land. The 
 
 first puts in $2350, the second $5820.35, the third 
 $7640.75. How much must the fourth man pay ? 
 
 12. What will be the cost of uniforms for a base-ball nine 
 
 at $2.87 for each uniform ? 
 
 13. At $15.87 a ton, what will be the value of 637 tons 
 
 of hay ? 
 
 14. If peaches are worth $1.25 a basket, and it takes 3 
 
 dozen for a basket, what is the value of 2892 dozen 
 peaches ? 
 
 15. If 964 baskets of peaches are sold for $1301.40, what 
 
 is the price per basket ? 
 
 16. If 324 men contribute together $2647.08, what is the 
 
 contribution of each ? 
 
 17. A boy picks blueberries in a pasture, giving to the 
 
 owner of the pasture for the privilege 1 quart out 
 of every 8 quarts. In 2 days he picks 48 quarts, 
 and sells his share of the berries for $3.78. What 
 did he get a quart? 
 
 18. If 150 men work on a railroad at the same price per 
 
 day, and if, at the end of the week, they all together 
 receive $1575, what price per day does each man 
 receive? 
 
 19. If a kite-string is 213.86 feet long, and the kite breaks 
 
 away and carries off 94.38 feet of the string, how 
 much will be left ? How much more must be 
 bought to make up 1000 feet? 
 
 20. At $9.17 a barrel, how many barrels of flour can be 
 
 bought for $876.35, and how much money will be 
 leftover? 
 
96 DECIMALS. 
 
 Ex. 60. 
 
 If the length of the diameter of a circle is multiplied by 
 3.1416, the product is the length of the circumference. 
 1. Find the length in inches of the circumference of a 
 circle if the diameter is 6 inches. 
 
 2. Find the length in inches of the circumference of a 
 
 circle if the diameter is 17 inches. 
 
 3. If a carriage wheel is 4 feet in diameter, what is its 
 
 circumference in feet? 
 
 4. If the carriage wheel in Example 3 rolls on the ground 
 
 without slipping, how many feet will it go in turn- 
 ing 27 times? 
 
 5. How many times will the carriage wheel in Example 3 
 
 turn in going 1160 feet? 
 
 If the length of the circumference of a circle is divided by 
 3.1416, the quotient is the length of the diameter. 
 
 6. Find the diameter in feet of a circle if its circumfer- 
 
 ence is 1000 feet. 
 
 7. Find the diameter in feet of a wheel which revolves 
 
 19.5 times in going 253.5 feet. 
 
 8. If the circumference of a circle is 198 yards, what is 
 
 its diameter in yards ? 
 
DECIMALS. 97 
 
 9. How deep is a well if the wheel whose diameter is 2 
 feet makes 10 revolutions in raising the bucket? 
 
 10. If a carriage wheel makes 440 revolutions in travelling 
 
 a mile (5280 feet), what is its diameter in feet ? 
 
 11. If 1964.52 bushels of corn are to be put into bags hold- 
 
 ing 2.14 bushels each, how many bags will it take? 
 
 12. A boy has 3 pieces of twine: one is 58.74 feet long, 
 
 another is 97.86 feet, and a third 57.26 feet. How 
 long a kite-string can he make, making no allow- 
 ance for knots ? 
 
 13. Boys in playing hare and hound run 3.876 miles. 
 
 The hares drop a piece of paper every 4.75 feet on 
 the average. How many pieces do they drop ? A 
 mile is 5280 feet. 
 
 14. If a man earns $23.25 a day, how many days will it 
 
 take him to earn $1964.87? 
 
 15. A grain merchant bought corn at 60 cents and rye at 
 
 75 a bushel. He bought the same number of bushels 
 of both kinds of grain and paid for both together 
 $607.50. How many bushels of each kind did he 
 buy? 
 
 16. When potatoes are worth $0.77 per bushel, and corn 
 
 $1.10, how many bushels of corn should a farmer 
 receive in exchange for 50 bushels of potatoes ? 
 
 17. How many gallons of 231 cubic inches are contained 
 
 in a cubic foot (1728 cubic inches) ? 
 
 18. How many cubic feet are contained in a bushel, a 
 
 bushel containing 2150.42 cubic inches? 
 
 19. For $7624.13 how many tons of hay can be bought at 
 
 $18.75 a ton? 
 
 20. The large wheel of a bicycle is 14.37 feet around. 
 
 How many times will it turn in going a mile (5280 
 feet) ? 
 
CHAPTER VII. 
 
 MULTIPLES AND MEASURES. 
 
 85, When the multiplier is an integral number, the 
 product is called a multiple of the multiplicand ; and, in 
 division, when the quotient is an integral number, the 
 divisor is called a measure of the dividend. Thus, 8 X 7 = 56 ; 
 the number 56 is a multiple of 7. Again, 56 -f- 7 = 8 ; the 
 number 7 is a measure of 56. 
 
 86, A number which cannot be divided by any other 
 number except unity without remainder is called a prime 
 number, 
 
 Thus, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc., are prime 
 numbers. 
 
 87, Other numbers are each the product of a fixed set of 
 prime numbers, and are called composite numbers, 
 
 88, Numbers which can be divided by 2 without re- 
 mainder are called even numbers ; and all other numbers 
 are called odd numbers, Even numbers end in 2, 4, 6, 8, 
 or 0; odd numbers end in 1, 3, 5, 7, or 9. 
 
 89, By way of distinction, when a number is used with- 
 out reference to any designated unit, it is called an abstract 
 number ; and, when used with reference to a specified unit, 
 it is called a concrete number, 
 
 Thus, 5, 7, 8 are abstract numbers, and 5 horses, 7 chairs, 
 8 dollars are called, by way of distinction, concrete numbers. 
 
MULTIPLES AND MEASURES. 99 
 
 90, To factor a composite number is to separate the 
 number into its factors. 
 
 Find the prime factors of 144. 
 2144 
 
 72 
 36 
 
 9 
 
 3 
 
 That is, 144 - 2 x 2 x 2 x 2 x 3 x 3. 
 
 91. To avoid the necessity of writing long rows of equal 
 factors, a small figure called the exponent is written at the 
 right of a number to show how many times the number is 
 taken as a factor. 
 
 Thus, 2x2x2x2x3x3 is written 2 4 X 3 2 . 
 The expression 2* is called the fourth power of 2, and 3 2 
 is called the second power of 3. 
 
 92. It is evident from 90 that the method of separating 
 a composite number into its prime factors is, 
 
 Divide the given number by any prime number that is 
 contained in it without remainder ; then the quotient by any 
 prime number that is contained in it without remainder ; and 
 so on until the quotient is itself a prime number. The 
 several divisors and the last quotient are the prime factors. 
 
 If no prime factor is found before the quotient becomes 
 equal to or less than the divisor, the number is prime. 
 
 03, The following tests are useful for determining with- 
 out actual division if a number contains certain factors : 
 
 1. A number is divisible by 2 if its last, or right hand, 
 digit is even. 
 
 2. A number is divisible by 4 (2 2 ) if the number denoted 
 by the last two digits is divisible by 4. 
 
100 MULTIPLES AND MEASURES. 
 
 3. A number is divisible by 8 (2 3 ) if the number denoted 
 by the last three digits is divisible by 8. 
 
 4. A number is divisible by 3 if the sum of its digits is 
 divisible by 3. 
 
 5. A number is divisible by 9 (3 2 ) if the sum of its digits 
 is divisible by 9. 
 
 6. A number is divisible by 5 if its last digit is either 
 5 orO. 
 
 7. A number is divisible by 25 (5 2 ) if the number de- 
 noted by the last two digits is divisible by 25. 
 
 8. A number is divisible by 125 (5 3 ) if the number de- 
 noted by the last three digits is divisible by 125. 
 
 9. A number is divisible by 6 if its last digit is even, 
 and the sum of its digits is divisible by 3. 
 
 10. A number is divisible by 11 if the difference between 
 the sum of the digits in the even places and the sum of the 
 digits in the odd places is either or a multiple of 11. 
 
 Ex. 61. 
 
 Find the prime factors of: 
 
 1. 32; 48: 56; 60; 75; 63; 92; 44; 88; 72; 84; 85. 
 
 2. 51; 69; 68; 87; 54; 98; 74; 90; 86; 70; 42; 62. 
 
 3. 112; 140; 132; 216; 162; 176; 252; 240; 360; 384. 
 
 4. 484; 476; 512; 525; 560; 572; 632; 648; 696; 720. 
 
 5. 748; 775; 824; 876; 888; 948; 960; 925; 117; 119. 
 
 94, The number 1.56 may be put in the form of 156 X .01, 
 and thus separated into 2 2 X 3 X 13 X .01. 
 
 Ex. 62. 
 
 Find the prime factors of: 
 
 'l. 1.05; 12.5; 14.3; 1.65; 19.2; 2.42; 62.4; 27.5. 
 2. 34.3; 5.39; 62.1; 118.8; 1.331; 1.452; 1.584; 92.4. 
 
MULTIPLES AND MEASURES. 101 
 
 GREATEST COMMON MEASURE. 
 
 95. The measures of 12 are 1, 2, 3, 4, 6, 12, and the 
 measures of 18 are 1, 2, 3, 6, 9, 18. These two numbers 
 have the measures 1, 2, 3, G in common, and of these 
 measures 6 is the greatest. 
 
 The measures that two or more numbers have in common 
 are called their common measures, and the greatest of these 
 is called their Greatest Common Measure, which, for the sake 
 of brevity, is denoted by the letters G. C. M. 
 
 If two or more numbers have no common measure they 
 are said to be prime to each other. Thus, 27 and 125 are 
 prime to each other . 
 
 96, The prime factors of 12 are 2 2 , 3. 
 The prime factors of 18 are 2, 3 2 . 
 
 The prime factors common to 12 and 18 are 2, 3. The 
 G.C.M. of 12 and 18, namely 6, is 2 X 3. 
 
 That is, the G. C. M. of two or more numbers is, 
 
 The product of the prime factors common to the numbers, 
 each prime factor having the least exponent that it has in 
 any one of the numbers. 
 
 Hence, to find the G.C.M. of two or more numbers, 
 
 Separate the numbers into their prime factors. 
 
 Select the lowest power of each factor that is common to 
 the given numbers, and find the product of these powers. 
 
 Find the G.C.M. of 84, 105, 63. 
 
 84 
 42 
 21 
 
 105 
 35 
 
 7 
 
 7 v 
 
 84 = 2 2 X < 7x 105 = \ x 5 x 7. 63 = x 7. 
 Hence, G.C.M. = 3 X 7 or 21> 
 
 01 \ v \ 
 
102 MULTIPLES AND MEASURES. 
 
 97, Common factors of two or more numbers may be 
 taken out of the numbers simultaneously, as follows : 
 
 84 
 
 105 
 
 63 
 
 28 
 
 35 
 
 21 
 
 4 
 
 5 
 
 3 
 
 The number 3 is seen to be a factor of all the numbers, and 7 of 
 the resulting quotients 28, 35, 21. The quotients 4, 5, and 3 have no 
 common factor. Therefore, 3 and 7 are the only common factors, and 
 the G.C.M. is 3 x 7, or 21. 
 
 Ex. 63. 
 Find the G.C.M. of: 
 
 1. 48, 128. 15. 216, 360. 28. 336, 884. 
 
 2. 36, 90. 16. 279, 403. 29. 352, 364. 
 
 3. 64, 256. 17. 294, 378. 30. 1344, 1536. 
 
 4. 24, 105. 18. 210, 294. 31. 21, 35, 56. 
 
 5. 125, 600. 19. 182, 196. 32. 42, 133, 56. 
 
 6. 56, 138. 20. 225, 375. 33. 32, 48, 128. 
 
 7. 63, 108. 21. 195, 299. 34. 27, 36, 108. 
 
 8. 40, 600. 22. 288, 360. 35. 96, 48, 60, 108, 
 
 9. 65, 91. 23. 133, 152. 36. 33, 297, 198. 
 
 10. 39, 273. 24. 23, 111. 37. 56, 63, 315. 
 
 11. 56, 126. 25. 352, 384. 38. 75, 225, 500. 
 
 12. 232, 493. 26. 123, 579. 39. 232, 290, 493. 
 
 13. 365, 511. 27. 960, 1536. 40. 365, 511, 803. 
 
 14. 148, 592. 
 
 98, When it is required to find the Gr.C.M. of two or 
 more numbers that cannot readily be separated into factors, 
 the method to be employed is as follows : 
 
MULTIPLES AND MEASURES. 103 
 
 Find the G.C.M. of 63 and 217. 
 
 OPERATION. 
 
 63)217(3 
 189 
 
 28)63(2 
 56 
 
 7)28(4 
 28 
 
 Therefore, the G.C.M. is 7. 
 Hence, by this method, 
 
 Divide the greater number by the less, and then the divisor 
 by the remainder left, and so on till there is no remainder. 
 The last divisor will be the G. C. M. required. 
 
 To find the G.C.M. of several numbers, find the G.C.M. 
 of two of the numbers, then of that result and a third num- 
 ber, and so on. The last G.C.M. is the one required. 
 
 Ex. 64. 
 Find the G.C.M. of: 
 
 1. 342, 665. 6. 1131, 2639. 11. 3927, 5049. 
 
 2. 841, 899. 7. 9889, 986. 12. 1287, 1551. 
 
 3. 961, 1178. 8. 1792, 1832. 13. 1537, 1802. 
 
 4. 1243, 1469. 9. 1847, 1792. 14. 3056, 3629. 
 
 5. 1001, 1287. 10. 1850, 1517. 15. 2108, 3813. 
 
 16. 4844, 5536. 22. 216, 105, 405. 
 
 17. 696, 1305. 23. 112, 192, 128. 
 
 18. 232, 3219. 24. 168, 132, 352. 
 
 19. 949, 1387. 25. 198, 495, 209, 660. 
 
 20. 1081, 1311. 86. 146, 730, 365, 219. 
 
 21. 4067, 2573, 27. 924, 378, 612, 246. 
 
104 MULTIPLES AND MEASURES. 
 
 LEAST COMMON MULTIPLE. 
 
 99, The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 
 30, etc. 
 
 The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, etc. 
 The multiples common to 3 and 5 are 15, 30, etc., and of 
 these 15 is the least. 
 
 100, The multiples that two or more numbers have in 
 common are called their common multiples, and the least of 
 these is called their Least Common Multiple, which is denoted 
 by the letters L.C.M. 
 
 Find the L.C.M. of 7, 8, 9, 21. 
 
 The L.C.M. of 7, 8, 9, 21, must contain the factor 7, or 
 it would not be a multiple of 7. It must also contain 2 3 to 
 be a multiple of 8, and 3 2 to be a multiple of 9. It must 
 contain the factors 3 and 7 to be a multiple of 21. That is, 
 the L.C.M. of 7, 8, 9, 21, is the product of the factors 7, 2 s , 
 3 2 ; therefore, it is 7 X 8 X 9 = 504. Hence, 
 
 To find the L. C. M. of two or more numbers, 
 Separate each number into its prime factors. 
 Select from these the highest power of each factor, and find 
 the product of these powers. 
 
 Find the L.C.M. of 16, 21, 24, 30, 32. 
 16 = 2 4 , 
 21 = 3 x 7, 
 
 30 = 2 x 3 x 5, 
 32 = 2 5 . 
 
 Hence, the L, C. M, = 2 5 x 3 x 5 X 7 3360, 
 
MULTIPLES AND MEASURES. 105 
 
 The L.C.M. of 16, 21, 24, 30, 32, may be found as fol- 
 lows : 
 
 21 24 30 32 
 
 21 12 15 16 
 
 21 6 15 8 
 
 21 g 15 4 
 
 7 54 
 
 Hence, the L.C.M. = 2 3 x 3 x 7 X 5 x 4 = 3360. 
 
 Since 16 is a measure of 32, it is elided, for any multiple of 32 is 
 also a multiple of 16. The even numbers are divided by 2 ; the quo- 
 tients and the odd numbers are written below the horizontal line. This 
 operation is repeated so long as 2 is a measure of more than one 
 number. In the fourth line 3, a measure of 15, is elided. The divi- 
 sion by 3 leaves in the fifth line the numbers 7, 5, 4, which are prime 
 to each other. 
 
 Therefore, the factors contained in the numbers are 2, 2, 2, 3, and 
 7, 5, 4. 
 
 Hence, the L.C.M. = 2x2x2x3x7x5x4 = 3360. 
 
 When two or more numbers are prime to each other, their 
 L.C.M. is their product. Thus, the L.C.M. of 3, 5, 7, is 
 3x5x7. 
 
 Ex. 65. 
 Find the L.C.M. of: 
 
 1. 3, 9, 27, 54. 9. 22, 44, 88, 108. 
 
 2. 6, 9, 24, 40. 10. 15, 30, 45, 60. 
 
 3. 144, 12, 18, 96. 11. 8, 16, 24, 32. 
 
 4. 3, 8, 12, 22. 12. 13, 15, 26, 39. 
 
 5. 16, 30, 48, 15. /""13. 7, 17, 51, 119. 
 
 6. 12, 24, 63, 84. . - 14. 8, 6, 28, 32. 
 
 7. 9, 27, 33, 54. 15. 4, 21, 42, 63. 
 
 8. 12, 20, 36, 54, 16. 3, 6, 18, 22, 
 
106 MULTIPLES AND MEASURES. 
 
 17. 5, 15, 24, 30. 27. 16, 24, 13, 7. 
 
 18. 7, 2, 3, 5. 28. 5, 9, 14, 96, 128. 
 
 19. 13, 5, 2, 26. 29. 32, 36, 49, 56, 42. 
 
 20. 5, 10, 20, 100. 30. 20, 24, 25, 27, 45. 
 
 21. 19, 38, 2, 76. 31. 28, 30, 32, 36, 42. 
 
 22. 3, 9, 27, 81. 32. 35, 40, 42, 49, 28. 
 
 23. 6, 18, 22, 99. 33. 14, 18, 21, 32, 28. 
 
 24. 18, 26, 117, 312. 34. 24, 27, 32, 36, 56. 
 
 25. 13, 26, 39, 65. 35. 21, 24, 27, 28, 35. 
 
 26. 9, 36, 3, 45. 36. 28, 32, 56, 72, 96. 
 
 101, If the given numbers are large, and contain no 
 prime factors that can readily be detected, the common 
 factors may be obtained by the process of finding the 
 G.G.M. under like circumstances. 
 
 Find the L.C.M. of 1189 and 2117. 
 
 1189)2117(1 ) 
 
 1189 
 
 928)1189(1 
 928 
 
 261)928(3 
 783 
 
 145)261(1 
 145 
 
 116)145(1 
 116 
 29)116(4 
 
 116 
 
 Hence, the G.C.M.=29. 
 Therefore, 1189 = 29 X 41 ; 2117 = 29 X 73. 
 Therefore, the L.C.M. = 29 X 41 X 73 = 1189 X 73. 
 
MULTIPLES AND MEASURES. 107 
 
 From this process it will be seen that : 
 
 The L.C.M. of two numbers may be found by dividing 
 one of the numbers by their G.C.M., and multiplying the 
 quotient by the other number. 
 
 Ex. 66. 
 
 Find the L.C.M. of: 
 
 1. 510 and 595. 7. 187 and 255. 
 
 2. 217 and 643. 8. 1261 and 663. 
 
 3. 506 and 308. 9. 255 and 357. 
 
 4. 296 and 407. 10. 432 and 840. 
 
 5. 645 and 275. 11. 949 and 2920. 
 
 6. 468 and 923. 12. 1247 and 1769. 
 
 Ex. 67. 
 
 1. A farmer owns 132 acres of wood-land, and 99 acres 
 
 of pasture ; he wishes to divide them into equal lots 
 of the largest possible size. How many lots will 
 there be, and what number of acres in each one ? 
 
 2. A merchant has 75 yards of one kind of silk, 225 of a 
 
 second, and 200 of a third ; if he cut them into dress 
 patterns of equal size, what is the largest number 
 of yards which each pattern can contain ? 
 
 3. Simeon Jones has 260 bushels of rye, 384 of oats, and 
 
 416 of wheat. He sends his grain to market in bags 
 of equal size. What is the greatest number of 
 bushels which each bag can hold, provided there is 
 no mixture of the different kinds of grain ? 
 
 4. What width of carpet will fit three rooms, the first 15 
 
 feet wide, the second 21 feet, and the third 33 feet ? 
 
108 MULTIPLES AND MEASURES. 
 
 5. A milk-man has four different measures, holding 2, 3, 
 
 5, and 6 quarts respectively. What is the smallest 
 vessel that can be exactly filled by each of them ? 
 
 6. Find the length of the greatest line that exactly meas- 
 
 ures the sides of an enclosure 216 yards long and 
 111 broad. 
 
 7. Find the contents of the smallest vessel that may be 
 
 filled by using a 4-quart, a 5-quart, or a 6-quart 
 measure. 
 
 8. Two apprentices carry 1147 and 961 ivory balls, re- 
 
 spectively, from the workshop to the showroom. 
 The balls are carried in baskets of equal size, which 
 are filled and emptied several times. How many 
 balls in a basketful ? 
 
 9. Find the shortest distance that three lines, 8 feet, 9 
 
 feet, and 12 feet long will exactly measure. 
 
 CANCELLATION. 
 
 102. Cancellation is the operation of striking out equal 
 factors from the dividend and the divisor. 
 
 12 X 3 
 
 Find the quotient of - 
 
 o X ^ 
 
 The 3 in the divisor cancels the 3 in the dividend. Then the fac- 
 tor 2 of the divisor is cancelled, and 2 is cancelled from 12 in the 
 dividend. The resulting dividend is 6x1, and the divisor 1x1, 
 and therefore the quotient is 6. 
 
 Cancellation is used to shorten arithmetical work, 
 
MULTIPLES AND MEASURES. 109 
 
 Ex. 
 l 9 X 18 X 24 
 
 68. 
 11. 
 
 12. 
 13. 
 14. 
 15. 
 16. 
 17. 
 18. 
 19. 
 20. 
 
 39 X 18 X 9 
 
 12 X 3 X 6 
 15 X 36 X 27 
 
 13 X 27 X 2 
 50 X 9 X 84 
 
 9x12x5 
 12 X 6 X 25 
 
 12 X 3 X 75 
 
 144 X 81 x 2 
 
 5 X 12 X 3 
 d 19 X 27 X 30 
 
 27x96 
 625x9 
 
 9x15x3 
 45 x 16 X 9 
 
 75x3 
 75 x 9 X 96 
 
 8x15x3 
 32 x 49 X 6 
 
 12 x 15 x 9 
 87 x 15 x 9 
 
 3 x 16 X 7 
 7 1728 
 
 5 x 9 x 29 
 84 X 91 x 8 x 20 
 
 ' 12x12x12 
 8 25 x 6 X 28 
 
 32 X 60 x 13 
 169 X 196 X 16 
 
 6 X 4 X 35 
 9 6 X 54 x 7 X 24 
 
 26 X 14 X 56 
 60 X 9 X 90 X 7 
 
 7 X 8 X 9 X 12 
 10 64 x 105 x 12 
 
 42 x 27 X 15 
 13 X 19 X 17 X 20 
 
 21 X 12 X 8 X 8 
 
 17 X 260 X 4 X 361 
 
 PRACTICAL APPLICATIONS. 
 
 A man carried to a store 49 bushels of potatoes, which 
 he sold at 35 cents a bushel, and took his,pay in sugar at 7 
 cents a pound. How many pounds of sugar did he receive ? 
 SOLUTION. 1 bushel of potatoes is sold for 35 cents. 
 49 bushels are sold for 49 X 35 cents. 
 1 pound of sugar is bought for 7 cents. 
 Number of pounds bought for 49x35 cents is 49 * 35 - 
 
 04* 
 
 Ans. 245 pounds, 
 
110 MULTIPLES AND MEASURES. 
 
 Ex. 69. 
 
 1.* How many yards of cloth, at $3 a yard, can be bought 
 for 12 tons of hay, at $ 15 per ton. 
 
 2. How many pairs of boots, at $4 a pair, can be bought 
 
 for 40 pounds of butter, at 40 cents per pound ? 
 
 3. How many jars of lard of 36 pounds each, at 8 cents 
 
 per pound, must be given for 16 pieces of cloth con- 
 taining 24 yards each, at 48 cents a yard? 
 
 4. How many coats, at $4 each, can be bought for 32 
 
 yards of broadcloth, at $2.50 a yard? 
 
 5. A milkman having 30 cows which daily give 8 quarts 
 
 each, sells the milk at 5 cents per quart. How many 
 pieces of cloth containing 40 yards each, at 12 cents 
 per yard, ought he to receive for the milk of 6 days? 
 
 6. A market gardener sold 16 lots of celery, 120 bunches 
 
 in each, at 28 cents per bunch ; how many 240-pound 
 barrels of sugar, at 8 cents a pound, will the celery 
 pay for? 
 
 7. John Peters sold 9 firkins of butter weighing 78 pounds 
 
 each, at 25 cents per pound ; how many pieces of 
 matting having 45 yards in a piece, at 30 cents per 
 yard, should he receive? 
 
 8. A workman has received for 15 days' work of 7 hours 
 
 each, 21 dollars. How much would he receive for 
 19 days' work of 5 hours each ? 
 
 9. Thirty workmen have made in 9 days 215 yards of 
 
 wall. At the same rate, how much would 36 work- 
 men make in 15 days? 
 
 10. A telegraph operator transmits 50 words, averaging 4 
 letters each, in the space of 5 minutes. At the same 
 rate, how many minutes will be required to send a 
 dispatch of 120 words, averaging 5 letters each ? 
 
CHAPTER VIII. 
 
 COMMON FRACTIONS. 
 
 103, What is the name of one of the parts when a unit 
 is divided into : 
 
 1. Two equal parts? 6. Eight equal parts? 
 
 2. Three equal parts? 7. Ten equal parts? 
 
 3. Four equal parts? 8. Twelve equal parts ? 
 
 4. Five equal parts? 9. Sixteen equal parts? 
 
 5. Six equal parts? 10. Twenty equal parts? 
 
 A unit contains how many : 
 
 1. Halves? 5. Sixths? 9. Sevenths? 
 
 2. Thirds? 6. Eighths? 10. Ninths? 
 
 3. Fourths? 7. Tenths? 11. Elevenths? 
 
 4. Fifths? 8. Twelfths? 12. Thirteenths? 
 
 13. Twentieths? 15. Thirtieths? 
 
 14. Twenty-fourths? 16. Thirty-seconds? 
 
 When a unit is divided into twelve equal parts, what is 
 the name of: 
 
 1. One part? 4. Two parts? 7. Eight parts? 
 
 2. Three parts? 5. Four parts? 8. Nine parts? 
 
 3. Five parts? 6. Six parts? 9. Twelve parts? 
 
112 
 
 COMMON FRACTIONS. 
 
 Express in figures 
 
 1. Three-sevenths. 
 
 2. Five-ninths. 
 
 3. Seven-eighths. 
 
 4. Five-twelfths. 
 Read: ft, ft, A 
 
 , A, 
 
 5. Seven-sixteenths. 
 
 6. Five-eighteenths. 
 
 7. Four-elevenths. 
 
 8. Nine-twentieths. 
 
 H- tt- ii A- A- 
 
 104, The expression means : 
 
 I. Seven of the parts when a unit has been divided into 
 nine equal parts. 
 
 II. One-ninth of seven units ; for, if seven units be divided 
 into nine equal parts, one of these parts will be seven times 
 as great as one of the parts obtained by dividing one unit 
 into nine equal parts. 
 
 III. The quotient of seven divided by nine. 
 
 105, In the fraction , the lower figure shows the num- 
 ber of equal parts into which the whole has been divided, 
 and is therefore a divisor ; but, since it shows the number 
 of parts into which the whole has been divided, it shows 
 the name of each part, and is therefore called the denom- 
 inator, 
 
 The upper figure shows the number of these parts taken, 
 and is therefore called the numerator, 
 
 The figure, then, above the line denotes number, the fig- 
 ure below the line name. 
 
 106, The numerator and denominator are called the 
 terms of a fraction. 
 
 107, A proper fraction is one of which the numerator is 
 less than the denominator ; as -. 
 
COMMON FRACTIONS. 113 
 
 108, An improper fraction is one of which the numerator 
 equals or exceeds the denominator ; as -|, *-. 
 
 When the numerator is greater than the denominator, more than 
 one unit must be regarded as divided into equal parts ; thus, f 
 
 I i i i I i i i I i i i I 
 
 means that three units have been divided each into four equal parts, 
 and that all the parts of two units and one part of the third unit are 
 taken. 
 
 109, A mixed number is an expression consisting- of a 
 whole number and a fraction ; as 4-f-, 5.35. These expres- 
 sions are read four and three-sevenths, five and thirty-five 
 hundredths. 
 
 Every mixed number means that some entire units are taken, and 
 the fraction of another unit. 
 
 Select the proper fractions, the improper fractions, and 
 mixed numbers from the following expressions : 
 
 f , . H, i. 9f W. i if . sf , 
 
 . 6f , ^, 18f, f. 
 
 110, An improper fraction represents a quantity which 
 can also be represented by a whole number or else by a 
 mixed number. Thus, -^ 2f . 
 
 For, if we suppose several units to he divided each into seven 
 equal parts, and we take 19 of these parts, 14 (that is, 2x7) will 
 make 2 units, and the five remaining parts will be five-sevenths of 
 another unit. 
 
 111, To reduce an improper fraction to a whole or mixed 
 number, 
 
 Divide the numerator by the denominator. 
 
 The quotient will be the whole number, and the remainder, if 
 any, will be the numerator of the fractional part, of which the de- 
 nominator is the same as the denominator of the improper fraction. 
 
114 COMMON FRACTIONS. 
 
 
 
 Ex. 70. (Oral.) 
 
 
 
 Reduce 
 
 to whole 
 
 or mixed numbers : 
 
 
 
 1. J- 
 
 7. 
 
 -V*. 13. f. 
 
 19. 
 
 
 
 2. ty. 
 
 8. 
 
 _2. 14. 2^ 
 
 20. 
 
 - 
 
 3- 
 
 9. 
 
 15 - if- 
 
 21. 
 
 
 
 4- 
 
 10. 
 
 J^. 16. ^l. 
 
 22. 
 
 tf 
 
 5. ^. 
 
 11. 
 
 14 17 . .3^0 
 
 23. 
 
 
 
 6. tt. 
 
 12. 
 
 V- 18. Y- 
 
 24. 
 
 
 Ex. 71. 
 Reduce to whole or mixed numbers : 
 
 1. fj. 9. 3^. 17. ^. 25. 
 
 2. ^^.. 10. ^L. 18. ^Z-. 26. 
 
 3. -^tV* 11. AJM 19. ^234 27. 
 
 4. ^3, 12. ^. 20. - 3 4 5 2 y. 28. 
 
 5. 4^L. 13. m. 21. -HrfP* 29 - 
 
 6. 4.^.. 14. |!$. 22. ^^2. 30. 
 
 7. ^.. 15. 3^. 23. ^-. 31. 
 
 8. 300 16. AJJ1. 24. HP-. 32. 
 
 112, A whole number or a mixed number represents a 
 quantity which can also be represented by an improper 
 fraction. Thus, $ 3| = $J^. 
 
 For each dollar contains 4 fourths; therefore 3 dollars contain 
 3 X 4 fourths or 12 fourth*; which, together with the 3 fourths, make 
 15 fourths. Hence, 
 
 113, To reduce a mixed number to an improper fraction, 
 Multiply the whole number by the denominator of the 
 
 fraction, and to the product add the numerator ; under this 
 sum write the denominator. 
 
COMMON FRACTIONS. 115 
 
 114, A whole number may be expressed as a fraction 
 with any given denominator. Thus, 9 4^. 
 
 For, as 1 unit contains 7 sevenths, 9 units contain 9x7 sevenths, 
 or 63 sevenths. 
 
 A whole number may be written in the form of a frac- 
 tion with 1 for a denominator. Thus, 9 = -f-. 
 
 Ex. 72. (Oral.) 
 Reduce to improper fractions : * 
 
 1. 4*. 
 
 7. 12f 
 
 13. llf 
 
 19. 25f 
 
 2. 7f. 
 
 8. lOf 
 
 14. 13 T V 
 
 20. 30|. 
 
 3. 8f 
 
 9. 7|. 
 
 15. 7A- 
 
 21. 17 T V 
 
 4. Of 
 
 10. 3^. 
 
 16. 9&. 
 
 22. 40. 
 
 5. 5f 
 
 11. 8/ T . 
 
 17. 20-&. 
 
 23. 50f 
 
 6. 6f 
 
 12- 3&. 
 
 18. 15J. 
 
 24. 80|. 
 
 25. Change 12 to thirds; 8 to fourths; 7 to fifths; 9 to 
 halves ; 12 to ninths ; 13 to sixths ; 11 to sevenths ; 
 14 to eighths. 
 
 Ex. 73. 
 Change to improper fractions : 
 
 1. 15-^j. 
 
 6. 45^-. 
 
 1 1 . 5^2 7 
 
 16. 155|^. 
 
 2. 36H- 
 
 7. 56&. 
 
 12. 46^. 
 
 17. 17||. 
 
 3P\6 1 
 . U-Jr-A-. 
 
 8. 77^. 
 
 1 o cy 7 
 
 18. 167H. 
 
 4 < 318 
 
 9. 183^-. 
 
 14. 9f|. 
 
 19. 29||. 
 
 5. 12Jf , t 
 
 10. 72H- 
 
 15. 10^. 
 
 20. 29ff. 
 
 21. Change 25 to 94ths; 218 to 23ds; 375 to 87ths. 
 
116 COMMON FRACTIONS. 
 
 REDUCTION OF FRACTIONS TO LOWER TERMS. 
 
 115, If the numerator and denominator of a fraction be 
 both multiplied or both divided by the same number, the 
 value of the fraction is not altered. 
 
 J i i I i i I i i I i i t i If 
 
 ' D B P 
 
 Thus, if the line AB be divided into 5 equal parts at 
 the points (7, D, E } and F, then ^.Fis of AB. 
 
 Now, if each of the parts be sub-divided into 3 equal 
 parts, AB will contain 15 of these sub-divisions, and AF 
 12 of these sub-divisions. Therefore AF is -J-J of AB. 
 
 Since AFis $ of AB and also -J-J of AB, it follows that 
 Y$ . But $ is obtained from -J-J by dividing both 
 numerator and denominator by 3. Therefore, 
 
 To reduce a fraction to lower terms, 
 Divide the numerator and denominator by any common 
 factor. 
 
 A fraction is expressed in its lowest terms when both the numer- 
 ator and denominator are divided by the greatest common divisor. 
 
 Eeduce ff to its lowest terms. 
 
 336 84 21 8 
 T8T TinT flF 7' 
 
 The common factors cancelled are 4, 4, and 7. 
 
 Reduce |-f to its lowest terms. 
 
 Since no common factor can readily be detected, we find theG.C.M 
 
 259)333(1 
 259 
 74)259(3 
 
 222 
 37)74(2 
 
 74 
 
 Divide 259 and 333 each by 37, their G.C,M. Then f ff = f 
 
COMMON FRACTIONS. 117 
 
 Ex. 74. (Oral.) 
 
 Reduce to lowest terms by inspection : 
 
 1. 
 
 A- 
 
 8. 
 
 H- 
 
 15. f. 
 
 22. 
 
 if 
 
 2. 
 
 & 
 
 9. 
 
 H 
 
 16. |f 
 
 23. 
 
 
 
 3. 
 
 !i 
 
 10. 
 
 H- 
 
 17. if. 
 
 24. 
 
 If 
 
 4. 
 
 
 
 11. 
 
 A- 
 
 18. if 
 
 25. 
 
 H. 
 
 5. 
 
 H 
 
 12. 
 
 A- 
 
 19. A. 
 
 26. 
 
 H 
 
 6. 
 
 tt 
 
 13. 
 
 if- 
 
 20. |f 
 
 27. 
 
 H 
 
 7. 
 
 H. 
 
 14. 
 
 |f. 
 
 21. if. 
 
 28. 
 
 If. 
 
 Ex. 75. 
 
 Reduce to lowest terms by the method of inspection or 
 
 by the method of finding the G.C.M. 
 
 ! m- i3 - AV 2s - * 37. 
 
 2. ^ n o 14 | o 26 i 3 38 
 
 3- Vft- l g - Ht- 27. T?^V 39. 
 
 4. 1 ? ^ 16. ^ 1 7 28. ^ f %- 40 
 
 5- -^r- 17- iff. 29. fff. 41. |ff. 
 
 6. iff 18. |JJ. 30. T^. 42. ^/ft. 
 
 7- T^T- 19- Iff 31. Ml- 43. ||f. 
 
 8. ||f. 20. |f 32. ||f. 44. ^V. 
 
 9- -Ms- 21. ^j.. 33. f||. 45. iff. 
 
 10- t%- 22. f|. 34. |^. 46. |^|. 
 
 11. ^. 23. ff|. 35. iff 47. 
 
 12. i*4. 24. m. 36. 444. 48. 
 
 NOTE. In the answers to all examples, fractions should be left in 
 their lowest terms. 
 
118 COMMON TRACTIONS. 
 
 MULTIPLICATION OF FRACTIONS. 
 
 116, 7x3 horses = 21 horses. 
 7 X 3 fifths = 21 fifths. 
 
 If three like quantities are taken 7 times, the result will 
 be 7 times 3 of the same quantities. 
 
 f X 15 means of 15, which equals 9. Hence, 
 
 117, To find the product of a whole number and a 
 fraction, 
 
 Find the product of the numerator and whole number, 
 and divide the result by the denominator. 
 
 A factor common to the whole number and the denominator of 
 the fraction may be cancelled. For, cancelling a factor common to 
 the whole number and the denominator of the fraction before the 
 multiplication, is evidently equivalent to dividing the numerator 
 and denominator of the resulting fraction by that factor after the 
 multiplication. Which may be done by 115. 
 
 Ex. 76. (Oral.) 
 Find the products of : 
 
 1. 18 xf 10. 14 Xf 19. Sxft. 28. ISxfa. 
 
 2. 25 X f 11. 20 X f 20. 16 X fa. 29. 25 X fa. 
 
 3. 27 Xf 12. 16 Xf 21. 36X-&. 30. fa X 26. 
 
 4. 10 Xf- 13. 7xf 22 - 24 Xf 31. x9. 
 
 5. 24 X f 14. 16 X f . 23. 32 X fa. 32. T 7 7 X 12. 
 6- 12 X f 15. ft X 10. 24. 12 X fa. 33. 22 X fa. 
 7 21 X f 16. ft X 12. 25. 27 X f 34. fa X 28. 
 8.-30xf 17. 27 Xf 26. 18 Xf 35. fa X 21. 
 9. 16 X f 18. ft X 15. 27. 12 X f 36. fa X 35. 
 
COMMON FRACTIONS. 119 
 
 118, To multiply a fraction by a fraction. 
 Multiply -fby f . 
 
 I . , I , . IP 
 
 |- multiplied by -| means -| of f. 
 
 If the line AB be divided into 5 equal parts at the 
 points <7, D, E, and F, AF will be \ of AB. Now, if 
 each part be sub-divided into three equal parts, there will 
 evidently be 15 such parts in the whole line, and each part . 
 will be ^5- of the line. 
 
 That is, ^ of -J- is y 1 ^ of the whole. 
 
 J of f will be ^ + -^ + ^4^, or ^ of the whole. 
 And | of f will be twice -fe ; that is, -^ of the whole. 
 Therefore, 
 
 To multiply a fraction by a fraction, 
 
 Find the product of the numerators for the required 
 numerator, and of the denominators for the required denom- 
 inator. 
 
 Mixed numbers must first be reduced to improper fractions. 
 Any factor common to a numerator and denominator should be 
 cancelled before the multiplication. 
 
 (2) tf 
 
 Reducing the mixed number to an improper fraction, we 
 have, 
 
 H x f| x H. 
 
 By cancellation, 
 
 , 2 1 
 
 3 
 
120 COMMON FRACTIONS. 
 
 Ex. 77. (Oral.) 
 Find the products of : 
 
 I- t X f . 4. } X f . 7. f X J. 10. ^ X 
 
 2. 1- X f . 5. t X f 8. ^ X f 11. i- X Y- 
 
 3- J X f 6. T V X f 9. ^ X 10. 12. iJ X f 
 
 Ex. 78. 
 Find the products of: 
 
 1. tX. 18. Si of A of A- 
 
 2. f X4f 19. -ft of 2$ of 21. 
 
 3. 2 X -ft X f 20. J of l^j of 3f 
 
 4. ^XS-ft-Xf 21. 8|Xfoff 
 
 5. | of H of If 22. ICx^of^. 
 
 6. | X | X f 23. 3 X 7^ X H X 3^. 
 
 7- |fX|Xf 24. A f fX* XHX77. 
 
 8. 3 X 6J X H. 25. H X f X ft X J. 
 
 9- Hx9iX^. 26. ^x^XifXii. 
 
 10. 2 X f X &. 27. 3^ X 4J X 15. 
 
 11. 2f of f of -fa. 28. T % of 7 X H of 87^-. 
 
 12. 2^of^of^f. 29. llf x 16^ r of ^ of 
 
 13. f X 7* X A- 30. ^ of | of 2| X 15f 
 
 14. |x7|xA- 81. 8^ of AX 2} off 
 
 15. 2fc of 3$ of ^. 32. 3| of f^ of 6| of f. 
 
 16. ^ of T T of 7f 33. I X 2J X 4^ X A- 
 
 17. 3| of | of H. 34. f X 2f X 7| X 4t X f 
 
COMMON FRACTIONS. 121 
 
 35. 2ixxfx3fx5i. 38. ^ 
 
 36. | X | X 7 X 5J X 6f 39. 3| of -ft of of 10. 
 
 37. 51 X ff X 7 X M X 7J. 40. ft X 21 X 3f X 2J X ft. 
 
 119, When the product of a mixed number and a whole 
 number is required, it is generally best to find the product 
 of the whole number and the fractional part of the mixed 
 number, then the product of the whole number and the 
 integral part of the mixed number, and combine the results. 
 Thus, 
 
 The product of 9 times 7 is found as follows : 
 
 OPERATION. 
 
 1 
 
 64| 
 
 Here 9 times J equals 1 J, the J is written, and the 1 is carried to 
 the product of 9 x 7, making 64. 
 
 Ex. 79. 
 Find the products of : 
 
 1. 
 
 3x4f 
 
 11. 
 
 5f 
 
 X9. 
 
 21. 
 
 20 x 5. 
 
 2. 
 
 5x7|. 
 
 12. 
 
 2^ 
 
 frXl5. 
 
 22. 
 
 4f x 17. 
 
 3. 
 
 21 x 18f 
 
 13. 
 
 2f 
 
 ^X20. 
 
 23. 
 
 5| x 18. 
 
 4. 
 
 22 X 29ft. 
 
 14. 
 
 H 
 
 Xl2. 
 
 24. 
 
 6| x 15. 
 
 5. 
 
 25 x 12f . 
 
 15. 
 
 n 
 
 X8. 
 
 25. 
 
 9| X 21. 
 
 6. 
 
 6x2f 
 
 16. 
 
 6f 
 
 X9. 
 
 26. 
 
 10J- X 41. 
 
 7. 
 
 7x2f. 
 
 17. 
 
 9x3f. 
 
 27. 
 
 11| x 32. 
 
 8. 
 
 8x2f 
 
 18. 
 
 12 
 
 X2J. 
 
 28. 
 
 15f X 36. 
 
 9. 
 
 6|x9. 
 
 19. 
 
 13 
 
 x3f 
 
 29. 
 
 16x40. 
 
 10. 
 
 3ft x 10. 
 
 20. 
 
 16 
 
 x 9f 
 
 30. 
 
 13| x 27. 
 
122 
 
 COMMON FRACTIONS. 
 
 DIVISION OF FRACTIONS. 
 
 120, When a product of two numbers is equal to 1, each 
 of these two numbers is called the reciprocal of the other. 
 
 Thus, 5 x J = 1. Hence, the reciprocal of J is 5, and the recipro- 
 cal of 5 is . Again, j x f = 1. Therefore, the reciprocal of is $, 
 and the reciprocal of ^ is J. 
 
 121, To multiply by the reciprocal of a number is 
 the same as to divide by the number. 
 
 Thus, to multiply by J, means to separate the multiplicand into 
 three equal parts, and to take one of the parts for the required pro- 
 duct ; and, to divide by 3 means to separate the dividend into three 
 equal parts, and to take one of the parts for the required quotient. 
 
 To multiply by means to separate the multiplicand into three 
 equal parts, and to take two of these parts ; and to divide by J, the 
 reciprocal of , means to divide the dividend into three equal parts. 
 and to take two of these parts. 
 
 Hence, to divide by a whole number or a fraction, 
 
 Multiply by its reciprocal. 
 
 Thus, *-*-2 = ixf = A- 
 
 Mixed numbers must first be reduced to improper fractions. 
 
 Ex. 8O. (Oral.) 
 Find the quotients of : 
 
 2. ftn-4. 
 
 3. f -r-2. 
 
 4. ft -3. 
 
 5. i^-f-2. 
 
 6. f-5. 
 
 8. H- 
 
 9. ff--f- 
 
 10. A+ 
 
 11. I 
 12. 
 
 ia. -^--6. 
 
 14. -^--3. 
 
 15. ff--6. 
 
 16. ^ -3. 
 
 17. ft -4. 
 
 19. 
 20. 
 21. 
 22. 
 23. 
 
 18. ft -12. 24. -8--- 
 
COMMON FRACTIONS. 123 
 
 Ex. 81. 
 Find the quotients of : 
 
 1. J|-12. 12. f^7 T V 23. f ^lf- 
 
 2. ff^25. 13.3^21. 24.^ + ^. 
 
 3. M-12- 14 - l + f 25. 3 ii^fr 
 
 4. ft H- 13. 15. f *f 26. 3|^9f 
 
 5. tf$-f. 19. 16. ioffn-f 27. 9 -+- 3f 
 
 6. AH- TV 17- ioff-A- 28. 8^^. 
 
 7. I-*-!. 18. iofSjH-H. 29. 19+1J. 
 
 8. 2J + J. 19. if-f-jf 30. 3f^3f 
 
 9- i+f 20. Jl-^f 31. ioff-frA- 
 
 10. |-n2f 21. 8|H-6|. 32. Ijof7i^-2f. 
 
 11. Gl-4f 22. 7|^8|. 33. 2Joflf-H2J. 
 
 34. 1 -H ^. of |. 40. 3| of 5J of 7| ^- 63. 
 
 35. f of 2J -:- H of 6f 41. 3} of 7| of If - Si. 
 
 36. |- of 4 H- -J of 3f. 42. ?iof S^H-l^of If 
 
 37. 2| of 1^ -* 5i of 3|. 43. 9 H- ^ of 1^. of 4|. 
 
 38. 2iof2i^f of3|. 44. lC 
 
 39. 1'i-s-liof A-of A- 45. 3f of 
 
 122, When a mixed number is to be divided by a whole 
 number, it is best to divide the integral part of the dividend 
 first, and then the fractional part. If there is a remainder 
 from dividing the integral part, this remainder may be 
 put with the fraction, and the result reduced to an improper 
 fraction, and then divided by the divisor. 
 
124 COMMON FRACTIONS. 
 
 Divide 16|- by 4 ; 16$ by 7. 
 
 OPERATION. OPERATION. 
 
 In the first problem we simply divide the whole number 16 by 4, 
 and then the fraction | by 4, and obtain the result at once, 4J. 
 
 In the second problem we divide the 16 by 7, and obtain the 
 quotient 2 and a remainder 2. The remainder is joined with the f , 
 making 2J = f , and J -j- 7 - /p 
 
 Ex. 82. 
 Find tbe quotients of : 
 
 1. 19$-*- 3. 
 
 6. 34$-*- 17. 
 
 11. 65^-^-9. 
 
 2. 12$-*- 5. 
 
 7. 31$ +-11. 
 
 12. 147$ -f- 13. 
 
 3. 24-f-8. 
 
 8. 371-7-18. 
 
 13. 76$-*- 19. 
 
 4. 19f-6. 
 
 9. 45^+7. 
 
 14. 124^-^6. 
 
 5. 17$-*- 9. 
 
 10. 57|-*- 16. 
 
 15. 326^15. 
 
 Ex. 83. 
 
 1. What must be paid for 24 yards of cloth, at $f per 
 
 yard? 
 
 2. A farmer bought 327 sheep, at $4 a head ; required 
 
 the cost of the flock. 
 
 3. At 25 cents a pound, what must be paid for 82J pounds 
 
 of butter? 
 
 4. A merchant sold 15 yards of silk, at $4S per yard ; 
 
 what change should he give back from 8 ten-dollar 
 bills? 
 
 5. If beefsteak cost 22 cents per pound, and mutton chops 
 
 21 cents, how much will a man pay for meat, who 
 eats J pound of beefsteak for breakfast, and li 
 pounds of mutton chops for dinnpr ? 
 
COMMON FRACTIONS. 125 
 
 6. At $|- per yard, how much cloth can be bought for 
 
 $25? 
 
 7. If $19} be paid for 9 yards of silk, what is the cost 
 
 per yard ? 
 
 8. A man walks 3 7^ miles in 6 hours ; how many miles 
 
 does he walk an hour ? 
 
 9. A farmer sells 19} acres of land for $375 ; what is the 
 
 price per acre ? 
 
 10 A lady pays $3 for -| of a yard of silk ; what is the 
 price per yard ? 
 
 11. A man in one year pays $45.26 for cigars, the average 
 
 price of which is 6-J- cents apiece. How many does 
 he smoke in a year ? 
 
 12. If - of an acre of tillage, land cost $125, what is the 
 
 price per acre? How many acres can be bought 
 for $1297? 
 
 13. Gideon Lyford earns $30 per week ; what will remain 
 
 at the close of the week when he has paid for 6 
 pounds of butter, at 33 cents a pound, 10} pounds 
 of mutton, at 20 cents a pound, 8$ pounds of beef, 
 at 25 cents, 3 boxes of strawberries, at 16 cents, 150 
 pounds of ice, at } cent, 20 loaves of bread, at 10 
 cents, fuel $2, vegetables $3? 
 
 14. Find the product of 17f X 8 of 6^. 
 
 15. If a man build ^ of a rod of wall in one hour, how 
 
 much will he build in of an hour? 
 
 16. If a ship costs $16,785, what will f of it be worth ? 
 
 17. If a water-pipe discharges 16 J barrels of water in an 
 
 hour, how many barrels will it discharge in 9^- hours? 
 
 18. For 4 sheep $25f are paid; what is the price per 
 
 head? 
 
126 COMMON FRACTIONS. 
 
 19. A coal dealer paid $375 freight for transporting coal 
 
 from Scranton to Hudson. If the price was $ per 
 ton, how many tons were transported ? 
 
 20. How many pounds of beef, at 18} cents per pound, 
 
 can be bought for $17.48? 
 
 21. A farmer hires an equal number of men and boys, and 
 
 pays for a man and boy $2f a day. If the pay roll 
 is $84 a day, how many men and boys does he hire ? 
 
 22. When Spff^ acres of land cost $1297, what will of 
 
 an acre cost? 
 
 23. A city speculator in land divided } of an acre into lots 
 
 of -jJg- of an acre each, and sold them all for $ 13,426f . 
 What was the average price per lot ? 
 
 24. For of J of a ship the sum of $6394 was received ; 
 
 what is the value of the ship ? 
 
 25. A vessel sails 17$ miles per hour; how many miles 
 
 will she sail in 26$ hours? 
 
 26. George is 13 1 years old, Henry is } as old as George, 
 
 and John's age is 1-J that of Henry ; what is the 
 age of John ? 
 
 27. There are 16J feet in one rod; how many feet are 
 
 there in 84|- rods? 
 
 28. How many feet around a field, each one of whose four 
 
 sides measures 7-| rods ? 
 
 29. A schooner sails on the average 175-| miles a day; 
 
 how far will she sail in a week ? 
 
 30. At the rate of 8f miles per hour, how many miles will 
 
 a ship sail from a quarter past three A.M. to a 
 quarter before six P.M. ? 
 
 31. Reduce f of ^ of -^ of |-| to a simple fraction in its 
 
 lowest terms. 
 
COMMON FRACTIONS. 127 
 
 32. George Ward inherited from his father -J-J- of a farm 
 
 containing 377 acres. He divided his share equally 
 among his four sons ; how many acres would each 
 one of the sons receive ? 
 
 33. How many pounds of sugar, at 9} cents a pound, can 
 
 be bought for $1.52? 
 
 34. For 231 baskets of peaches, a grocer gave $20.59; 
 
 what was the price per basket ? 
 
 35. A farmer sold 42 bushels of potatoes for $26.58 ; what 
 
 was the average price per bushel ? 
 
 36. At 37} cents per yard, how many yards of lace can be 
 
 bought for $5i? 
 
 37. A farmer sold 6} bushels of apples for $4.87} ; what 
 
 was the price per bushel ? 
 
 38. When 6} bushels of apples bring $3.90, what are they 
 
 worth a peck ? (Four pecks make a bushel.) 
 
 39. At 60 cents a pound, how many pounds of tea can be 
 
 bought for $4.65? 
 
 40. If i^j- of a yard of cloth cost 80 cents, what should be 
 
 paid for -j^- of a yard ? 
 
 41. The cost of fencing a lot 8} rods in circuit is $6, 
 
 what is the rate per rod ? 
 
 42. A roll of carpeting containing 202 yards is cut into 
 
 pieces of 25 1 yards each, and each piece is sold for 
 $32}. Required the number of pieces, and the 
 price per yard. 
 
 43. When 35} bushels of turnips cost $28.60, what should 
 
 be paid for } of a bushel ? 
 
 44. How many yards of cloth can be bought for $10.80, 
 
 if $ of a yard cost 63 cents ? 
 
128 
 
 COMMON FRACTIONS. 
 
 LEAST COMMON DENOMINATOR. 
 
 123. A fraction is changed to higher terms by multipli- 
 cation. 
 
 Reduce to twelfths. 
 Multiply both terms by 3 ; thus, 
 
 3x8 24 
 3x4 = 12 
 
 In either of the two forms f or f J the value of the fraction is 2. 
 
 124. Hence, to reduce a fraction to higher terms, 
 
 Multiply both terms of the fraction by that number which 
 will change the given denominator to the required denomi- 
 nator. 
 
 The required multiplier is found by dividing the required denom- 
 inator by the denominator of the given fraction. 
 
 Reduce : 
 
 Ex. 84. (Oral.) 
 
 1. 
 
 | to 
 
 20ths. 
 
 8. 
 
 f to 
 
 27ths. 
 
 15. 
 
 fV 
 
 to 
 
 26ths. 
 
 2. 
 
 H 
 
 lOths. 
 
 9. 
 
 rr to 
 
 33ds. 
 
 16. 
 
 TV 
 
 to 
 
 36ths. 
 
 3. 
 
 I to 
 
 9ths. 
 
 10. 
 
 ^ to 
 
 28ths. 
 
 17. 
 
 f 
 
 to 
 
 Slsts. 
 
 4. 
 
 ^to 
 
 14ths. 
 
 11. 
 
 A to 
 
 36ths. 
 
 18. 
 
 A 
 
 to 
 
 96ths. 
 
 5. 
 
 | to 
 
 18ths. 
 
 12. 
 
 ft to 
 
 20ths. 
 
 19. 
 
 & 
 
 to 
 
 44ths. 
 
 6. 
 
 f to 
 
 12ths. 
 
 13. 
 
 fa to 
 
 45ths. 
 
 20. 
 
 1 
 
 to 
 
 16ths. 
 
 7. f to 24ths. 14. - 2 ^ to lOOths. 21. ^ to 72ds. 
 
 125, Similar fractions are fractions that have a common 
 denominator. 
 
 All fractions must be expressed as similar fractions before they 
 can be added or subtracted, and in all cases it is best to express them 
 with the least common denominator. (L.C.D.) 
 
COMMON FRACTIONS. 129 
 
 126. The least common denominator of two or more 
 fractions is the least common multiple of their denominators. 
 
 Reduce -J-, J, -J- to similar fractions. 
 
 The least common multiple of the denominators 2, 3, 4 is 12. It 
 is therefore necessary to reduce J, J, J to 12ths, by the method ex- 
 plained in \ 124, and we have ^, T \, -fa. 
 
 127, Hence, to reduce fractions to similar fractions, 
 Find the least common multiple of the denominators ; 
 
 this will be the required denominator. Divide this denomi- 
 nator by the denominator of each fraction. 
 
 Multiply the first numerator by the first quotient, the sec- 
 ond numerator by the second quotient, and so on. 
 
 The products will be the numerators of the equivalent 
 fractions. 
 
 Ex. 85. (Oral.) 
 Reduce to similar fractions : 
 
 1- i, i 5. i, f 9. i, *, A- 13. |, f, J. 
 
 2- *. i 6- i f 10. |, |, TV 14. f, |, ft. 
 
 3- i, i 7. f f. 11. f , f . 15. |, f A. 
 4. i f 8. |, f. 12. f f f 16. f f f 
 
 Ex. 86. 
 Keduce to similar fractions : 
 
 1- if, If, T^r- 6. |, ^ if, ff. 
 
 2- A, A- H- 7. f f ,-A, A- 
 
 3 - A. A- f 8 - i, f if- , AT- 
 
 4- A, A- f 9- t, A, , If - A- 
 
 5 - f A. A- 10. |, f, f A, if 
 
130 COMMON FRACTIONS. 
 
 ADDITION OF FRACTIONS. 
 
 128. Add |, f,f 
 
 These fractions reduced to similar fractions become ^y, y 9 ^, T , and 
 A + ft + ii-H-2A-2J- 2 J. An*. 
 
 129. Hence, to add fractions, 
 
 Reduce the fractions to similar fractions, and write the 
 sum of the numerators over the common denominator. 
 
 Add |, A, A- 
 
 ~ 8 12 15 
 
 6 15 
 
 2 3 15 
 
 2 1 5 
 Hence, L. C. D. - 2 s x 3 x 5 = 120. 
 
 Numerators . . . 
 
 Sum of numerators = 
 Therefore, sum of fractions = f $ = J$ = IfJ. Iff Ans. 
 
 130, If any of the expressions are integers or mixed 
 numbers, add together separately the integers and the frac- 
 tions, and find the sum of the results. 
 
 Find the sum of 2&, !&, 5|. 
 
 L. C. D. of the fractions = 2 2 X 3 x 5 = 60. 
 
 f 9 
 
 Numerators . . . . < 28 
 
 I 
 
 Sum of numerators = 92 
 
 Sum of fractions = f f = f | - 1^ 
 
 Sum of integers = 2 + 1 + 5 = 8 
 
 9A. Ans. 
 Ex. 87. (Oral.) 
 Find the sum of : 
 
 i- A> A- 3 - !> f 5 - tt. A- 7 - 7 i 3 i 
 
 2- A, A- 4. A, A- 6. ft, 8. 8|, 4f 
 
COMMON FRACTIONS. 131 
 
 9. 5f , 4f 13. f , |. IT. i, A. 21. 8ft, 7f 
 
 10. 9fV, 5 T V 14. |, |. 18. |, ||. 22. 6f , 5J. 
 
 11. 8f, 5f. 15. -1%, f. 19. SjV, 4A- 23. 7f, 4f 
 
 12. TTV, 3iV 16. |, ^. 20. 2-f, 8f. 24. Oft, 8f 
 
 Ex. 88. 
 Find the sum of: 
 
 1- i, i f 13- A, f, f ft- 
 
 2- i A- f 14. f , |, H, 'TV- 
 S' I. f. i 15. -fr, A, H. f 
 
 4- I f, f 16- H. M- i T 3 ?- 
 
 5- f f f 17. i A- A. A- 
 
 6- I i 18- A. A- H- 
 
 7. 4f 3|, 6A- 19- I i, f, I, I H- 
 
 8. 7|, 8H, 9|. 20. |, A, f, ||, A, 
 8A. 4|. S|. 21. |, A, H, If, , 
 
 10. 4|, 5|, 6|. 22. H, , A. if, A- fi- 
 
 ll. 9f, 4f, 8f. 23. |f, tf. If, , i|, . 
 
 12. 4|f, 5|f, 6^. 24. |, |, ^, ff, , |i. 
 
 25. 24|, 13f 36|, 60|. 47f. 
 
 26. 35|, 17f 25f, 48|, 18J. 
 
 27. 54|, 28|, 16|, 36f, 64|. 
 
 28. 36f 87f 59f, 54f 16|. 
 
 29. 23f, 32f, 18f, 27|, 28|. 
 
 30. 74^, 64H, 4S&, 2SH, 
 
132 COMMON FRACTIONS. 
 
 SUBTRACTION OF FRACTIONS. 
 
 131. From | take -ft. 
 
 24 = 2 3 x 3, 
 18 = 2 X 3 2 . 
 
 Hence, die L.C.D. = 2 3 X 3 2 = 72. 
 
 132, Hence, to subtract one fraction from another, 
 
 Reduce the fractions to similar fractions. 
 Subtract the numerator of the subtrahend from the numer- 
 ator of the minuend. 
 
 Write the result over the common denominator. 
 
 133, If the terms are mixed numbers, subtract separately 
 the integers and fractions, and unite the results. 
 
 Subtract 5f from 15f . 
 Here theL.C.D. = 8. 
 
 15f - 5f = 10*^ = 10f 10f Ans. 
 
 Subtract 3f from 
 
 = Hi 
 
 Ans. 
 
 The difference between 5 T \ and 3| is 2^-^-- Since if cannot be 
 subtracted from JJ, 1 is taken from 2, and added to JJ, making f J. 
 
 From 9 take ||. 
 
COMMON FRACTIONS. 133 
 
 
 
 Ex 
 
 89. 
 
 
 
 Find the value of : 
 
 i- H-A. 
 
 25. 
 
 14- 
 
 iV- 
 
 49. 
 
 24^-16|i. 
 
 3. n-n- 
 
 26. 
 
 21- 
 
 if- 
 
 50. 
 
 92| - 73f . 
 
 3. ii -M- 
 
 27. 
 
 20- 
 
 H 
 
 51. 
 
 19 T \-14|f. 
 
 4- fi-4i 
 
 28. 
 
 42 
 
 tt 
 
 52. 
 
 23| - 16f. 
 
 5- tt-W- 
 
 29. 
 
 25 -|f. 
 
 53. 
 
 15|-12|. 
 
 6. $-f. 
 
 30. 
 
 21- 
 
 tt- 
 
 54. 
 
 42|-14|. 
 
 7. f-f. 
 
 31. 
 
 14- 
 
 If. 
 
 55. 
 
 24-^- - 15f . 
 
 84 1 
 * "5" IT* 
 
 32. 
 
 13- 
 
 A 
 
 56. 
 
 72f-28f 
 
 9 - A -A- 
 
 33. 
 
 24- 
 
 13f. 
 
 57. 
 
 19|-13|. 
 
 10. |-f. 
 
 34. 
 
 42- 
 
 15ft. 
 
 58. 
 
 26f-19|. 
 
 11. |i f. 
 
 35. 
 
 20- 
 
 1%. 
 
 59. 
 
 45^-26ft. 
 
 12. f-f. 
 
 36. 
 
 84- 
 
 37|f 
 
 60. 
 
 34f-16f. 
 
 1 Q 8 5 
 
 1O. ^- T^J-. 
 
 37. 
 
 21 A 
 
 -11- 
 
 61. 
 
 34|-18f. 
 
 14- A -A- 
 
 38. 
 
 27f-l. 
 
 62. 
 
 64f-28|f. 
 
 15. H-A. 
 
 39. 
 
 42 T\ 
 
 -A- 
 
 63. 
 
 48f-19f 
 
 16- M-li- 
 
 40. 
 
 26 ^ 
 
 -H 
 
 64. 
 
 76|-72ft. 
 
 i n 2 i 
 "5" "S"* 
 
 41. 
 
 43 li 
 
 -H- 
 
 65. 
 
 97|-32||. 
 
 is. tf-f 
 
 42. 
 
 27A 
 
 -if- 
 
 66. 
 
 90^ -9f. 
 
 19. If ~|f 
 
 43. 
 
 1A 
 
 -f 
 
 67. 
 
 78ft -56f. 
 
 20. H-^. 
 
 44. 
 
 3 -H 
 
 -'Jl- 
 
 68. 
 
 96| - 49|. 
 
 QI l 9 2 3 
 
 45. 
 
 83ft 
 
 1 3 
 T6"' 
 
 69. 
 
 47| _ 43||. 
 
 oo l 1 1 3 
 
 46. 
 
 26ft 
 
 -A- 
 
 70. 
 
 55f-54f 
 
 23. |f T \. 
 
 47. 
 
 74A 
 
 -ri 
 
 71. 
 
 69ft -67f| 
 
 24 - U-H- 
 
 48. 
 
 68|- 
 
 -f 
 
 72. 
 
 69^-23|J. 
 
134 COMMON FRACTIONS. 
 
 Ex. 90. 
 
 1. A country merchant received on Monday $25f, on 
 
 Tuesday $19J, on Wednesday $231, on Thursday 
 $32f, on Friday $29J, on Saturday $37. What 
 had he left after paying a freight bill of $19f, and 
 to his clerk $ 12 J? 
 
 2. A farmer sold two loads of hay, one for $13i and the 
 
 other for $16f , and received $25 down. How much 
 is still due ? 
 
 3. A miner digs 17$, 19^, 18f ounces of gold. In wash- 
 
 ing there is a loss of 3$ ounces. How much gold 
 has he left? 
 
 4. Henry Cameron had three wheat-fields; the first pro- 
 
 duced 217f bushels, the second 309f , the third 419$. 
 He sent 516$ bushels to a flour mill, and sold 193 
 bushels. How many bushels had he left ? 
 
 5. From a piece of cloth containing 47-J- yards, 22$ yards 
 
 were sold, and then 5| yards were sold. How many 
 yards remained ? 
 
 6. A grocer sold 2| pounds of tea to one man, li pounds 
 
 more to a second man than to the first, and to a 
 third man 1} pounds less than the amount he sold 
 the first and second together. How many pounds 
 did he sell to the second man, and how many to 
 the third man ? 
 
 7. Of the prismatic spectrum red occupies $, orange 1 %, 
 
 and yellow -^. What part of the whole do these 
 three colors together occupy ? 
 
 8. What part of a piece of cloth has a merchant sold, who 
 
 has cut off and sold -f$, ^, -fo, and ^ of it ? 
 
COMMON FRACTIONS. 135 
 
 9. A treasurer has expended --, ^-, ^f, --$, and -$ of a 
 
 given sum. What part of the whole has he left ? 
 
 
 
 10. Of a pole % is blue, -f- red, and the rest white. What 
 
 part of it is white ? 
 
 11. A jeweller has used -j^-, -^, and -^ of an ingot of gold. 
 
 What part of it still remains ? 
 
 12. A student has read -j 5 T , ^, and of a certain book. 
 
 What part of it has he yet to read ? 
 
 13. A traveller has gone -J- of a journey on foot, ^ on 
 
 horseback, ^ by rail, and the rest by coach. What 
 part of the journey has he gone by coach? 
 
 14. Of the component elements of albumen ^ is carbon, 
 
 y^ hydrogen, and % nitrogen. What part of the 
 whole do these elements constitute? 
 
 15. Add together the greatest and least of the fractions, 
 
 f , , -J-J-, , and subtract this sum from the sum of 
 the other two fractions. 
 
 16. How many tons of ore must be raised from a mine so 
 
 that, on losing ^J in roasting, and -fe of the remain- 
 der in smelting, there may be obtained 506 tons of 
 pure metal? 
 
 17. A man invested $ of his capital in bank stock, f of the 
 
 remainder in real estate, and had left $6000. Find 
 his capital. 
 
 18. A man invests $ of his money in land, in bank stock, 
 
 -J- in railroad stock, and has $8000 left. What is 
 his fortune ? 
 
 19. A owns of a ship, and B the remainder; and f of 
 
 the difference between their shares is $1500, What 
 is the value of the ship ? 
 
136 COMMON FRACTIONS. 
 
 COMPLEX FRACTIONS. 
 
 134. The quotient f-^-f may be written in the form 
 
 $, in which the dividend is the numerator, and the divisor 
 
 I 
 
 the denominator of a complex fraction. 
 
 135. A complex fraction has a fraction in either its 
 numerator or denominator, or in both of them. 
 
 The reduction of complex to simple fractions is similar to division 
 of fractions. 
 
 Reduce t to a simple fraction. 
 
 Multiply both terms by 4 and we have -fc. 
 
 Reduce | to a simple fraction. 
 
 D 
 
 Multiply both terms by 12, which is the L.C. M. of 4 and 
 6, and we have at once $. 
 
 Reduce * to a simple fraction. 
 ^v 
 
 Multiply both terms by 12 and we have |ff = ^f . 
 
 Ex. 91. 
 Reduce to simple fractions : 
 
 4 -5j- 5 -f ' 
 
 7 11 1 10 
 
 2. . 4. 6. I. 8. It 
 
 llf 13* * H 
 
COMMON FRACTIONS. 137 
 
 9. 
 
 I 
 
 14. ?i 19. ^ f $ . 
 
 
 it 
 
 A 
 
 10. 
 
 tt 
 
 QQ 7 2 
 
 
 If 
 
 f 
 
 11. 
 
 H. 
 
 lfi f of 3^- 20 | 
 
 
 12. 
 
 5f 
 
 17 t of 18*, 
 
 4 of 7-J 21 . 
 
 13. 
 
 H 
 
 19f 
 
 1ft ^y Of 12f 
 
 . ^r 
 
 3 of ^ 
 
 136, To express one number as a fraction of another. 
 
 What fraction of 8 is 5? 
 
 Since 1 = J of 8, 
 
 5 = 5 x J of 8. 
 That is, 5-| of 8. 
 
 The number which follows " of" is the denominator, and the other 
 number the numerator of the required fraction. 
 
 Ex. 92. 
 What fraction of : 
 
 1. 8 is 7? 7. 2|is|? 13. 3f isf? 
 
 2. 7 is 8? 8. | is 4? 14. 5Jis4f? 
 
 3. 6 is 2? 9. 2fisli? 15. llf is5|? 
 
 4. 5 is 3? 10. 2is5f? 16. 
 
 5. 7 is 15? 11. 2|is5? 17. 
 
 6. 15 is 7? 12. 5Jis2t? 18 14fis4? 
 
138 
 
 COMMON FRACTIONS. 
 
 19. 7f is 2|? 
 
 20. 7|is|f ? 
 
 21. | of 10isf|? 
 
 23. f ofl2fisfi? 
 
 24. f of 3|is Jf? 
 
 137. To reduce a decimal to a common fraction. 
 Reduce 0.25 to a common fraction. 
 0.25 = ,&&. = i 
 
 25. Hf 
 
 26. ^ is i 
 
 27. isof 2? 
 
 28. 33 is 2 J of 
 
 29. 27|f is 2 of 
 
 30. 36 is 3f of 
 
 138, Hence, to reduce a decimal to a common fraction, 
 
 TFh'fe the figures of the decimal for the numerator ; and 
 1 , with as many zeros as there are figures in the decimal, 
 for the denominator. 
 
 Ex. 93. 
 
 Reduce to common fractions : 
 
 1. 
 
 0.5. 
 
 9. 
 
 0.015. 
 
 17. 
 
 0.7168. 
 
 25. 
 
 1.6125. 
 
 2. 
 
 0.06. 
 
 10. 
 
 0.18. 
 
 18. 
 
 3.02. 
 
 26. 
 
 8.0396. 
 
 3. 
 
 0.15. 
 
 11. 
 
 0.125. 
 
 19. 
 
 5.85. 
 
 27. 
 
 2.18375. 
 
 4. 
 
 0.025. 
 
 12. 
 
 0.004. 
 
 20. 
 
 7.075. 
 
 28. 
 
 1.0725. 
 
 5. 
 
 0.7. 
 
 13. 
 
 0.032. 
 
 21. 
 
 0.15625. 
 
 29. 
 
 22.848. 
 
 6. 
 
 0.19. 
 
 14. 
 
 0.3125. 
 
 22. 
 
 0.46875. 
 
 30. 
 
 1.30125, 
 
 7. 
 
 0.135. 
 
 15. 
 
 0.0625. 
 
 23. 
 
 0.00256. 
 
 31. 
 
 17.875. 
 
 8. 
 
 0.005. 
 
 16. 
 
 0.0425. 
 
 24. 
 
 0.00375. 
 
 32. 
 
 2.9375, 
 
COMMON FRACTIONS. 139 
 
 139, To reduce a common fraction to a decimal. 
 Change 1 to a decimal. 
 
 8)3.000 
 0.375 
 
 140, Hence, to reduce a common fraction to a decimal, 
 Divide the numerator by the denominator. 
 
 141, If a fraction, when reduced to its lowest terms, con- 
 tains in the denominator any other factor than 2 or 5 (the 
 prime factors of 10), the division of the numerator by the 
 denominator will not terminate. In general, it will be 
 sufficient to obtain five decimal places in the quotient. 
 But the number in the fifth place of the quotient must 
 be increased by 1 if the number in the next place of the 
 quotient is five, or greater than five. 
 
 Ex. 04. 
 Change the following fractions to decimals : 
 
 1. f. 5. llff. 9. If 13. 
 
 2. dft. 6. y. 10. 14. 
 
 3- Tttr- 7 - Tffcr- n - H- 15 - 
 
 4. 1G. 8. 5. 12. ii. 
 
 Express the following as decimals to five places : 
 
 16. f. 19. TV 22. if- 25. 
 
 17. f 20. if. 23. iff. 26. 
 
 18. f. 21. f 24. jff. 27. 
 
140 COMMON FRACTIONS. 
 
 Ex. 95. 
 
 Solve the following problems, first changing the common 
 fractions to decimals : 
 
 1. A person owed $24,560. When he has paid $8345^, 
 
 $7234^, $6472^, how much does he still owe? 
 
 2. A man sold 46| acres of land, at the rate of $9i an 
 
 acre, and 54 J acres at the rate of $2^-J. How much 
 did he receive for the whole ? 
 
 3. A merchant purchases 346 pieces of cloth, each con- 
 
 taining 32f yards, at $1J a yard, and sells the 
 whole for $2^ a yard. What does he gain ? 
 
 4. A merchant purchased 8 yards of cloth at $6i a yard. 
 
 What sum will he gain per yard if he sells the whole 
 piece for$56f$? 
 
 5. A man bought a piece of land for $1046} at the rate 
 
 of $ 15 J an acre. He sells it for $ 17 an acre. How 
 much does he gain on the whole? 
 
 6. A merchant purchased 15 casks of wine of 25 gallons 
 
 each. He paid $980 for the wine, $78J tax, $33f 
 for transportation. He sold it for $3 a gallon. 
 How much did he gain ? 
 
 7. A speculator purchased 738 acres of land for $21,294. 
 
 He sells -| of his land at the rate of $34f an acre, 
 and the rest at the rate of $35 per acre. What does 
 he gain? 
 
 8. A piece of cloth is 29f yards in length. How many 
 
 pieces, each containing Iff yards, can be cut from 
 it? 
 
COMMON FRACTIONS. 141 
 
 9. How many postage-stamps, each containing f of a 
 square inch, are in a sheet of 172 square inches? 
 
 10. Of a boat worth $5600, A, who has |, sells f of nis 
 
 share to B, and B sells of his share to C. Find 
 the value of C's share. 
 
 11. From Montreal to Toronto, by the Grand Trunk Rail- 
 
 way, the distance is 332 miles. One-half a mile 
 more than -| of this distance was opened in Novem- 
 ber, 1855, and the remainder in November, 1856. 
 Find the number of miles opened in 1856. 
 
 12. The 36 Israelites who fell in the first assault on Ai, 
 
 were y|^- of the force sent by Joshua. How many 
 were sent by Joshua? 
 
 13. What number multiplied by 8| equals 3++f|+f-f ? 
 
 14. Multiply the sum of -^j- and by their difference. 
 
 15. Of the distance from Edinburgh to London by rail, 
 
 that from Edinburgh to Carlisle is J, from Carlisle 
 to Preston -fo, while that from Preston to London is 
 210 miles. Find the distance from Edinburgh to 
 London. 
 
 16. How many times can a measure holding -J of a pint be 
 
 filled from a vessel containing 63i pints? 
 
 17. Of a consignment of guano |-g-J consisted of carbonate 
 
 of lime and phosphates of lime and magnesia, and the 
 phosphates made up ^-| of the guano. How many 
 parts in a hundred of the guano was carbonate of 
 lime? 
 
 18. Of the water of the Dead Sea yff^ is muriate of lime, 
 
 -fo muriate of magnesia, -fffo muriate of soda, 2 fa g 
 sulphate of lime. What part of the whole do these 
 ingredients constitute? 
 
142 COMMON FRACTIONS. 
 
 Ex. 96. 
 ORAL EXERCISE IN FRACTIONS. 
 
 1. From a piece of cloth J of it and i of it have been 
 
 cut. What fraction of the cloth is left ? 
 
 2. To make a yard of cloth, what fraction of a yard must 
 
 be added to the sum of J and J of a yard ? 
 
 3. A boy gave to his sister J of an apple, to his brother 
 
 i as much as to his sister, and kept the rest himself. 
 What part of the apple did he keep ? 
 
 4. A grocer sold } of a dozen eggs, and carried home the 
 
 rest of the dozen. How many did he carry home? 
 
 5. What is meant by J, i, J of a unit? 
 
 6. How is J, J, J of a unit found? 
 
 7. How is J, J, i of a number found? 
 
 8. At ? of a dollar per yard, what is the cost of 6 yards 
 
 of cloth ? 
 
 9. At $7 per ton, what is the cost of i of a ton of coal? 
 
 10. Three packages of sugar weigh respectively 2i, 3}, 4J 
 
 pounds. What is the weight of the whole ? 
 
 11. When poultry is worth 20 cents per pound, what must 
 
 be paid for a turkey weighing 8J pounds, and a 
 chicken weighing 3i pounds? 
 
 12. From a jar of butter containing 15J pounds there have 
 
 been sold 7} pounds. How many pounds remain 
 in the jar? 
 
 13. Change to mixed numbers Jf, y, if-, *f- t *, ff. 
 
COMMON FRACTIONS. 143 
 
 14. Express in lowest terms |f , T \, $, ff , ff , -nrfr- 
 
 15. Change to improper fractions 5^-, 6-|, 8-|, 9^-, 13^. 
 
 16. Reduce f to lOths ; to 15ths ; to 20ths ; to 25ths. 
 
 17. A lady gave % a dollar to her daughter, and ^ of a 
 
 dollar to her son. What fraction of a dollar did 
 the daughter receive more than the son ? 
 
 18. At |- of a dollar per bushel, how many bushels of 
 
 apples can be bought for $3? 
 
 19. Four pecks make a bushel. If 21 pecks be sold from 
 
 a bushel of cranberries, how many pecks remain ? 
 
 20. A gentleman bought 2 pairs of gloves at $1J a pair, 
 
 and 3 pairs of slippers at $1J per pair. He gave a 
 ten -dollar bill in payment. What change should 
 he receive? 
 
 21. What part of 2 is 1? of 7 is 3? of 9 is 2? of 12 is 4? 
 
 22. A farmer planted 3 bushels of potatoes, and harvested 
 
 50 bushels. What fraction of the crop was the 
 seed? 
 
 23- From a piece of cloth containing 81 yards there were 
 sold 45 yards. What part of the piece was sold ? 
 
 24. What part of f is f ? of | is ? of | is |? 
 
 HINT. Reduce the fractions to similar fractions. 
 
 25. What part of 8 is | ? of 7 is -f ? 
 
 26. In a year there are 365 days. What part of a year 
 
 are 30 days? 50 days? 75 days? 105 days? 
 
 27. Three-fourths of a cord of oak wood costs $6. What 
 
 is the cost of of a cord ? of a cord ? 
 
144 COMMON FRACTIONS. 
 
 28. Seven-eights of a yard of cloth cost 42 cents. Find 
 
 the cost of i of a yard ; of 1 yard ; of 2i yards. 
 
 29. Four-fifths of a load of wood is sold for $8. Required 
 
 the cost of of the load ; of the whole load ; of 4f 
 such loads. 
 
 ;0. What is the price of a bushel of turnips, when } of a 
 bushel are sold for 30 cents ? 
 
 31. A farmer divided among his 4 sons f of his farm. 
 
 What part of the farm did each son receive ? 
 
 32. At the rate of $10 per week, what is the cost of board 
 
 per day? 
 
 33. How many bushels of carrots, at $5 per bushel, can 
 
 be bought for $3i? 
 
 34. How many cows are of 20 cows? 16 sheep are 
 
 of how many sheep ? 
 
 35. Five-sixths of 12 hens are -| of how many hens? 
 
 36. Three-fourths of a cord of wood, at $7 per cord, will 
 
 pay for what part of a ton of coal, at $9 per ton ? 
 
 37. The captain of a vessel owns } of it, the first mate I, 
 
 and the captain's wife \ of the remainder. What 
 part of the vessel does she own ? 
 
 38. John Rogers sold to Henry Cook -J- of his woodland, 
 
 and then bought back J of what he had sold. What 
 part of the land did each have then ? 
 
 39. From a bin of potatoes containing 30 bushels, 5 
 
 bushels, 2 bushels, 4f bushels were sold. How 
 many bushels were left in the bin ? 
 
COMMON FRACTIONS. 145 
 
 40 A bushel of wheat weighs 60 pounds. If a miller 
 takes 3 pounds from each bushel for toll, what part 
 of a bushel does he take ? 
 
 41. At 20 cents per yard, how many yards of ribbon can 
 
 be bought for $2.20? 
 
 42. Four-fifths of $20 is - of how much money? 
 
 43. Two-thirds of a yard of silk can be bought for $|. 
 
 What is the price per yard ? How many yards can 
 be bought for $31? 
 
 44. If 5 bushels of oats cost $2, what will be the cost of 
 
 9 bushels at the same rate ? 
 
 45. If $lj are paid for f of a yard of velvet, what will be 
 
 the cost of -| of a yard ? 
 
 46. If -| of the distance between two towns is 6| miles, 
 
 what is the whole distance ? 
 
 47. If 2J bushels of apples make a barrel, how many 
 
 barrels will 11 bushels make? 
 
 48. A carpet dealer sold of of a roll of carpet. What 
 
 part of the roll was left ? 
 
 49. How many pigs can be bought for $20, at $2J 
 
 each? 
 
 50. Four quarts make a gallon. When 2J quarts have 
 
 been taken from a gallon of vinegar, what part of 
 the gallon has been taken ? 
 
 51. A drover puts - of his cattle in a field, f of them in 
 
 another field, and 10 in a barn. How many cattle 
 has he ? 
 
146 COMMON FRACTIONS. 
 
 52. A merchant sold ^ of a chest of tea, then of it, and 
 
 took the rest home. If he took home 12 pounds, 
 how many pounds were there in the chest at first r 
 
 53. A cistern has two pipes. By one pipe, 3 gallons of 
 
 water run into the cistern in a minute, and by the 
 other, 5 gallons run out in a minute. If the cistern 
 contains 42 gallons, and both pipes are open, in how 
 many minutes will it be emptied ? 
 
 54. A man can perform a certain piece of work in 4 hours, 
 
 and a boy can do the same work in 6 hours. What 
 part of the work can the man do in 1 hour ? What 
 part can the boy do ? What part can both together 
 do? How many hours will it take both together 
 to do the work ? 
 
 65. C can plant an acre of corn in 6 hours, C and D 
 together in 4 hours. What part of an acre can 
 C and D together plant in 1 hour ? What part can 
 plant in 1 hour ? What part then can D plant 
 in 1 hour ? How many hours will it take D to 
 plant the acre of corn ? 
 
 56. A fox is 90 rods in advance of a greyhound. The 
 
 fox runs 60 rods a minute, the greyhound 65. In 
 how many minutes will the fox be overtaken ? 
 
 57. By selling cigars at $7 a hundred -fa of their cost is 
 
 gained. Find the price per hundred at which they 
 must be sold, in order to gain f of their cost. 
 
 58. By selling a farm for $2400, the owner lost of what 
 
 it cost him. How much did he pay for the farm ? 
 
 59. How many flowers can be planted along the borders 
 
 of a flower-bed 12 feet long and 10 feet wide, if the 
 flowers are of a foot apart ? 
 
COMMON FRACTIONS. 147 
 
 Ex. 97. 
 
 1. Reduce to simple fractions, 81 of I of 1, - , - ;-|. 
 
 f 01 7 5 OI f 
 
 2. Find the values of 
 
 169 14$; ! T 9 T --|-of4; 76} f of 19. 
 
 3. Six pieces of cloth measure respectively 23^- yards, 
 
 19| yards, 21 yards, 24 yards, 35f yards, 18f 
 yards. After 39^ yards have been sold from their 
 sum, how much remains? 
 
 4. The remainder being 4, the quotient 51, and the 
 
 divisor 25, it is required to find the dividend. 
 
 5. Find the value of f of a chest of tea weighing 57 i 
 
 pounds, at $1J per pound. 
 
 6. If a man work 82 hours in a day he can finish a piece 
 
 of work in 12 2 days. How many hours per day 
 must he work to complete it in 10 1 days? 
 
 7. A confectioner sells f of % of a bushel of walnuts. 
 
 What part of the bushel remains, and what will it 
 bring at 15 cents per quart? 
 
 8. What is the value of a basket of 588 eggs, worth 25 
 
 cents per dozen ? 
 
 9 A man starts on a journey 5 hours before the mail 
 coach. How many miles will the coach be ahead 
 of the man after it has run for 12 hours, supposing 
 that he travels at the rate of 3 miles an hour, and 
 the coach 10 miles an hour? 
 
 10. If f of | of a piece of land cost $420, what is the 
 value of the whole ? 
 
148 COMMON FRACTIONS. 
 
 11. A farmer sold at market 15 sheep at $2|- each, and 
 
 bought 7 yards of cloth at $1 per yard. How 
 much money did he take home ? 
 
 12. A man walked a distance of 60 miles ; for the first 5 
 
 hours, at the rate of 3 miles an hour, and during 
 the remainder of the journey he walked at the rate 
 of 4 miles an hour. In how many hours did he 
 complete the journey ? 
 
 13. The circumference of a fore wheel of a wagon is 6-| 
 
 feet ; that of the hind wheel 8f feet. In a distance 
 of 20 miles, of 5280 feet each, how many more turns 
 will be made by the former than by the latter ? 
 
 14. A young man received $1200 from his father. He 
 
 spent -J- of the money for clothes, % of it in travelling, 
 and invested the remainder in a mortgage. What 
 fraction of the whole was the sum invested ? 
 
 15. A baker paid $32 for ^ of a hogshead of molasses. 
 
 What was the value of -J- of the remainder ? 
 
 16. A gentleman paid $125 for keeping 2 horses 12 weeks. 
 
 What would it cost, at the same rate, to keep one 
 horse - of a week ? 
 
 o 
 
 17. If -| of a yard of ribbon cost $ -J, what will be the value 
 
 of 5f yards? 
 
 18. Reduce f, , -f% to decimal fractions, and add the 
 
 results. 
 
 19. The contents of a chest of tea weighing 87.5 pounds 
 
 are made up into 1 pound, ^ pound, ^ pound pack- 
 ages, an equal number of each, How many pack- 
 ages of each kind ? 
 
COMMON FRACTIONS. 149 
 
 20. In five successive days a farmer puts into his bin 371 
 
 bushels of potatoes, and on each of these days he 
 sells 19f bushels. How many bushels have been 
 put into the bin ? How many more are in the bin 
 at the end than at the beginning of the five days? 
 
 21. A man's weekly income is $18i, and his weekly 
 
 expenses are $23?. If he have $75} in reserve, 
 how many weeks can he live without incurring 
 debt? 
 
 22. By a leak 87-f barrels of water enter the hold of a 
 
 boat in 1 hour; the pumps will discharge 58f 
 barrels in an hour. If she can carry only 875 
 barrels, in how many hours will she sink? 
 
 23. A can mow a field in 10 days, B in 8 days, and C in 
 
 5 days. When wonting together, how many days 
 will they need? 
 
 24. A carpenter alone can build a shop in 15 days, and 
 
 with the help of his son he can build it in 10 days.' 
 In how many days will the son alone build the 
 shop? 
 
 25. Wales Edwards and George Peters hire a pasture for 
 
 $14. Edwards puts in 8 horses ; Peters puts in 50 
 sheep. If 21 sheep will eat as much as 2 horses, 
 what must each pay ? 
 
 26. A flour dealer bought 125 barrels of flour at $61. 
 
 He sold 97 barrels at $7f, and the remainder, being 
 injured, brought only $5. What did he gain? 
 
 27. A lady bought -| of -| of a yard of ribbon for 
 
 What was the cost per yard? 
 
150 COMMON FRACTIONS. 
 
 28. From two fields 482 bushels of corn are gathered. 
 
 The first field yields $ as much as the second. How 
 many bushels does each field yield ? 
 
 29. A farmer brought to market 3 jars of butter, weighing 
 
 26 pounds, 37 pounds, 19 pounds. The empty jars 
 weighed 3^ pounds, 4J- pounds, 5| pounds. The 
 butter brought $30. What was the price per pound ? 
 
 30. From 120 acres of land 32 acres are sold to one man, 
 
 and -J- of the remainder to another. How many 
 acres are unsold ? 
 
 31. If the rent of 3 acres of land for -| of a year be $9, 
 
 what will be the rent of 45 acres for 1 year ? 
 
 32. If -| of a ton of coal cost $4, how many tons can be 
 
 bought for $145? 
 
 33. If 12 horses eat 653 bushels of oats in 3 months, how 
 
 many bushels will 7 horses eat in 2 years ? 
 
 34. The agent for a line of steamers sells ^ of a steamship 
 
 to one company, J of the remainder to a second, 
 and of what is left to a third. What part of the 
 whole ship has the third company ? 
 
 35. A farmer exchanged 13 loads of oats, of 18 bags each, 
 
 every bag containing 2J bushels, for 150 sheep, at 
 $2.925. What was the price of the oats per bushel ? 
 
 36. Two men 95.784 miles apart approached each other 
 
 until they met. One travelled 7.476 miles more 
 than the other. How many miles did each travel ? 
 
 37. A teacher spent - of his salary in board for himself 
 
 and family, and ^ of it in clothing for himself. 
 The clothing of his wife and child cost f as much as 
 his own. At the end of the year $187 remained, 
 What was the salary ? 
 
COMMON FRACTIONS. 151 
 
 38. A road to the top of a hill has a rise of -^ of a foot in 
 
 100 feet. How many feet is the total elevation of 
 the hill, if the length of the road is 2 miles? 
 
 39. A man bequeathes to his wife -J- of his estate ; to his 
 
 daughter, of it; to his son, ^ of the daughter's 
 share ; he divides the remainder equally between a 
 hospital and a public library. What part is received 
 by the hospital ? 
 
 40. If the above estate is worth $150,784, T^hat is the 
 
 amount received by the hospital ? 
 
 41. A can build a wall in 7 days, B in 6 days, and C in 
 
 5 days. A and B worked together for 2 days, 
 when they were joined by C. How many days 
 will they need to complete the remainder of the 
 work? 
 
 42. Find the cost of 75,849 bricks, at $9.75 per M. 
 
 43. A lumberman exchanged 50,495 feet of round timber, 
 
 at $4 per M, for pork, at $20f per barrel. How 
 many barrels of pork did he receive ? 
 
 44. For % of a bushel of apples $|^ are paid. What will 
 
 4f bushels be worth ? 
 
 45. Henry Jones bought at a saw-mill 3485 ft. boards, at 
 
 $7.50 per M ; 9872 feet laths, at $0.25 er C ; 6492 
 feet flooring, at $85 per M; 8975 feet cherry 
 boards, at $15.05 per M. He paid $152.75 in 
 cash, and the balance in flour, at $9.25 per barrel. 
 Required the number of barrels of flour. 
 
 46. A merchant mixed 7 pounds of black tea at 68 cents 
 
 with 9 pounds of green tea at 75 cents. At what 
 price per pound must he sell the mixture to gain 
 $3.69? 
 
152 COMMON FRACTIONS. 
 
 47. Nine men working 10 hours per day will harvest a 
 piece of grain in 8 days. How many days will be 
 needed for the same work by G men working 9 hours 
 per day ? 
 
 48. 
 
 r v * j ' 
 
 At $8.75 per M, how many bricks can be bought for 
 $393.75? 
 
 49. When 1000 bricks cost $7.20, what is the cost of a 
 
 single brick ? 
 
 50. If $437.645 be paid for 6500 feet of rosewood, what is 
 
 the cost per M ? 
 
 51. A sea captain who owned $ of a ship and cargo, gave 
 
 to his wife -^ of his share, to his daughter -J- of what 
 his wife received, to his son -| of the remainder, and 
 equally divided what was still left between two 
 nieces. What part of the whole had each niece ? 
 
 52. Peter Knowlton sold a farm for $9786, which was | 
 
 of the sum paid for it. Required the original cost 
 of the farm. 
 
 53. A merchant bought a bag of coffee, containing 60 
 
 pounds, for $15. At what advance per pound must 
 he sell it to buy with the gain on the coffee 3 yards 
 of velvet at $3 per yard? 
 
 54. After selling -| of his sheep to a drover, and -J- of the 
 
 remainder to his neighbor, a farmer has 150 left. 
 How many were there in the flock at first? 
 
 55. A stock broker bought 9 shares in the Northern Pacific 
 
 Railroad, at $99}, and 12 shares in the Illinois Cen- 
 tral Railroad, at $ 102}. He sold them all at $ 103 i 
 How much did he gain ? 
 
COMMON FRACTIONS. 153 
 
 56. A bankrupt's available property can be sold for 
 
 $19,780, which will pay 62J cents on every dollar 
 he owes. How much does he owe ? 
 
 57. A loaf of bread weighing 2 pounds, when flour is worth 
 
 $ 9.80 per barrel, is sold for 10 cents. What should 
 it bring when flour is worth $7.84? 
 
 58 Divide 0.75 of 17f by f of 0.035. 
 
 59. An army of 7844 men has 490,250 pounds of beef. If 
 
 for every man 11 pounds daily be allowed, in how 
 many days will the beef be consumed ? 
 
 60. A seedsman bought 373 bushels of lawn grass-seed for 
 
 $226. He sold 25 bushels at a profit of $lf per 
 bushel. For what price per bushel must he sell the 
 remainder to make his whole gam $73? 
 
 61. The cost of 50 gallons of molasses is $25. By leakage 
 
 -J- of it is lost ; 20 gallons are sold at 62 J cents. At 
 what rate must the remainder be sold to gain $5 on 
 the whole? 
 
 62. For | of a yard of broadcloth at $6-} per yard, 1 J yards 
 
 of cassimere and 50 cents in money were given in 
 exchange. What was the price per yard of the 
 cassimere ? 
 
 63. A owns f of a ship and cargo worth $25,748, B of 
 
 the remainder, C of the amount belonging to A 
 and B, and D owns what is still left. Eequired the 
 amount of D's share ? 
 
 64 o A farmer gives to his eldest son ^-| of a farm, and the 
 remainder to his daughter. The difference between 
 their shares is 780 acres. How many acres does the 
 daughter receive? 
 
154 COMMON FRACTIONS. 
 
 65. If 1200 pounds can be carried 36 miles for $14, bow 
 
 many pounds can be carried 24 miles for $ 14 ? 
 
 66. If 2J- acres of land cost $500, wbat will 460 acres cost? 
 
 87. Four and four-sevenths tons of cannel coal cost $64. 
 Required tbe cost of 13f tons. 
 
 68 Of a certain estate is pasture, -| land suitable for 
 cultivation, and the remainder, woodland, is 50 
 acres. How many acres in the estate ? 
 
 69. If 1.4 bushels of walnuts cost $1.50, find the value of 
 
 7 bushels. 
 
 70. If a man breathes 17 times a minute, and takes in at 
 
 each breath -f- of a quart of air, how many quarts of 
 air does he need in 1 hour? 
 
 71. If the crop of potatoes from an acre is on the average 
 
 255 bushels, but the potato beetle destroys -| of the 
 crop, how many bushels will 3 acres produce ? 
 
 72. If a miller takes -^ for toll, and a bushel of wheat pro- 
 
 duces 40 pounds of flour, how many bushels must be 
 carried to the mill to obtain 196 pounds of flour ? 
 
 73. An expressman carried 100 vases, on the condition 
 
 that he was to receive of a dollar for every one he 
 carried without breaking, and pay 1J dollars for 
 every one he broke. He received Ib dollars. How 
 many did he break? 
 
 HINT. The expressman loses $9 on the lot, and he loses 
 $1J on each vase broken. 
 
 74. A man who rows 4 miles an hour in still water takes 1^ 
 
 hours to row 4 miles up a river. How many minutes 
 will it take him to row 4 miles down the river ? 
 HINT. The man rows 4f miles in 1J hours. Hence the cur- 
 rent sets him back f of a mile in 1 = f hours, or f -*- f = f of 
 a mile in I hour In rowing down the river he rows 4 miles 
 an hour, and the current carries him f of a mile an hour 
 
CHAPTER IX. 
 
 COMPOUND QUANTITIES. 
 
 142, A quantity expressed with reference to a single unit 
 is called a simple quantity; but a quantity expressed with 
 reference to different units is called a compound quantity, 
 
 Thus, 201 pounds is a simple quantity, but 20 pounds 4 
 ounces is a compound quantity. 
 
 143, The process of changing the unit in which a quan- 
 tity is expressed, without changing the value of the quantity, 
 is called reduction, 
 
 144, If the change be from a higher denomination to a 
 lower, it is called reduction descending ; if from a lower to 
 a higher, it is called reduction ascending, 
 
 Thus, 1 yard = 36 inches is an example of reduction 
 descending ; and 24 inches = 2 feet is an example of reduc- 
 tion ascending. 
 
 UNITS OF LENGTH. 
 
 145, 12 inches (in.) = 1 foot (ft.) 
 
 3 feet = 1 yard (yd.). 
 
 5} yards, or 16J feet, = 1 rod (rd.). 
 
 320 rods, 1760 yards, or 5280 feet, = 1 mile (mi.). 
 
 NOTE. A line = ^ in. ; a barleycorn = J in. ; a hand (used in 
 measuring the height of horses) = 4 in. ; a palm = 3 in. ; a span = 
 9 in. ; a cubit = 18 in. ; a military pace = 2J ft. ; a chain = 4 rds. ; 
 a link = T J 7 chain ; a furlong = J mi. ; a knot (used in navigation) 
 = 6086 ft. ; a nautical league = 3 knots ; a fathom (used in measuring 
 depths at sea) = 6 ft. ; a cable length = 120 fathoms. 
 
156 COMPOUND QUANTITIES. 
 
 Ex. 98. (Oral.) 
 
 1. How many inches in 1 yd. ? in J yd. ? in J yd. ? 
 
 2. How many yards in 180 in. ? in 48 in. ? in 45 in. ? 
 
 3. How many yards in 3 rds. ? in 4 rds. ? in 5 rds. ? 
 
 4. How many feet in 2 yds. ? in 2 rds. ? in 2 rds. 2 yds.? 
 
 5. How many rods in 33 ft. ? How many yards in 33 ft. ? 
 
 6. In \ mi. how many rods ? yards ? feet ? 
 
 7. How many rods in 0.4 of a mile? in 0.3? in 0.7 ? 
 
 8. What part of a mile are 160 rds. ? 80 rds. ? 40 rds. ? 
 
 9. What part of a foot are 4 in. ? 3 in. ? 6 in. ? 8 in. ? 
 10. What part of a yard are 2 ft. ? 1 ft. 6 in. ? 2 ft. 6 in. ? 
 
 REDUCTION DESCENDING. 
 146, Change 10 mi. 40 rds. to feet. 
 
 10 mi. 40 rds. 
 320 
 
 3200 
 
 40 10 X 320 rds. = 3200 rds., to which the 40 rds. 
 
 3240 are added. 
 
 Again, 3240 X 16J ft. = 53,460 ft. 
 
 1620 The multiplicand and multiplier are inter- 
 
 19440 changed in the operation. 
 
 3240 
 
 53460 
 
 Ex. 99. 
 
 Reduce to feet : Reduce to inches : 
 
 1. 3 mi. 5 yds. 2 ft. 4. 18 mi. 252 rds. 2 yds. 
 
 2. 40 mi. 5 rds. 2} yds. 5. 11 mi. 6 rds. 4 yds. 
 
 3. 2 mi. 52 rds. 1 ft. 6. 18 mi. 230 rds. 8 ft. 
 
COMPOUND QUANTITIES. 157 
 
 7. 2 yds. 1 ft. 9 in. 10. 8 mi. 96 rds. 4 yds. 
 
 8. 5 yds. 2 ft. 7 in. 11. 2 mi. 80 rds. 2 ft. 
 
 9. 170 rds. 3 yds. 9 in. 12. 200 rds. 115 yds. 5 in. 
 
 REDUCTION ASCENDING. 
 
 147, Change 53,463 ft. to a compound quantity. 
 
 16 }) 53463 ft. 
 
 2 
 
 33) 106926 half-feet. 
 
 320) 3240 rds. . . 6 half-feet = 3 ft. 
 10 mi. . . 40 rds. 
 
 10 mi. 40 rds. 3 ft. Ans. 
 
 There are 16J ft., qr 33 half-feet, in a rod ; so the 53,463 ft. are 
 changed to half-feet, and the half-feet to rods, by dividing by 33. 
 The remainder is 6 half-feet = 3 ft. 
 
 3240 rds. are changed to miles by dividing by 320, the number of 
 rods in a mile. The remainder is 40 rds. 
 
 Reduce 376,985 in. to higher denominations. 
 
 12 
 3 
 5J 
 
 2 
 
 376985 in. 
 
 yds. 
 
 31415 ft. . . 5 in. 
 
 10471 yds. . 2 ft. 
 
 2 
 
 11 
 320 
 
 20942 half-yards. 
 
 1903 rds. . 9 half-yards = 4} 
 
 5 mi. . . 303 rds. 
 
 The J yd. of the 4J yds. should be reduced to lower denominations, 
 and the result, 1 ft. 6 in., added to the 2 ft. 5 in. Thus, 
 
 mi. rds. yds. ft. in. 
 
 5 303 4 2 5 
 
 1 6 
 
 5 303 5 11 
 5 mi. 303 rds. 5 yds. ft. 11 in. Ans. 
 
158 COMPOUND QUANTITIES. 
 
 Ex. 100. 
 Reduce to higher denominations : 
 
 1. 211 in. 6. 125,899m. 9. 348,164 in. 
 
 2. 33,777 in. 6. 179,875 in. 10. 247,391 in. 
 
 3. 142,737 in. 7. 87,476 ft. 11. 99,204 ft. 
 
 4. 33,000ft. 8. 97,378yds. 12. 11,220ft. 
 
 COMPOUND ADDITION AND SUBTRACTION. 
 148. Add: 
 
 ml. 
 
 rds. 
 
 yd.. 
 
 n. 
 
 In 
 
 6 
 
 120 
 
 3 
 
 2 
 
 2 
 
 18 
 
 15 
 
 1 
 
 1 
 
 6 
 
 3 
 
 215 
 
 2 
 
 2 
 
 8 
 
 28 
 
 31 
 
 2J 
 
 
 
 4 
 
 
 
 
 
 =1 
 
 6 
 
 28 31 2 1 10 
 
 28 mi. 31 rds. 2yds. 1 ft. 10 in. Ans. 
 
 Write the numbers so that units of the same denomination shall 
 De in the same column. The sum of the inches is 16. Divide the 
 16 in. by 12 (12 in. = 1 ft.). The result is 1 ft. 4 in. Write 4 under 
 the column of inches, and add 1 to the column of feet 
 
 The sum of the feet, including the 1 ft. from the 16 in., is 6. Divide 
 by 3 (3 ft = 1 yd.). The result is 2 yds. ft. Write under the 
 column of feet, and add 2 to the yards. 
 
 The sum of the yards, including the 2 yds. from the 6 ft., is 8. 
 Divide by 5} (5J yds. = 1 rd.). The result is 1 rd. 2J yds. Write 2J 
 under the column of yards, and add 1 to the rods. 
 
 The sum of the rods, including the 1 rd. from the 8 yds., is 351. 
 Divide by 320 (320 rds. => 1 mi.). The result is 1 mi. 31 rds. Write 
 31 under the column of rods, and add 1 to the miles. 
 
 The sum of the miles, including the 1 mi. from the 351 rds., is 28. 
 
 The J yd. is changed to 1 ft, 6 in. and added to ft. 4 in. 
 
COMPOUND QUANTITIES. 159 
 
 149, Take 4 mi. 110 rds. 5 yds. 2 ft. from 6 mi. 25 rds. 
 4 yds. 2 ft. 
 
 mi. rds. yds. ft. 
 
 6 25 4 2 
 4 110 5 1 
 
 1 234 4 1 
 
 f=l 6 in. 
 
 1 234 4 2 6 
 
 1 mi. 234 rds. 2 ft. 6 in. Ans. 
 
 Write the numbers so that units of the same denomination shall 
 be in the same column. 
 
 Since 5 yds. are more than 4 yds., 1 rd. is taken from the 25 rds., 
 and reduced to yards, and the result added to 4 yds., making 9J yds. 
 Then 9 yds. - 5 yds. = 4J yds. 
 
 The 4J is written under the column of yards. 
 
 Since 110 rds. are more than 24 rds., 1 mi. is taken from the 6 mi., 
 and reduced to rods, and the result added to 24 rds., making 344 rds. 
 Then 344 rds. - 110 rds. = 234 rds. 
 
 The 234 is written under the column of rods. The 4 mi. are sub- 
 tracted from 5 mi. and the J yd. is changed to 1 ft. 6 in, 
 
 Ex. 101 
 Add: 
 
 yds. ft. in. rds. yds. ft. mi. rds. yds 
 
 1. 15 1 7 2. 23 3 1 3. 17 23 4 
 
 23 2 9 18 4 2 9 17 2 
 
 35 6 27 2 23 3 
 
 7 2 11 640 11 35 1 
 
 
 mi. 
 
 rds. 
 
 yds. 
 
 mi. 
 
 rds. 
 
 ft. 
 
 mi. 
 
 rds. 
 
 ft. 
 
 in. 
 
 4. 
 
 37 
 
 14 
 
 2 
 
 5. 23 
 
 119 
 
 15 
 
 6. 7 
 
 95 
 
 8 
 
 9 
 
 
 28 
 
 16 
 
 2 
 
 19 
 
 173 
 
 11 
 
 8 
 
 96 
 
 7 
 
 8 
 
 
 19 
 
 10 
 
 4 
 
 8 
 
 65 
 
 12 
 
 3 
 
 98 
 
 9 
 
 9 
 
 
 10 
 
 56 
 
 3 
 
 32 
 
 147 
 
 8 
 
 6 
 
 87 
 
 8 
 
 7 
 
160 COMPOUND QUANTITIES. 
 
 Find the difference between : 
 
 yds. ft. In. rds. yds. ft. ml. rds. ft. 
 
 7. 14 1 4 8. 22 2 9. 23 76 1 
 
 10 2 11 19 3 2 6 157 2 
 
 ml. rds. ft. in. mi. rds. yds. ft. ml. rds. yds. 
 
 10. 17 125 1 10 11. 7 000 12. 13 33 2 
 8 187 2 11 3 64 3 2 9 32 4 
 
 COMPOUND MULTIPLICATION AND DIVISION. 
 
 150, Multiply 37 yds. 2 ft. 11 in. by 4. 
 
 4 X 11 in. = 44 in. = 3 ft. 8 in. Write the 
 yda , in 8 in. under the column of inches. 
 
 37 2 ll 4 X 2 ft. - 8 ft.; 8 ft with the 3 ft. added 
 
 4 are 11 ft. = 3 yds. 2 ft. Write the 2 ft. under 
 
 151 2 8 ^ e c l umn f f ee t- 
 
 4 X 37 yds. = 148 yds. ; and 148 yds. with 
 the 3 yds. added = 151 yds. 
 
 151 yds. 2 ft. 8 in. Ans. 
 
 NOTE. When the multiplier is the product of two factors, multiply 
 by one of the factors, and the resulting product by the other. 
 
 151. Divide 121 yds. 2 ft. by 73. 
 
 73)121 2 (lyd. 2ft. 
 _73 
 
 48 
 3 The remainder from dividing 121 yds. by 73 is 
 
 TT1 48 yds., which are reduced to feet by multiplying 
 
 o by 3 (3 ft. = 1 yd.). The result with the 2 ft. 
 
 r^ added is 146 feet. 
 
 - . There is no remainder from dividing 146 ft. by 
 
 iS2 73. 
 
 1 yd. 2 ft. Ans. 
 
COMPOUND QUANTITIES. 161 
 
 Divide 10 ft. 11 in. by 2 ft. 8 in. 
 Reduce both quantities to inches. 
 
 10 ft, 11 in. = 131 in. jiai = 4 8 
 2 ft. 8 in. = 32 in. 4^. Ans. 
 
 Ex. 102. 
 
 1. Multiply 33 yds. 2 ft. 11 in. by 17. 
 
 2. Multiply 23 rds. 3 yds. 2 ft. by 100. 
 
 3. Divide 15 yds. 1 ft. 9 in. by 3. 
 
 4. Divide 289 yds. 2 ft. 9 in. by 213. 
 
 5. Divide 150 mi. 178 rds. 3 yds. by 9. 
 
 6. Multiply 3 mi. 72 rds. 9 ft. by 11. 
 
 7. Multiply 150 rds. 2 yds. 1 ft. by 235. 
 
 8. Divide 33 mi. 40 rds. by 200. 
 
 9. Divide 200 mi. 56 rds. 3 yds. 2 ft. by 121. 
 
 10. Multiply 11 mi. 200 rds. by 14. 
 
 11. Multiply 52 mi. 1021 yds. by 47. 
 
 12. Divide 43 mi. 280 rds. by 24. 
 
 FRACTIONS OF SIMPLE AND COMPOUND QUANTITIES. 
 
 152, Express -| of a mile in rods, feet, and inches. 
 | mi. = - of 320 rds. = 213 rds. 
 rd. ="|of 16^ ft. -5^ ft. 
 ^ ft. =$ of 12 in. = 6 in. 
 
 213 rds. 5 ft. 6 in. Ans. 
 
162 COMPOUND QUANTITIES. 
 
 Express 0.6275 of a mile in rods, feet, and inches. 
 
 0.6275 
 320 
 
 12 5500 0.6275 mi. = 0.6275 of 320 rds. = 200.8 rds. 
 
 18825 0.8 rd. = 0.8 of 16J ft. = 13.2 ft. 
 
 200.8 rds. 0.2 ft. =0.2 of 12 in. = 2.4 in. 
 
 13.2 ft. 200 rds. 13 ft. 2.4 in. Am. 
 
 12 
 2.4 in. 
 
 Find the value of f of 3 rds. 14 ft. 7 in. 
 
 o 14 tj Here we multiply by the numerator of the 
 
 c fraction, and divide the product by the denomi- 
 nator. 
 
 9)19 6 11 
 
 227-| 2 rds. 2 ft. 7f in. Ans. 
 
 NOTE. When the multiplier is a mixed number, multiply by the 
 integer and the fraction separately, and add the resulting products. 
 
 Ex. 103. 
 Find the value of : 
 
 1. | of a mile. 4. ^ mi. + of 40 rds. + f yd. 
 
 2. % of a mile. 5. 0.475 of a mile. 
 
 3. | mi. -f rd. 6. 0.3975 of a mile. 
 
 7. 0.01284 of 14 miles. 
 
 8. 3.726 mi. - 33.57 rds. 
 9. Find | of 5 mi. 89 rds. 3 yds. 2 ft. 
 
 10, Take of 4 mi. from f of 3 mi. 18 rds. 3 yds. 2 ft. 
 
 11. Add 0.525 mi., 0.125 rd, 0.5 yd, 0.16 ft. 
 
COMPOUND QUANTITIES. 163 
 
 To EXPEESS ONE QUANTITY AS THE FRACTION OF ANOTHER, 
 
 153, Express 145 rds. 2 yds. 1 ft. 6 in. as the fraction 
 of a mile. 
 
 6i n . = 6 ? ft - = & 
 
 2^ yds. = ras.=.&rcL 
 
 Of 
 
 1600 
 
 ^P of a mile. -4ns. 
 
 Express 120 rds. 3 yds. 1 ft. 6.72 in. as the decimal of a 
 
 mile. 
 
 6.72 in. -i- 12 = 0.56 ft., and this added to 
 12 
 3 
 
 5.5 
 320 
 
 6.72 in. the 1 ft. gives 1.56 ft. 1.56 ft. -i- 3 = 0.52 yds., 
 
 1.56 ft. and this added to 3yds. gives 3.52 yds. 3.52 
 
 3.52 yds. yds. -i- 5.5 gives 0.64 rds., and this added to 
 
 120.64 rds. 120 rds. gives 120.64 rds. 120.64 rds. -*- 320 
 
 0.377 mi. gives 0.377 mi. 
 
 0.377 of a mile. Ans. 
 
 NOTE. The quotient in any case need not be carried beyond the 
 fifth decimal place, and the required answer will be sufficiently accu- 
 rate for all practical purposes. 
 
 J54, Express 1 yd. 2 ft. 3 in. as the fraction of 5 yds. 
 1 ft. 3 in. 
 1 yd. 2 ft. 3 in. : 5 yds. 1 ft. 3 in. 
 
 Sin. =^ ft. ==Jft. Sin. = ^ ft. = ft. 
 
 2i ft- = f yd. -| yd. H ft. = ii yd. = & yd. 
 
 If yds. 5-j^- yds. 
 
 ft. Ans. 
 
164 COMPOUND QUANTITIES. 
 
 NOTE. If the answer to the last problem is to be expressed as a 
 decimal fraction, first find the answer as a common fraction, and 
 reduce this common fraction to a decimal fraction. 
 
 Ex. 1O4. 
 Express : 
 
 1. 125 rds. 4 yds. 2 ft. 6 in. as the fraction of a mile. 
 
 2. 1 yd. 2 ft. 3 in. as the fraction of 5 yds. 
 
 3. 51 rds. 1 yd. 3.6 in. as the decimal of a mile. 
 
 4. % rd. + yd. as the fraction of a mile. 
 
 5. o mi. 53 rds. 4 yds. 1.2 ft. as the decimal of 5 mi. 
 
 89 rds. 3 yds. 2 ft. 
 
 6. 2 mi. 138 rds. 1 yd. as the fraction of 3 mi. 265 rds. 
 
 3 yds. 
 
 7. 233 rds. 9 ft. 10.8 in. as the decimal of a mile. 
 
 8. 3 mi. 242 rds. 2| yds. as the decimal of 7 mi. 160 rds. 
 
 9. 2 ft. 7| in. as the decimal of 100 yds. 
 
 10. 11 rds. 4 yds. 4^- in. as the fraction of a mile. 
 
 11. -J rd. + f yd. + -fo ft. as the fraction of a rod. 
 
 12. 195 yds. 1 ft. 8 in. as the fraction of -J- of a mile. 
 
 13. 1 mi. 232 rds. 4 yds. 1 ft. 6 in. as the fraction of 8 mi. 
 
 204 rds. yd. 1 ft. 6 in. 
 
 14. 127 rds. 3 ft. 3.6 in. as the decimal of a mile. 
 
 15. 261 rds. 4 yds. 1 ft. 6 in. as the fraction of a mile. 
 
 16. f of the difference between 3 yds. 2 ft, 11 in. and 10 
 
 yds. 7 in. as the fraction of 16 yds. 
 
 17. 7 rds. 1 ft. 3.17 in. as the decimal of 76 rds. 2 yds. 5 in. 
 
 18. 248 rds. 4 yds. 2 ft. 8 in. as the fraction of 2 mi. 
 
COMPOUND QUANTITIES. 
 
 165 
 
 MEASURES OF SURFACE. 
 
 155. A surface has two dimensions, length and breadth. 
 
 156. If a surface is flat and has four square corners, it is 
 called a rectangle, 
 
 157. If a rectangle has its four sides equal, it is called a 
 square. 
 
 Rectangle, 
 
 158. The unit of surface is a square each side of which 
 is a linear unit. 
 
 159. The area of a surface is the number of square units 
 it contains. 
 
 160. Suppose the rectangle in the margin is 3 in. long 
 
 and 2 in. wide. If lines are drawn 
 
 as represented in the figure, the sur- 
 face will be divided into square inches. 
 
 There will be 2 horizontal rows of 3 
 square inches each; that is, in all, 
 2x3 square inches. Hence, 
 
 Express the length and breadth of a rectangle in the same 
 linear unit ; the product of these two numbers will express 
 its area in square units of the same name as the linear unit 
 of the sides. 
 
 Conversely, the number of square units in a rectangle 
 divided by the number of linear units in one side will give 
 the number of linear units in its adjacent side. 
 
166 COMPOUND QUANTITIES. 
 
 UNITS OF SURFACE. 
 
 161, 144 square inches (sq. in.) = 1 square foot (sq. ft.). 
 
 9 square feet = 1 square yard (sq. yd.). 
 
 30} square yards, or j = l d ( rf } 
 
 272} square feet, 
 
 160 square rods, or j = 1 acre , A . 
 
 10 square chains, ) 
 
 640 acres = 1 square mile (sq. mi.). 
 
 A square of flooring or roofing 100 sq. ft. 
 A section of land = 1 mile square. 
 
 A township = 36 sq. mi. 
 
 The units of surface measure are obtained by squaring the units 
 of linear measure. Thus, 
 
 144 = 12"; 9-3'; 30} -(5J) 2 ; 272} = (16J) 2 
 
 Ex. 105. (Oral.) 
 
 1. How many square feet of surface in a blackboard 4 ft. 
 
 wide and 9 ft. long ? 
 
 2. If a slate is 8 in. wide and has a surface of 80 sq. in., 
 
 what is the length of the slate ? 
 
 3. How many square inches in J of a square foot? in f ? 
 
 in |? in |? 
 
 4. How many square feet in 3 sq. yds. ? in 5 sq. yds. ? 
 
 5. How many square inches in a board 4 in. long and 
 
 3 in. wide ? 
 
 6. A square yard of carpet is 3 ft. long and 3 ft. wide ; 
 
 how many square feet in it ? 
 
COMPOUND QUANTITIES. 167 
 
 7. How many square feet in a yard of carpet 2 ft. wide ? 
 
 2J ft. wide? 
 
 8. How many square feet in a room 12 ft. by 15 ft. ? 
 
 9. How many yards of carpet 2 ft. wide will be required 
 
 to cover the floor of the above room, if the strips 
 run lengthwise of the room ? 
 
 10. How many square yards in 81 sq. ft. ? 
 
 11. How many square rods in f of an acre? 
 
 12. What part of an acre are 40 sq. rds. ? 80? 100? 
 
 Ex. 106. 
 
 1. Reduce 5 A. 147 sq. rds. to square rods. 
 
 2. How many square inches in 9 sq. yds. 7 sq. ft. ? 
 
 3. Reduce 33,796 sq. in. to square yards. 
 
 4. Reduce 153 A. 87 sq. rds. to square inches. 
 
 5. In 67,413 sq. yds. how many acres? 
 
 6. In a rectangular field 49 yds. long and 16 yds. wide, 
 
 how many square feet ? 
 
 7. How many tile.s 1 ft. square will be needed to pave a 
 
 hall 20 ft. long and 9 ft. wide? 
 
 8. How much greater is the area of a lot 50 rds. square 
 
 than that of a lot containing 50 sq. rds. ? 
 
 9. How many square yards in a square lot measuring 
 
 142 ft. on a side ? 
 
 10. Ingrain carpet is 3 ft. wide. How many yards will 
 be required for a room 27 ft. long and 18 ft, wide? 
 
168 COMPOUND QUANTITIES. 
 
 11. From each corner of a square, the side of which is 
 
 2 ft. 5 in., a square measuring 5 in. on a side is cut 
 out. Find the area of the remainder of the figure. 
 
 12. Find the value of 0.45 of an acre. 
 
 13. Reduce ^J of a square mile to lower denominations. 
 
 14. Reduce 80 sq. rds. 2.42 sq. yds. to the decimal of an 
 
 acre. 
 
 15. Add of an acre, $ sq. rd., and -| sq. yd. 
 
 16. Add -| of an acre and f of a square rod. 
 
 17. From -j^- of a square rod take of a square yard. 
 
 18. Find -f of 9 A. 70 sq. rds. 15 sq. yds. 7 sq. ft. 19 sq. in. 
 
 19. A side of Russell Square in London is 660 ft. How 
 
 many acres does it contain ? 
 
 20. The area of a rectangular field is 33 sq. rds. 1 sq. yd. 
 
 6 sq. ft. 108 sq. in., and the length is 9 rds. 1 ft. 
 Gin. What is the width? 
 
 The area of a circle'is found by multiplying the square of 
 its radius by 3.1416. (The radius is half the diameter.) 
 
 21. Find the area of a circular pond if its radius is 300 ft. 
 
 22. Find the area of the bottom of a round cistern if its 
 
 diameter is 11 ft. 
 
 23. The radius of the rotunda of the Pantheon of Rome is 
 
 71 ft. 6 in. Find the area of the floor in square 
 feet. 
 
 24. The two dials of the clock of St. Paul's, London, are 
 
 each 18| ft. in diameter. Find the area of each 
 in square feet, 
 
COMPOUND QUANTITIES. 169 
 
 CAEPETING ROOMS. 
 
 In determining the number of yards of carpeting required for a 
 room, we first decide whether the strips shall run lengthwise or across 
 the room, and then find the number of strips needed. The number 
 of yards in a strip, including the waste in matching the pattern, mul- 
 tiplied by the number of strips will give the required number of yards. 
 
 25. How many yards of carpet 21 ft. wide will cover a 
 
 floor 18 ft. by 15 ft., if the strips run across the room ? 
 
 HINT. 18-*- 2J- = 8. Hence 8 strips are required. 15ft. = 5 yds., 
 and 8x5 yds. = 40 yds. 
 
 26. How many yards of carpet I of a yard wide will be 
 
 required for a floor 26 ft. by 15J ft., if the strips run 
 lengthwise? If the strips run across the room? 
 How much will be turned under in each case? 
 
 27. How many yards of carpeting I of a yard wide will be 
 
 required for a room 8J yds. long and 17 ft. wide, if 
 the strips run lengthwise and there is a waste of -fa 
 of a yard in each strip in matching patterns? 
 
 28. Find the cost of carpet 30 inches wide, at $1.25 per 
 
 yard, for a room 18 ft. by 14 ft., if the strips run 
 lengthwise. If the strips run across the room. 
 
 29. Find the cost of carpeting J of a yard wide, at $2.75 
 
 per yard, for a room 34 ft. 8 in. by 13 ft. 3 in., if 
 the strips run lengthwise, and if there is a waste of 
 i of a yard on each strip in matching the pattern. 
 
 30. Which way must the strips of carpet J of a yard wide 
 
 run in order to carpet most economically a room 20 
 ft. 6 in. long and 19 ft. 6 in. wide, if there is no 
 waste for matching the pattern ? 
 
170 COMPOUND QUANTITIES. 
 
 PAPERING AND PLASTERING. 
 
 The area of the four walls of a room is equal to that of a rectangle 
 whose length is equal to the perimeter of the room, and whose 
 breadth is equal to the height of the room. 
 
 31. How many yards of plastering in the four walls of a 
 
 room 14 ft. 3 in. by 13J ft., and 7 ft. high, if no 
 allowance is made for doors and windpws ? 
 
 HINT. Perimeter = 2 X 14J + 2 x 13J = 55J ft. 
 
 (7 X 55J) sq. ft. = 386i sq. ft. ; 386i -H 9 = 43 8q. y<ls. 
 
 32. Find the yards of plastering in the walls of a room 
 
 211 ft. long, 16} ft. wide, and 11 ft. high, if 12 sq. 
 yds. be allowed for doors, windows, and base- 
 boards? 
 
 33. How many square yards of plastering in the walls and 
 
 ceiling of a room 30 ft. 8 in. long, 26 ft. 5 in. wide, 
 10 ft. 6 in. high, if 24 sq. yds. be allowed for doors, 
 windows, and base-boards? 
 
 34. What will be the cost of plastering the walls and ceil- 
 
 ing of a room 27 ft. 4 in. long, 20 ft. wide, and 12 ft. 
 6 in. high, at 27 cents per square yard, if 20 sq. yds. 
 be deducted for doors, windows, and base-boards ? 
 
 35. Find the cost of whitening the ceiling and walls of a 
 
 room 14 ft. 4 in. wide, 15 ft. 6 in. long, and 10 ft. 
 6 in. high, at 5 cents per square yard, allowing 9 
 sq. yds. for doors and windows. 
 
 36. Find the cost of papering a room 32 ft. long, 22 ft. 
 
 wide, 13 ft. high, with paper 18 in. wide, 8 yds. in 
 a roll, at $1.25 a roll, if 50 sq. yds. be allowed for 
 doors, windows, and base-boards. 
 
COMPOUND QUANTITIES. 171 
 
 i 
 
 SURVEYORS' CHAIN. 
 
 162, Surveyors use, in measuring distances, a chain 4 
 rds. long and containing 100 links. 
 
 The links can be written as decimals of a chain. 
 
 Ex. 107. (Oral.) 
 
 1. How many chains make a mile? 
 
 2. In 10 mi. how many chains? 
 
 3. How many rods in 9 chains? 
 
 4. How many yards in 1 chain? how many feet? 
 
 5. How many chains in 48 rods? 
 
 6. In 0.75 of a mile how many chains? 
 
 7. How many chains in f of a mile? 
 
 8. If a square field measures 1 chain on a side, how many 
 
 square rods does it contain ? How many square 
 rods then in a square chain? 
 
 9. If a square chain contains 16 sq. rds., how many square 
 
 chains make an acre ? 
 
 10. What part of an acre are 4 sq. chains? 5 sq. chains? 
 
 8 sq. chains ? 
 
 11. What part of a chain are 50 links? 25 links? 20 links? 
 
 12. How many square rods in a square garden-plot meas- 
 
 uring 75 links on a side ? 
 
 13. How many inches long is a link ? 
 
 14. The distance between two places is found to be 320 
 
 chains. Express the distance in miles. 
 
172 COMPOUND QUANTITIES. 
 
 Ex. 108. 
 
 1. Redupe 38 chains 80 links to the decimal of a mile. 
 
 2. The four sides of a field are 23 chains 19 links, 17 
 
 chains 34 links, 6 chains 85 links, and 24 chains G2 
 links. How many yards around the field ? 
 
 3. One field contains 3 sq. chains, and another is 3 
 
 chains square. How many acres in both fields 
 together ? 
 
 4. A field is crossed by a driveway 15 links wide, and 
 
 13 chains 43 links long. How many square rods in 
 the driveway? 
 
 5. From a field of 4 A. a rectangular piece 3 chains 25 
 
 links long and 2 chains 75 links wide is reserved. 
 How much of the field is left ? 
 
 6. The sides of a triangular field measure 21 chains, 
 
 14 chains 11 links, and 8 chains 10 links respec- 
 tively. By how many yards is the longest side less 
 than the sum of the other two ? 
 
 BOARD MEASURE. 
 
 163, Boards one inch or less in thickness are sold by 
 the square foot. 
 
 Boards more than one inch in thickness, and all squared 
 lumber, are sold by the number of square feet of boards 
 one inch in thickness to which they are equivalent. 
 
 Thns, a board 16 ft. long, 1 ft. wide, and 1 in. thick, contains 16 
 ft. board measure. If only J, f , or J of an inch thick, it still contains 
 16 ft. ; but, if 1 J in. thick, it contains 20 ft. board measure. 
 
COMPOUND QUANTITIES. 
 
 173 
 
 Ex. 109. 
 How many feet board measure in : 
 
 1. A board 18 ft. long, 6 in. wide, in. thick? 
 
 2. A board 16 ft. long, 12 in. wide, 1 in. thick? 
 
 3. Forty boards 14 ft. long, 10 in. wide, J in. thick ? 
 
 4. A stick of square timber 8 in. by 9 in., and 30 ft. longf 
 
 5. Six joists, each 3 in. by 4 in., and 11 ft. long? 
 
 6. Ten joists, each 6 in. by 4 in., and 14 ft. long? 
 
 7. A stick of square timber 10 in. by 12 in., and 36 ft. 
 
 long? 
 
 8. Ten 2-in. planks, each 13 ft. long, 15 in. wide? 
 
 9. Thirty 3-in. planks, each 12 ft. long, 10 in. wide? 
 
 10. A board 24 ft. long, 23 in. wide at one end, and 17 in. 
 at the other, and 1 in. thick? 
 
 HINT. The average width is 
 
 23 + 1 7 
 
 in. 
 
 11. In a stick of timber 40 ft. long and 15 in. square? 
 
 12. Ten 4-in. planks 16 ft. long and 10 in. wide? 
 
 MEASURES OF VOLUME. 
 
 164, A rectangular solid is a solid bounded by six rect 
 angles. 
 
 165, If the rectangles are all squares, the solid is called 
 a cube, 
 
174 
 
 COMPOUND QUANTITIES. 
 
 166, In the figure represented in the margin, let the 
 length contain 5, the breadth 3, and the 
 height 7 in. 
 
 The base may be divided into square 
 inches ; there will be three rows of 5 sq. in. 
 each ; in all 15 sq. in. Upon each square 
 inch may be placed a pile of 7 cu. in., so 
 that the solid will contain 15 X 7 cu. in. ; 
 that is, 3 X 5 X 7 cu. in. 
 Hence, to find the volume of a rectangular solid, 
 Express its length, breadth, and height in the same linear 
 unit; the product of these numbers will express its volume 
 in cubic units of the same name as the linear unit. 
 
 UNITS OF VOLUME. 
 
 167. 1728 cubic inches (cu. in.) = 1 cubic foot (en. ft.). 
 
 27 cubic feet = 1 cubic yard (cu. yd.). 
 
 The units of volume are cubes of the linear units. Thus, 1 728 = 12 3 , 
 27 = 3 s . 
 
 168, In measuring wood, a pile 8 ft. long, 4 ft. wide, and 
 4 ft. high, is called a cord. 
 
 X 
 
 Hence, 
 
 A cord foot is one foot in length of such a pile. 
 
 1 cord foot (cd. ft.) 16 cu. ft. 
 8 cord feet 1 cord fed.), 
 
COMPOUND QUANTITIES. 175 
 
 Ex.110. (Oral.) 
 
 1. A brick is 2 by 4 by 8 in. Find its volume. 
 
 2. How many cubic feet in 2 cu. yds. ? 
 
 3. How many cubic yards in 81 cu. ft. ? 
 
 4. Twelve cubic feet are what part of a cubic yard ? 
 
 5. How many cords in 40 cd. ft. ? in 16 cd. ft. 
 
 6. How many cords in a pile of wood 32 ft. long, 4 ft. 
 
 wide, and 4 ft. high? 
 
 7. How many loads of earth, each 1 cu. yd., must be 
 
 removed in digging a ditch 21 ft. long, 3 ft. wide, 
 and 3 ft. deep ? 
 
 8. How many cubic feet in a stick of timber 12 in. wide, 
 
 9 in. thick, and 24 ft. long? 
 
 9. How many feet board measure in a cubic foot? 
 
 10. How many cubic feet in a stick of timber 16 in. wide, 
 9 in. thick, and 21 ft. long? 
 
 Ex. 111. 
 
 1. Reduce 15 cu. yds. 13 cu. ft. to cubic inches. 
 
 2. Reduce 150,000 cu. in. to cubic yards. 
 
 3. Subtract 28 cu. yds. 25 cu. ft. 1500 cu. in. from 47 
 
 cu. yds. 13 cu. ft. 1236 cu. in. 
 
 4. Multiply 17 cu. yds. 17 cu. ft. 187 cu. in. by 11. 
 
 5. Divide 22 cu. yds. 10 cu. ft. 933 cu. in. by 7. 
 
 6. How many cubic inches can be cut out of a cubic foot? 
 
 7. How many cubic feet of water will a cistern hold whose 
 
 three dimensions are each 4 ft. ? 
 
176 COMPOUND QUANTITIES. 
 
 8. How many cubic inches in a rectangular stone post 
 
 3 ft. high, 1 ft. wide, and 1 ft. thick ? 
 
 9. Find the value of 0.975 of a cubic yard. 
 
 10. Reduce 13 cu. ft. 864 cu. in. to the decimal of a cubic 
 
 yard. 
 
 11. A man bought 52 cu. yds. 18 cu. ft. 984 cu. in. of stone 
 
 for the cellar of a house, and | as much for the cellar 
 of a second house. How much did he buy for both ? 
 
 12. How many cords of wood in a pile 50 ft. long, 6 ft. 
 
 high, and 4 ft. wide ? 
 
 13. How many cords of wood in a pile 42 ft. long, 6|- ft. 
 
 high, and 8 ft. wide ? 
 
 14. What must be the length of a load of wood that is 
 
 4 ft. wide, 5 ft. high, to contain 2 cds. ? 
 
 15. A cubic foot of wood weighs 20 pounds. Find the 
 
 weight of 10 boards, each 30 ft. long, 1 ft. wide, and 
 1 in. thick. 
 
 UNITS OF CAPACITY. 
 Dry Measure. 
 
 170, 2 pints (pt.) = 1 quart (qt.). 
 8 quarts = 1 peck (pk.). 
 
 4 pecks = 1 bushel (bu.). 
 
 Liquid Measure. 
 
 171, 4 gills (gi.) = lpint(pt.). 
 2 pints = 1 quart (qt.). 
 
 4 quarts = 1 gallon (gal.). 
 
 31J gallons = 1 barrel (bbl.). 
 2 barrels = 1 hogshead (hhd.). 
 
 NOTE. The gallon of liquid measure contains 231 cu. in. The 
 bushel of dry measure contains 2150.42 cu. in. Therefore the quart 
 of liquid measure contains 57f cu. in., and the quart of dry measure 
 67 cu. in. 
 
COMPOUND QUANTITIES. 177 
 
 Ex. 112. (Oral.) 
 
 1. How many pints in 16 gi. ? in 37 gi. ? 
 
 2. How many pints in 2 qts. ? in 7 qts. 1 pt. ? 
 
 3. How many quarts in 2 pks. ? in 3 pks. 3 qts. ? 
 
 4. How many quarts in 3 bu. ? 
 
 5. How many baskets holding 2^- pks. each will 10 bu. 
 
 of apples fill ? 
 
 6. If a pint of milk cost 4 cts., what will a gallon cost? 
 
 7. How many times will a gallon of water fill a half-pint 
 
 cup? 
 
 8. How many pint bottles will be required to hold 5 gals. 
 
 2 qts. of cider ? 
 
 9. If 4 qts. of blueberries cost 32 cts., what will a bushel 
 
 cost at the same rate ? 
 
 10. A 2-gal. measure of molasses lacks 3 pts. of being full. 
 
 What is the molasses worth at 80 cts. a gallon ? 
 
 11. If a horse eats 4 qts. of corn a day, how many days 
 
 will a bushel last him ? 
 
 12. If a quart of berries is worth 10 cts., what is a peck 
 
 worth ? 
 
 Ex. 113. 
 
 1. Reduce 440 pts. to pecks, and 109 pts. to gallons. 
 
 2. Reduce 2024 pts. to bushels. 
 Add: 
 
 gals. qts. pts. bu. pks. qts. bu. pks. qts. 
 
 3. 13 2 1 4. 5 1 3 5. 17 2 4 
 * 2 3 611 11 34 
 
 15 00 200 300 
 
 711 302 18 3 4 
 
178 COMPOUND QUANTITIES. 
 
 Subtract : 
 
 gals. qts. pts. bu. pks. qts. bu. pks. qta. 
 
 6. 4 1 7. 56 1 ' 8. 27 1 1 
 
 321 27 3 Q 18 1 3 
 
 9. Multiply 15 gals. 2 qts. 1 pt. by 130|. 
 
 10. Divide 34 bu. 3 pks. 4 qts. by 9. 
 
 11. How many pint bottles will be required to hold 63 
 
 gals, of wine ? 
 
 12. How many pint packages can a seedsman make from 
 
 3bu. 3 pks. 6 qts. of peas? 
 
 13. In one season a market-gardener sold 3758 baskets 
 
 of strawberries averaging 1 pt. each. How many 
 bushels did he sell ? 
 
 14. A farmer having a flock of 87 fowls feeds them daily 
 
 2 bu. 1 pk. 1 pt. of grain. What is the average 
 amount for each fowl ? 
 
 15. A lady in one month gave to her 4 canaries f of a 
 
 quart of seed, to her parrot 2-J- qts., and to 3 mocking- 
 birds If qts. How much seed did she give to all 
 the birds together ? 
 
 16. A gardener raised f of a bushel of Lima beans, and 
 
 -$ of a bushel of Caseknife beans. He sold 3 pks. 
 6 qts. 1 pt. ; how many had he left ? 
 
 17. A merchant receives 37 boxes of oranges, amounting 
 
 to 25 bu. 3 pks. 7 qts. Only $ of the fruit was fit 
 to sell ; how many bushels had to be thrown away ? 
 
 18. A tank is 30 ft. 3 in. long, 16 ft. 4 in. wide, and 6 ft. 
 
 4 in. deep. Find how many gallons it will hold. 
 
 19. Find the number of bushels in a bin that is 6 ft. long, 
 
 5 ft. wide, 4 ft. deep. $ 
 
 20. How many gallons will a cistern hold that is 5 ft. 
 
 square and 6 ft. deep? 
 
COMPOUND QUANTITIES. 179 
 
 UNITS OF WEIGHT. 
 Avoirdupois Weight. 
 
 172. 16 drams (drs.) = 1 ounce (oz.). 
 
 16 ounces = 1 pound (lb.). 
 
 100 pounds = 1 hundred-weight (cwt.), 
 
 20 hundred-weight => 1 ton (t.). 
 
 112 pounds = 1 long hundred- weight. 
 
 2240 pounds = 1 long ton. 
 
 NOTE. Avoirdupois weight is used for weighing all articles except 
 gold, silver, and jewels. 
 
 In the United States custom house and in wholesale transactions 
 in coal and iron the long ton is used. 
 
 The pound avoirdupois contains 7000 grains. 
 
 Ex. 114. (Oral.) 
 
 1. How many ounces in 2 Ibs. ? in 5 Ibs. ? 
 
 2. How many ounces in \ of a pound? in f of a pound? 
 
 in -J- of a pound ? in -f- of a pound ? in % of a pound? 
 
 3. What part of a pound are 4 oz. ? 2 oz. ? 8 oz. ? 6 oz. ? 
 
 12 oz. ? 
 
 4. How many pounds in 48 oz. ? in 36 oz. ? in 24 oz. ? 
 
 5. How many hundred-weight in 2 t. ? in 3 t. ? 
 
 6. How many 4-oz. packages of nutmegs can be put up 
 
 from 2-J Ibs. of nutmegs ? 
 
 7. If hay is $20 a ton, how many pounds can be bought 
 
 for $5? $7? $10? 
 
 8. If hay is $16 a ton, what are 750 Ibs. worth? 
 
 9. What part of a pound is $ of an ounce ? 
 
180 COMPOUND QUANTITIES. 
 
 10. If butter is 25 cts. a pound, and hay is $16 a ton, how 
 many pounds of butter will it take to pay for 1500 
 Ibs. of hay ? 
 
 Ex. 115. 
 
 1. Reduce 12,484 oz. to higher denominations. 
 
 2. Reduce 7 cwt. 64 Ibs. to ounces. 
 
 3. Reduce 95,784 oz. to higher denominations. 
 
 4. A bushel of wheat weighs 60 Ibs. How many bushels 
 
 in lit.? 
 
 5. A cubic foot of water weighs 1000 oz. In 1800 cu. ft. 
 
 of water how many tons ? 
 
 6. What is the difference in pounds between 27 long tons 
 
 of coal and 27 short tons of coal ? 
 
 7. Find the value of -^ of a ton. 
 
 8. What fraction of a pound is 0.00006 of a ton ? 
 
 9. Add | 1., cwt., Ib. 
 
 10. Reduce 8 cwt. 34 Ibs. to the decimal of a ton. 
 
 11. Find the value of 0.472875 of a ton. 
 
 12. Reduce 12 cwt. 80 Ibs. 6 oz. to the decimal of a ton. 
 
 13. A farmer sold in one week 5.825 t. of hay. On 
 
 Monday he sold 1350 Ibs. ; on Tuesday, t. ; on 
 Wednesday, 1|- t. ; on Thursday, 1.415 t. ; on Fri- 
 day, If t. What part of a ton did he sell on 
 Saturday ? 
 
 14. A grocer sold in one day 17 cwt. 83 Ibs. 6 oz. of loaf 
 
 sugar, 13 cwt. 95 Ibs. 12 oz. of coffee sugar, 15 cwt. 
 78 Ibs. 15 oz. of brown sugar. Required the whole 
 amount sold. 
 
COMPOUND QUANTITIES. 181 
 
 15. A grocer has 7 cwt. 57 Ibs. 12 oz. of Java coffee, 5 cwt. 
 
 39 Ibs. 10 oz. of Mocha. After mixing the two 
 kinds of coffee, he sells from the mixture 10 cwt. 
 97 Ibs. 9 oz. How much coffee has he left ? 
 
 16. A butcher receives from the West every day, Sundays 
 
 excepted, 9 cwt. 81 Ibs. 7 oz. of beef. How much 
 does he receive per week ? 
 
 17. A man puts into his cellar 17 loads of coal, averaging 
 
 1 t. 387 Ibs. a load. Required the whole amount. 
 
 18. Divide 19 t. 17 cwt. 58 Ibs. by 9. 
 
 19. A farmer sells 4 oxen whose united weight is 2 t. 
 
 7 cwt. 29 Ibs. 13 oz. What is their average weight ? 
 
 20. Find f of 8 t. 16 cwt. 24| Ibs. 
 
 21. Divide 15 t. 17 cwt. 29 Ibs. 7 oz. by 
 
 Troy Weight. 
 
 173, 24 grains (grs.) = 1 pennyweight (dwt.). 
 
 20 pennyweights = 1 ounce (oz.). 
 12 ounces = 1 pound (lb.). 
 
 NOTE. Troy weight is used for weighing gold, silver, and jewels. 
 The pound Troy contains 5760 grs. 
 
 Ex. 116. (Oral.) 
 
 1. How many grains in 2 dwt.? in 2 dwt. 9 grs.? in 
 
 3 dwt. 7 grs. ? 
 
 2. How many pennyweights in 24 grs. ? 
 
 3. How many pennyweights in 1 oz. ? in 2 oz. ? in 2 oz. 
 
 8 dwt.? in 5 oz. 17 dwt.? 
 
 4. How many ounces in 40 dwt. ? in 100 dwt. ? in 60 
 
 dwt. 
 
182 COMPOUND QUANTITIES. 
 
 5. How many ounces in 1 Ib. ? in 5 Ibs. ? in 10 Ibs. ? in 
 
 3 Ibs. 6 oz. ? in 4 Ibs. 9 oz. ? 
 
 6. How many pounds in 12 oz. ? in 48 oz. ? in 72 oz. ? 
 
 in 80 oz. ? in 90 oz. ? 
 
 7. If 1 dwt. of gold is worth $1.50, find the value of 1 oz. 
 
 of gold ; 1 Ib. of gold. 
 
 8. How many spoons weighing 25 dwt. each can be made 
 
 from 1 Ib. 3 oz. of silver ? 
 
 9. If 10 dwt. of silver are worth 70 cts., find the value 
 
 of 1 Ib. of silver. 
 
 Ex. 117. 
 
 1. Reduce 3 Ibs. 9 oz. 18 dwt. 17 grs. to grains. 
 
 2. Reduce 25 Ibs. 9 oz. 5 dwt. to pennyweights. 
 
 3. Reduce 3420 dwt. to higher denominations. 
 
 4. What is the difference in weight between 3 doz. silver 
 
 tablespoons weighing 5 Ibs. 9 oz. 8 dwt. and 3 doz. 
 silver teaspoons weighing 1 Ib. 9 oz. 16 dwt. 18 grs. ? 
 
 5. Required the weight of 8 silver teapots, each weighing 
 
 3 Ibs. 9 oz. 18 dwt. 13 grs. 
 
 6. When 12 tankards weigh 36 Ibs. 8 oz. 14 dwt. 16 grs., 
 
 what is their average weight? 
 
 7. Find the value of f of a pound. 
 
 8. Reduce -| of a grain to the fraction of an ounce. 
 
 9. Reduce 7 oz. 10 dwt. to the fraction of a pound. 
 
 10. Add 0.475 Ibs., 0.75 dwt., 0.125 oz, 0.374 Ibs. 
 
 11. From 0.675 Ibs. subtract 5.25 oz. 
 
 12. Reduce 1 oz. 7 dwt. 18 grs. to the decimal of a pound. 
 
COMPOUND QUANTITIES. 183 
 
 13. Reduce f dwt. to the fraction of a pound. 
 
 14. Reduce 4 oz. 4 dwt. to the fraction of a pound. 
 
 15. What decimal of a pound is -% Ib. -| oz. ? 
 
 174, In preparing medicines, apothecaries use the fol- 
 lowing : 
 
 Apothecaries 1 Weight. 
 
 20 grains (grs.) = 1 scruple (3). 
 
 3 scruples = 1 dram ( 3 ) 
 
 8 drams = 1 ounce ( ). 
 
 12 ounces = 1 Ib. 
 
 Apothecaries' Measure. 
 
 60 minims (fi\,) = 1 dram (fl^ l x -)- 
 
 8 drams = 1 ounce (fl. drm. viij.). 
 
 16 ounces = 1 pint (fl. oz. xvj.). 
 
 Ex. 118. 
 
 1. In 4 Ibs. 854323 how many grains? 
 
 2. In 7864 grs. how many pounds? 
 
 3. A patient is required to take daily 2 3 2 3 of bark. 
 
 How many weeks will 7 Ibs. of bark last him ? 
 
 4. Find the amount of 0.4 Ib. 0.25 5 0.375 3 0.648 3 
 
 2.147 grs. 
 
 5. Subtract 3 5 7 3 12 grs. from 9 5 6 3 1 3 16 grs., 
 
 and reduce the result to the decimal of a pound. 
 
 6. How many grains in 1 Ib. of apothecaries' weight? 
 
 7. What part of a pound avoirdupois is a pound troy or 
 
 a pound apothecaries' weight ? 
 
 8. What part of an ounce avoirdupois is an ounce troy or 
 
 an ounce apothecaries' weight? 
 
184 COMPOUND QUANTITIES. 
 
 UNITS OF TIME. 
 
 175, 60 seconds (sec.) = 1 minute (min.). 
 
 60 minutes = 1 hour (hr.). 
 
 24 hours = 1 day (dy.). 
 
 7 days -= 1 week (wk.). 
 
 365 days (or 52 wks. 1 dy.) = 1 common year (yr.). 
 
 366 days = 1 leap-year. 
 100 years -= 1 century. 
 
 The names of the months called calendar months, and the number 
 of days in each are : 
 
 1. January (Jan.) . 
 
 dy.. 
 
 . . . 31 
 
 7. July . . . 
 
 dy.. 
 
 . . 31 
 
 2. February (Feb.) 
 3 March 
 
 . 28 or 29 
 31 
 
 8. August (Aug.) . 
 9 September (Sept.) . 
 
 . . 31 
 
 30 
 
 4 April 
 
 . . . 30 
 
 10 October (Oct ) . . 
 
 . . 31 
 
 5 May .... 
 
 . . . 31 
 
 11. November (Nov.) . 
 
 . . 30 
 
 6. June 
 
 30 
 
 12. December (Dec.} . 
 
 31 
 
 NOTE. The number of days in each month may be easily remem- 
 bered by committing the following lines: 
 
 " Thirty days hath September, 
 April, June, and November ; 
 All the rest have thirty-one, 
 Except the second month alone, 
 Which has but twenty-eight, in fine, 
 Till leap-year gives it twenty-nine." 
 
 A solar year is 365 dys. 5 hrs. 48 min. 50 sec. ; that is, nearly 
 365J days. As there are 365 days in a common year, a common year 
 lacks nearly J of a day of being a solar year, and this defect is made 
 up by reckoning for some years (leap-years) 366 days. 
 
 Whenever the number representing the year is divisible by 4 and 
 not by 100, or is divisible by 400, that year is a leap-year. Thus, 
 1884, a leap-year ; 1885, not a leap-year; the year 1800, not a leap- 
 year ; the year 2000, a leap-year. 
 
COMPOUND QUANTITIES. 185 
 
 Ex. 119. (Oral) 
 
 1. How many seconds in 2 min. ? in 3 min. ? 
 
 2. How many minutes in 2 hrs. ? in 3 hrs. ? in 60 sec. ? 
 
 in 120 sec. ? 
 
 3 How many hours in 2 dys. ? in 3 dys. ? in 120 min. ? 
 in 360 min. ? 
 
 4. How many days from Jan. 1 to Feb. 17, both days 
 
 inclusive ? 
 
 5. How many months from Aug. 9 to Nov. 9? from 
 
 March 5 to Sept. 5 ? from April 4 to Oct. 4? 
 
 6. If a man can do a piece of work in 30 min., how many 
 
 hours will it take him to do four times as much ? 
 
 7. If a man can walk a mile in 15 min., how many hours 
 
 will it take him to walk 24 mi. ? 
 
 80 At the rate of 3 mi. an hour, how far will a man walk 
 in 45 min. ? 
 
 9. If a man earns $12 a week, and pays for expenses $12 
 per month of 4 wks., how much will he save in 
 20 wks. ? 
 
 10. If a man walks -J- of a mile in 5 min., how many hours, 
 at that rate, will it take him to walk 4 mi. ? 
 
 Ex. 120. 
 
 1. Reduce 4 yrs. 39 dys. 17 hrs. 22 min. 18 sec. to sec- 
 
 onds. 
 
 2. In 48,967,349 sec. how many years? 
 
 3* Find the exact length of the lunar month which con- 
 tains 2,551,443 sec. 
 
186 COMPOUND QUANTITIES. 
 
 4. How many seconds more are there in the 3 spring 
 
 months than in the 3 autumn months ? 
 
 5. Reduce f of a year to days. 
 
 6. Find the value of 0.375 yr. 0.142 dy. 0.27 min. 
 
 7. What part of a day are 12 hrs. 15 min. 25 sec. ? 
 
 8. What part of 2 dys. 7 hrs. 18 min. are 1 dy. 3 hrs. 
 
 15 min. ? 
 
 9. How much greater is the quotient of 100 yrs. 25 dys. 
 
 12 hrs. 27 min. 28 sec. divided by 4 than the product 
 of 4 yrs. 17 dys. 9 hrs. 12 min. 18 sec. multiplied 
 by 5? 
 
 10. Find the number of days, reckoning from noon of the 
 
 one to noon of the other, between Feb. 24 and 
 June 23, 1884; also between Dec. 25, 1884, and 
 May 25, 1885. 
 
 11. How many hours from noon of the 4th to midnight of 
 
 the 7th of July, 1885? 
 
 12. Divide 11 wks. 6 dys. 18 hrs. by 9. 
 
 13. Divide 2 yrs. 135 dys. 17 hrs. by 72. 
 
 14. From 5 yrs. 17 hrs. take 2 yrs. 138 dys. 22 hrs. 
 
 15. Find the value of 3.1725 dys. 
 
 16. Find the value of 21.325 of a year. 
 
 17. Express 9 dys. 3 hrs. as the decimal of a week. 
 
 18. Express 13 hrs. 15 min. 17 sec. as the fraction of 
 
 6 dys. 1 hr. 48 min. 7 sec. 
 
 19. Express 3 dys. 20 hrs. 35 min. 33 sec. as the decimal 
 
 of 27 dys. 13 hrs. 22 min. 30 sec. 
 
 20. Find the value of 5.58 yrs. 
 
COMPOUND QUANTITIES. 187 
 
 DIFFERENCE BETWEEN Two DATES. 
 
 176, Find the difference between April 3, 1885 and 
 rs. mos. v&. May 7, 1837. 
 
 K 7 ^ p filing the Difference between long dates, 
 
 30 davs are considered a month. As April is the 
 
 . 
 
 47 10 2fo f our th and May the fifth, month, we write 4 and 
 5 instead of the names of these months. 
 
 In finding the difference between short dates, the exact number of 
 days is generally counted. 
 
 Ex. 121. 
 
 1. On the 1st of January, 1885, how much time had passed 
 
 since the discovery of the Island of San Salvador, 
 Oct. 12, 1492? 
 
 2. At the birth of Lafayette, Sept. 6, 1757, what was 
 
 the age of George Washington, born Feb. 22, 1732? 
 
 3. If a note is dated March 5, 1885, and has 3 mos. 3 dys. 
 
 to run, when is the note due ? 
 
 4. If a note is discounted Feb. 1, and is due April 22, how 
 
 many months and days has it to run ? 
 6. Find the exact number of days from Sept. 23 to 
 Jan. 11. 
 
 NOTE. In finding the difference of these dates, the 23d of 
 September is not counted, but the llth of January is. 
 
 6. Find the exact number of days between March 5 and 
 
 July 4. 
 
 7. Find the exact number of days between June 3 and 
 
 Nov. 1. 
 
 8. Find the exact number of days between Feb. 3 and 
 
 June 3, of a common year. 
 
 9. Find the difference between June 7, 1885, and July 4, 
 
 1776. 
 
188 COMPOUND QUANTITIES. 
 
 ANGULAR MEASURE. 
 
 177, A circle is a plane figure bounded by a curved line 
 called the circumference, all points of which are equally dis- 
 tant from a point within called the centre. A part of the 
 circumference is called an arc, 
 
 178, A line drawn through the centre and terminated 
 by the circumference is called a diameter; and half the 
 diameter is called the radius, 
 
 If a straight line fixed at one end is revolved, the other end will 
 describe the circumference of a circle ; and the amount of rotation 
 of the straight line from its position at the start to any other given 
 position, is the angular magnitude described by the moving straight 
 line. Thus, if OA revolve about as a fixed point, the extremity A 
 will describe the circumference ABC. When OA has reached the 
 position OB, the part of the circumference 
 between A and B is described by A, and 
 the part of the angular magnitude about 
 the point 0, between OA and OB, is de- 
 scribed by OA. The angle A OB is such 
 a part of the angular magnitude about as 
 AB is of the circumference. 
 
 The circumference of every circle is divided 
 into 360 equal parts, called arc-degrees, and 
 corresponding to every one of these equal parts is an angle at the 
 centre of the circle. Hence the whole angular magnitude about any 
 point in a plane is divided into 360 equal parts called angle- degrees, 
 and the number of degrees in the angle formed by two lines drawn 
 from the centre of a circle is the same as the number of degrees in 
 the arc which is intercepted between these two lines. 
 
 179, An angle described by a line making one-fourth of 
 a revolution contains 90 and is called a right angle, as 
 AOB ; and OA and OB are said to be perpendicular to 
 
 each other. 
 
COMPOUND QUANTITIES. 189 
 
 UNITS OP ANGULAR MEASURE. 
 
 180, 60 seconds (") = 1 minute ('). 
 
 60 minutes = 1 degree (). 
 
 360 degrees = 1 revolution. 
 
 NOTE. A degree of the circumference of the earth at the equator 
 contains 60 geographical miles, or 69.16 statute miles. 
 
 Ex. 122. 
 
 1. Keduce 49 37' 29" to seconds. 
 
 2. In 13,978" how many degrees? 
 
 3. Find the value of of 360. 
 
 4. What part of the whole angular magnitude about a 
 
 point is -| of a second ? 
 
 5. Find the sum of 45.425, 0.115', 0.255". 
 
 6. Change 0.471 of a minute to the decimal of a degree. 
 
 7. What part of 7 35' 15" are 3 20' 45" ? 
 
 8. Divide 17 27' 13" by 5 ; multiply 8 19' 47" by 8 ; 
 
 and find the difference between the results. 
 
 9. From 7 0' 18" subtract 3 47' 36". 
 
 10. The latitude of New York is 40 42' 43" North ; the 
 
 latitude of Boston is 42 21' 30" North. Find their 
 difference in latitude. 
 
 11. The latitude of New Orleans is 29 57' 46" North ; the 
 
 latitude of Rio Janeiro is 22 56' South. Find their 
 difference in latitude. 
 
 HINT. Their difference in latitude is found by taking the 
 sum of their latitudes. 
 
190 COMPOUND QUANTITIES. 
 
 CURRENCY. 
 
 181, The coins of the United States are : 20-dollar, 10- 
 dollar, 5-dollar, 3-dollar, 2^-dollar, and 1-dollar gold coins ; 
 1-dollar, 50-cent, 25-cent, and 10-cent silver coins ; 5-cent 
 and 3-cent nickel coins ; and 2-cent and 1-cent bronze coins. 
 
 182, As any sum of money can be expressed in United 
 States currency as dollars and decimal fractions of a dollar, 
 it is always best to treat United States money as a simple 
 quantity. 
 
 183, The same is true of French, Italian, German, Rus- 
 sian, and Austrian currency. 
 
 184. 
 
 French Currency : 100 centimes - 1 franc (fr.) = $0.193 
 Italian Currency . 100 centissimi = 1 lira = $0.193. 
 
 German Currency : 100 pfennigs = 1 mark = $0.238. 
 Russian Currency . 100 kopecks - 1 rouble = $0.734. 
 Austrian Currency : 100 kreutz ere = 1 florin (fl.) - $0.453. 
 
 English Currency. 
 
 185, 4 farthings = 1 penny (d.). 
 
 12 pence = 1 shilling (s.). 
 20 shillings = 1 pound (). 
 
 A guinea = 21 . A crown = 5 s. 
 
 A sovereign = 20 s. A florin = 2 s. 
 
 A sovereign = $4.866J. 
 
 Ex. 123. (Oral.) 
 
 1. How many shillings in 48 d. ? in 60 d. ? 
 
 2. How many pence in 5s. ? in 10s.? in a sovereign? 
 
 3. How many shillings in 3? in 2f ? 
 
COMPOUND QUANTITIES. 191 
 
 4. How many pounds in 95 s. ? in 100 5. ? 
 
 5. How many pence in a crown ? in a florin? 
 
 6. How many shillings in a guinea ? in a half-sovereign ? 
 
 7. What part of a pound are 4 s. ? 5s.? 8s.? 
 
 8. What part of a shilling are 6 d. ? 4 d. ? 3 rf. ? 
 
 9. How many pence in 1 s. 3 d. ? in 2 s. 6 d.? in 3 s. 2 c?.V 
 
 Ex. 124. 
 
 1. Reduce 432 15s. 10 d. to pence. 
 
 2. Change 4238 farthings to higher denominations. 
 
 3. Express in dollars the value of 18. 
 
 4. Express in English money $60.83. 
 
 5. Express in United States money 3 16 s. 
 
 6. How many sovereigns are equal in value to $389.32 ? 
 
 7. Reduce s. to the fraction of a guinea. 
 
 8. What part of 13s. 2 d. 1 farthing are 9s. 10 d. 2 farthings? 
 
 9. Find the value of 5.375. 
 
 10. Reduce 6s. 5 d. 3.04 farthings to the decimal of a pound. 
 
 11. Express in pounds 5 9 s. 3 d. 
 
 12. Add 0.75, 0.125 guineas, 0.54s., 0.55 d. 
 
 MISCELLANEOUS. 
 186, 
 
 Numbers. 
 
 12 units = 1 dozen. 
 12 dozen = 1 gross. 
 12 gross = 1 great gross. 
 20 units = 1 score. 
 
 Paper. 
 
 24 sheets = 1 quire. 
 20 quires = 1 ream. 
 
 2 reams = 1 bundle. 
 
 5 bundles = 1 bale. 
 
192 COMPOUND QUANTITIES. 
 
 tnn Weights. 
 
 A bushel of corn or rye = 56 Ibs. 
 A bushel of corn meal, -\ 
 
 rye meal, or cracked > = 50 Ibs. 
 
 corn, 
 
 ) 
 
 A bushel of wheat = 60 Ibs. 
 
 A bushel of potatoes = 60 Ibs. 
 
 A bushel of beans = 60 Ibs. 
 
 A bushel of oats = 32 Ibs. 
 
 A bushel of barley = 48 Ibs. 
 
 i 
 
 . 45 Ibs. 
 
 - 14 Ibs. 
 
 A bushel of timo- 
 thy-seed 
 
 A stone of iron or 
 lead 
 
 A pig of iron or lead = 21J stone. 
 
 A fother of iron or ) = g 
 lead J 
 
 The weight of a bushel of potatoes, corn, etc., varies slightly in 
 different States, but the weights here given are those generally 
 adopted in business transactions. 
 
 A barrel of flour = 196 Ibs. 
 
 A barrel of pork or beef = 200 Ibs. 
 A cask of lime = 240 Ibs. 
 
 A cental of grain = 100 Ibs. 
 
 A quintal of fish = 100 Ibs. 
 
 Books. 
 188, A book formed of sheets folded in 
 
 2 leaves is a folio ; 
 
 4 leaves is a quarto ; 
 
 8 leaves is an octavo ; 
 12 leaves is a duodecimo ; 
 16 leaves is a 16mo. 
 
 Ex. 125. 
 
 1. How many barrels in 75 t. of beef? 
 
 2. In a car-load of 36,000 Ibs. of wheat, how many 
 
 bushels ? 
 
 3. Find the weight of 27 bu. of potatoes. 
 
 4. How much paper will be used by an author who sends 
 
 to a semi-weekly paper 6 sheets of manuscript twice 
 a week for a year ? 
 
COMPOUND QUANTITIES. 
 
 193 
 
 5. Reduce -J of a quire to the fraction of a bundle. 
 
 6. Reduce 2 bundles 6 quires 6 sheets to the fraction of 
 
 2 bales 1 bundle. 
 
 7. Reduce 3 bundles 7 quires 18 sheets to the decimal 
 
 fraction of a bale. 
 
 8. A button manufactory makes 96 dozen buttons a day. 
 
 How many great gross will it make in 24 wks. ? 
 
 9. Find the weight of 103 bu. 3 pks. 4 qts. of barley. 
 10. In 5 t. 624 Ibs. of oats, how many bushels? 
 
 LONGITUDE AND TIME. 
 
 189, A meridian is any line drawn straight around the 
 earth, and passing through both poles. 
 
 190, The longitude of a place is the 
 angle of inclination of the two planes 
 which are supposed to pass through 
 the centre of the earth, and contain, 
 the one the meridian of that place, 
 and the other the standard meridian. 
 Thus, the longitude of (7, reckoned 
 from meridian ABE, is the angle 
 
 BOO, OB and GO being both perpendicular to the diam- 
 eter of the earth AE at the point 0. Places on the Eastern 
 Hemisphere are in East Longitude ; on the "Western Hem- 
 isphere, in West Longitude. 
 
 191, As the earth turns upon its axis once in twenty- 
 four hours, a point on the earth's surface will describe 
 a circumference (360) in twenty-four hours. Therefore 
 longitude may be reckoned in time as well as in degrees. 
 
 In one hour a point on the earth's surface describes -fa of 
 
194 COMPOUND QUANTITIES. 
 
 360 = 15 ; in one minute -fa of 15 = 15' ; and in one 
 second ^ of 15' =15". 
 
 Again, since it requires one hour (60 min.) for a point to 
 pass over 15, to pass over 1 it requires -fa of 60 min. = 4 
 min. ; and to pass over 1' it requires -fa of 4 min. = 4 sec. 
 
 192, Express 20 36' 15" of longitude in time. 
 
 15) 20 36' 15" Since 15 longitude give 1 hr. in 
 
 Ihr. 22 min. 25 sec. time > I5f longitude 1 min., and 15" 
 longitude 1 sec., divide 20 36' 15" by 
 
 15, as in compound division, and the quotient will be the time 
 required. 
 
 193, Express 1 hr. 4 min, 4 sec. in degrees. 
 
 1 hr. 4 min. 4 sec. Since 1 hr. of time equals 15 of longitude, 
 15 1 min. of time 15 f , and 1 sec. of time 15", mul- 
 
 16 1' 0" ^Pty 1 kr. ^ m ^ n - 4 8ec - hy 15, as in compound 
 
 multiplication, and the product will be the lon- 
 gitude required. 
 
 194, Hence, if longitude is expressed in degrees, divide by 
 15 ; the quotient gives the longitude in hours, minutes, and 
 seconds. 
 
 195, If longitude is expressed in time, multiply by 15 ; 
 the product gives the longitude in degrees, minutes, and 
 seconds. 
 
 Ex. 126. 
 
 1. The difference in time between New York and Paris 
 
 is 5 hrs. 5 inin. 20 sec. What is the difference in 
 longitude ? 
 
 2. Boston is 71 3' and San Francisco 122 26' west of 
 
 Greenwich. What is the difference in clock-time 
 between the two cities ? 
 
COMPOUND QUANTITIES. 195 
 
 3. The difference in clock-time between New York arid 
 
 Canton is 12 hrs. 28 min. 12 sec. Find the differ- 
 ence in longitude. 
 
 4. The difference in longitude between Cincinnati and 
 
 Boston is 13 26'. Find the difference in time. 
 
 5. New York is 74 and Cincinnati is 84 30' west longi- 
 
 tude. Find the difference in time. 
 
 6. The difference in time between Canton and Cincinnati 
 
 is 13 hrs. 10 min. 8 sec. Find the difference in 
 longitude. 
 
 7. The difference in longitude between New York and 
 
 Canton is 187 3'. What is the difference in time ? 
 
 8. Find the difference in time between Philadelphia, lon- 
 
 gitude 75 10 f West, and Buenos Ayres, longitude 
 58 22 f West. 
 
 9. The difference of time between St. Petersburgh and 
 
 New Orleans is 8 hrs. 1 min. 16 sec. What is the 
 difference in longitude ? 
 
 10. Find the difference in time between the Cape of Good 
 Hope, longitude 18 28' East, and Halifax, longitude 
 63 36' West? 
 
 196, Since the sun appears to move from east to west, 
 sunrise will occur earlier at all points east, and later at all 
 points west, of a given place. Hence, clock-time will be 
 later in all places east, and earlier in all places west, of a 
 given meridian. 
 
 Therefore, if the time of a place be given, 
 
 To find the time of a place east, add to the given time 
 the difference of time between the two places. 
 
 To find the time of a place west, subtract from the given 
 time the difference of time between the two places. 
 
196 COMPOUND QUANTITIES. 
 
 To FIND THE DIFFERENCE IN CLOCK-TIME WHEN THE DIFFERENCE 
 IN LONGITUDE is KNOWN. 
 
 When it is noon at Boston (long. 71 3 f 30" West), what 
 is the time at Paris (long. 2 20' 22" East) ? 
 71 3 f 30"W. 
 2 20' 22" E. 
 73 23' 52" . . . difference in longitude. 
 
 15) 73 23' 52" 
 
 4 hrs. 53 min. 35.47 sec. 
 
 G min. 24} sec. before 5 P.M. Ans. 
 
 Since Boston is west and Paris is east of the meridian of Green- 
 wich, the difference between their longitudes is found by taking the 
 sum of their longitudes. 
 
 Their difference in longitude, 73 23' 52", is equivalent to 4 hrs. 
 53 min. 35.47 sec., and as Paris is east of Boston, the time at Paris is 
 found by adding the 4 hrs. 53 min. 35.47 sec. to the time at Boston. 
 
 Ex. 127. 
 
 1. When it is noon at Chicago, what is the hour at 
 
 New York, the difference in longitude being 13 37'? 
 
 2. What is the time in London when it is half-past 3 in 
 
 the afternoon at Constantinople, Constantinople be- 
 ing 29 east of London ? 
 
 3. The longitude of New York is 74 West, that of Paris 
 
 is 2 20' East, When it is 15 minutes past 10 A.M. 
 in New York, what is the time in Paris? 
 
 4. The longitude of Boston is 71 3' and that of New 
 
 York 74 West. What is the time in Boston when 
 it is midnight in New York ? 
 
 5. The difference in longitude between San Francisco and 
 
 Chicago is 34 49 f . What time is it at San Francisco 
 when it is 9 o'clock P.M. at Chicago? 
 
COMPOUND QUANTITIES 197 
 
 6. Paris is 45 10' east of Rio Janeiro. What time is it 
 
 at Rio Janeiro when it is 7 o'clock P.M. at Paris? 
 
 7. If the sun rises at half-past 4, when it is sunrise at 
 
 Richmond, Va., what is the time at Rouen, France, 
 the difference of longitude being 78 46' ? 
 
 8. The French residents in Calcutta wish to unite with 
 
 the people of Paris in a celebration to occur at 
 3 o'clock P.M. Paris is 2 20' East, Calcutta, 
 88 27 ' East. At what hour must the festivities 
 begin in Calcutta ? 
 
 NOTE. Standard Time is the clock-time of some selected meridian. 
 Eastern standard time is the clock- time of the meridian 75 west of 
 Greenwich, and is five hours slow of Greenwich time. Central 
 standard time is the clock-time of 90 west of Greenwich, and is 
 just one hour slow of Eastern standard time. Mountain standard 
 time is the clock-time of the meridian of 105, and is one hour slower 
 than that of 90. Western standard time is the clock-time of the 
 meridian of 120, and is one hour slower than that of 105. The 
 railroads and many cities and towns of the United States have 
 adopted standard time. 
 
 MISCELLANEOUS EXAMPLES. 
 
 Ex. 128. 
 1. Find the amount of the following bill : 
 
 /f >3 -M-ftd. dflstsM- (cb. 
 7 
 
 @ 
 
 
 How much change out of a 20-dollar bill should the 
 purchaser receive? 
 
198 COMPOUND QUANTITIES. 
 
 Make out the bills for : 
 
 2. 27 yds. of flannel at 80 cts. a yard ; 
 32 yds. of calico at 11 cts. a yard ; 
 
 3 doz. of stockings at $2 per dozen ; 
 6 pairs of gloves at 84 cts. a pair ; 
 
 4 collars at 35 cts. each. 
 
 3. 10 Ibs. of sugar @ 10 cts. a pound ; 
 
 6 Ibs. of tea @ 88 cts. a pound ; 
 
 8 Ibs. of coffee @ 32 cts. a pound ; 
 12 Ibs. of currants @ 11-J- cts. a pound ; 
 
 10 Ibs. of rice @ 9 cts. a pound. 
 
 4. 18| Ibs. of beef @ 22 cts. a pound ; 
 10^ Ibs. of mutton @ 21 cts. a pound ; 
 
 7-J- Ibs. of pork @ 17 cts. a pound ; 
 
 16 Ibs. of veal @ 16 cts. a pound ; 
 
 14f Ibs. of ham @ 20 cts. a pound. 
 
 5. 5 Ibs. of soap @ 9^ cts. a pound ; 
 3^- Ibs. of candles @ 13 cts. a pound ; 
 2 Ibs. of butter @ 35 cts. a pound ; 
 
 56 Ibs. of rice @ 4^- cts. a pound. 
 
 6. 7 doz. and 4 eggs @ 18 cts. per dozen ; 
 19 Ibs. of soap @ 11 cts. per pound ; 
 18 Ibs. of butter @ 28 cts. per pound ; 
 13^- Ibs. of cheese @ 15 cts. per pound ; 
 
 ^ Ib. pepper @ 2^- cts. per ounce. 
 
 7. 12| t. of hay @ $18 a ton ; 
 66 bu. of rye @ $1.26 a bushel ; 
 
 102 bu. of barley @ 78 cts. a bushel ; 
 
 5 bbls. of flour @ $6.60 a barrel. 
 
COMPOUND QUANTITIES. 199 
 
 8. A man walks 1 mi. 47 rds. in 20 min. How many 
 
 hours will it take him to walk 41 mi. 92 rds. ? 
 
 9. Kequired the cubic feet of a box 6 ft. 6 in. long, 4 ft. 
 
 9 in. wide, and 3 ft. 3 in. deep. 
 
 10. What will be the weight of a wall of brick- work 10 ft. 
 
 long, 1-J- ft. thick, 4 ft. high, if each cubic foot 
 weighs 120 Ibs. ? 
 
 11. How many cubic yards of earth will be cut out of a 
 
 drain 420 ft. long, 2 ft. wide, and 4 ft. deep ? 
 
 12. What will be the expense of glazing a window of 16 
 
 squares, each LJ- ft. long and -f ft. wide, at $1.08 
 per square foot ? 
 
 13. What length must be cut off an inch-board 9 in. wide 
 
 to obtain 4 ft. board measure ? 
 
 14. How many boards each 11^- ft. long, and 10 in. wide 
 
 will be required for the flooring of a room 23 ft. long 
 and 17 ft. Gin. wide? 
 
 1.5. A farm of 22^- acres is divided into house-lots measur- 
 ing 75 yds. in length by 33 yds. in breadth. How 
 many lots are there ? 
 
 16. At 9 cts. per cubic foot, what will be the cost of a 
 
 block of stone 9 ft. long, 5^- ft. wide, and 4 ft. thick? 
 
 17. At 50 cts. an ounce, what is the value of a silver cup 
 
 weighing 15 oz. 12 dwt. 12 grs. ? 
 
 18. If the cost of making a barrel of flour into bread is 
 
 $2.20, and flour is worth $9 a barrel, what should a 
 baker receive for a loaf containing If Ibs. of flour ? 
 
 19. At $30 per M., what is the value of a stick of timber 
 
 24 ft. long, and 2 ft. square at the end ? 
 
200 COMPOUND QUANTITIES. 
 
 20. A schoolroom is 44 ft. long, 28-^ ft. wide, and 13 ft. 
 
 high. What will be the cost of painting the four 
 walls and the ceiling, at the rate of 18 cents a square 
 yard, making no allowance for doors and windows ? 
 
 21. A druggist pays 50 cts. a pound avoirdupois for 
 
 chloride of potash, and retails it in powders contain- 
 ing 135 grs., at 5 cts. each. How much will he 
 gain on 5 Ibs. ? 
 
 22. Find the entire surface of a block of marble 8 ft. long, 
 
 2 ft. wide, 1 ft. thick. 
 
 23. How many revolutions will be made by a wheel 3| 
 
 yds. in circumference in passing over 198 mi. ? 
 
 24. When an ounce of gold is worth $16.25, what must be 
 
 paid for -^ of a pound ? 
 
 25. If candles 8 in. long are worth 9 cts. a half-dozen, 
 
 and candles 10J in. long are worth 11 cts. a half- 
 dozen, which is the better kind to buy ? 
 
 26. How many silver spoons weighing 1 oz. 18 dwt. 12 
 
 grs. each can be made from 23 oz. 2 dwt. of silver ? 
 
 27. An apprentice 14 yrs. 11 mos. 14 dys. old is to serve 
 
 his employer until he is 21 yrs. of age. How long is 
 he to stay with his employer ? 
 
 28. What is the rate per hour of a horse that travels 18 mi. 
 
 1620 yds. in 3 hrs. 45 min. ? 
 
 29. At 15 cts. a yard, what will be the cost of fencing a 
 
 rectangular field 325 yds. long and 215 yds. wide ? 
 
 30. What will be the width of carpeting, if 120 yds. are 
 
 necessary to cover a floor 30 ft. long and 22-J- ft. 
 wide? 
 
COMPOUND QUANTITIES. 201 
 
 31. When the mercury in the tube of a barometer is 30 in. 
 high, the pressure of the atmosphere is about 15 Ibs. 
 for every square inch. What will be its pressure 
 when the mercury stands 25 in. high? 
 
 32- A cistern containing 60 hhds. of water has two pipes 
 open, by one of which 3 gals, of water per minute 
 run in, and by the other 9 gals, run out. In how 
 many hours will the cistern be emptied ? 
 
 33. How many pounds of cement will be required to plas- 
 
 ter an open cistern whose dimensions are 4^- it. 
 long, 3^- ft. wide, and 2| ft. deep, if the cement on 
 a square foot weighs 6 J Ibs. ? 
 
 34. How many tons of water will the cistern in example 
 
 33 hold, if a cubic foot of water weighs 1000 oz. ? 
 
 35. What will be the cost of covering with paper % yd. 
 
 wide the four walls of a room 21 ft. long, 16 ft. wide, 
 and 10 ft. high, if the cost of the paper is 12 cts. 
 per yard, and no allowance is made for doors and 
 windows ? 
 
 36. What is the breadth of a rectangular field containing 
 
 7 A., if the length is 242 yds. ? 
 
 37. A certain watch gains 3|- sec. in 24 hrs., and another 
 
 loses 2-z 1 - sec. in the same time. If both be set right 
 on Monday at noon, what will be the difference 
 between them at 6 o'clock (true time) the next 
 Saturday evening ? 
 
 38. A milkman paid a farmer $3.20 for ten 2-gal. cans of 
 
 milk. He lost 20 qts. At what price per quart 
 must he retail the remainder to gain 8 cts. a gallon? 
 
 39. A man who had f of a square mile of woodland sold 
 
 12^- sq. rds. How much had he left ? 
 
202 COMPOUND QUANTITIES. 
 
 40. How many yards of Florence silk -| yd. wide will be 
 
 required to line 19 yds. of camel's hair cloth 1-j- yds. 
 wide ? 
 
 41. How many days from Sept. 16, 1882, to Feb. 12, 1884? 
 
 42. A miller makes 154 bbls. of flour from 885 bu. of 
 
 wheat. How many bushels on the average are 
 required for a barrel of flour ? 
 
 43. What must be the length of a walk 2-J- ft. wide to con- 
 
 tain 38 sq. ft. ? 
 
 44. A cistern 7 ft. long and 5 ft. wide contains 105 cu. ft. 
 
 What is its depth ? how many gallons of water will 
 it hold ? 
 
 45. What must be paid for a pile of wood 25 ft. long, 3 ft. 
 
 high, and 4 ft. wide at $5.50 per cord? 
 
 46. Each person on the average breathes 28 cu. ft. of air 
 
 in an hour. How many hours will the air in a room 
 14 ft. long, 12 ft. wide, and 6 ft. high last 12 men ? 
 
 47. Sound travels at the rate of 1130 feet a second. How 
 
 long after the flash will the clap of thunder come 
 when the cloud is 2 mi. 1000 yds. distant ? 
 
 48. There are 9 oz. of iron in the blood of 1 man. How 
 
 many men would furnish iron enough in their veins 
 to make a ploughshare weighing 22| Ibs. ? 
 
 49. The fore wheel of a carriage is 4 ft. 7 in. in circum- 
 
 ference, and the hind wheel 5 ft. 6 in. How many 
 more times will the fore wheel turn than the hind 
 wheel in going a distance of 1 mi. ? 
 
 50. A boarding-house uses 3 pks. of potatoes daily. At 
 
 87-J- cts. per bushel, what will be the expense for pota- 
 toes during October, November, and December ? 
 
CHAPTER X. 
 
 PERCENTAGE. 
 
 197, 1. A boy gave away 6 marbles out of every hundred 
 he had. How many did he give away out of 400? 
 of 600? of 1000? 
 
 2, A man had 300 sheep, and sold 10 out of every hun- 
 
 dred. How many did he sell ? 
 
 3, A man sold 7 tons of coal out of every hundred he had. 
 
 How many did he sell out of 900 tons ? 
 
 4, A man sells 11 Ibs. of sugar out of every hundred he 
 
 has. How many pounds will he sell out of 500? 
 How many pounds will he have left? 
 
 198, In considering the increase or decrease of quantities, 
 we usually employ the number 100 as the representative of 
 the quantity considered, 
 
 199, Instead of using the phrases 6 in every hundred, 10 
 in every hundred, 7 in every hundred, 11 in every hundred, 
 we say 6 per cent, 10 per cent, 7 per cent, 11 per cent. The 
 words per cent therefore mean hundredths. 
 
 200, The symbol % is used for the words per cent. 
 
 How many hundredths of a number are : 
 
 20%? 
 25%? 
 
204 PERCENTAGE. 
 
 How many per cent of a number is : 
 
 0.20? 0.75? 0.12J? 1.40? 
 
 0.15? 0.06^? 0.50? 2.25? 
 
 What common fraction of a number (in its lowest terms) is 
 
 10%? 25%? 8|%? 12|%? 125%? 
 
 20%? 50%? 6J%? 66|%? 160%? 
 
 16|%? 75%? 334%? 100%? 175%? 
 
 Express as hundredths and as per cent : 
 
 i; I' *; f: A; A; A; 
 
 8 > S > TS" TO" ' SO i 
 
 4 5> T i Toi Ai ^5' 
 
 201. Express \ % as hundredths and as a common frac- 
 
 tion. 
 
 In like mariner express as hundredths and also as com- 
 mon fractions - 
 
 tyo : I7o ; f* ; f> ; 
 
 3 _1_ fij. . 5 C/ n JL 0) 
 
 %; 1%; 1%; 1%; &% 
 
 202, Find 8% of 250 bu. of corn. 
 
 8% of a number is yj^ of the number ; and T |^ of 250 bu. = 20 bu. 
 
 20 bu. Ans. 
 Find 20% of 80 yds. of cloth. 
 
 20 % = iSfr = i J and J of 80 yds. is 16 yds. 
 
 16 yds. Ans. 
 
PERCENTAGE. 
 
 205 
 
 Ex. 129. (Oral) 
 
 8. 80% of 400 A. 
 
 9. 6J% of 320 rds. 
 
 10. 121% O f 400 melons. 
 
 11. 66f% of 300 oranges. 
 
 12. 8-^% of 1 doz. eggs. 
 
 Find: 
 
 1. 4% of 400 sheep. 
 
 2. 5% of 1000 bricks. 
 
 3. 8% of 200 ft. of board. 
 
 4. 6% of 90 dys. 
 
 5. 10% of 150 cds. of wood. 
 
 6. 20%of250prs.ofgloves. 13. 75% of 40 hens. 
 
 7. 25% of 120 horses. 14. 60% of 20 girls. 
 
 15. 16|% of 60 Ibs. of butter. 
 
 16. 37|% of 120 gals, of syrup. 
 
 17. 62|% of 800 soldiers. 
 
 18. |% of 500 bu. of wheat. 
 
 19. \ r % of 4000 yds. of cloth. 
 
 20. |% of 100 dollars. 
 
 Ex. 130. 
 
 Find : 
 
 1. 9% of 1297. 
 
 2. 2|% of 4300. 
 
 3. | of 1% of 1346. 
 
 4. 12% of 6072. 
 
 5. 1% of 150,975. 
 
 6. 1 T V% of 1984. 
 
 7. 150% of 1050. 
 
 8. 100% of 7968. 
 
 9. A farmer having a flock of 1200 sheep lost 37% of 
 them. What per cent of them, and how many sheep, 
 had he left ? 
 
 10, If copper ore yields 6% of pure metal, how many 
 pounds of copper will be obtained from 1 t. of ore ? 
 
206 PERCENTAGE. 
 
 11. If a man buys 24 A. of land at $84 an acre, what 
 
 must be the annual income that the investment may 
 yield 10% ? 
 
 12. A grocer bought 40 cwt. of sugar for $240. 4% of 
 
 it is wasted, and the remainder is retailed so that 
 there is neither loss nor gain. What is the retail 
 price per pound ? 
 
 13. A stone-mason contracted to dig a cellar 45 ft. long, 
 
 36 ft. wide, and 6 ft. deep at 25 cts. a cubic yard, 
 He lost 5% of his contract price. What was his 
 loss? 
 
 14. A coal-dealer bought 25,784 t. of coal at $5 a ton. 
 
 He sold 40% of it at $7, 20% at $8.50, and the 
 remainder at $4.50. How much did he gain? 
 
 15. A gentleman owns 2 farms. The first contains 360 A., 
 
 and the number of acres in the second is 150% of 
 the number of acres in the first. Find the number 
 in the second farm. 
 
 203, What per cent of 20 is 5 ? 
 5 = &or Jof20; and J - -flfr = 25%. 
 Therefore 5 = 25% of 20. 25 % . Ans. 
 
 Ex. 131. (Oral.) 
 What per cent of: 
 
 1. 16 is 8? 9. $72 are $18? 
 
 2. 20 is 5? 10. $52 are $39? 
 
 3. 25 is 15 ? 11. 50 qts. are 5 qts. ? 
 
 4. 48 is 8? 12. 66 gals, are 6 gals. ? 
 
 5. 100 is 12? 13. 480 dys. are 24 dys. ? 
 
 6. 2is? 14. 90 cds. are 9 cds. ? 
 
 7. 3 is |? 8. fisf? 15. 80 men are 50 men ? 
 
PERCENTAGE. 207 
 
 Ex. 132. 
 
 1. From a school of 150 scholars, 50 are absent. What 
 
 per cent of the whole is the number present ? 
 
 2. In a school numbering 200 the daily average atten- 
 
 dance is 160. What is the per cent of attendance ? 
 The number absent on the average is what per cent 
 of the number present ? 
 
 3. A person bought a house and lot for $6000, paying 
 
 $5000 for the house. The value of the lot is what 
 per cent of the value of the house ? 
 
 4. From a peck of corn a crop of 48J bu. was raised. 
 
 What per cent was the increase ? 
 
 5. From 67-^- bu. of corn, 6 bu. 3 pks. are sold. What 
 
 per cent of the whole is sold ? 
 
 6. A house worth $8000 rents for $720 a year. What 
 
 per cent of its value does it rent for ? 
 
 7. From Delhi to Bombay the distance is 720 miles, and 
 
 from Delhi to Madras 1080 miles. What per cent 
 of the distance to Madras is the distance to Bombay ? 
 
 8. Westminster Hall is 270 ft. long and 75 ft. broad. 
 
 What per cent of the length is the breadth? 
 
 9. The Peak of Teneriffe is 12,232 ft. high. What per 
 
 cent of a mile is its height ? 
 
 10 The Danube is 1630 miles long, and the Missouri from 
 its source to the Gulf of Mexico is 4000 miles long 
 What per cent of the length of the Missouri is the 
 length of the Danube ? 
 
 11. What per cent of 7 hrs. and 30 min. are 6f min. ? 
 
 12. What per cent of 3 wks. and 4 dys. are 3 dys. and 
 
 10 hrs. ? 
 
208 PERCENTAGE. 
 
 204, What is the number of which 15 is 5% ? 
 If 15 is 5%, then 15 is y^y or ^V of the number, and, if 15 is -fa of 
 the number, the number itself will be 20 x 15 = 300. 
 
 300. Ans. 
 Ex. 133. (Oral.) 
 What is the number of which : 
 
 1. 10 is 20%? 6. |is!6%? 11. J is 175%? 
 
 2. 3 is 10%? 7. f is 50%? 12. 17 is 34% ? 
 
 3. 8 is 25% ? 8. f is 75% ? 13. 50 is 62% ? 
 
 4. 4 is 6%? 9. 60 is 60%? 14. 300 is 0.3%? 
 
 5. 5 is 8% ? 10. 50 is 40% ? 15. 20 is 
 
 Ex. 134. 
 
 1. 10.08 is 16% of what number? 
 
 "ps 2. 24 is 7|% of what number? 
 
 3. 10.94 is i% of what number ? 
 
 4. 2500 is 12|% of what number? 
 
 5. 960 is 33% of what number ? 
 
 6. 6000 is 20% of what number? 
 ~f 7. 990 is 110% of what number? 
 
 8. 810 is 90% of what number? 
 
 9. 980 is 175% of what number? 
 
 10. A city in 5 yrs. increased 12,000 in population, a gain 
 
 of 25%. What was the population at the beginning 
 and end of the 5 yrs. ? 
 
 11. A schoolboy in one week read 450 lines of Latin, 
 
 which was 75% of the number in the book. How 
 many lines had he still to read ? 
 
PERCENTAGE. 209 
 
 12. A boy sold chestnuts at 12-J- cts. a quart, which was 
 
 200% of their cost. What did they cost a bushel? 
 
 13. A clerk spent 60% of his salary for board, 20% of it 
 
 for clothes, 11% for books, and saved $117. What 
 was his salary ? 
 
 14. At Christmas a lady gave her daughter an atlas worth 
 
 $27, and -f of the cost of the atlas was 90% of the 
 sum paid for an engraving. What was the sum paid 
 for the engraving ? 
 
 15. A sea-captain owning 60% of a vessel gave to his NL, 
 
 son 50% of his share, which was worth $6000. 
 What was the value of the vessel ? 
 
 16. A gentleman worth $50,000 gave 30% of his property 
 
 to his son, and this gift was 80% of the property 
 which the son already owned. Find the amount 
 the son was worth after receiving his father's gift. 
 
 205, By selling a horse for $90, a man gains 20% of its 
 cost. Find the cost. 
 
 He gets the cost (100%) and 20% of the cost, or 120% of the cost. 
 The question therefore is, $90 is 120% of how many dollars? 
 
 A man sold a horse for $90, and lost 25% of the cost. 
 What did the horse cost ? 
 
 He got the cost (100%) minus 25% of the cost, or 75% of the cost. 
 The question therefore is, $ 90 is 75 % of how many dollars ? 
 
 Ex. 135. (Oral) 
 
 1. 36 is 12-% more than what number? 
 
 2. 65 is 6^% less than what number? 
 
 3. 68 is 6^% more than what number? 
 
 4. 75 is 12^-% less than what number? 
 
 5. By selling a hat for $5.40, I sell it for 20% more 
 
 than the cost. What was the cost ? 
 
210 PERCENTAGE. 
 
 6. A manufacturer sells mowing-machines at $ 125 apiece, 
 
 and gains 40%. What do they cost? 
 
 7. Sold a carriage for $240, which was 20% more than 
 
 the cost. What was the cost ? 
 
 8. 64 is 33^% more than what number? 
 
 9. What number diminished by 5% of itself equals 190? 
 10. What number diminished by 10% of itself equals 180 ? 
 
 Ex. 136. 
 
 1. 874 is 33% less than what number? 
 
 2. 1740 is 20% more than what number? 
 
 3. 40% of 4000 is 20% less than what number? 
 
 4. What number diminished by 15% of the number 
 
 equals 5100? 
 
 5. What fraction increased by 25% of itself equals j-f ? 
 
 6. 7500 is 33% less than what number? 
 
 7. A drover sold 250 sheep for $1150, which was 15% 
 
 more than they cost. Find the cost of the sheep 
 per head. 
 
 8. At a forced sale, a bankrupt sold his house for $8000, 
 
 which was 20% less than its real value. If the 
 house had been sold for $12,000, what per cent 
 above its real value would it have brought ? 
 
 9. A flock of sheep has been increased by 250% of its 
 
 number, and now numbers 1050. What is the origi- 
 nal number? 
 
 10. If 20% be lost on a ton of rye-straw sold for $19.20, 
 what is the cost of the straw ? 
 
PERCENTAGE. 211 
 
 PROFIT AND Loss. 
 
 206. The difference between the buying and selling prices 
 of goods is called profit or loss, according as the selling-price 
 is more or less than the buying-price. 
 
 Ex. 137. 
 
 1 A horse which cost $80 was sold for $60. Find the 
 
 actual loss and the loss per cent. 
 NOTE. Gain or loss is so much per cent on the cost of the goods. 
 
 2. Flour that cost $10 per barrel was sold for $12 per 
 
 barrel. Find the gain per cent. 
 
 3. If milk is bought at 4 cts. a quart, and sold at 6 cts., 
 
 what is the gain per cent ? 
 
 4. Goods that cost $40 were sold at 20% below cost. 
 
 What was the actual loss ? 
 
 5. Velvet is sold for $3.75 per yard, at a gain of 25%. 
 
 Find the cost of the velvet. 
 
 6. By selling cloth at $1.60 a yard, a merchant loses 
 
 20%. What is the cost? 
 
 7. Five cords of wood costing $20 were sold at $7 per 
 
 cord. What was the gain per cent ? 
 
 8. A carpenter paid $5000 for a house ; spent in repairs 
 
 a gum equal to 80% of the purchase-price ; and 
 then sold the house for $12,000. How much did lie 
 gain, and what per cent of the whole cost ? 
 
 9. In selling 32 yds. of cloth, a merchant made $6.40, 
 
 which was 16% of the cost. What did the cloth 
 cost a yard ? 
 
212 PERCENTAGE. 
 
 10. Goods were sold for $1615.12, at a gain of 
 
 What did they cost ? ' 
 
 11. If tea sold at 84 cts. a pound gives a profit of 20%, 
 
 what would be the profit per cent if it were sold at 
 75 cts. a pound ? 
 
 12. A trader's profits were $1980 in the year 1880. This 
 
 sum was 20% more than his profits in 1881. Find 
 his profits in 1881. 
 
 13. A cord of wood costing $4.50 sold for $9. What was 
 
 the gain per cent ? 
 
 14. A house-lot was sold for $1850, at an advance of 15% 
 
 on its cost. What would have been the gain per 
 cent if it had been sold for $2210? 
 
 15. A manufacturer owning of a factory sold 12^% of 
 
 his share, at 10% above cost, for $1100. What is 
 the cost of the factory ? 
 
 16. Wliat per cent is made in buying coal by the long ton, 
 
 at $5 a ton, and selling it by the short ton, at the 
 same price ? 
 
 17. Corn cultivated at an expense of 28 cts. a bushel is 
 
 sold at 1 ct. a pound. What is the gain per cent ? 
 
 18. What per cent advantage is there in buying opium by 
 
 the pound avoirdupois, and selling it by the pound 
 apothecaries' weight? 
 
 19. A grocer lost 5% in selling a 50-lb. tub of butter for 
 
 $ 15.20. What did the butter cost per pound ? 
 
 20. Ten cows were sold for $690, at a gain of 15%. For 
 
 how much per head, on the average, should, they 
 have been sold to ^ ji 30% ? 
 
PERCENTAGE. 213 
 
 21. For what price per dozen must gloves be bought in 
 
 order that, by selling them at $1.75 per pair, there 
 may be a gain of 25% ? 
 
 22. A merchant lost 25% by selling flour at $6 per barrel. 
 
 If he had sold it at $9 per barrel, what would have 
 been the gain per cent? 
 
 .23. A fruit-grower sent to New York 300 peck baskets of 
 peaches, valued at 75 cts. each. Sixty baskets were 
 spoiled on the journey. At what rate per basket 
 must he sell the remainder to make 20% profjTon 
 the entire value of his fruit ? 
 
 24. Sold goods at a loss of 20%, and actual loss of $57.50. 
 
 What was the prime cost ? 
 
 25. Find the selling-price of goods by which there is a loss 
 
 of 2% and an actual loss of $54.50. 
 
 26. How many pounds of cheese bought at 9 cts. a pound 
 
 must be sold at 12 cts. a pound to gain $30? 
 
 27. Sold steel at $25.44 a ton with a profit of 6% and a 
 
 total profit of $ 103.32. What quantity was sold ? 
 
 COMMISSION AND BROKERAGE. 
 
 207, The commission paid to an agent for his services is 
 generally reckoned at a rate per cent. 
 
 208, The sum left after the payment of the commission 
 and other expenses is called net proceeds, 
 
 209, Commission paid to a broker is called brokerage, 
 
 210, In selling, the commission is reckoned on the money 
 received; in buying, the commission is reckoned on the 
 money paid, 
 
214 PERCENTAGE 
 
 (1) A real-estate agent sold a house for $7000. Find 
 the amount of his commission, at \\%. 
 
 $7000 
 0.01} 
 
 3500 
 7000 
 
 1105.00 $105. .47W. 
 
 (2) A jorkey receives $32 as his commission, at 4%, for 
 purchasing a pair of horses. What did he pay for the 
 horses ? 
 
 Commission on $1 invested at 4% is $0.04. Therefore the sum 
 invested to obtain a commission of $32 is 
 
 ? a^ = ?80a $800. Ans. 
 
 (3) If a commission of $212.94 is paid for selling wool 
 to the amount of $6552, what is the rate per cent allowed? 
 
 ^.f the commission on $6552 is $212.94, the commission on $1 will be 
 
 $212^ 03*. 
 
 * 6552 
 Therefore the commission is at the rate of 3J%. 
 
 (4) A speculator in New York sent $ 18,360 to his agent 
 in Chicago, with which to buy wheat. If the agent charges 
 2% for buying, how many bushels of wheat can he buy 
 at 90 cents a bushel ? 
 
 Commission on $1, at 2%, is $0.02. Hence out of every $1.02 
 sent, there is invested in wheat $ 1. 
 
 Hence, out of $18,360 sent there is invested in wheat 
 $18360 
 
 -I Q(V)(~) 
 
 and the number of bushels of wheat bought is = 20,000. 
 
 20,000. Ans. 
 
PERCENTAGE 215 
 
 Ex. 138. 
 
 1. A commission-merchant sold 90 bbls. of flour at $6 a 
 
 barrel, and received 5% commission. What was his 
 commission ? 
 
 2. A commission of $ 121.29 was charged for selling $1866 
 
 worth of goods. What was the rate of commission ? 
 
 3. A grain-dealer charged 7% commission for selling a 
 
 quantity of wheat, and received for his commission 
 $109.20. What was the total amount received for 
 the wheat? 
 
 4. A real-estate broker sold a house on \% commission, 
 
 and sent to the owner as net proceeds $3060. What 
 was the broker's commission, and what sum was 
 received for the house? 
 
 HINT. The broker received f>]%, and the owner 93^%, of the 
 sum the house sold for. Hence the question is, $3060 is 933% 
 of what sum ? 
 
 5. A New York merchant sent $1295.32 to New Orleans 
 
 to be expended in cotton. The broker in New 
 
 Orleans charged 6% commission. What sum was 
 
 paid for cotton ? 
 
 HINT. The broker received C% commission on the money 
 invested in cotton. Therefore, the question is: $1295.32 is 
 106% of what sum? 
 
 6. If $5125 include the amount expended for wool and 
 
 2^% commission to the purchasing agent, how much 
 money does the agent lay out in wool ? 
 
 7. A lawyer collected 75% of a debt of $1260, and charged 
 
 5% commission on the sum collected. What did the 
 creditor receive? 
 
216 PERCENTAGE. 
 
 8. An agent sold 420 bu. of corn at 60 cts. a bushel, and 
 
 the commission was $7.56. What rate of com- 
 mission was charged for selling ? 
 
 9. A land agent charged 4% for selling 750 A. of land 
 
 at $ 20 an acre. What was his commission ? 
 
 10. How many yards of cloth, at 45 cts. a yard, can an 
 agent buy with the commission received from the 
 sale of 180 bu. of potates at 50 cts. a bushel, his rate 
 of commission for selling the potatoes being 
 
 11. A man bought a horse for $225, which sum was half 
 
 of his commission, at %%%, on the sale of a farm. 
 What did the farm bring ? 
 
 12. A young man selling tea on 2|% commission sent to 
 
 his employer $875.25 as the net proceeds of one 
 week's sales. What did the average daily sales 
 amount to ? 
 
 13. A St. Louis merchant received $150 as his commission, 
 
 at 2J% for purchasing 1200 bbls. of flour. What 
 was the price paid per barrel ? 
 
 14. A broker sold for a farmer 12,000 Ibs. of pork, at 8| 
 
 cts. per pound. He charged 3% commission for 
 selling, and paid $37.60 for freight. How many 
 feet of pine boards, at $25 per M., can the broker 
 buy with the net proceeds, if he charges 1% com- 
 mission for buying ? 
 
 15. A broker is offered a commission of 5^% for selling 
 
 wool and guaranteeing payment, or a commission of 
 3% without guaranteeing payment. He accepts 
 the $>\/o commission, and guarantees the payment. 
 The sales amount to $8500, and the bad debts to 
 $147.75. How much did he gain by his choice ? 
 
PERCENTAGE. 217 
 
 INSURANCE. 
 
 211. In insurance a payment called a premium of- insur- 
 ance is made for a guaranty of a specified sum of money 
 in the event of loss from fire or accident, and is reckoned 
 at a rate per cent on the amount insured. 
 
 212. In life-insurance an annual payment is made in 
 order to secure a specified sum of money in the event of 
 death, or at the end of a fixed period of time. 
 
 213. The written contract is called the policy of insurance, 
 
 (1) A house worth $8000 is insured for three years for f 
 of its value, at 1%. Find the premium. 
 
 J of $ 8000 - $ 6000, and 1% of $ 6000 = $ 60. $ 60. Ans. 
 
 (2) The premium for insurance on a store, at 1^%, is 
 $150. Find the amount of the insurance. 
 
 The premium on $1 insurance, at 1J%, is $0.015. 
 
 Of 
 $10,000. Ans. 
 
 150 
 
 Hence the amount of insurance is $ = $10,000. 
 
 0.015 
 
 (3) A man pays $27.50 premium for having his house 
 insured for five years, at \\/o on \ of its value. What is 
 the value of the house ? 
 
 The premium on $1 insured at 1J% is $0.0125. 
 
 9*7 CA 
 
 Hence the amount of the insurance is $-^ ' = $2200, and the 
 value of the house is $ 2200 -*- = $ 3300. $ 3300. Ans. 
 
218 PERCENTAGE. 
 
 (4) For what sum must a cargo worth $24,500 be in- 
 sured, at 2%, so that, in case of loss, the owner may recover 
 both the value of the cargo and the premium paid ? 
 
 Premium on $1 at 2% is $0.02. 
 
 Insurance on $0.98 worth of cargo = $1. 
 
 94.^00 
 
 Hence insurance on $24,500 worth of cargo = $ - - = $ 25,000. 
 
 u.yo 
 
 $25,000. Ans. 
 Ex. 139. 
 
 1. Find the cost of insuring property worth $15,000, if f 
 
 of the value is insured at $>. 
 
 2. Find the cost of insuring |- of the value of 6000 bbls. 
 
 of flour worth $9.60 a barrel, the insurance being 
 reckoned at \Jo. 
 
 3. A stock of goods worth $12,000 was insured for \ of 
 
 its value at f %. If the whole stock were burned, 
 what would be uie loss to the owner, including the 
 premium paid for insurance? 
 
 4. After three annual payments of $337.50, premium at 
 
 \\fo on f of the value of a mill, it was burned. 
 Find the loss to the insurance company. 
 
 5. At \/o, how much insurance can be effected upon a 
 
 store for $108? 
 
 6. What annual premium at \\/o must be paid on a life- 
 
 insurance of $6000? 
 
 7. At the rate of $ 17 upon $ 1000, what annual premium 
 
 will be paid on a life-insurance of $6700 ? 
 
 8. The annual premium paid for life-insurance at If % 
 
 is $70. What is the sum insured ? 
 
 9. For what sum should a cargo worth $74,496 be in- 
 
 4sured, at 3%, so that, in case of loss, the owner may 
 recover both the value of the cargo and the premium 
 paid? 
 
PERCENTAGE. 219 
 
 TAXES AND DUTIES. 
 
 214, Taxes on property are reckoned at a rate per cent 
 on the assessed value of the property ; and duties on im- 
 ported goods are sometimes reckoned at a rate per cent on 
 the cost in the country from which they are imported. 
 
 Ex. A tax of $18,000 is levied upon a town which contains 
 800 polls, assessed at $1.50 each, and which has 
 taxable property valued at $1,100,000. It is esti- 
 mated that the town will receive from the state 
 $3600 as its share of the railroad tax. Find the 
 rate of taxation and the tax paid by Brown, whose 
 property is assessed at $5960, and who pays for 1 
 poll. 
 
 The amount of poll-taxes = 800 X $1.50 = $1200 
 
 The amount from the state = $3600 
 
 The sum from state and polls = $4800 
 
 Sum levied on property = $18,000 - $4800 = $13,200. 
 
 The rate = $13,200 -4- $1,100,000 = $0.012. 
 
 That is, the tax is 12 mills on a dollar, or $12 on $1000. 
 
 Therefore Brown's property-tax is 0.012 of $5960 = $71.52. 
 
 Total tax - $71.52 +$1.50 = $73.02. 
 
 Ex. 14O. 
 
 1. If the assessed valuation of a town is $784,750, and 
 
 the town has 260 polls, paying $1.25 each, what is 
 the rate when the tax levy is $16,020 besides the 
 estimated amount to be received from the state ? 
 
 2. A district schoolhouse is to cost $3500, and the prop- 
 
 erty of the district is assessed at $210,000. What 
 is the rate, and what tax must be paid on property 
 assessed at $3798.60? 
 
220 PERCENTAGE. 
 
 3. In a city of 2000 polls, each paying $1.50, the sum of 
 
 $111,000 is to be raised by taxation on property 
 assessed at $9,000,000. What is the tax of a man 
 who pays for 4 polls, and tax on property assessed 
 at $25,670? 
 
 4. What is the rate of taxation when $710.92 is the tax 
 
 upon $50,780? 
 
 5. If a tax of $12,350 is to be raised, and the collector 
 
 receives 5% for collecting the taxes, what sum must 
 be levied? 
 
 6. A town-hall is to be built at a cost of $ 11,400. What 
 
 sum must be assessed if the collector receives 5% 
 for collecting the taxes, and what will be the rate 
 if the assessed valuation of the town is $800,000? 
 
 7. Find the duty, at 15%, on 95 cases of indigo, each 
 
 weighing 190 Ibs., and invoiced at 75 cts. per pound. 
 
 8. After deducting 20% for leakage, what will be the 
 
 duty on 40 hhds. of molasses, of 84 gals, each, if the 
 molasses is invoiced at 90 cts. a gallon, and the duty 
 is 30% ? 
 
 9. On 15 doz. bottles of sherry wine there is paid $1.25 
 
 per dozen for transportation, and $1.50 per dozen 
 for duty. What is the whole cost of importation ? 
 
 10. A Boston merchant received from Paris : 
 325 yds. of silk @ $2.25 a yard ; 
 296 yds. of lace @ 1.50 a yard ; 
 480 yds. of ribbon @ 0.50 a yard ; 
 
 45 doz. gloves @ 15.00 a doz. 
 If the duty en silk, ribbon, and lace is 35%, and on 
 gloves 25%, what is the whole amount of the 
 duties 9 
 
PERCENTAGE. 221 
 
 11. If the duty is $2.50 a gallon on cologne- water, what 
 
 must be paid on 75 doz. pint bottles, if there is an 
 allowance of 5% for breakage? 
 
 12. What is the invoice cost of goods upon which $625 
 
 duty is paid, if the duty is reckoned at 25% ? 
 
 13. What will be paid by a grocer importing 120 chests 
 
 of tea, containing 79 Ibs. each, invoiced at 75 cts. 
 per pound, if the duty is 
 
 Ex. 141. 
 MISCELLANEOUS EXAMPLES. 
 
 1. Of what number is 450 nine per cent? 
 
 2. What is the excess of 5% of 1500 over \% of 7000 ? 
 
 3. What per cent of 9000 is 45? 
 
 4. Five hundred and sixty is 12% more than what 
 
 number ? 
 
 5. Seven hundred and fifty-two is 6% less than what 
 
 number ? 
 
 6. There is a difference of 893 between a certain number 
 
 and 6% of the number. Find the number. 
 
 7. What per cent of 25 Ibs. are 3 Ibs. 4 oz. ? 
 
 8. The difference between 50% and 75J% of a number 
 
 is 99. Find the number. Let the example be 
 proved. 
 
 9. A merchant sold cloth at $4.20 per yard, and gained 
 
 20%. If it had been sold at $3.60, what actual 
 gain, and what gain per cent, would have been made ? 
 
 10. By how much does % exceed % ? 
 
222 PERCENTAGE. 
 
 11. At an average price of 55 cts. per bushel, and a charge 
 
 of 2^-% commission, how many bushels of grain can 
 
 be bought for $4510? 
 
 HINT. First find the cost of 1 bu. f including commission. 
 
 12. A landau was sold for $488, at a gain of 22%. 
 
 Kequired the cost. 
 
 13. A milkman's gallon measure was too small by ^ gi. 
 
 What was the rate per cent of fraudulent gain ? 
 
 14. A merchant paid $112.50 for 75 yds. of silk, of which 
 
 15 yds. were worthless. At what price per yard 
 must the remainder be sold to gain 20% on the 
 purchase-price of the whole ? 
 
 15. For selling goods, an agent received $106.83 commis- 
 
 sion, 2J% for selling, 2f% for guaranteeing pay- 
 ment. What sum was received for the goods? 
 
 16. A dealer bought 70 bags of wool at $32 a bag ; 10% 
 
 of it proved unsalable. For what price per bag 
 must he sell the rest to realize 15% on his pur- 
 chase ? 
 
 17. A lady paid for investing money $9.37| brokerage, 
 
 rate |%. Required the amount invested. 
 
 18. From a stack of hay, 7 t. 11 cwt. were sold, which was 
 
 75% O f the whole. What did the stack contain 
 before the sale ? 
 
 19. A carriage worth $250 was bought for $50 less, and 
 
 sold for $25 more, than its value. What was the 
 rate of gain on the price paid ? 
 
 20. A man left 30% of his estate to his wife, 50% of the 
 
 remainder to his son, 75% of the residue to his 
 daughter, and the balance, $546, to a family servant. 
 Required the value of the estate. 
 
PERCENTAGE. 223 
 
 21. What per cent of ^ is ^ ? of T 7 is $* ? 
 
 22. A man sold 36 horses for $200 each : on half of them 
 
 he gained 20%, and on half he lost 10%. What 
 was his gain per cent on the whole sale ? 
 
 23. A gentleman sent to a broker $1281.25 to be invested 
 
 in land at $62.50 an acre. A commission of 2^-% 
 being charged for buying, how many acres were 
 bought? 
 
 24. The dimensions 10, 8, and 6, of a rectangular bin being 
 
 increased 10%, what will be the rate per cent of 
 increase in capacity? 
 
 25. One-half of a stock of goods valued at $612.60 was 
 
 sold for - of the value of the whole stock. What 
 was the gain per cent ? 
 
 26. A roll of 140 yds. of carpet was sold for $72, at a loss 
 
 of 10%. What should it have brought per yard to 
 insure a gain of 15% ? 
 
 27. A railroad company with $9,000,000 capital declares 
 
 a dividend of $360,000. What sum will be received 
 on 120 shares of $100 each? 
 
 28. Ten per cent of a roll of carpet having been sold to 
 
 one man, 10% of the remainder to another, 30.375 
 yds. are left. How many yards were there at first? 
 
 29. At an annual premium of $405, rate 1$%, f of the 
 
 value of a mill is insured. What is the entire value 
 of the mill ? 
 
 30. A broker buying cotton at f % commission retained 
 
 $75 for his commission, and paid $25 for storage. 
 What sum was sent by his employers to cover the 
 whole expense of investment ? 
 
224 PERCENTAGE. 
 
 31. What sum must be insured upon a library to cover its 
 
 entire value, $18,000, and the premium at If % ? 
 
 HINT. If 100 be taken to represent the sum to ba insured, 
 then If will represent the premium ; and 100 -If, that is, 98 J, 
 will represent the value of the library. Hence the sum to be 
 insured will be $ 18,000 H- 0.98$ = f 18,329.94. 
 
 32. A merchant placed 80% of his year's profits in a bank ; 
 
 having drawn out 20% of this deposit, $2880 
 remained. What were his profits for the year ? 
 
 33. Required the tax-rate, in a city appropriating for pub- 
 
 lic expenses $147,000, to be assessed on property 
 worth $35,000,000. 
 
 34. A lady bought a house for $7965, which rented for 
 
 $841.85. The taxes were $50; repairs, $75. What 
 rate per cent did the investment yield ? 
 
 35. A premium of $960 was paid for full insurance on a 
 
 ship and cargo, at 1%. The cost of the cargo was 
 60% of the cost of the ship. What was the value 
 of each ? 
 
 36. Find the entire cost of 4000 bbls. of flour purchased 
 
 by an agent, at $7 a barrel, who charged 3% com- 
 mission, and paid $315 for freight. 
 
 37. How many barrels of flour can be bought for $5924,38 
 
 by an agent who pays $7 a barrel for the flour, 
 charges 3% commission, and pays $315 for the 
 freight? 
 
 38. The insurance on -| the value of a hotel and furniture 
 
 cost $300. The rate being 75 cts. on $100, what 
 was the value of the property ? 
 
 39. What is the duty, at 25f cts. per gallon, on 48 bbls. 
 
 of turpentine, 31 gals, making a barrel, and b% 
 being allowed for leakage ? 
 
CHAPTER XL 
 
 INTEREST AND DISCOUNT. 
 
 215. Interest is the payment made for the use of money. 
 The interest to be paid for the use of a given sum of money differs 
 
 from the payments considered in the last chapter, in that it depends 
 upon the time for which the sum is loaned as well as on the rate per 
 cent charged. 
 
 216. The sum loaned is called the principal. The princi- 
 pal and interest together is called the amount. 
 
 SIMPLE INTEREST. 
 
 217. If 100 be taken as the representative of the princi- 
 pal, the rate will represent the interest for one year; the 
 product of the rate by the number of years will represent 
 the whole interest. 
 
 Thus, if the time be 4 yrs., and the rate per cent 5, the interest will 
 be represented by 20, and the amount by 120. 
 
 Find the interest on $512 for 2 yrs. 4 mos., at 6%. 
 
 $512 
 0.06 
 
 $30.72 - interest for 1 yr. 
 2-^ = 2 yrs. 4 mos. 
 
 1024 
 6144 
 $71.68 $71.68. Am. 
 
 218. In most business transactions the time for which 
 interest is required is 1, 2, 3, or 4 months (30 dys. being 
 
226 INTEREST AND DISCOUNT. 
 
 reckoned 1 mo.), and the rate of interest is 6%, that is, 
 \} a month. 
 
 Hence the interest at 6% on a given sum for 2 mos. (or 
 60 dys.) is found by moving the decimal-point two places 
 to the left; for 1 mo., 3 mos., 4 mos., by moving the deci- 
 mal-point two places to the left, and multiplying by \, \\, 
 and 2 respectively. 
 
 Thus, the interest on $2500 for 2 mos. is $25.00 ; for 1 mo., $ 12.50 ; 
 for 3 mos., $37.50 ; for 4 mos., $50 
 
 Find the interest on $1120 for 3 yrs. 2 mos. 18 dys., 
 at 6%. 
 
 The interest at 6% for 1 yr. - 0.06 of the principal. 
 The interest for 1 mo. is ^ of 0.06 = 0.005 of tho principal. 
 The interest for 1 dy. is ^ of 0.005 - i of 0.001 of the principal. 
 Hence the interest for . 
 
 3 yrs. =3x0.06 =0.18 
 
 2 mos. = 2 x 0.005 = 0.01 
 18 dys. = 18 X J of 0.001 = 0.003 
 
 3 yrs 2 mos. 18 dys. - 0.193 of the principal. 
 And 0.193 of $1120 = $216.16. 
 
 $216.16. Ans. 
 
 219. The six per cent method may be employed for any 
 rate per cent by first finding the interest at 6%, and then 
 taking such a part of the interest as the given rate is of six 
 per cent. 
 
 Thus, the interest at 4 % = ^ = f of the interest at 6 %. In this 
 
 6 
 
 case, we should diminish the interest at 6% by J of itself. The inter- 
 est at 8% is -| = f of the interest at 6%. In this case, we increase 
 the interest at 6% by J of itself. 
 
 220, To compute interest for days at 6%, we move the 
 decimal-point in the principal three places to the left, and 
 multiply by one-sixth of the number of days. 
 
INTEREST AND DISCOUNT. 227 
 
 Find the interest for $8080 for 93 Ays., at 6%. 
 
 $3.08 By moving the decimal-point three places to the 
 
 151 left, we have $8.08 ; and of 93 dys. = 15 J. There- 
 
 404 fore, multiplying $8.08 by 15, we obtain the 
 
 4040 required interest. 
 
 $125.24. Ans. 
 
 221. For any other rate, find the interest 
 at 6%, and then increase or diminish this interest by such 
 a fraction of itself as the given rate is greater or less 
 than 6%. 
 
 Ex. 142. 
 
 Find the interest of : 
 
 1. $51.25 for 30 dys., at 6% 
 
 2. $2581 for 60 dys., at 6%. 
 
 3. $1261 for 90 dys, at 6%. 
 
 4. $ 1250.60 for 4 mos, at 6%. 
 
 5. $3020 for 3 mos., at 6%. 
 
 6. $2300 for 3 mos, at 6%. ^ 
 
 7. $275 for 2 mos, at 6%. 
 
 8. $5000 for 1 mo, at 6%. 
 
 9. $1361 for 2yrs, at 5%. 
 
 10. $675.90 for 5 yrs, at 3|-%. 
 
 11. $775.83 for 3 yrs. 9 mos, at 
 
 12. $533.33^ for 10 mos, at 4|$> 
 
 13. $250.60 for 3 yrs. 6 mos, at 
 
 14. $575.87| for 1 yr. 10 mos. 15 dys, at 5%. 
 
 15. $760 for 2 yrs. 11 mos. 27 dys, at \%. 
 
228 INTEREST AND DISCOUNT 
 
 16. $725.40 for 5 mos. 27 Ays., at 
 
 1.7. $547.60 from Feb. 20 to Dec. 5, at 
 
 18. $1750 from May 5, 1884, to June 21, 1885, at 
 
 19. $1517 from Jan. 5 to July 1, at 4|%. 
 
 20. $476.50 from July 5, 1884, to Feb. 9, 1885, at 
 
 21. $319.20 from April 7 to Aug. 31, at 3J%. 
 
 22. $6460 from June 15, 1883, to May 7, 1885, at 
 
 23. $150 from Aug. 5, 1883, to March 17, 1885, at 
 
 24. $527.20 from Jan. 1 to Nov. 20, at 4|%. 
 
 25. $1250 from Nov. 15, 1884, to March 1, 1885, at 
 
 26. $624.36 from March 5 to Dec. 20, at 
 
 Find the amount of : 
 
 27. $ 1100 for 3 yrs. 4 mos., at 5%. 
 
 28. $1290.50 for 60 dys., at 6%. 
 
 29. $1275 for 3 yrs. 2 mos. 15 dys., at 8%. 
 
 30. $250.80 for 10 mos. 10 dys., at 
 
 31. $377.65 for 1 yr. 3 mos., at 5 
 
 32. $7234.25 for 22 yrs. 2 mos. 20 dys., at 
 
 33. $6130 from May 6 to Oct. 24, at 3}%. 
 
 34. $258.85 from March 6 to June 24, at 5%. 
 
 35. $25.62 for 33 dys, at 6%. 
 
 36. $85.85 for 1 yr. 7 mos. 21 dys, at 6%. 
 
 37. $600 for 93 dys, at 4%. 
 
 38. $350 from Sept. 21, 1884, to March b, 1885, at 
 
INTEREST AND DISCOUNT. 229 
 
 39. $1226 from Oct. 4, 1884, to May 6, 1885, at 5%. 
 
 40. $342.42 from Feb. 5, 1884, to March 15, 1885, at 7%. 
 
 41. $360.50 from Aug. 1, 1884, to March 3, 1885, at 6|%. 
 
 42. $504.25 from Jan. 8 to March 10, at 6J%. 
 
 43. $1240 from Mar. 3 to Aug. 28, at 7%. 
 
 NOTE. In business, a year is reckoned at 360 days in computing 
 interest for a time less than a year expressed in months and days ; 
 hence the interest is y^ or ^ too great. But national governments 
 take the number of days between the two given dates, and reckon for 
 the interest such a part of a year's interest as this number of days is 
 of 365 days. 
 
 222. It is often required to find the rate, time, or prin- 
 cipal, when two of these and the interest (or amount) are 
 given. 
 
 223. When the principal, interest (or amount), and time 
 are given, to find the rate per cent. 
 
 At what rate per cent will $320 produce $48 in 3 yrs.? 
 Interest on $320 for 3 yrs. is $48. 
 Interest on $320 for 1 yr. is J of $48. 
 Interest on $ 1 for 1 yr. is ^ of J of $48 = $0.05. 
 But $0.05- 5% of fl. 5^ Am 
 
 At what rate per cent will $8000 amount to $9277.78 
 in 2 yrs. 6 mos. 20 dys. ? 
 
 Interest is $9277.78 - $8000 = $1277.78. 
 Time is 2 yrs. 6 mos. 20 dys. = 2f yrs. 
 ' Interest on $8000 for 2f yrs. = $1277.78. 
 Interest on $8000 for 1 yr. - f 1277.7a 
 
 Interest on $ 1 for 1 yr. - 1^- $- 06 l- 
 
 But $0.06i -6J% of |1. 
 
230 INTEREST AND DISCOUNT. 
 
 Ex. 143. 
 Find the rate per cent : 
 
 1. When the interest on $500 for 1 yr. 6 mos. is $67.50. 
 
 2. When the interest on $250 for 2 yrs. is $52.50. 
 
 3. When $500 amount to $754 in 9 yrs. 
 
 4. When the interest on $725 for 12 yrs. is $141.37f 
 
 5. When $880 amount to $899.25 for 7 mos. 
 
 6. When the interest on $424 for 2 yrs. 6 mos. is $26.50. 
 
 7. When the interest on $255.50 from April 1 to June 
 
 20 is $2.80. 
 
 8. When $175 amount to $203.35 for 3 yrs. 7 mos. 6 dys. 
 
 9. When a sum of money is doubled in 16 yrs. 
 
 10. When an investment for 6 yrs. produces a sum equal 
 to of the capital. 
 
 224, When the principal, interest (or amount), and rate 
 per cent are given, to find the time. 
 
 In what time will the interest on $793.87 be $11.96, 
 
 Interest on $793.875 at 5J% for 1 yr. - $43.663. 
 Therefore the number of yea 
 And 0.274 yr. = 3 mos. 9 dys. 
 
 Therefore the number of years will be 11 - 965 = 0.274. 
 
 43.663 
 
 3 mos. 9 dys. Ans 
 
 Ex. 144. 
 Find the time in which : 
 
 1. The interest on $225 will be $36, at 4%. 
 
 2. $440 will amount to $505.45. at 
 
INTEREST AND DISCOUNT. 231 
 
 3. $2ioO will double itself, at 
 
 4. $225 will amount to $256.50, at 
 
 5. $50 will amount to $85, at 6%. 
 
 6. The interest on $4260 will be $873.30, at 
 
 7. $1005.34 will amount to $1156.14, at 
 
 8. $1587.75 will amount to $1611.68, at 
 
 9. A sum of money will double itself, at 6%. 
 10. $1000 will amount to $1125, at 4%. 
 
 225, When the interest, time, and rate are given, to find 
 the principal. 
 
 What principal will in 8 yrs. 6 mos. produce $100 in- 
 terest, at 5%? 
 
 8 yrs. G mos. = 8.5 yrs. 
 Interest for 1 yr. = $199 = $11.705. 
 Interest on $1 for 1 yr. at 5% = 0.05 of $1. 
 
 >3I 
 
 $235.30. Ans. 
 
 Hence principal required - ^ 1L765 = $235.30 
 0.05 
 
 Ex. 145. 
 
 Find the principal that will : 
 
 1. Produce $180 interest in 3 yrs., at 
 
 2. Produce $189 interest in 3 yrs., at 
 
 3. Produce $3493.20 interest in 3 yrs. 5 mos., at 
 
 4. Produce $10.70 interest in 5 mos., at 4%. 
 
 5. Produce $75.40 interest in 3 yrs. 4 mos., at 
 
 6. Produce $75.05 interest in 3 mos. 2 dys., at 
 
232 INTEREST AND DISCOUNT. 
 
 7. Produce $1746.60 interest in 3 yrs. 5 mos., at (j%. 
 
 8. Produce $64.46 interest in 6 yrs., at 4%. 
 
 9. Produce $80.62 interest in 3 yrs. 9 mos., at 4%. 
 
 10. Produce $669.64 interest in 2 yrs. 7 mos. 24 dys., 
 at 6%. 
 
 226, When the amount, time, and rate are given, to find 
 the principal. 
 
 Find the principal that will amount to $748.12| in 3 yrs. 
 6 raos., at 4%. 
 
 3 yrs. 6 mos. = 3J yrs. 
 
 Let the principal be represented by 100. 
 
 The interest will be represented by 3 J X 4 = 14. 
 
 The amount will be represented by 100 + 14 = 114. 
 
 Hence the principal = fJJ of $748.125 = $656.25. 
 
 $656.25. Ans, 
 
 Ex. 146. 
 
 Find the principal that will amount : 
 
 1. To $1680 in 3 yrs., at 4%. 
 
 2. To $962 in 4 yrs., at 4%. 
 
 3. To $725.47 in 2 yrs. 3 mos., at 3|%. 
 
 4. To $3215.83 in 4 yrs. 6 mos., at 3%. 
 
 5. To $595.20 in 8 mos., at 6%. 
 
 6. To $1275.75 in 1 yr. 1 mo., at 5%. 
 
 7. To $2053. 32 in 3 yrs. 5 mos., at 6%. 
 
 8. To $131.88 in 2 yrs. 11 mos. 15 dys., at 
 
 9. To $37.02 in 2 yrs. 3 mos. 18 dys., at 5 
 10. To $2359.38 in 2 yrs. 7 mos. 24 dys., at 
 
INTEREST AND DISCOUNT. 233 
 
 BANK DISCOUNT. 
 
 227, When the holder of a promissory note sells the 
 note to a bank, or other purchaser, the sum paid by the 
 bank is called the proceeds or avails of the note, and 
 the difference between the sum named in the note and the 
 proceeds is called the discount, 
 
 228, Discount is reckoned at so much per cent, and the 
 per cent is called the rate of discount, 
 
 229, Questions in bank discount are calculated like ques- 
 tions in simple interest, the terms used being discount 
 instead of interest, and rate of discount instead of rate of 
 interest. 
 
 NOTE. The sura named in the note should he written in words, and 
 is called the face of the note. The person signing the note is called 
 the maker ; a person who writes his name on the back of the note is 
 called an indorser, and is responsible for the payment of the note. 
 
 A note should contain the words " value received. 1 ' A note, to be 
 negotiable, must be made payable to the bearer, or to the order of 
 some person who must indorse the note. 
 
 When a note bears interest, the discount is computed on the amount 
 of the note. 
 
 A note is nominally due at the expiration of the time named in 
 the note, but it does not mature, that is, become legally due, until 
 three days after this time. These three days are called days of 
 grace. And the discount is computed on the time between the day 
 the note is discounted and the day of its maturity. 
 
 When the time is expressed in days, the day of maturity is found 
 by counting forward from the date of the note the number of days 
 named in the note, and the three days of grace. When the time is 
 in months, the day of maturity is found by counting the number of 
 calendar months, and the three days of grace. When a note falls due 
 on Sunday, or a legal holiday, it is payable on the day previous. 
 
 A protest is a notice in writing by a notary public to the indorsers 
 that a note has not been paid. If a note be not protested on the 
 last day of grace the indorsers are released from their obligation. 
 
234 INTEREST AND DISCOUNT. 
 
 230, Find the day of maturity, the time to run (from the 
 day the note is discounted), the discount, and the proceeds 
 of the following notes : 
 
 $610.25. BOSTON, June 12, 1885. 
 
 Sixty days after date I promise to pay to the order ol 
 Edwin Ginn six hundred ten and -j 2 ^ dollars, for value 
 received. 
 
 Discounted at 6%, July 1. SAMUEL HALE. 
 
 Counting 60 dys. from June 12, we have 18 in June, 31 in July, 
 and 11 in August. 
 
 Therefore the note becomes due Aug. n /u (11 denotes the day it 
 is nominally due, and 14 the day it is legally due). 
 
 The time to run is 30 dys. in July and 14 in August, that is, 
 44 dys. 
 
 The discount is the interest on $610.25 for 44 dys., at 6%. There- 
 fore (J 220) the discount is 7 X $0.61025 = $4.48. 
 
 T^e proceeds - $ 610.25 - $4.48 = $605.77. 
 
 Due Aug. 14; discount, $4.48; proceeds, $605.77. Ans. 
 
 $ 1050. CHICAGO, Feb. 13, 1885. 
 
 Six months from date we jointly and severally promise 
 to pay to the order of George Hall ten hundred and fifty 
 dollars, for value received, with interest at six per cent. 
 
 Discounted at 8%, May 13. JAMES BLAKE. 
 
 HENRY SHAW. 
 
 interest on note for 6 mos. 3 dys. = $32.03. 
 
 Amount of note when due is $ 1050 + $32.03 = $ 1082.03. 
 
 Day of maturity, Aug. 13 / 16 . 
 
 Time to run, 3 mos. 3 dys. 
 
 Discount on $1082.03, at 8%, for 3 mos. 3 dys. = $22.36. 
 
 Proceeds, $ 1082.03 - $ 22.36 = $ 1059.67. 
 
 Due Aug. 16; discount, $22.36 ; proceeds, $1059.67. An* 
 
INTEREST AND DISCOUNT. 235 
 
 Ex. 147. 
 
 Find the day of maturity, the time to run, the discount, 
 and the proceeds, on the following notes : 
 
 1. $2250. CONCORD, N.H., Jan. 1, 1885. 
 Four months from date I promise to pay to the order of 
 
 George Marston twenty-two hundred and fifty dollars, for 
 value received. 
 
 Discounted at 1%, Jan. 12. SIMON STEVENS. 
 
 2. $432.55. NEW YORK, Jan. 3, 1885. 
 Sixty days from date I promise to pay James Wilson, or 
 
 order, four hundred thirty-two and ^j- dollars, value 
 received. 
 
 Discounted at 6^%, Jan. 6. JOHN ALLEN. 
 
 3. $670.35. ST. Louis. Jan. 6, 1885. 
 Ninety days from date I promise to pay to the order of 
 
 Peter Holmes six hundred severity and -j^j- dollars, value 
 received. 
 
 Discounted at 7%, Jan. 26. ROBERT DAY. 
 
 4. $1304.90. CINCINNATI, Jan. 25, 1885. 
 Five months after date I promise to pay to the order of 
 
 John Shannon thirteen hundred four and -ffo dollars, for 
 value received, with interest at six per cent. 
 
 Discounted at 4^-%, March 15. CHARLES HILLMAN. 
 
 5. $2260. BALTIMORE, MD., June 19, 1885. 
 
 Sixty days from date I promise to pay to the order of 
 John Morrison twenty-two hundred and sixty dollars, value 
 received. 
 
 Discounted at 5%, July 16. FRANK HOWE. 
 
236 INTEREST AND DISCOUNT. 
 
 6. $645. AUSTIN, TEX., July 28, 1885. 
 Thirty days from date I promise to pay to the order of 
 
 John Moses six hundred and forty-five dollars, value 
 received. 
 
 Discounted at 6%, Aug. 3. RICHARD SMITH. 
 
 7. $1000. SAVANNAH, GA., Oct. 4, 1884. 
 Six months after date I promise to pay to John Proctor, 
 
 or order, one thousand dollars, value received, with interest 
 at seven per cent. 
 
 Discounted at 8%, Dec. 31. JAMES WHITRIDGE. 
 
 8. $2912.60. PHILADELPHIA, Feb. 19, 1885. 
 Ninety days after date I promise to pay to the order of 
 
 George Wright twenty-nine hundred twelve and -ffo dol- 
 lars, value received. 
 
 Discounted at 6$, March 1. PETER BURKE. 
 
 9. $455.04. CHARLESTON, S.C., Sept. 2, 1885. 
 Four months from date I promise to pay to the order of 
 
 Edmund Home four hundred fifty-five and y^j- dollars, 
 value received. 
 
 Discounted at 5|$>, Sept. 16. PAUL WEST. 
 
 10. $1140. NEW ORLEANS, LA., July 1, 1885. 
 Ninety days after date I promise to pay to the order of 
 
 William Whitridge eleven hundred and forty dollars, 
 value received. 
 
 Discounted at 7%, Aug. 15. JOHN CLEMENT. 
 
 11. $10,089.25. DENVER, COL., Oct. 14, 1885. 
 
 Ninety days after date I promise to pay to the order of 
 John Higgins ten thousand eighty-nine and -ffo dollars, 
 value received. 
 
 Discounted at 10%, Dec. 1. JOHN KELLEY, 
 
INTEREST AND DISCOUNT. 237 
 
 231. To determine the face of a note that will yield a 
 given sum when discounted. 
 
 For how much must a four-months' note without interest 
 be made that it may yield $1000 when discounted at a 
 bank at 6% ? 
 
 The discount on $1 for 4 mos. 3 dys. is $0.0205. 
 Proceeds of $ 1 is $ 1 - ? 0.0205 = $ 0.9795 = 0.9795 of $ 1. 
 Therefore the face required is $ 1000 -*- 0.9795 = $ 1020.93. 
 
 Ex. 148. 
 
 1 . Find the face of a note for 30 dys. that will realize 
 
 $600 when discounted at 6J%. 
 
 2. Find the face of a note for 60 dys. that will realize 
 
 $8000 when discounted at 8%. 
 
 3. Find the face of a four-months* note that will realize 
 
 $800 when discounted at 5J%. 
 
 4. Find the face of a note for 90 dys. that will realize 
 
 $1700 when discounted at 1%. 
 
 6. Find the face of a two-months' note that will realize 
 $900 when discounted at 7 T 8 5 %. 
 
 6. Find the face of a three-months' note that will realize 
 $2200 when discounted at 7%. 
 
 PRESENT WOBTH AND DISCOUNT. 
 
 232, The present worth of a sum of money due at the 
 end of a fixed time is the sum that, put at interest for the 
 fixed time, will amount to the given sum. 
 
 Thus, $100 will in 2 yrs., at 6%, amount to $112. And $112 to 
 be paid at the end of 2 yrs. is equal in value to $ 100 paid now. 
 Hence $100 is regarded as the present worth of $112 to be paid in 
 2 yrs. 
 
238 
 
 INTEREST AND DISCOUNT. 
 
 COMMERCIAL DISCOUNT. 
 
 233, Commercial discount is a reduction from the nominal 
 price or value of anything. 
 
 234. Price-lists of articles manufactured and sold are 
 issued by manufacturers and wholesale dealers. These 
 prices are subject to many and various discounts. 
 
 The following bill will afford a good illustration of this 
 discount : 
 
 JTew York, Feb. 1, 1889. 
 
 of 8. L. JtfOIflSOJ? $ CO. 
 
 8 doz. olts, ^3.00 
 (Discount, 40, 5, 25, 
 11 gro. Screws, 92.25 
 f and 30, 
 6 doz. Qkest Handles, 91-50 
 40, 5, 25, 17$, 
 
 '- '4 
 13 
 
 00 
 74 
 
 $10 
 5 
 3 
 
 26 
 78 
 18 
 
 24 
 18 
 
 75 
 
 97 
 
 9 
 5 
 
 00 
 82 
 
 
 $19 
 
 22 
 
 In the first item, $3.00 is the list prico per dozen of the bolts, $24 
 is the gross price of the 8 dozen bolts, $10.26 is the net price. This 
 net price is found by taking 40% from $24, which leaves $14.40; 
 then taking 5% from $14.40, which leaves $13.68, and then taking 
 25% from $13.68, which leaves $ 10.26. 
 
 In the second item, the gross price is $24.75. The f means a dis- 
 count of of the gross price, which leaves $ 8.25, and the 30 means 
 there is a discount of 30% from $8.25, which leaves $5.78. 
 
 In the third item, 40% is taken off the gross price, 5% taken from 
 the remainder thus found, then 25% from the second remainder, and 
 then 17 J% from the third remainder. 
 
 NOTE. In finding 17J%, first find 10%, and then add to it J for 5% 
 and i for 2}%. 
 
INTEREST AND DISCOUNT. 239 
 
 To combine two discounts so as to form one discount, from their 
 sum take one per cent of their product. Thus, 50% and 10% = 50 + 10 
 
 Ex. 149. 
 
 Find the net amount of a bill of : 
 
 1. $320 subject to a discount of 10% for cash. 
 
 2. $1680 subject to discounts of 15 and 10. 
 
 3. $980 with 15 off. 
 
 4. $1620 with 20 and 15 off. 
 
 5. $1440 with 25, 10, and 5 off. 
 
 6. $587.50 with 35 and 15 off. 
 
 7. $1920 with 25 and 7 2 l off. 
 
 8. $1530 with 25, 10, and 5 off. 
 
 9. $500 with 25, 15, and 12| off. 
 
 10. $870.40 with 30, 22 J, and 121 off. 
 
 11. Find the net cash amount of a bill of $1088, discounts 
 
 being 50 and 10, and an additional discount of 5% 
 for cash. 
 
 12. Find the difference between a single discount of 55, 
 
 and successive discounts of 40 and 15. 
 
 13. Find the net cash amount of a bill of $136, discounts 
 
 being 50, 10, and 5. Find a single discount equiv- 
 alent to these three successive discounts. 
 
 14. Find the net cash amount of a bill of $164.50, dis- 
 
 counts being f and 30. 
 
 15. Find the net cash amount of a bill of $15, discounts 
 
 being 40, 5, 25, and 17J. Find a single discount 
 equivalent to these four successive discounts. 
 
240 INTEREST AND DISCOUNT. 
 
 PABTIAL PAYMENTS. 
 
 235. When settlements of accounts are made at the expi- 
 ration of a year or less, it is customary to reckon interest 
 on each item from the time it is due to the time of settle- 
 ment. 
 
 And when partial payments are made and indorsed on 
 a note that contains the words with interest, provided the 
 note is paid in full within a year, it is usual to compute 
 the interest on the principal, and on each of the payments 
 to the time of settlement. 
 
 Samuel Paine buys of Edgar Smith $400 worth of goods 
 at 30 dys. At the end of 3 mos. he pays $200, and the 
 balance 2 mos. later. Find the balance. 
 
 The time between the end of 30 dys. and the time of settlement is 
 4 mos. Therefore interest is reckoned on the $400 for 4 mos., and 
 on the $ 200 for 2 mos. 
 
 $400 + 4 mos. interest - $ 408 
 
 $200 + 2 mos. interest - 202 
 
 Therefore balance due is $206 
 
 A man holds a note for $1000, dated Jan. 1, 1885, on 
 which are indorsed payments as follows: March 1, 1885, 
 $100; Oct. 1, 1885, $50; Nov. 1, 1885, $800. What is 
 due Jan. 1, 1886, interest at 6% ? 
 
 Amount of $ 1000 for 1 yr. f at 6%, is $1060.00 
 
 Amount of $ 100 for 10 mos., at 6% = $ 105.00 
 Amount of $50 for 3 mos., at 6% = 50.75 
 Amount of $800 for 2 mos., at 6% = 808.00 
 
 963.75 
 
 Balance due, $96.25 
 
 This method is in accordance with what is called the 
 Merchant's Kule, 
 
INTEREST AND DISCOUNT. 241 
 
 Ex. 150. 
 
 1. A note for $3000, dated April 1, 1884, payable on 
 
 demand, with interest at 7%, bears the following 
 indorsements : May 6, $600 ; July 5, $676.11 ; Oct. 
 18, $966. What is due Jan. 1, 1885 ? 
 
 2. A note for $1237.50, dated April 17, 1884, payable on 
 
 demand, bears the following indorsements : June 5, 
 $253 ; Aug. 20, $274.50 ; Nov. 17, $420. What is 
 due Jan. 1, 1885, reckoning interest at 6% ? 
 
 3. A note for $775.50, dated May 15, 1884, payable on 
 
 demand, bears the following indorsements: July 21, 
 $150 ; Oct. 10, $250; Feb. 24, 1885, $100. What 
 is due May 15, 1885, reckoning interest at 6% ? 
 
 4. A note for $1670.50, dated July 1, 1884, payable on 
 
 demand, with interest at 6^-%, bears the following 
 indorsements: Aug. 20, $315; Sept. 21, $360.50; 
 Oct. 5, $400; Dec. 1, $160. What is due Jan. 1, 
 1885? 
 
 236. When a note that contains the words " with interest" 
 runs longer than a year, and partial payments have been 
 made, the interest is computed by a rule adopted by the 
 Supreme Court of the United States, and therefore called 
 
 THE UNITED STATES RULE. 
 
 Find the amount of the principal to the time when the 
 payment, or sum oj the payments, equals or exceeds the 
 interest. 
 
 From this amount deduct the payment or sum of the pay- 
 ments. 
 
 Consider the remainder as a new principal, and proceed 
 as before. 
 
242 INTEREST AND DISCOUNT. 
 
 Ex. A note of $1520, dated May 20, 1884, and drawing 
 interest at 6%, had payments indorsed upon it as 
 follows: Oct. 2, 1884, $300; Feb. 26, 1885, $25 ; 
 April 2, 1885, $570; Aug. 8, 1885, $600. Find 
 the amount due Dec. 6, 1885. 
 
 yr. mo*, djs. 
 
 1884 10 2 $1520 let principal. 
 
 1884 6 20 .022 
 
 4 12 .022 $33.44 1st interest. 
 
 1520.00 
 1 300. $1553.44 
 
 300.00 1st payment. 
 
 1885 2 26 $1253.44 2d principal 
 
 1884 10 2 .024 
 
 4 24 .024 $25 $30.08 2d interest. 
 
 $1253.44 2d principal. 
 $25. .006 
 
 $570 $7.52 3d interest 
 
 1885 4 2 30.08 2d interest 
 1885 2 26 1253.44 
 
 1 6 006 $1291.04 
 
 595.00 2d & 3d payments 
 f 570 $696.04 3d principal. 
 
 .021 
 
 1885 8 8 $14.62 4th interest 
 
 1885 4 2 696.04 
 
 4 6 .021 $710.66 
 
 600.00 4th payment 
 $600. $110.66 4th principal. 
 
 .019| 
 
 1885 12 6 $2.18 5th interest 
 
 1885 8 8 110.66 
 
 3 28 .019$ $112.84 $112.84. Ana. 
 
 In the first place, find the difference in time between each pair of 
 consecutive dates. At the right of the result in each case put the 
 corresponding decimal multiplier for the interest at 6%, and put the 
 corresponding payment below. 
 
INTEREST AND DISCOUNT. 243 
 
 Generally, it can be determined mentally whether one or more 
 payments must be taken to make a sum equal to or greater than the 
 interest. If two or more payments are required, the corresponding 
 decimal multipliers may be added, and the result taken for the mul- 
 tiplier. Thus, it is evident that .024 of $1253.44 is more than $25; 
 therefore .024 4- .006 .03 may be taken for the multiplier, which 
 will give for the interest $37.60. To this the principal is added, and 
 from the amount the sum of the payments is subtracted. 
 
 When the rate is greater or less than 6%, the several interests 
 must be increased or diminished according to the given rate* 
 
 Ex. 151. 
 
 1. A note of $1000, dated Jan. 22, 1884, and drawing 
 
 interest at 6%, had payments indorsed upon it as 
 follows : May 20, 1884, $50 ; July 20, 1884, $ 162.50 ; 
 Dec. 23, 1884, $ 72.50. Find the balance due March 
 1, 1885. 
 
 2. A note of $3325, dated Jan. 15, 1884, and drawing 
 
 interest at 6^-%, had payments indorsed upon it as 
 follows : June 24, 1884, $100; Sept. 2, 1884, $1250; 
 Jan. 31, 1885, $1400. Find the balance due May 
 12, 1886. 
 
 3. A note of $2280, dated Jan. 22, 1883, and drawing 
 
 interest at 7%, had payments indorsed upon it as 
 follows: Jan. 10, 1884, $1000; Aug. 31, 1884, $250; 
 Jan. 15, 1885, $600; March 4, 1885, $430. Find 
 the balance due June 15, 1885. 
 
 COMPOUND INTEREST. 
 
 237, When a note contains the words "with interest 
 annually," and ^the interest is not paid at the time it is due, ' 
 the inter-ast is ,miaUy. added to the principal; and new 
 principals are thus formed at regular intervals of time T 
 
244 INTEREST AND DISCOUNT. 
 
 238. The interest may be compounded with the principal 
 (that is, made a part of the principal) annually, semi- annu- 
 ally, quarterly, etc., according to agreement. 
 
 Ex. Find the compound interest of $800 for 2 yrs. 3 mos. 
 15 dys., at 7%. 
 
 $800 
 .07 
 
 $56 1st interest 
 800 
 $856 2d principal 
 
 .07 
 
 $59.92 2d interest 
 856.00 
 
 $915.92 3d principal. 
 , 0175 
 
 3 mos. 15 dys. \ 6)16.03 
 ( 2.67 
 
 $18.70 3d interest 
 915.92 
 
 $934.62 amount 
 
 800.00 
 $134.62 interest. 
 
 $134.62. Ans. 
 
 239. If the given time be not an integral number of years, 
 the amount is found for the number of entire years, and 
 then the amount of this for the fractional part of a year. 
 
 Ex. 152. 
 
 I. Find the compound interest on $125 for 3 yrs., 
 at 
 
 2. Find the amount of $87.50 for 3 yrs., at 4% per annum, 
 
 at compound interest. 
 
 3. Compare the simple and compound interest on $21.50, 
 
 at the end of 4 yrs., at 5%. 
 
INTEREST AND DISCOUNT. 245 
 
 4. What will a debt of $4250 amount to, if left standing 
 
 for 2 yrs. 6 mos., at 5% per annum, compound 
 interest ? 
 
 5. Find the compound interest on $104 for 1 yr. 9 mos., 
 
 at 5%. 
 
 6. Find the compound interest on $1800 for 2 yrs. 3 mos. 
 
 15 dys., at 3|%. 
 
 7. Find the compound interest on $4500 for 3 yrs. 6 mos., 
 
 at 4%. 
 
 If the interest be payable semi-annually, quarterly, etc., the 
 half, quarter, etc., of the rate per cent, must be used, and the 
 amount obtained for each half-year, quarter-year, etc. 
 
 8. Find the compound interest on $4000 for 2 yrs. 6 
 
 mos., at 5% per annum, interest payable semi- 
 annually. 
 
 9. Find the compound interest on $1001.50 for 1 yr. 
 
 3 mos., at 6%, interest payable semi-annually. 
 
 10. Find the compound interest on $4000 for 1 yr. 3 mos., 
 
 at 4% per annum, interest payable quarterly. 
 
 Ex. What principal will produce in 2 yrs. $650.14, com- 
 pound interest at 6% ? 
 
 Amount of $1 for 1 yr., at 6%, is 1.06 X $1. 
 
 Amount of $ 1 for 2 yrs. is 1.06 x 1.06 X $ 1 = (1.06)* x $ 1. 
 
 That is, the amount of $1 for 2 yrs., at 6%, is $1.1236. 
 
 Interest is $ 1.1236-$! - $0.1236 - 0.1236 of $1. 
 
 The principal required is $650.14 -* 0.1236 = $5260. 
 
 $5260. Ans. 
 
 11. What principal will amount to $275.62 in 2 yrs., at 
 
 5% compound interest? 
 
 12. What principal will amount to $620.32 in 3 yrs., at 
 
 6% compound interest ? 
 
246 INTEREST AND DISCOUNT. 
 
 ANNUAL INTEREST. 
 
 240, Annual Interest is simple interest on the principal 
 and on each year's interest from the time each interest is due 
 until settlement. 
 
 (1) Find the interest due Aug. 4, 1885, on a note dated 
 June 4, 1881, for $1700, with interest payable 
 annually, at 6%. 
 
 yrs. moi. dy. 
 
 1885 8 4 $1700.00 
 
 1881 6 4 0.25 
 
 4 2 $425.00 Interest for 4 yrs. 2 mos. 
 
 Annual interest. 
 
 4.08 
 36.72 
 
 $40.80 Interest on annual int. 
 425.00 
 
 3 2 
 2 2 
 1 2 
 2 
 
 $1700.00 
 .06 
 
 $102.00 
 .06 
 
 6 8 = 6f yrs. 
 
 $6.12 
 
 $465.80 Total interest due. 
 
 $465.80. Ans. 
 
 The first year's interest, $ 102, remains overdue 3 yrs. 2 
 mos., the second year's 2 yrs. 2 mos., the third year's 1 yr. 
 2 mos., and the fourth year's 2 mos. Now the interest on 
 $ 102 for the sum of these periods, 6f yrs., is $ 40.80. Hence 
 the total interest is $465.80. 
 
 13. Find the amount due May 17, 1885, on a note dated 
 
 May 17, 1881, for $700, at 6% annual interest. 
 
 14. Find the amount due May 27, 1885, on a note dated 
 
 Jan. 4, 1883, for $431, at 5$% annual interest. 
 
 15. Find the amount due May 19, 1885, on a note dated 
 
 Dec. 26, 1881, for $612.30, at 5% annual interest. 
 
INTEREST AND DISCOUNT. 247 
 
 16. Find the amount due Jan. 16, 1885, on a note dated 
 
 Jan. 8, 1883, for $623.04, at 5% annual interest. 
 
 17. Find the amount due Jan. 18, 1885, on a note dated 
 
 Jan. 8, 1881, for $575, at 6% annual interest. 
 
 STOCKS AND BONDS. 
 
 241, The name stock is applied to the capital of banks, 
 railroads, and other incorporated companies. 
 
 The capital of a company is usually divided into shares, 
 of which the original value is $100, or some other fixed 
 sum ; but the market-value at any time is the price per 
 share at that time. 
 
 When the market-value of stock is equal to its origi- 
 nal value, it is said to be at par. In quotations of 
 stocks, par is generally represented by 100 ; and when stock 
 is quoted at above 100, it is said to be at a premium ; below 
 100, at a discount. The premium or discount is the dif- 
 ference between the quotation and 100. 
 
 Thus, when the price of a stock on a given day is 91, or, as it is 
 commonly expressed, when the stock is at 91, the meaning is, that 
 $ 100 stock costs on that day $91 money ; or that, if 100 be the repre- 
 sentative of any quantity of stock, 91 will represent the correspond- 
 ing value in money. In this case the stock is said to be 9% discount. 
 
 The buying and selling of stocks is conducted through the agency 
 of stock-brokers, who receive a commission on the stock. The com- 
 mission is generally reckoned at \ of 1% on the par value of the stock. 
 Thus, if a broker buy stock for a person at 91, that person pays 91 J. 
 
 (1) How much would be received for 52 shares of stock, 
 $100 each, at 89? 
 
 $ J per share will represent the commission. 
 52x$89 =$4654. 
 Commission = 6.50 
 
 Proceeds =$4647.50 $4647.50. Ans. 
 
248 INTEREST AND DISCOUNT. 
 
 (2) What amount of stock, at 84f, including brokerage, 
 
 may be bought for $9393.37? 
 
 Since $0.84f, or 0.84 of $1, buys $1 stock, the amount bought 
 for $9393.37J will be $ 9393 - 37 * = $11,100. 
 
 $11,100. Ans. 
 
 (3) What is the quoted price of stock when $42,464.25 is 
 
 paid for $46,600 stock? 
 
 $46,600 stock costs $42,464.25. 
 
 $1 stock costs ^J^ of $42,464.25 = $0.91J. gl ^ 
 
 Ex. 153. 
 
 1. Find the cost of $5000 stock, at 98. 
 
 2. Find the cost of $7800 stock, at 78$. 
 
 3. Find the cost of $20,000 stock, at 109f 
 
 4. Find the cost of $5000 United States 4% bonds, at 
 
 121. 
 
 5. Mr. Jones owns 20 United States 4% bonds of $1000 
 
 each. The interest on these bonds is paid quarterly. 
 How much interest does Mr. Jones receive every 
 quarter ? 
 
 6. Find the cost of 20 shares of Boston and Maine Kail- 
 
 road stock, at 174. 
 
 7. How much of United States 4% bonds may be bought 
 
 for $6305, at 1211? 
 
 8. How much of Northern Pacific 6% bonds, selling at 
 
 102|, may be bought for $10,275? 
 
 9. How many shares ($100 each) of Old Colony Railroad, 
 
 at 137, may be bought for $1650? 
 
INTEREST AND DISCOUNT. 249 
 
 10. How many shares of railroad stock, at 91^, may be 
 
 bought for $8474.62|? 
 
 11. What must be the price of stock in order that $9200 
 
 stock may be bought for $8970? 
 
 12. What must be the price of stock in order that $11,600 
 
 stock may be bought for $8729? 
 
 13. If $3000 stock is bought for $2748.75, what is the 
 
 price of the stock ? 
 
 14. What income will be derived from $15,000 of 5% 
 
 bonds ? 
 
 15. Find the income from $9000 of 6% stock. 
 
 16. How much will a person receive from $18,800 railroad 
 
 stock, if a dividend of 1% be declared? 
 
 17. What income will be derived from $30,000 of 4% 
 
 bonds? 
 
 Ex. How much 4% stock must be bought to give an 
 income of $320? 
 
 Since $0.04 is derived from $1 stock, $320 will be derived 
 from as many times $ 1 as $0.04 is contained in $320. $320 ^ 
 $0.04 = $8000. 
 
 $8000. Am. 
 
 18. How much 4% stock must be bought to give an 
 
 income of $2400? 
 
 19. A person receives $343 as his quarterly dividend from 
 
 a 7% stock. How much stock does he hold? 
 
 20. Find the entire income of a person whose property 
 
 consists of $6000 of 6% stock and $16,400 of 7% 
 stock. 
 
 21. Find the rate of dividend paid by a railroad when a 
 
 holder of 246 shares receives $1722. 
 
250 INTEREST AND DISCOUNT. 
 
 22. Find the rate per cent at which $22,200 will yield a 
 
 semi-annual return of $990. 
 
 Ex. If $5125 is invested in 6% stock, at 102J, what income 
 will be obtained ? 
 
 $1 stock costs 1.025 of $1. 
 
 Hence $5125 will be the cost of $5125-*- 1.025 = $5000 
 stock. And 6% of $ 5000 = $ 300. 
 
 $300. Ans. 
 
 23. Find the income on $39,000 invested in 4% stock, at 
 
 91. 
 
 24. Find the income on $7000 invested in 4% stock, at 
 
 25. Find the income on $13,600 invested in 7% stock, at 
 
 130. 
 
 26. A person invests $14,280 in railroad stock, at 127^. 
 
 What will he receive if a dividend of 3J% be 
 declared ? 
 
 27. Find the income on $14,000 when invested in 8% 
 
 stock, at 103J. 
 
 Ex. If a person buys 5% stock at 120, what rate of interest 
 does he receive on his money invested ? 
 
 $100 stock costs $120. $100 stock pays $5. Hence the 
 $120 invested yields $5. 
 
 Therefore, the rate of interest is = 0.04 J, or 4J% 
 
 . Ans. 
 
 28. If an 8% stock is worth 150, what rate of interest will 
 
 a purchaser receive on his money ? 
 
 29. If a 10% stock is worth 175, what rate of interest will 
 
 a purchaser receive on his money ? 
 
INTEKEST AttD DISCOUNT. 251 
 
 30. If a 9% stock is worth 170, what rate of interest will 
 
 a purchaser receive on his money ? 
 
 31. If a 4% stock is worth 70, what rate of interest will a 
 
 purchaser receive on his money ? 
 
 32. If a 3% stock is worth 65, what rate of interest will a 
 
 purchaser receive on his money ? 
 
 Ex. Find the sum required for an investment in a 4% stock, 
 at 98, to produce an income of $200 a year. 
 $4 are received from $100 stock. 
 
 Hence $200 will be received from X $100 stock = $5000 
 stock. 
 
 $100 stock costs $ 98 J. 
 
 Therefore $5000 stock will cost 50 x $ 98 J = $ 4925. 
 
 $4925. Ana. 
 
 33. How much money must be invested in 8% stock, at 
 
 92, to produce $400 income? 
 
 34. How much money must be invested in a 3% stock, at 
 
 87^-, to produce an income of $250? 
 
 35. A person bought some bank stock at 107, and received 
 
 $265 when a 5% dividend was declared by the bank. 
 How much money had he invested ? 
 
 36. A person buys some 6% railroad stock at 75, and 
 
 receives $750 income. How much money has he 
 invested ? 9 
 
 Ex. What must be the price of a 5% stock in order that a 
 buyer may receive 6% on his investment ? 
 
 $100 must be invested to produce $6. 
 
 Hence \ of $100 = $83J must be invested to produce $5. 
 
 Therefore the price of the 5% stock must be 83 J. 
 
 Ans. 
 
252 INTEREST AND DISCOUNT. 
 
 37. What must be the price of a 6% stock in order that a 
 
 buyer may receive 7% on his investment? 
 
 38. What must be the price of an 8% stock in order that 
 
 a buyer may receive 6% on his investment? 
 
 39. A person invested $5710 in bank stock when the stock 
 
 was at 142. What per cent dividend is declared, 
 if he receives $300? 
 
 40. A person receives 5% interest on his money by invest- 
 
 ing in some six per cent, stock. At what price did 
 he buy it? 
 
 41. What must be the price of a 7% stock in order that a 
 
 buyer may receive 6% on his investment? 
 
 EXCHANGE. 
 
 242. A bank draft or bill of exchange is a written order 
 directing one person to pay a specified sum of money to 
 another. 
 
 243. A commercial draft is a draft payable at a specified 
 time after sight (or date). 
 
 When the person on whom a commercial draft is drawn accepts 
 the draft, ho writes the word " Accepted," with the date, across the 
 fact, and signs his name. The draft is then called an acceptance, 
 and the acceptor is responsible for its payment. 
 
 An acceptance is of the nature of a promissory note, the acceptor 
 and maker having respectively the same responsibility for payment 
 as the maker and indorser of a promissory note. 
 
 244. The system of paying money to persons at a dis- 
 tance by transmitting bank drafts or bills of exchange 
 instead of money is called exchange. 
 
 When a bank draft can be bought for its face, it is said to be at par. 
 When the cost is less than the face, it is said to be at a. discount; and 
 when the cost is more than the face, it is said to be at a premium. 
 
INTEREST AND DISCOUNT. 253 
 
 Ex. 154. 
 
 Ex. Find the cost of a draft on New York for $1000, at \ 
 of 1% premium. 
 
 % of $1000 = $2.50 (premium). 
 $1000 + $2.50 = $1002.50 (cost). 
 
 $1002.50. Ans. 
 
 1. Find the cost of a draft on New York for $1200, at \ 
 
 of 1% discount. 
 
 2. Find the cost of a draft on St. Louis for $2000, at J of 
 
 1% premium. 
 
 3. Find the cost of a draft on New Orleans for $2400, at 
 
 J-% premium. 
 
 4. Find the cost of a draft on Chicago for $3200, at f % 
 
 discount. 
 
 Ex. Find the cost of a draft on Cincinnati for $1000, pay- 
 able in 30 dys. after sight, exchange being % 
 premium, and interest 6%. 
 
 $1000.00 
 0.0055 of $1000 = $5.50*di8count for 33 dys. 
 
 $994.50 cost of draft at par. 
 0.005 of $1000- ^ 5.00 premium. 
 
 $999.50 cost of draft. 
 
 5. Find the cost of a draft for $800, payable 30 dys. after 
 
 sight, when exchange \^\/o premium, and interest 
 6%. 
 
 6. Find the cost of a draft for $1900, payable in 30 dys., 
 
 when exchange is at par, and interest 4|-%. 
 
 7. Find the cost of a draft for $1450, payable in GO dys., 
 
 when exchange i-s ^% discount, and interest 5%. 
 
 8. Find the cost of a draft for $1000, payable 60 dys. 
 
 after sight, when exchange is -|% discount, and 
 interest 7%. 
 
CHAPTER XII. 
 
 PROPORTION. 
 
 245, The relative magnitude of two numbers is called 
 their ratio when expressed by the fraction which has the 
 first number for numerator, and the second number for 
 denominator. 
 
 Thus the ratio of 2 to 3, commonly written 2 : 3, is expressed by 
 the fraction J. 
 
 246, The first term of a ratio is called the antecedent, 
 and the second term the consequent. 
 
 247, If both terms of a ratio be multiplied or divided by 
 the same number, .the ratio is not altered. 
 
 Thus, if both terms of the ratio 2J : 3J be multiplied by 6, the 
 resulting ratio is 15 : 20, and fhe two ratios are equal, for -* = JJ. 
 
 ^3 
 
 Since J$ = f , the simplest expression for 2| : 3J is 3 : 4. 
 
 248, If the numerator and denominator are interchanged, 
 the fraction is said to be inverted; likewise, if the ante- 
 cedent and consequent of a ratio are interchanged, the 
 resulting ratio is called the inverse of the given ratio. 
 
 Thus, if the fraction is inverted, the resulting fraction is J, and 
 the inverse of the ratio 4 : 5 is 5 : 4. 
 
 249, If two quantities are expressed in the same unit, 
 their ratio will be the same as the ratio of the two numbers 
 by which they are expressed. 
 
 Thus the quantity $ 7 is the same fraction of $ 9 as 7 is of 9. 
 
PROPORTION. 255 
 
 250. Since ratio is simply relative magnitude, two quan- 
 tities different in kind cannot form the terms of a ratio ; 
 and two quantities of the same kind must be expressed in a 
 common unit before they can form the terms of a ratio. 
 
 Thus no ratio exists between $5 and 20 dys. ; and the ratio of 3 t. 
 to 5000 Ibs. can be expressed only when both quantities are written 
 as tons or pounds. 
 
 251. When two ratios are equal, the four terms are said 
 to be in proportion, and are called proportionals. 
 
 Thus 6, 3, 18, 9 are proportionals ; for f = - 1 /. 
 
 252. A proportion is written by putting the sign = or a 
 double colon between the ratios. 
 
 Thus 6 : 3 = 18 : 9, or 6 : 3 : : 18 : 9, means, and is read, the ratio of 
 6 to 3 is equal to the ratio of 18 to 9. 
 
 253. The jrs and last terms of a proportion are called 
 the extremes, and the two middle terms are called the means. 
 
 254. Test of a proportion. When four numbers are pro- 
 portionals, the product of the extremes is equal to the 
 product of the means. 
 
 This is seen to be true by expressing the ratios in the 
 form of fractions, and multiplying both by the product of 
 the denominators. 
 
 Thus the proportion 5 : 3 : : 15 : 9 may be written $ = -^ ; and, if 
 both be multiplied by 3 X 9, the result will be 5 X 9 = 3 X 15. 
 
 255. Either extreme, therefore, will be equal to the prod- 
 uct of the means divided by the other extreme ; and either 
 mean will be equal to the product of the extremes divided 
 by the other mean. Hence, if three terms of a proportion 
 be given, the fourth may be found. 
 
256 PROPORTION. 
 
 (1) What number is to 4 as 3 is to 6? 
 
 This may be written What number = ? ? 
 Multiply both sides of the equation by 4. 
 
 The result is, What number 
 
 6 
 
 2. Ans. 
 
 (2) 20 is to 24 as what number is to 30? 
 
 This may be written gO a What number ? 
 24 30 
 
 Multiply by 30, 20x30 - What number ? 
 
 25. Ans. 
 
 (3) 18 is to 32 as 45 is to what number? 
 
 This may be written = ? 
 
 32 What number 
 
 As these fractions are equal, their reciprocals are equal ; 
 
 that is, = What number ^ 
 
 18 45 
 
 Multiply by 45, 32x45 
 
 18 
 
 80. Ans. 
 
 256, When three terms of a proportion are given, the 
 method of finding the fourth term is called the Eule of Three, 
 
 It is usual to arrange the quantities (that is, to state the 
 question) so that the quantity required for the answer may 
 be the fourth term. Hence the quantity which corresponds 
 to that of the required answer must be the third term. 
 
 (1) If 5 t. of hay cost $87.50, what will 21 t. cost? 
 
 Since the cost of 21 t. is required, $ 87.50 is the third term. 
 Since 21 t. will cost more than 5 t., 21 t. is the second term 
 and 5 t. the first term. 
 
 That is, 5 t : 21 t. : : $87.50 : What quantity? 
 
PROPORTION. 257 
 
 A difficulty presents itself here, inasmuch as no meaning can 
 be given to the product of the means ($87.50 multiplied by 21 1.). 
 Since, however, the ratio of 5 t. : 21 t. = the ratio of 5 : 21, the 
 ratio 5 : 21 may be substituted for 5 t. : 21 t. 
 
 Then 5 : 21 : : $87.50 : What quantity ? 
 
 That is, What quantity = gl.X$ 87.50 ? 
 
 $367.50. Ans. 
 
 (2) When a post 11.5 ft. high casts a shadow on level 
 ground 17.4 ft. long, a neighboring steeple casts a 
 shadow 63.7 yds. long. How high is the steeple? 
 
 Height is required ; the height 11.5 ft. is therefore the third 
 term. 
 
 Since the shadow of the steeple is the longer, the height of the 
 steeple must be the greater ; therefore the second term must be 
 the greater of the two remaining quantities expressed in the 
 same unit. 63.7 yds. = 191.1 ft. 
 
 Shadow. Shadow. Height. Height. 
 
 17.4 ft. : 191.1 ft. : : 11.5 ft. : What? 
 or, 17.4 : 191.1 :: 11.5 ft. : What? 
 
 That is. height of steeple = 19L1 * 1L5 ft - 126.3 ft. 
 
 126.3 ft. Ans. 
 
 257, In solving problems by the Bule of Three, 
 
 Make that quantity which is of the same kind as the 
 required answer the third term. 
 
 The numbers by which the other two quantities are ex- 
 pressed, when expressed in a common unit, will be the first 
 and second terms. 
 
 If, from the nature of the question, the answer will be 
 greater than the third term, make the greater of these two 
 numbers the second term ; if less, make the smaller of these 
 numbers the second term, and the other the first term. 
 
 Divide the product of the second and third terms by the 
 first term, and the quotient will be the answer required. 
 
258 PROPORTION, 
 
 Ex. 155. 
 
 1. An express-train runs 40 mi. in 64 mm. At the same 
 
 rate, how many miles will it run in 24 min. ? 
 
 2. If 110 A. produce 200 hhds. of sugar, how many hogs- 
 
 heads will 176 A. produce? 
 
 3. If 48 reapers cut 20 A. in a given time, how many 
 
 acres will 156 reapers cut in the same time ? 
 
 4. If 20 reapers can cut a field in 6 dys., in how many 
 
 days will 30 reapers do it? 
 
 5. The number of copies in the first edition of the " Lady 
 
 of the Lake " was 2050, and was to the number in 
 the second as 41 to 69. Find the number in the 
 second edition. 
 
 6. The length of the steamer-track from Liverpool to 
 
 Quebec is 2502 mi., and is to that from Liverpool 
 to Boston as 139 is to 155. Find the length of the 
 track from Liverpool to Boston. 
 
 7. If a steamer from Liverpool to Portland makes the 
 
 passage of 2750 mi. in 5-f- dys., in how many days, 
 at the same rate, would the passage of 2980 mi. from 
 Liverpool to New York have been made ? 
 
 8. If a person can walk 8^- mi. in 2J- hrs., how many miles 
 
 can he walk in 3 J hrs. ? 
 
 9. If the shadow of a staff 3 ft. 7 in. high is 4 ft. 9 in., 
 
 find the height of a steeple whose shadow, is 158 ft. 
 4 in. 
 
 10 A train, at the rate of 25f mi. an hour, goes a certain 
 distance in 3 hrs. In how many hours will one at 
 the rate of 24J mi. an hour go the same distance ? 
 
PROPORTION. 259 
 
 11. The ratio of the diameter to the circumference of a 
 
 circle was given by Metius as 113 : 355. Find the 
 circumference of a fly-wheel 10 ft. in diameter. 
 
 12. Find the horse-power of an engine that can raise 
 
 11,200 Ibs. of coal in an hour from a pit whose depth 
 is 396 ft. 
 
 NOTE. The labor necessary to raise 1 Ib. through 1 ft. is called the 
 unit of work ; and a horse can do 33,000 units of work a minute. 
 
 Therefore one horse-power = 33,000 units of work, and 396 X 112QQ 
 
 33000 X 60 
 = the horse-power required. 
 
 13. If 1000 sq. yds. of a field produce a load of hay, how 
 
 many such loads will 25 A. of the field produce ? 
 
 14. If a train runs 177 mi. 120 rds. in 3 hrs. 56^- min., 
 
 what is the rate per hour? 
 
 16. If 136 masons can build a fort in 28 dys., how many 
 men will be required to build it in 8 dys. ? 
 
 16. There are provisions in a fort sufficient to support 
 
 4000 soldiers for 3 mos. How many must be sent 
 away to make them last 8 mos. ? 
 
 17. A coach travels 7-J- mi. an hour. How many miles will 
 
 it go between a quarter past ten A.M. and a quarter 
 to six P.M. ? 
 
 18. The expense of making the hay on 5 A. 135 sq. rds. is 
 
 $29.08. What is the expense per acre? 
 
 19. If 300 laborers can make an embankment in 48 dys., 
 
 how many more days would be required if the num- 
 ber of men is diminished by 60 ? 
 
 20. If 2.45 tons of straw cost $22.75, how many tons can 
 
 be bought for $11.70? 
 
260 PROPORTION. 
 
 COMPOUND PROPORTION. 
 
 258, A ratio is said to be compounded of two or more 
 given ratios, when it is expressed by a fraction which is the 
 product of the fractions representing the given ratios. 
 
 Thus the ratios 2:3 and 7 : 11 are represented by the fractions 
 f and -fa ; and the ratio 14 : 33, which is represented by Jf (the prod- 
 uct of J and -fa\ is said to be compounded of the ratios 2 : 3 and 7:11. 
 
 259, A proportion which has one of its ratios a compound 
 ratio is called a compound proportion, 
 
 In stating problems in compound proportion the quantity 
 which corresponds to the answer required is made the third 
 term. Each pair of the remaining quantities is then con- 
 sidered separately with reference to the answer required. 
 The process will be understood by the following example : 
 
 If 4 men mow 15 A. in 5 dys. of 14 hrs., in how many 
 days of 13 hrs. can 7 men mow 19^- A. ? 
 
 As the answer is to be in days, make 5 dys. the third term. 
 
 I. Will it require more or less days for 7 men to mow 15 A. than 
 it did for 4 men ? Evidently less. 
 
 Therefore make 7 the first term and 4 the second. 
 
 II. Will it require more or less days for the same number of men 
 to mow 19} A. than it did to mow 15 A. ? Evidently more. 
 
 Therefore make 15 the first term and 19} the second. 
 
 III. Will it require more or less days of 13 hrs. to mow the same 
 number of acres than it did of 14 hrs.? Evidently more. 
 
 Therefore make 13 the first term and 14 the second. 
 Hence the statement is 
 
 7:4 
 
 15: 19.5: : 5 days: what? 
 13:14 
 
 4 x 19.5 x 14 x 5 days 
 7 X 15 x 13 
 
 This, simplified by cancellation, gives 4 days. 
 
PROPORTION. 261 
 
 Ex. 156. 
 
 1. If 13 bu. of oats serve 3 horses for 11 dys., how many 
 
 bushels will serve 7 horses for 12 dys. ? 
 
 2. If a traveller walks 140 mi. in 8 dys., walking 7 hrs. a 
 
 day, how many miles can he walk in 12 dys. of 8 
 hrs. each ? 
 
 3. If 4 masons build 27 yds. of wall in 5 dys., working 9 
 
 hrs. a day, in how many days will 32 masons build 
 81 yds. of a similar wall, if they work 10 hrs. a day? 
 
 4. A bootmaker who employs 15 men fills an order for 
 
 25 doz. pairs of boots in 4 wks. In how many days 
 can he make 45 pairs if he employs 18 men ? 
 
 5. If a family, by using 2 gas-burners 7 hrs. a day, pays 
 
 $6 a quarter when gas is $2.40 per 1000 cu. ft., 
 what will a family using 3 burners 4 hrs. a day pay 
 per quarter when gas is $1.80 per 1000 cu. ft. ? 
 
 6. If 330 slices T \ of an inch thick are obtained from 12 
 
 rounds of beef, how many similar rounds will be 
 required for 495 slices of an inch thick ? 
 
 7. If 5 horses eat 8 bu. 14 qts. of oats in 9 dys., how 
 
 many days, at the same rate, will 66 bu. 30 qts. last 
 17 horses? 
 
 8. If a man walks 600 mi. in 25 dys., walking 8 hrs. a 
 
 day, in how many days will he walk 330 mi., walk- 
 ing 10 hrs. a day ? 
 
 9. If a pane of glass 18 in. long and 12^ in. wide costs 20 
 
 cts., what will be the cost, at the same rate, of a 
 pane 22-1- i n . ] on g an( j 15 | n w id e ? 
 
 10. If 18 men can dig a trench 200 yds. long, 3 yds. wide, 
 and 2 yds. deep, in 6 dys. of 10 hrs. each, in how 
 many days of 8 hrs. each will 10 men dig a trench 
 100 yds. long, 4 yds. wide, and 3 yds. deep? 
 
262 PROPORTION. 
 
 PROPORTIONAL PARTS. 
 
 260. If it be required to divide a quantity into parts 
 proportional to 3, 4, 5, the numbers 3, 4, 5 may be taken 
 as representatives of the parts, and then the whole quantity 
 will be represented by 3 + 4 + 5 ; that is, by 12. 
 
 (1) Divide $391 into parts proportional to 5, 7, and 11. 
 
 The whole quantity will be represented by 5 + 7 + 11 = 23. 
 Therefore the respective parts will be ^, ^, J of $391. 
 
 $85, $119, $187. Ans. 
 
 (2) Divide $248 into parts proportional to fa, -fa, fa. 
 
 Multiply the fractions by 150, the L.C.M. of their denomina- 
 tors. The results are 15, 10, 6. Hence the parts will be repre- 
 sented by the numbers 15, 10, 6, and the whole by 31. 
 
 Therefore the respective parts will be jf, J, -ft of $248. 
 
 $120, $80, $48. Ans. 
 
 Ex. 157. 
 
 1. Divide 1200 into parts proportional to 11, 12, 13, 14. 
 
 2. Divide 390 into parts proportional to --, , . 
 
 3. Divide a profit of $689 among 3 partners, of whom the 
 
 first owns -fa, the second -fa, and the third ^ of the 
 joint stock. 
 
 4. Four men invest $450, $230, $190, $110 respectively 
 
 in a joint business. Find their respective liabilities 
 in a loss of $313.60. 
 
 5. Three partners claim respectively -J-, -J-|-, and -fa of 
 
 $1260. Give to each his proportional share. 
 
 6. An analysis of dissolved bones gives the following 
 
 results for every 100 parts. Water, 13.97 ; organic 
 matter, 15.71; soluble phosphates, 21.63; insoluble 
 phosphates, 11.43; sulphate of lime, 15.83; sulphuric 
 acid, 15.63; alkaline salts, 1.10; silica, etc., the 
 remainder. Find the number of pounds of each in 
 a ton of dissolved bones. 
 
PROPORTION. 263 
 
 PAETNERSHIP. 
 
 261, Partnership is separated into simple and compound. 
 In simple partnership the capital of each partner is invested 
 for the same time. In compound partnership the time for 
 which the capital of each partner is invested is taken into 
 account, as well as the amount of the capital ; and the divi- 
 sion of profits and losses is made proportionally to the 
 amount of the capital and the time it is invested. 
 
 A and B enter into partnership. A puts in $2000 for 
 2 yrs., and B puts in $3000 for 1 yr. Their profits are 
 $ 1400. What is the share of each ? 
 
 The use of $2000 for 2 yrs. is equivalent to 2 X $2000 for 1 yr. 
 Hence their profits must be divided in the ratio $4000 to $3000; 
 that is, 4 : 3. 
 
 Ex. 158. 
 
 1. Three drovers rent a field of 9 A., at $5 an acre. A 
 
 puts in 6 cows for 2 mos ; B, 9 cows for 1 mo. ; and 
 0, 12 cows for 2 mos. How much should each pay ? 
 
 2. In a co-partnership A contributed $400 for 9 mos. ; B, 
 
 $350 for 8 mos. ; and C, $600 for 2 mos. Divide a 
 gain of $570 among them. 
 
 3. At the end of 12 mos. A, B, and C, having a joint 
 
 capital of $6000, find they have lost $625. As 
 capital of $2500 has been in the business for 12 mos., 
 B's of $1500 for 8 mos., and C's of $2000 for 4 mos. 
 Divide the loss among them. 
 
 4. A and B enter into partnership, A with $1800, and 
 
 B with $900. At the end of 8 mos. B adds $300 to 
 his capital. Divide a profit of $840 between them, 
 at the end of the year. 
 
264 PROPORTION. 
 
 AVERAGES. 
 
 262, If a dozen eggs weigh 1 Ib. 8 oz., what is their 
 average weight? 
 
 Since the 12 eggs weigh 1 Ib. 8 oz., that is, 24 oz., the average 
 weight of an egg will be ^j of 24 oz. = 2 oz. 
 
 Ex. 159. 
 
 1. A merchant mixes 3 Ibs. of coffee worth 27 cts. a 
 
 pound, 2 Ibs. worth 35 cts., and 1 Ib. worth 41 cts. 
 What is the mixture worth a pound ? 
 
 2. What is the cost of a gallon of a mixture containing 
 
 7 gals, worth $1.35 a gallon, 5 gals, worth $1.05 a 
 gallon, and water enough to make the whole mix- 
 ture 15 gals. ? 
 
 3. Of 32 candidates for office, 3 were 20 yrs. old, 4 were 
 
 21, 12 were 22, 12 were 23, and 1, 24. What was 
 the average age of the candidates ? 
 
 4. A bankrupt owes A $962.50, B, $3487, and C, 
 
 $12,686.50. His estate, after paying expenses of 
 settlement, is $3427.20. How much can he pay on 
 a dollar ? 
 
 5. A grocer buys 106 Ibs. of tea, at 80 cts. per pound, 
 
 75 Ibs., at $1.24 per pound, and 94 Ibs., at $1.30 
 per pound, and mixes the three lots together. At 
 what price per pound must he sell the mixture so 
 as to make 10% on his outlay? 
 
 6. In what proportions must oils worth $1.25 a gallon 
 
 and 80 cts. a gallon be mixed to make a mixture 
 
 worth $1.00 a gallon? 
 
 HINT. The loss on the $1.25 oil is 25 cts. a gallon. The gain 
 on the 80 ct. oil is 20 cts. a gallon. Therefore there must be 
 more of the 80 ct. oil taken than of the $1.25 oil, and in the 
 ratio of 25 : 20 or 5 : 4. 
 
PROPORTION. 265 
 
 7. In what proportion must oils worth $1.20 and 60 cts. 
 
 a gallon be mixed, so that the mixture may be worth 
 70 cts. a gallon ? 
 
 8. Solder is composed of tin and lead. If a solder weighs 
 
 10.44 times as much as an equal bulk of water, 
 while tin weighs 7.29, and lead 11.35 as much, find 
 the weight of each metal in a pound of solder. 
 
 AVERAGE OF PAYMENTS. 
 
 A has given to B notes as follows : $ 250, due in 3 mos. ; 
 $400, due in 6 mos. ; $700, due in 8 mos. He wishes to 
 pay them all at one time. In how many months shall the 
 entire payment be made ? 
 
 The use of $250 for 3 mos. equals the use of $750 for 1 mo. 
 
 The use of $400 for 6 mos. equals the use of $2400 for 1 mo. 
 
 The use of $700 for 8 mos. equals the use of $5600 for 1 mo. 
 $1350 $8750 for 1 mo. 
 
 The question is, for how many months is the use of $1350 equal 
 to the use of $8750 for 1 mo. ? 
 
 The answer required is {Jf $ mos. = 6J mos. 
 
 ty% mos. Ans. 
 
 9. Find the equated time for the payment of $300 due 
 
 in 3 mos., $500 due in 6 mos., $200 due in 9 mos. 
 
 10. A owes B $50 payable in 6 mos., $60 payable in 8 
 
 mos., and $90 payable in 4 mos. Find the equated 
 time of payment. 
 
 11. A owes B $1000, payable at the end of 9 mos. He 
 
 pays $200 at the end of 3 mos. and $300 at the end 
 of 8 mos. When is the balance due ? 
 
 12. On the first day of January, A purchases of B $200 
 
 worth of goods on 3 mos. credit, and $500 worth on 
 4 mos. credit, and gives one note in payment. When 
 does the note become due? 
 
CHAPTER XIII. 
 
 POWERS AND ROOTS. 
 
 263. The square of a number is the product of two fac- 
 tors, each equal to this number. 
 
 Thus the squares of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 
 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 
 
 264. The square root of a number is one of the two equal 
 factors of the number. 
 
 Thus the square roots of 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 
 are 1,2,3, 4, 5, 6, 7, 8, 9, 10. 
 
 265. The square root of a number is indicated by the 
 radical sign -^/, or by the fraction -J- written above and to 
 the right of the number. 
 
 266. Since 35 = 30+5, the square of 35 may be obtained 
 as follows : 
 
 30+5 
 
 30+5 30'= 900 
 
 30'+ (30x5) 2(30x5)= 300 
 
 (30x5) + 5* 5'= 25 
 
 30'+ 2 (30x5) + 5* =1225 
 
 267. Hence, since every number consisting of two or more 
 figures may be regarded as composed of tens and units, 
 
 The square of a number will contain the square of the 
 tens + twice the tens X the units + the square of the units. 
 
POWERS AND ROOTS. 267 
 
 SQUARE ROOT. 
 
 The first step in extracting the square root of a 
 number is to mark off the figures of the number in groups. 
 
 Since 1 = I 2 , 100 = 10 2 , 10,000 = 100 2 , and so on, it is evident that 
 the square root of any number between 1 and 100 lies between 1 and 
 10 ; of any number between 100 and 10,000 lies between 10 and 100. 
 In other words, the square root of any number expressed by one or 
 two figures is a number of one figure ; of any number expressed by 
 three or four figures is a number of two figures, and so on. 
 
 If, therefore, an integral number be divided into groups of two fig- 
 ures each, from the right to the left, the number of figures in the root 
 will be equal to the number of groups of figures. The last group to 
 the left may consist of only one figure. 
 
 Find the square root of 1225. 
 
 The first group 12, contains the square of the tens' 
 12 25 (35 number of the root. 
 
 9 The greatest square in 12 is 9, and the square root 
 
 65) 3 25 of 9 is 3. Hence 3 is the tens* figure of the root. 
 
 3 25 The square of the tens is subtracted, and the 
 
 remainder, contains twice the tens X the units + the 
 square of the units. Twice the 3 tens is 6 tens, and 6 tens is con- 
 tained in the 32 tens of the remainder 5 times. Hence 5 is the units' 
 figure of the root. Since twice the tens X the units + the square of 
 the units is equal to (twice the tens + the units) X the units, the 5 
 units are annexed to the 6 tens, and the result, 65, is multiplied by 5. 
 
 269, The same method will apply to numbers of more than 
 two groups of figures, by considering the part of the root al- 
 ready found as so many tens with respect to the next figure. 
 
 Extract the square root of 7890481. 
 
 7 89 04 81 (2809 When the third group, 04, is brought 
 
 4 down, and the divisor, 56, formed, the next 
 48) 3 89 figure of the root is 0, because 56 is not con- 
 
 3 34 tained in 50. Therefore, is placed both 
 
 5 04 81 * n ^ e r00 ^ an( ^ ^ e Divisor, and the next 
 
 5 04 81 two fig ures 81, are brought down. 
 
268 POWERS AND ROOTS. 
 
 270. If the square root of a number have decimal places, 
 the number itself will have twice as many. 
 
 Thus, if 0.11 be the square root of some number, the number will 
 be (O.ll) 2 - 0.11 X 0.11 = 0.0121. Hence, if a given number contain 
 a decimal, we divide it into groups of two figures each, by beginning 
 at the decimal point and marking toward the left for the integral 
 number, and toward the right for the decimal. We must be careful 
 to have the last group on the right of the decimal point contain two 
 figures, annexing a cipher when necessary. 
 
 Extract the square root of 52.2729. 
 
 '52.2729(7.23 
 49 
 
 149)3 9 7 Jt wil1 te 8een frcm tte ? roups of fi g ures 
 
 9 4 that the root will have ene integral and two 
 
 1443)1329 ^imal places. 
 
 4329 
 
 271. If a number is not a perfect square, ciphers may be 
 annexed, and an approximate value of the root found. 
 
 Extract to six places of decimals the square root of 19. 
 19000000(4.358899 
 
 16 
 
 83)3 00 
 
 o 40 In this example, after finding four 
 
 r figures of the root, the other three are 
 
 865) 51 00 foun(i by common division. The rule 
 
 ^ ^ in such cases is, that one less than the 
 
 8708/ 7 75 00 number of figures already obtained may 
 
 6 96 64 be found without error by division, the 
 
 8716) 78 360 divisor to be employed being twice the 
 
 69 728 part of the root already found. 
 86320 
 78444 
 78760 
 
 272. The square root of a common fraction is found by 
 extracting the square roots of the numerator and denomi- 
 
POWERS AND ROOTS. 
 
 269 
 
 nator. But, when the denominator is not a perfect square, 
 it is best to reduce the fraction to a decimal and then 
 extract the root. 
 
 Ex. 160. 
 Find the square roots of : 
 
 1. 4225. 
 
 5. 15.7609. 
 
 9. 0.025. 
 
 13. T 6 ^-. 
 
 2. 31.36. 
 
 6. 0.180625. 
 
 10. 28.75. 
 
 14- f 
 
 3. 50625. 
 
 7. 0.001296. 
 
 11. 0.009. 
 
 15. f. 
 
 4. 401956. 
 
 8. 0.042849. 
 
 12. 0.081. 
 
 16. 4. 
 
 The side of a square is found by extracting the square 
 root of its area. 
 
 17. A rectangle is 972yds. long and 432 yds. wide. Find 
 
 the side of a square which has the same area as the 
 rectangle. 
 
 18. Find in yards the length of 
 
 the side of a square field 
 
 containing 27 A. 12 sq. rds. 
 
 1 sq. yd. 
 
 In a right triangle, the square 
 on the hypotenuse {AC) is equal 
 to the sum of the squares on the 
 two legs. 
 
 Hence hypotenuse = square root of 
 sum of squares on the legs ; and one leg = square root of difference 
 of squares on the other two sides. 
 
 19. Base = 39, perpendicular 52 ; find hypotenuse. 
 
 20. Base = 35, hypotenuse 91 ; find perpendicular. 
 
 21. Perpendicular = 72, hypotenuse = 75 ; find base. 
 
 22. A cord 287 ft. long is stretched from the top of a flag- 
 
 pole 63 ft. high ; find the distance of the end in 
 contact with the ground from the base of the pole. 
 
270 POWERS AND ROOTS. 
 
 TJie length of the diagonal of a room is the square root of 
 the sum of the squares of the length, breadth, and height. 
 
 23. Find the diagonal of a room 28 ft. long, 21 ft. wide, 
 
 and 12 ft. high. 
 
 24. Find the diagonal of a hall 50 ft. long, 30 ft. wide, and 
 
 15 ft. high. 
 
 CUBE ROOT. 
 
 273, The cube of a number is the product of three factors, 
 each equal to the number. 
 
 The cubes of 1, 2, 3, 4, 5, G, 7, 8, 9, 10, 
 are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. 
 
 274, The cube root of a number is one of the three equal 
 factors of the number. 
 
 Thus the cube roots of 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 
 are 1,2, 3, 4, 5, 6, 7, 8, 9, 10. 
 
 275, The cube root of a number is indicated by {/, or by 
 the fraction -^ written above and to the right of the 
 number. 
 
 Thus, v/343, or 343$, means the cube root of 343, 
 
 276, Since 35 = 30 + 5, the cube of 35 may be obtained 
 thus: 
 
 30+5 
 30+5 
 
 30'+ (30x5) 30 s - 27,000 
 
 + (30x5) + 5 2 3(30 2 x5)=13,500 
 
 30 2 +2(30x5) + 5 2 3(30 X 5 2 ) = 2,250 
 
 30+5 _ 5 s - _ 125 
 
 30 3 +2(30 2 x5)+ (30 x5 2 ) 42,875 
 
 (30 2 x5) + 2(30x5 2 ) 
 
 30 3 + 3(30 2 x 5) + 3(30 x 5 2 ) + 5 s 
 
POWERS AND ROOTS. 271 
 
 Hence the cube of any number composed of tens and 
 units contains four parts : 
 
 I. The cube of the tens. 
 
 II. Three times the product of the square of the tens by 
 the units. 
 
 III. Three times the product of the tens by tht square of 
 the units. 
 
 IV. The cube of the units. 
 
 277, In extracting the cube root of a number, the first 
 step is to mark off the figures of the number in groups. 
 
 Since 1 = 1 s , 1000 = 10 3 , 1,000,000 = 100 3 , and so on, it follows that 
 the cube root of any number between 1 and 1000, that is, of any num- 
 ber that has one, two, or three figures, is a number of one figure ; and 
 that the cube root of any number between 1000 and 1,000,000, that 
 is, of any number that has four, five, or six figures, is a number of two 
 figures, and so on. 
 
 . If, therefore, an integral number be divided into groups of three 
 figures each, from right to left, the number of figures in the root will 
 be equal to the number of groups. The last group to the left may 
 consist of one, two, or three figures. 
 
 Extract the cube root of 42875. 
 
 42 875 (35 Since 42875 consists of two 
 27 groups, the cube root will 
 
 15 875 consist of two figures. 
 
 The first group, 42, contains 
 
 the cube of the tens' number 
 
 of the root. 
 15 875 The greatest cube in 42 
 
 3 X 30* = 2700 
 
 3 X (30x5)= 450 
 
 5'= 25 
 
 3175 , 
 
 is 27, and the cube root of 
 27 is 3. Hence 3 is the tens' figure of the root. 
 
 The remainder, 15875, resulting from subtracting the cube of the 
 tens, will contain three times the product of the square of the tens by 
 the units + three times the product of the tens by the square of the 
 units + the cube of the units. 
 
 Each of these three parts contains the units' number as a factor. 
 
272 POWERS AND ROOTS. 
 
 Hence the 15875 consists of two factors, one of which is the units' 
 number of the root ; and the other factor is three times the square of 
 the tens + three times the product of the tens by the square of the 
 units -f the square of the units. The larger part of this second factor 
 is three times the square of the tens. 
 
 And, if the 158 hundreds of the remainder be divided by the 
 3 x 30 2 =- 27 hundreds, the quotient will be the units' number of the 
 root. 
 
 The second factor can now be completed by adding to the 2700 
 3 x (30 X 5) = 450 and 5* = 25. 
 
 278. The same method will apply to numbers of more 
 than two groups of figures, by considering the part of the 
 root already found as so many tens with respect to the 
 next figure of the root. 
 
 Extract the cube root of 57512456. 
 
 57512456(386 
 
 3 x 30' = 2700 
 
 3 X (30 x 8) = 720 
 
 8' = 64 
 
 3484 
 
 2 640 456 
 
 3 x 380 1 = 433200 ' 
 3 X (380 x 6) = 6840 
 6'= 36 
 
 440076 
 
 27872 
 
 2 640 456 
 
 279, If the cube root of a number have decimal places, 
 the number itself will have three times as many. 
 
 Thus, if 0.11 be the cube root of a number, the number is 0.11 X 
 0.11 x 0.11 = 0.001331. Hence, if a given number contain a decimal, 
 we divide the figures of the number into groups of three figures each, 
 by beginning at the decimal-point and marking toward the left for 
 the integral number, and toward the right for the decimal. We must 
 be careful to have the last group on the right of the decimal-point 
 contain three figures, annexing ciphers when necessary. 
 
POWERS AND ROOTS. 
 
 273 
 
 Extract the cube root of 187.149248. 
 
 187.149248(5.72 
 125 
 8x50' =7500 
 
 3 X (50 X 7) = 1050 
 
 7*= 49 
 
 8599 
 
 62149 
 
 60193 
 
 3 x570' = 974700 
 
 3 X (570 X 2) = 3420 
 
 2 a = 4 
 
 978124 
 
 1 956 248 
 
 1 956 248 
 
 It will be seen from the groups of figures that the root will have 
 one integral and two decimal places, and therefore the decimal-point 
 must be placed in the root as soon as one figure of the root is obtained. 
 
 280. If the given number be not a perfect cube, ciphers 
 may be annexed, and a value of the root may be found as 
 near to the true value as we please. 
 
 Extract the cube root of 1250.6894. 
 
 1250.689400(10.77 
 1 
 
 3x10*= 300 | 250 
 
 Since 300 is not contained in 200, the next figure of the root will 
 beO. 
 
 250 689 
 
 3 x 100 2 = 30000 
 3 X (100x7)= 2100 
 
 7 2 = 49 
 
 32149 
 
 225 043 
 
 3 x 1070 2 = 3434700 
 3 X (1070 x 7) = 22470 
 
 7 2 = 49 
 
 3457219 
 
 25 646 400 
 
 24 200 533 
 
 1 445 867 
 
274 
 
 POWERS AND ROOTS. 
 
 281. The following method very much shortens the work 
 in long examples. 
 
 Extract the cube root of 5 to five places of decimals. 
 
 5.000(1.70997 
 1 
 
 3 X 10 2 = 300 
 3 (10x7) = 210 
 7' = J9-j 
 559 [ 
 259 J 
 
 4000 
 
 3913 
 
 3 x 1700 2 = 8670000 
 3(1700x9)- 45900 
 9' = _ 81} 
 8715981 I 
 45981 J 
 
 3 x 1709' = 8762043 
 
 87 000 000 
 
 78 443 829 
 
 85561710 
 7 885 8387 
 
 670 33230 
 613 34301 
 
 After the first two figures of the root are found, the next trial divi- 
 sor is obtained by bringing down the sum of the 210 and 49 obtained 
 in completing the preceding divisor, then adding the three lines con- 
 nected by the brace, and annexing two ciphers to the result. 
 
 It is seen at a glance that, when the trial divisor is increased by 
 3 times the 17 tens of the root, it will be greater than 87000 ; so that 
 is placed in the root, and 3 X 1700 2 is obtained by annexing two 
 ciphers to the 86700. Again : the trial divisor is obtained by bringing 
 down the sum of the 45900 and 81, which was obtained in completing 
 the preceding divisor, then adding the three lines connected by the 
 brace, and annexing two ciphers to the result. 
 
 The last two figures of the root are found by division. The rule 
 in such cases is, that two less than the number of figures already 
 obtained may be found without error by division, the divisor to be 
 employed being three times the square of the part of the root already 
 found. 
 
POWERS AND ROOTS. 
 
 275 
 
 282. The cube root of a common fraction is found by 
 taking the cube roots of the numerator and denominator ; 
 but, if the denominator be not a perfect cube, it is best to 
 reduce the fraction to a decimal, and then extract the root. 
 
 Ex. 161. 
 
 Find the cube roots of: 
 
 9. 12396.8834. 
 
 10. 0.00027. 
 
 11. 0.00008. 
 
 12. 277.2738. 
 
 Find the 
 
 1 29791. 5. 53157376. 
 
 2 357911. 6. 62099136. 
 
 3. 148877. 7. 41.421736. 
 
 4. 103823. 8. 12.812904. 
 
 17. The liter contains 61.027 cu. in. 
 
 cube containing a liter. 
 
 18. The edges of a rectangular solid are 154 
 
 70 ft. 7 in., 53 ft 1 in. Find the edge 
 equivalent to it. 
 
 The square of (30 + 5) = 30* -f 2 (30 X 5) + 5 2 . 
 
 The 30 2 may be represented by a square (Fig. 1) 30 in. 
 
 The 2(30 X 5) may be represented by two strips 30 in. 
 in wide, of Fig. 2, which are added to two adjacent sides 
 
 The 5 2 may be represented by the small square of Fig. 
 to make Fig. 2 a complete square. 
 
 13. 
 
 15. U- 
 
 16. f. 
 
 side of a 
 
 ft. 11 in., 
 of a cube 
 
 ?266. 
 on a side, 
 long and 5 
 of Fig. 1. 
 3 required 
 
 Fig. 1. 
 
 Fig. 2. 
 
 Fig. 8. 
 
 In extracting the square root of 1225, the large square, which is 30 
 in. on a side, is first removed, and a surface of 325 sq. in. remains. 
 
 This surface consists of two equal rectangles, each 30 in. long, and 
 a small square whose side is equal to the width of the rectangles. 
 
 The width of the rectangles is found by dividing the 325 sq. in. by 
 the sum of their lengths, that is, by 60, which gives 5 ia. 
 
276 
 
 POWERS AND ROOTS. 
 
 Hence the entire length of the surfaces added is 30 in. + 30 in. 
 + 5 in. = G5 in., and the width is 5 in. 
 
 Therefore the total area is (65 X 5) = 325 sq. in. 
 
 The cube of (30 + 5) = 30 3 + 3 (30 2 X 5) + 3 (30 X 5 2 ) + 5 s . g 392. 
 
 The 30 3 may be represented by a cube whose edge is 30 in. (Fig. 1). 
 
 The 3 (30 2 X 5) may be represented by three rectangular solids, 
 each 30 in. long, 30 in. wide, and 5 in. thick, to be added to three 
 adjacent faces of Fig. 1. 
 
 The 3(30 X 5 2 ) may be represented by three equal rectangular 
 solids, 30 in. long, 5 in. wide, and 5 in. thick, to be added to Fig. 2. 
 
 The 5 3 may be represented by the small cube required to complete 
 the cube of Fig. 3. 
 
 Fig. 
 
 Fig. S. 
 
 Fig. 4. 
 
 In extracting the cube root of 42875, the large cube (Fig. 1), whose 
 edge is 30 in., is first removed. 
 
 There remain (42875 - 27000) cu. in. = 15875 cu. in. 
 
 The greater part of this is contained in the three rectangular sol- 
 ids which are added to Fig. 1, and which are each 30 in. long and 
 30 in. wide. 
 
 The thickness of these solids is found by dividing the 15875 cu. in. 
 by the sum of the three faces, each of which is 30 in. square ; that is, 
 by 2700 sq. in. The result is 5 in. 
 
 There are also the three rectangular solids which are added to 
 Fig. 2, and which are 30 in. long and 5 in. wide ; and a cube which 
 is added to Fig. 3, and which is 5 in. long and 5 in. wide. 
 
 Hence the sum of the products of two dimensions of all these 
 solids is 
 
 For the larger rectangular solids, 3(30 X 30) sq. in. = 2700 sq. in. 
 
 For the smaller rectangular solids, 3 (30 X 5) sq. in. = 450 sq. in. 
 
 For the small cube, (5 x 5) sq. in. = 25 sq. in. 
 
 3175 sq. in. 
 
 This number multiplied by the third dimension gives (5 x 3175) 
 cu. in. 15,875 cu. in. 
 
POWERS AND ROOTS. 277 
 
 283, In bodies of the same shape, 
 
 Two corresponding lines are in the same ratio as any other 
 two. 
 
 The ratio of two corresponding surfaces is the square of 
 the ratio of two corresponding lines. 
 
 The ratio of two corresponding volumes is the cube of the 
 ratio of two corresponding lines. 
 
 Conversely, 
 
 The ratio of two corresponding lines is the square root of 
 the ratio of two corresponding surfaces, and the cube root 
 of the ratio of two corresponding volumes. 
 
 Ex. 162. 
 
 1. The volume of a rectangular solid is 1728 cu. in. The 
 
 volume of a similar solid is 3375 cu. in. Find the 
 ratio of two corresponding edges. 
 
 2. The surface of a solid is 600 sq. in. What is the sur- 
 
 face of a similar solid whose edges are twice as great ? 
 
 3. If the volumes of two similar solids be 100 cu. in. and 
 
 1000 cu. in. respectively, find the ratio of their 
 heights to the nearest thousandth of an inch. 
 
 4. If two hills have the same shape, and one is 2700 ft. 
 
 high, while the other is 3600 ft. high, find the ratio 
 of their surfaces, and also the ratio of their volumes. 
 
 5. A bushel measure and a peck measure are of the same 
 
 shape. Find the ratio of their heights. 
 
 6. The surfaces of two hills having the same shape are 
 
 as 25 : 16. Find the ratio of their heights. 
 
 7. Of two similar solids, the volume of the larger is 1 
 
 of that of the smaller. Find the ratio of their 
 heights ; find also the ratio of their bases. 
 
 8. The equatorial diameter of the earth is 7926 mi. Find 
 
 that of Venus whose volume is 0.953 of the volume 
 of the earth. 
 
CHAPTER XIV. 
 
 MENSURATION. 
 
 (PRACTICAL RULES.) 
 
 284. A surface has two dimensions : length and breadth, 
 
 285. A solid has three dimensions : length, breadth, and 
 thickness. 
 
 286. The area of a surface is the number of units of sur- 
 face which it contains, the unit of surface being a square 
 which has a linear unit for each of its dimensions. 
 
 287. The volume of a solid is the number of units of 
 volume which it contains, the unit of volume being a 
 cube which has a linear unit for each of its three dimen- 
 sions. 
 
 288. In writing the dimensions of surfaces and solids, the 
 sign X is used for the word by, an accent (') for the word 
 feet, and two accents (") for the word inches. Thus, the 
 dimensions of the floor of a room, 15 feet 6 inches long, 
 13 feet 8 inches wide, are denoted by 15' 6" X 13 f 8". The 
 dimensions of a brick, 8 inches long, 4 inches wide, 2-J 
 inches thick, are denoted by 8" X 4" X 2". 
 
 289. Rectangle. The area of a rectangle equals the 
 product of its length arid breadth. (See page 165.) 
 
 290. The perimeter of a rectangle or of any other sur- 
 face figure is the sum of the lengths of the lines which 
 bound it. 
 
MENSURATION. 279 
 
 Ex. 163. 
 
 1. The floor of a room is a rectangle 15 f 6" X 18'. Find 
 
 its perimeter and its area. 
 
 2. The ceiling of a room is a rectangle 16 f X 20'. Find 
 
 its perimeter and its area. 
 
 3. A rectangular field is 60 rods X 80 rods. Find its 
 
 area in acres, and the cost of fencing it at $1.50 a rod. 
 
 4. How many boards 12 ft. long will be required to 
 
 inclose a square field 48 rds. on a side with a fence 
 4 boards high ? How many acres are there in the 
 field? 
 
 291. Carpeting. Carpeting is sold by the yard in length. 
 The common widths are a yard, and three-quarters of a 
 yard. It will be remembered that in determining the num- 
 ber of yards of carpet for a room, we first decide whether 
 the strips shall run lengthwise or across the room, and then 
 find the number of strips needed. The number of yards in 
 a strip, including the allowance for waste in matching the 
 pattern, multiplied by the number of strips will give the 
 required number of yards. (See page 169.) 
 
 5. How many yards of carpeting I of a yard wide will be 
 
 required for a floor 20 f X 17' 6", if the strips run 
 lengthwise, and if there is a waste of 9 in. a strip 
 in matching the pattern ? 
 
 6. How many yards of carpeting 1 yd. wide will be 
 
 required for a room 18' 4" X 17' 8", if the strips run 
 lengthwise of the room, and if there is a waste of 
 8 in. a strip in matching the pattern? Find the 
 cost of carpeting the room if the carpet is worth 
 85 cents a yard, and 10 cents a yard is paid for 
 making and laying. 
 
280 MENSURATION. 
 
 7. Find the cost of the carpet for a room 19' 8" X 17' 10", 
 if the carpet is J of a yard wide and costs $1.75 a 
 yard, the strips running across the room, and 9 in. a 
 strip being wasted in matching the pattern. 
 
 292. Plastering. The unit for measuring painting, plas- 
 tering, and paving is the square yard. The practice in 
 painting and plastering is to find the total area within the 
 bounding lines of the work, to deduct from this amount 
 half the area of all doors, windows, and other openings, 
 and to take as the net area the nearest whole number of 
 square yards in the remainder. 
 
 Ex. A rectangular room is 15' X 13' 4" X 9 f . The base- 
 board is 1 foot high ; there is a door 7' 4" X 4', and 
 two windows 6' X 4 f each. Find the cost of plaster- 
 ing the walls and ceiling at 18 cents a square yard. 
 
 Perimeter of room = 2 X 15' + 2 X 13' 4" = 56' 4". 
 Height of room above baseboard = 9' 1' = 8'. 
 Total area of walls = 8 X 56J = 450$ sq. ft. 
 Area of ceiling = 15 X 13 J = 200 sq. ft. 
 
 Area of walls and ceiling 
 Height of door above baseboard 
 Area of door above baseboard 
 Area of 2 windows = 2 X 6' X 4' 
 
 Area of door and windows 
 
 Half the area of door and windows = i 
 
 Area allowed is 
 
 [ sq. ft. - 36 sq. ft. = 614 sq. ft. = AJA = 68 sq. yds. 
 
 68 X 18 cents = $ 12.24. Am. 
 
 Find the cost of plastering the walls and ceiling of a 
 room 17' 4" X 15 f 8" X 10' 4", at 20 cents per square 
 yard, if 10 sq yds. are deducted for doors, window?, 
 and baseboard. 
 
MENSURATION 281 
 
 9. Find the cost of whitening the walls and ceiling of a 
 room 16' 6" X 15' 6" X 9' 6", at five cents per square 
 yard, deducting 12 sq. yds. for doors, windows, and 
 baseboard. 
 
 10. Find the cost of plastering a room 18' X 15' X 10 f , at 
 
 30 cents per square yard, if the room contains one 
 door 7' 6" X 4', three windows each 6' X 4', and a 
 baseboard one foot high around the room. 
 
 293. Wall Paper. Wall paper is 18 in. wide and is sold 
 in single rolls 8 yds. long, or in double rolls 16 yds. long. 
 In estimating the number of rolls of paper required for a 
 room of ordinary height, find the number of feet in the 
 perimeter of the room, leaving out the widths of the doors 
 and windows, and allow a double roll or two single rolls 
 for every 7 ft. 
 
 Ex. How many double rolls of paper will be required for 
 a room of ordinary height, 18' X 16', with one door, 
 and four windows, each 4 ft. wide ? 
 
 Perimeter of room = 2 x 18' + 2 x 16' = 68' 
 Width of door and windows = 5 x 4' = 20' 
 
 Deducting door and windows = 48' 
 
 ^- = 7. 7 double rolls. Ans. 
 
 11. How many double rolls of paper will be required for 
 
 a room of ordinary height, 18' 4" X 16' 6", with two 
 doors and three windows, each 4 ft. wide? 
 
 12. Find the cost of paper at 25 cents a single roll, and 
 
 bordering at 8 cents a yard, for a room of ordinary 
 height, 17' 9" X 17' 3", allowing for one door and 
 four windows, each 4' 2". (No allowance for doors 
 and windows is made for the bordering.) 
 
282 MENSURATION. 
 
 13. Find the cost of paper at 50 cents a single roll for a 
 
 room of ordinary height, 20' 8" X 17' 6", with two 
 doors and three windows, each 4' 2" wide. 
 
 294. Laths. Laths are put up in bundles, 100 pieces, 
 each 4 ft. long, and a bundle is estimated to cover 5 sq. 
 yds. In estimating the number of bundles of laths, deduct 
 the whole area of all openings. 
 
 Ex. How many bundles of laths will be required for the 
 ceiling of a room 36 ft. square ? 
 
 36 ft. = 12 yds. 
 
 Hence ceiling contains 12x12= 144 sq. yds. 
 
 ^ = 29. 29 bundles. Ans. 
 
 14. How many bundles of laths will be required for the 
 
 ceiling and walls of a room 26 ft. square, 14 ft. 
 high, allowing 20 sq. yds for doors, windows, and 
 baseboard ? 
 
 15. How many bundles of laths are required for the ceiling 
 
 and walls of a room 28' X 32' and 16' high, allowing 
 for three windows 8 r X 3' 6" each, and two doors 
 8' X 4 f 2" each, and a baseboard 1 ft. high ? 
 
 295. Clapboards. Clapboards are 4 ft. long and are laid 
 3^- or 4 in. to the weather. 
 
 Ex. Find the number of clapboards required for the front 
 of a house 42 ft. long and 22 ft. high, allowing 
 100 sq. ft. for doors and windows, and adding 10 % 
 for waste. 
 
 3J in. = 5i ft. = & ft. 
 
 1 L 
 
 4 X -ff = f f = 1J sq. ft. for each clapboard. 
 
 42 x 22 = 924 sq. ft. 
 
 924 sq. ft. - 100 sq. ft. = 824 sq. ft. 
 
MENSURATION. 283 
 
 = f of 824 = 706. 
 
 10% of 706 = 71. 
 
 706 + 71 = 777. Ans. 
 
 16. How many clapboards will be required for the front 
 
 of a house 40 ft. long and 20 ft. high, allowing 
 96 sq. ft. for doors and windows, and adding 10 % 
 for waste ? 
 
 296. Boofing and Flooring. The unit of measure for 
 roofing and flooring is a square containing 100 sq. ft. 
 Shingles are 16 in. long, and are estimated to average 
 4 in. wide, so that a shingle laid 4-J- in. to the weather 
 would cover 18 sq. in., and 8 shingles would be required 
 for 1 sq. ft. At this rate 800 shingles would cover a 
 square, but to allow for waste it is usual to reckon 1000 
 shingles to the square. It is found, however, in practice, 
 that 1000 shingles of the best quality, laid 4 in. to the 
 weather, will cover about 120 sq. ft. 
 
 17. Allowing 1000 shingles for 120 sq. ft., how many 
 
 thousand would be required to cover the pitched 
 roof of a house 44 ft. long, if the width of each side 
 of the roof is 24 ft. ? 
 
 18. Allowing 1000 shingles for 110 sq. ft., how many thou- 
 
 sand would be required to cover the pitched roof 
 of a building 54 ft. long, if the width of each side 
 of the roof is 28 ft. ? 
 
 19. How many slates at 3 to the square foot will be re- 
 
 quired to cover 28 squares of roof? 
 
 20. The floor of a gymnasium is 100' X 60'. Find the 
 
 cost of birch for the floor at $40 a thousand, adding 
 20% for waste. 
 
284 
 
 MENSURATION. 
 
 297. Triangles. 
 
 
 D 
 
 A triangle is a plane figure bounded by 
 three straight lines. Thus, the figure 
 ABC is a triangle. The side AB 
 is called the base ; the corner C oppo- 
 site the base, the vertex ; and the per- 
 
 . pendicular CD, drawn from Cto AB, 
 the altitude. 
 
 298. To find the area of a triangle. 
 Take one-half the product of the base by the altitude. 
 NOTE. The area of a right triangle equals one-half the product of 
 the base and the perpendicular. 
 
 If the lengths of the three sides of a triangle are given, 
 the area is found as follows : 
 
 From the half-sum of the sides subtract each side sepa- 
 rately. Find the continued product of the half-sum and 
 the three remainders. The square root of the product is 
 equal to the area of the triangle. 
 
 Ex. Find the area of a triangle having a base 16 ft. and 
 altitude 10 ft. 
 
 Area 
 
 - = 80. 80 sq. ft. Am. 
 
 Ex. Find the area of a triangle if the sides are 6, 8, and 
 12 ft. 
 
 Area = Vl3 x 7 X 5 x 1 = 21.3 21.3 sq. ft. Ans. 
 Find the area of a triangle, having given : 
 
 21. Base 30' 6", altitude 12' 6". 
 
 22. Base 148 rds., altitude 60 rds. 
 
 23. Base 10 chains 40 links, altitude 8 chains 50 links. 
 
 24. Sides 60 ft., 80 ft,, 90 ft. 
 
 25. Sides 100 ft., 110 ft., 120 ft. 
 
MENSURATION. 285 
 
 299. The area of any surface figure bounded by straight 
 lines can be found by dividing the figure into triangles, com- 
 puting the areas of these triangles, and talcing the sum of 
 these areas. 
 
 300. To find the area of a circle. 
 
 Multiply the square of the radius by 3.1416. (See page 
 168.) 
 
 Find the area of a circle, having given : 
 
 26. Radius 14 ft. 28. Diameter 32 yds. 
 
 27. Radius 7 yds. 29. Diameter 40 ft. 
 
 30. Diameter 100 rds. 
 
 301. To find the volume of a rectangular solid. 
 
 Multiply the area of the base by the altitude ; that is, take 
 the product of its three dimensions. (See page 174.) 
 
 Find the volume of: 
 
 31. A cube whose edge is 3 in. 
 
 32. A cube whose edge is 14 in. 
 
 33. A cube whose edge is 2 ft. 
 
 34. A rectangular solid 10" X 8" X 6". 
 
 35. A rectangular solid 6' x 5' X 4'. 
 
 36. A rectangular solid 4' 8" X 3' 10" X 3' 6". 
 
 37. If a cellar which measures 32 f X 28' is flooded to a 
 
 depth of 4 in., what is the weight of the water, 
 allowing 1 cu. ft. of water to weigh 1000 oz.? 
 
 302. To find the number of gallons that a cistern of given 
 dimensions will hold, 
 
 Find the number of cubic inches in the cistern, and divide 
 this number by 231. 
 
286 MENSURATION. 
 
 Ex. If a rectangular cistern is 4' X 3 f 4" X 3', how many 
 gallons of water will it hold ? 
 
 4 ft. = 48 in. 3 ft. 4 in. = 40 in. 3 ft. = 36 in. 
 Since a gallon is 231 cu. in., the number of gallons is 
 48 x 40 x 36 _ 9Qq 9 
 
 231 299.2 gals. Ans. 
 
 303. Since a cubic foot contains -y^ gals. = 7.48 gals., 
 we may find the number of gallons of water a cistern will 
 hold as follows : 
 
 Express the dimensions in feet, and multiply the continued 
 product of these dimensions by 7J, and take from the result 
 io/1% of it. 
 
 304. To find the number of barrels of water a cistern will hold. 
 Divide the number of cubic feet the cistern contains by 
 
 4.21. 
 
 Ex. Find the number of gallons in a cistern 6 ft. square 
 and 4 ft. deep. 
 
 6x6x4= 144 
 
 7} 
 
 1080 
 
 Jofl% = 5.4 
 
 1074.6 gals. Am. 
 
 Ex. Find the number of gallons that a round cistern 7 ft 
 
 in diameter and 7 ft. deep will hold. 
 Area of base = 3.1416 X 3.5 X 3.5 = 38.4846 sq. ft. 
 
 7 
 
 269.3922 cu. ft. 
 7.5 
 
 2020.44150 
 ofl%= 10.10 
 
 2010. gals. Am. 
 
MENSURATION. 287 
 
 38. How many gallons of water will a cistern hold that is 
 
 5J ft. long, 3f ft. wide, and 4 ft. deep? 
 
 39. How many barrels of water will a cistern hold that is 
 
 13 ft. long, 8 ft. wide, and 7 ft. deep ? 
 
 40. How many barrels of water will a cistern hold that is 
 
 12 ft. long, 9 ft. wide, and 6 ft. deep ? 
 
 41. Find the number of gallons that a round cistern will 
 
 hold, 8 ft. in diameter and 7 ft. deep. 
 
 42. Find the number of barrels in a round cistern 21 ft. in 
 
 diameter and 10 ft. deep. 
 
 305. To find the number of bushels of grain in a bin. 
 
 A bushel = 2150.42 cu. in., and 0.8 of this equals 
 1720.336 cu. in. If we add to 1720 i of 1% of 1720, 
 rejecting the decimals, we obtain 1728 cu. in. 
 
 Hence, take 0.8 of the number of cubic feet in the bin, and 
 add to the result i ofl% of it. 
 
 Ex. Find the number of bushels in a bin 12 ft. long, 8 ft. 
 wide, and 5 ft. high. 
 
 12x8x5= 480 
 0.8 
 
 384.0 
 }ofl% - 1.92 
 
 385.92 bu. Ans. 
 
 43. Find the number of bushels in a bin 20 ft. long, 6 ft. 
 
 wide, and 4 ft. high. 
 
 44. Find the number of bushels in a bin 8} ft. long, 5J ft. 
 
 wide, and 4 ft. high. 
 
 45. Find the number of bushels in a bin 8 ft. long, 6f ft. 
 
 wide, and 4} ft. high. 
 
 306. To express in cubic feet a given number of bushels. 
 
 To the number of bushels add } of the number, and sub- 
 tract from the sum %of\//) of it. 
 
288 MENSURATION. 
 
 Ex. To find the number of cubic feet required for 1200 bu. 
 
 4)1200 
 300 
 1500 
 Jofl% = 7.5 
 
 1492.5 cu. ft. Ans. 
 
 46. How many cubic feet in a bin that will hold 400 bu. ? 
 
 47. How many cubic feet in a bin that will hold 372 bu. ? 
 
 48. How many cubic feet in a bin that will hold 1326 bu. ? 
 
 307, To find the number of bushels in a load of charcoal. 
 Multiply the continued product of the length, width, and 
 
 height, expressed infect, by 0.8 and add to the result 1 of 
 1% of it. 
 
 49. How many bushels of charcoal in a load 8 ft. long, 4 
 
 ft. wide, and 6 ft. high ? 
 
 50. Find the number of bushels in a load of charcoal that 
 
 is 8 ft. long, 41 ft. wide, and 6 ft. high. 
 
 308, To measure wood. 
 
 Find the product of the length, width, and height, ex- 
 pressed in feet, and divide this product by 8x4x4. The 
 result is the number of cords. 
 
 Ex. Find the number of cords in a pile of wood 24 ft. long, 
 4 ft. wide, and 6 ft. high. 
 
 ^ 
 
 2 
 
 51. What part of a cord does a load of wood contain which 
 
 is 8 ft. long, 4 ft. wide, 3J ft. high? 
 
 52. What part of a cord does a load of wood contain which 
 
 is 8 ft. long, 3 ft. 8 in. high, if the average length 
 of the sticks is only 3 ft. 8 in. ? 
 
MENSURATION. 289 
 
 53. Find the number of cords in a pile of wood 120 ft. 
 
 long, 4 ft. wide, and 6 ft. high. 
 
 54. How much should be paid for a pile of 4-foot wood, 
 
 100 ft. long, and averaging 5 ft. high, at $5 a cord? 
 
 309. To measure coal. 
 
 A long ton of anthracite coal measures about 37 cu.ft. 
 A long ton of soft coal measures about 48 cu. ft. A bushel 
 of hard coal weighs about 75 Ibs. A bushel of soft coal 
 weighs about 60 Ibs. 
 
 55. How many long tons of hard coal will a rectangular 
 
 bin hold 9 ft. long, 6 ft. 6 in. wide, and 6 ft. high ? 
 
 56. How many long tons of hard coal can be put into a 
 
 rectangular bin 8 ft. long, 7 ft. wide, and 6 ft. high ? 
 
 57. How many long tons of soft coal can be put into a rec- 
 
 tangular bin 12 ft. long, 9 ft. wide, and 7 ft. high? 
 
 310. To measure sand, gravel, and earth. 
 A cubic yard of earth is called a load. 
 
 58. How many loads are there in a rectangular embank- 
 
 ment 200 ft. long, 15 ft. wide, and 10 ft, high ? 
 
 59. How many loads in an embankment 150 ft. long, 20 
 
 ft. wide, and 5 ft. high ? 
 
 311. To measure brickwork. 
 
 Brickwork is estimated by the thousand, reckoning 22 
 bricks laid in mortar to the cubic foot. 
 
 60. How many bricks will be required to build a wall 84 
 
 ft. long, 32 ft. high, and 1 ft. thick? 
 
 61. How many bricks will be required for the walls of a 
 
 house 42 ft. long, 32 ft. wide, and 21 ft. high, if the 
 
290 MENSURATION. 
 
 walls are 1 ft. thick, and there are deducted 2 doors 
 7' 6 ff X 4 r each, and 16 windows 5' X 4' each ? 
 
 NOTE. In finding the perimeter of the building, measure the exte- 
 rior. 
 
 312. To measure stone masonry. 
 
 Stone masonry is reckoned by the cubic foot, or by the 
 perch of 25 cu. ft. 
 
 62. How many cubic feet of stone masonry in the foun- 
 
 dation of a house 40 r X 30 r , if the foundation is to 
 be 4 ft, high and 2 ft. thick ? 
 
 63. How many perches of stone are required for the foun- 
 
 dation of a building 100' X 60', if the foundation is 
 6 ft. high and 2 ft. thick ? 
 
 313. To measure boards and dimension lumber. 
 
 Boards one inch or less in thickness are sold by the 
 square foot. Boards more than one inch in thickness and 
 all squared lumber are sold by the number of square feet 
 of boards one inch in thickness to which they are equivalent. 
 
 Thus, a board 12 ft. long, 1 ft. wide, and 1 in. thick, 
 contains 12 ft. board measure. If only i or I or J of an 
 inch thick, it still contains 12 ft. board measure; but if li 
 in. thick, it contains It X 12 = 15 ft. board measure. 
 Hence, 
 
 Express the length and width in feet, and the thickness in 
 inches. The product of these three numbers will be the 
 number of feet board measure. 
 
 In practice, the width of a board, unless sawed to order, is reck- 
 oned only to the next smaller half-inch. Thus, a width of llf inches 
 is reckoned 11 inches ; of 13f or 13J inches, is reckoned 13J inches. 
 
 Ex. How many feet in a 2-inch plank, 18 ft. long and 
 
 14 in. wide? 
 14 in. = 1J ft. 2 X 1J X 18 = 42 ft. board measure. Am. 
 
MENSUPwATION. 291 
 
 64. How many feet board measure in 8 planks, 4 in. thick, 
 
 18 ft. long and 16 in. wide? 
 
 65. How many feet board measure in a stick of timber 
 
 1 ft. square and 20 ft. long? 
 
 66. How many feet board measure in 40 joists 10" X 2" 
 
 and 12ft. long? 
 
 314, To measure round logs. Round logs are sold by the 
 amount of square lumber that can be cut from them, ac- 
 cording to calipers now in use. When logs do not exceed 
 16 ft. in length, the length and the diameter of the small 
 end are taken, and a table stamped upon the calipers gives 
 the number of feet board measure. This table may be 
 calculated as follows : 
 
 Express the diameter in inches, subtract twice the diameter 
 
 from the square of the diameter, and f ^ of the remainder 
 
 will be the number of feet board measure in a log 10 ft. long. 
 
 The formula is %$(d 2 2c?), in which d stands for the 
 
 diameter of the log in inches. 
 
 Ex. Find the number of feet board measure in a log 16 ft. 
 long and 20 inches in diameter. 
 20 2 - 2 x 20 = 400 - 40 = 360. 
 ft of 360 = 189. 
 jf of 189 = 302.4 ft. board measure. Ans. 
 
 By this rule find the number of feet board measure in : 
 
 67. A log 12 ft. long and 16 in. in diameter. 
 
 68. A log 13 ft. long and 12 in. in diameter. 
 
 69. A log 14 ft. long and 20 in. in diameter. 
 
 70. A log 15 ft. long and 15 in. in diameter. 
 
 315. Oak and other heavy timber. 
 
 Large heavy timber of hard wood is generally sold by the 
 ton, signifying 50 cu.ft. or 600 /. board measure. 
 
292 MENSURATION. 
 
 316. To find the contents of a cask. 
 
 Subtract the diameter of one of the heads from the bung 
 diameter expressed in inches, and multiply the difference by 
 0.65 ; to the product add the head diameter, and this will 
 give the mean diameter. 
 
 Square the mean diameter and multiply it by the length 
 in inches. Divide this product by 294. The quotient is the 
 number of gallons the cask will hold. 
 
 71. Find the number of gallons contained in a cask of 
 
 which the bung diameter is 24 in., head diameter 
 20 in., and the length 36 in. 
 
 72. Find the number of gallons contained in a cask of 
 
 which the bung diameter is 30 in., head diameter 
 26 in., and the length 38 in. 
 
 317. To find the volume of an irregular body. 
 
 Immerse the body in a vessel full of water. JRemove the 
 body and calculate the amount of water displaced. 
 
 318. To find the surface of a sphere. 
 Multiply the square of the diameter by 3.1416. 
 
 73. How many square inches on the surface of a ball 4 in. 
 
 in diameter? 
 
 74. How many square inches on the surface of a globe 18 
 
 in. in diameter? 
 
 319. To find the volume of a sphere. 
 
 Multiply the cube of the diameter by 0.5236 (that is, of 
 3.1416). 
 75 Find the volume of a globe 2 ft. in diameter. 
 
CHAPTER XV. 
 
 Ex. 164. 
 MISCELLANEOUS PROBLEMS. 
 
 1 . Fifteen men and eight boys together earn $ 342 a 
 
 week. If a boy's pay is half a man's pay, what are 
 the daily wages of a man, and also of a boy ? 
 
 2. A man divides $1622.50 among four persons so that 
 
 the first has $40 more than the second, the second 
 $60 more than the third, and the third $87.50 more 
 than the fourth. Find the part of the fourth. 
 
 3. A family of six persons makes $8.75 a day, and works 
 
 304 days in the year. At the end of the year each 
 member of the family puts $80 in a savings bank. 
 Find the daily expense of the family. 
 
 4. A man bought 5.5 yds. of cloth for $35. In having 
 
 a suit made from it, he found that he lacked 1.75 
 yds., which he procured at the price per yard of 
 his first purchase. What is the cost of the suit if 
 the trimmings cost $6.50 and the making $15? 
 
 5. A man has 76.25 yds. of linen, worth 44 cts. a yard, 
 
 made into shirts. It takes 3.05 yds. for a shirt, and 
 the price for making is 50 cts. a shirt. Find the 
 cost of a shirt, and the number he has made. 
 
 6. A man's expenses from the first of January to the end 
 
 of October 17 are $1845.50. How much must he 
 diminish his daily expense in order that the total 
 expense for the year shall not exceed $2200? 
 
294 MISCELLANEOUS PROBLEMS. 
 
 7. A quart contains 1600 beans of average size, and a 
 
 field is planted with 22 rows of 800 hills each, with 
 6 beans in a hill. The increase is tenfold. What 
 is the value of the crop at $3 a bushel? (There 
 are 32 quarts in a bushel.) 
 
 8. For making 25 gallons of ordinary beer 60 pounds of 
 
 barley and 0.5 of a pound of hops are needed. If 
 the barley costs $1.50 for 60 pounds, and the hops 
 cost 18 cents a pound, what is the profit to the 
 brewer on a cask of 42 gallons if he sells it for $5 
 and reckons his labor $1.50? 
 
 9. A person receives his income quarterly. The first 
 
 quarter he receives $533.25, the second $1535.20, 
 the third $856.44, the fourth $ 725.19. His expenses 
 for these quarters are respectively $686.60, $734.25, 
 $589.15, $849.65. How much does he save for the 
 year? 
 
 10. A hen lays on an average 120 eggs a year worth 24 
 
 cents a dozen. She eats a quart of barley every 5 
 days. The barley is worth 56 cents a bushel (32 
 quarts). What is the annual profit from this hen ? 
 
 11. A square garden measuring on each side 40.50 yards 
 
 is enclosed by three lines of galvanized iron wire. 
 Eight yards of this wire weigh a pound, and it is 
 worth 7.5 cents per pound. What is the cost of the 
 wire ? 
 
 12. A family composed of five persons consumes daily one 
 
 pound of stale bread for each person, or 1.15 pounds 
 of fresh bread. If bread is worth 5 cents a pound, 
 find the annual saving which this family will make 
 if it eats stale bread altogether. 
 
MISCELLANEOUS PROBLEMS. 295 
 
 13. Tfc is estimated that in France 240,000 women and 
 
 girls are employed in making lace. The annual 
 production has a value of $13,000,000, and the 
 value of the raw material is 0.27 of the value of 
 the lace. Find the average daily wages of these 
 women and girls, supposing that each works 240 
 days in the year. 
 
 14. The salt water which is obtained from the bottom of a 
 
 mine of rock salt contains 0.09 of its weight of pure 
 salt. What weight of salt water is it necessary to 
 evaporate in order to obtain 4734 pounds of salt T 
 
 15. The weight of ashes from the burning of oak wood is 
 
 0.03 of the weight of the wood, and the weight of 
 carbonate of potash contained in the ashes is 0.065 
 of the weight of the ashes. Find the weight of 
 carbonate of potash from 1170 pounds of wood. 
 
 16. The weight of sugar from the sugar beet is nearly 0.06 
 
 of the weight of the beet. If an acre produces 30,000 
 pounds of beets that are sold at the rate of $2 a 
 thousand pounds, how many acres of land is it 
 necessary to sow to furnish beets to a sugar factory 
 which produces 150,000 pounds of sugar a year, and 
 what will be the value of the crop obtained ? 
 
 17. If a workman has taken every day for the last 12 
 
 years two glasses of beer at 5 cents a glass, how 
 much could he have saved if he had not indulged 
 this habit, reckoning 365 days each year? 
 
 18. A woman has three children. She pays for each $15 
 
 a year for having their clothes made, $1.50 a month 
 for mending, and $0.35 a week for washing. How 
 much could she save in a year if she knew how to 
 wash, make clothes, and mend ? 
 
296 MISCELLANEOUS PROBLEMS. 
 
 19. A sheep raiser shears his sheep at an expense of 11 cts. 
 
 a head. The sheep average 8 Ibs. of wool which 
 he sells for 23 cts. a pound. He finds that his net 
 profit after paying for the shearing is $1297.50. 
 How many sheep has he ? 
 
 COMMON FRACTIONS. 
 
 20. Find the prime factors of 41,580. 
 
 21. Find the G.C.M. of 144, 126, 108. 
 
 22. Find the L.C.M. of 18, 90, 60, 24. 
 
 23. Find the L.C.M. of 14, 35, 343. 
 
 24. At 16 cts. a yard, what will 3 yds. of cloth cost? 
 
 25. A man has 376|- quarts of berries, which he wishes to 
 
 put into boxes holding 2J- qts. each. How many 
 boxes will be required, and what part of a box will 
 be left over? 
 
 26. If a man earns $2f a day, how many days will it take 
 
 him to earn $100? 
 
 27. A lady has 37-J- qts. of berries to can. If each can 
 
 holds 2f qts., how many cans of berries will she 
 have, and what part of another can will there be 
 over? 
 
 28. If a man walks 4|- miles an hour, how many hours 
 
 will it take him to walk 40f miles ? 
 
 29. Some boys wanted a long rope to use on the ice. They 
 
 made the rope by taking off their sled-ropes and 
 tying them together. The first sled-rope was 2f 
 yds. long, the second 3^- yds., the third 2 yds., the 
 fourth 5f yds., and the fifth 3^ yds. If the whole 
 length was shortened 1-|- yds. by the knots, from 
 
MISCELLANEOUS PROBLEMS. 297 
 
 tying the sled-ropes together, how long was the 
 rope? 
 
 30. A lady bought 3|- yds. of cotton cloth, 4J- yds. of 
 
 calico, 16| yds. of flannel, and 12-J- yds. of ging- 
 ham. How many yards did she buy in all ? 
 
 31. A boy went to a store with $5.75 in his purse. He 
 
 bought 3^ Ibs. of butter at 28 cts. a pound, 13^ Ibs. 
 of sugar at 11 cts. a pound, and 1^- Ibs. of coffee at 
 35 cts. a pound. How much money did he have 
 left? 
 
 32. Four boys went fishing, and caught 40 trout ; the first 
 
 caught -- of the whole, the second -J-, and the third 
 J. How many did the fourth boy catch ? 
 
 33. George has his choice to be one of 3 boys to receive 8 
 
 oranges, or one of 4 boys to receive 11 oranges. 
 Which shall he choose? 
 
 34. Five girls pick blueberries together ; the first picks 
 
 7| qts., the second 5^ qts., the third 12| qts., the 
 fourth 8% qts., and the fifth 3 qts. How much 
 will they all together get for their berries, at 12^ 
 cts. a quart? 
 
 35. A farmer puts the following lots of apples into 6 bins: 
 
 namely, 6f bu., 18 bu., 25 bu., 19f bu, 143| bu., 
 976^ bu., 25 bu. How many bushels will there be 
 for each bin ? 
 
 COMPOUND QUANTITIES. 
 
 36. How many rods are there in 4379 ft. ? 
 
 37. Reduce 9,627,834 ft. to yards, rods, etc. 
 
 38. Reduce 96,284 sq. in. to square feet. 
 
298 MISCELLANEOUS PROBLEMS. 
 
 39. Reduce 15 sq. rds. 3 sq. yds. 18 sq. ft. 3 sq. in. to 
 
 square inches. 
 
 40. What will 1000 sq. ft. of land cost at $67 an acre? 
 
 41. What will 20 sq. yds. of land cost at 75 cts. a square 
 
 foot? 
 
 42. How much less will 15 acres of land cost, at $16 an 
 
 acre, than 96,342.42 sq. ft. at 5 cts. a foot? 
 
 43. How many acres in a rectangular piece of land 963$ 
 
 ft. long and 3840 ft. wide? 
 
 44. A pile of four-foot wood is 4 ft. high and 75 ft. long. 
 
 How many cords of wood are there in the pile ? 
 
 45. In a woodshed there is a pile of wood 12 ft. long and 
 
 10 ft. high. If the sticks average a foot in length, 
 what part of a cord is there in the pile? 
 
 46. What will 7 bu. 3 pks. of blueberries bring at 9 cts. a 
 
 quart ? 
 
 47. How many gallons of milk, at 8 cts. a quart, can be 
 
 bought for $7.37? 
 
 48. How many quarts of water will a tin box hold that is 
 
 13 in. long, 6 in. wide, and 7 in. deep? 
 
 49. The total net weight of several loads of hay is 63,782 
 
 Ibs. How many tons in all the loads of hay ? 
 
 50. If an ounce of candy is worth 5 cts., what will 5 Ibs. 
 
 cost at the same rate? 
 
 51. Reduce 9 dys. 5 hrs. 16 min. to seconds. 
 
 52. Reduce 948,741 min. to higher denominations. 
 
 53. How many weeks between Jan. 1 and Nov. 1? 
 
 54. A boy has 10 mi. to go. After he has gone 6 mi. 48 
 
 rds. 12 ft., how much of his journey has he still to 
 go? 
 
MISCELLANEOUS PROBLEMS. 299 
 
 55. A lady bought 4 remnants of cloth; the first con- 
 
 tained 9J yds., the second 4 yds. 11 in., the third 
 6J yds., and the fourth 5-} yds. How much cloth 
 did she buy in all ? 
 
 56. A certain basket holds 1 bu. 3 pks. 7 qts. A farmer 
 
 raises enough of yellow-eyed beans to fill this bas- 
 ket 7 times. How many bushels does he raise ? 
 
 57. A farmer cuts 26 loads of hay, which average 1 t. 
 
 436 Ibs. How many tons does he cut in all ? 
 
 58. What is ^ of 9 mi. 5 rds. 13 ft. ? 
 
 59. Three men in company buy 175 t. 19 cwt. 36 Ibs. of 
 
 hay. What is each man's share? 
 
 GO. Seven boys together pick 4 bu. 3 pks. 7 qts. of berries. 
 What is each boy's share ? 
 
 61. Bought 9 Ibs. of sugar at 13 cts. a pound, 18 yds. of 
 
 cloth at 33 cts. a yard, 4 doz. eggs at 29 cts. a 
 dozen; and 5 Ibs. of butter at 32 cts. a pound. 
 What change should I receive from a ten-dollar bill 
 given in payment? 
 
 62. How many quarts of berries, at 12 cts. a quart, will 
 
 it take to pay for 8 yds. of cloth, at 16 J cts. a yard? 
 
 63. A basket of peaches is half a bushel ; how many 
 
 bushels are there in 250 car-loads of 500 baskets 
 each ? 
 
 64. A fast railway train in England went 186 mi. 240 rds. 
 
 in 3 hrs. What was the rate per hour ? 
 
 65. Tf a man could proceed to the moon at the same rate 
 
 per hour as the train went in example 64, how 
 many hours would it take him, reckoning the dis- 
 tance 239,000 miles? 
 
300 MISCELLANEOUS PROBLEMS. 
 
 66. In one bin there are 23 bu. 2.48 pks. of wheat, and in 
 
 another 141 bu. 2 pks. If J of the wheat in the first 
 bin is put into the second, how much wheat will 
 there be in the second bin ? 
 
 67. A load of four-foot wood is 3} ft. high and 7 ft. long. 
 
 What is it worth at the rate of $6.40 a cord? 
 
 68. In one field there are 17J A., in a second there are 
 
 49 sq. rds., and a third field is 1740 ft. long and 
 927 ft. wide. What is the area of the three fields 
 together ? 
 
 69. A bin contains 164 bu. 3 pks. 2 qts. of oats. How 
 
 long will these oats last if there are taken out 3 
 qts. of oats three times a day ? 
 
 70. From a barrel containing 27f gals, of oil, 3 qts. a day 
 
 were taken out for 3 weeks. How many gallons 
 were left in the barrel at the end of that time ? 
 
 71. If from a barrel of oil holding 27 gals. 2 qts. 1 pt. 
 
 there is drawn out a can full, holding 1 gal. 2 qts. 
 1 pt., every day, how many days will the oil last? 
 
 72. Reduce of -f- of {^ of a mile to rods. 
 
 73. Reduce $ of -j- of 3 in. to the fraction of a yard. 
 
 SPECIAL PROBLEMS. 
 
 If a man can do a piece of work in 5 dys., in one day he can do 
 J of the work ; and if another man can do the same work in 4 dys., 
 in one day he can do J of it. 
 
 Therefore, both men together can do + J = ^ in one day. 
 
 Hence they will do $ in ^ of a day, and therefore the whole 
 work in * days, that is, in 2f days. 
 
 74. If A can do a piece of work in 4 dys., B in 5 dys., 
 
 and in 7 dys., in how many days will they do it, 
 all working together ? 
 
MISCELLANEOUS PROBLEMS. 301 
 
 75. A can do a piece of work in 2 hrs., B in 2-J- hrs., and 
 
 C in 3^ hrs. How much of the work can they do 
 in 20 min., all working together? 
 
 76. If A and B can do a piece of work in 18 dys., A and 
 
 C in 12 dys., and B and C in 9 dys., find the num- 
 ber of days that it will take them, all working 
 together. 
 
 77. A can do a piece of work in 6 dys., B in 8 dys., and 
 
 G in 10 dys. How much of it can they do in 2 dys. 
 together ? 
 
 78. A cistern can be filled by means of a water-pipe in 30 
 
 min., and can be emptied by a waste-pipe in 20 min. 
 If the cistern is full, and both pipes are open, in 
 what time will it be emptied ? 
 
 79. From Paris to Berlin by railway it is 1308 km . A kilo- 
 
 meter is 1093.63 yds. Express the distance between 
 Paris and Berlin in miles and yards. 
 
 80. Mercury revolves around the sun in 87.9692580 dys. 
 
 Express the period of revolution in days, hours, min- 
 utes, and seconds. 
 
 81. The Roman foot was 0.97075 of our foot. The Greek 
 
 foot was -|| of the Roman foot. Find the length in 
 inches of the Greek foot. 
 
 82. The radius of a circle is 0.1591549 of its circumfer- 
 
 ence, which contains 360. Find the angle at the 
 centre whose arc is equal to the radius. 
 
 83. Find the L.C.M. of all the multiples of 3, from 6 to 
 
 27, inclusive. 
 
 84. Arrange -|, -|-J, and |-jj- in order of magnitude. 
 
 85. Subtract the sum of -f, , f , ||, ^ from 5. 
 
302 MISCELLANEOUS PROBLEMS. 
 
 86. Find the decimal which, when added to the difference 
 
 of -^-g- and 0.002775, produces the square of 0.215. 
 
 87. A, at the rate of 4*- miles an hour, walks a certain 
 
 distance in 3^ hrs. In what time will B walk the 
 same distance at the rate of | of 5J- miles an hour? 
 
 PERCENTAGE. 
 
 88. A house w r orth $15,000 sustains injury from fire to 
 
 the amount of $3840. What is the rate per cent 
 of loss? 
 
 89. A and B have each $350; A spends 1G% and B 
 
 spends 20%. A's expenditure is what per cent of 
 B's? 
 
 90. A gentleman having a court 20 ft. by 40 ft. enlarged 
 
 it 10% in each dimension. Find the per cent of 
 increase in area. 
 
 91. A young man buys a farm for $5200, which sum is 
 
 30% more than a legacy received from his grand- 
 father. Required the amount of the legacy. 
 
 92. A lady gave to her daughter 25% and to her son 20% 
 
 of her estate. The difference between the shares of 
 the son and daughter was $1500. What is the 
 value of the estate? 
 
 93. If a quart of Jersey milk is worth 10 cts., and pro- 
 
 duces 1 gi. of cream worth 25 cts. a pint, what per 
 cent of the value of the milk is the value of the 
 cream ? 
 
 94. A farmer raised 360 bu. of potatoes, and the crop was 
 
 2400% of the seed. How many bushels did he 
 plant ? 
 
 95. A man received from a bankrupt $937.50, which was 
 
 of the sum due. What was his loss? 
 
MISCELLANEOUS PROBLEMS. 303 
 
 96. What per cent of f is \ ? 
 
 97. If 200% of a number is \<J of 70, what is the num- 
 
 ber? 
 
 98. A grocer sold 10% of his stock of sugar, and then 
 
 25% of the remainder, after which he had 3 t. 
 1560 Ibs. How much sugar had he at first? 
 
 99. A man lost 37i% of his money. He then earned 
 
 $50, and had 125% of what he had at first. How 
 much did he have at first ? 
 
 100. A merchant bought a cask of molasses from which 
 
 20% of the molasses had been drawn. He sold 
 30 \ gals., and then the cask was one-quarter full. 
 Find the capacity of the cask in gallons. 
 
 101. What per cent of a common year is the time from 
 
 July 1 to November 23, both days included ? 
 
 102. A horse and chaise together are valued at $225 ; the 
 
 horse is worth 25% more than the chaise. Find 
 the value of the horse. 
 
 103. A man owning 30% of a mine sold 50% of his share 
 
 for $3000. What was the value of the mine ? 
 
 104. For what price per pair must shoes be sold to gain 
 
 25%, if 15% is lost when they are sold at $1.275 
 per pair ? 
 
 105. If - of goods valued at $1500 are sold at a loss of 
 
 10%, what must the remainder bring to gain 20% 
 on the whole ? 
 
 106. A fruit dealer bought 200 apples at the rate of 4 for 
 
 a cent, and 200 at 5 for a cent. He sold them all 
 at 5 for 3 cents. What per cent did he gain on 
 his investment? 
 
304 MISCELLANEOUS PROBLEMS. 
 
 107. If 75% of the price of a bushel of corn is 50% of the 
 
 price of a bushel of wheat, how many bushels of 
 corn can be bought for $24 when wheat is worth 
 $1.20 a bushel? 
 
 108. A horse dealer sold a horse for $90, and lost 25% of 
 
 the cost of the horse. He sold another horse at 
 an advance of 20% on the cost, and gained as much 
 as he lost on the first horse. What was the selling 
 price of the second horse ? 
 
 109. If 20 men can build a wall in 9 dys., what per cent 
 
 of the number of men could build the wall in 12 
 dys.? 
 
 110. If 7% of a ton of butter costs $42, what per cent of 
 
 a ton can be bought for $57 ? 
 
 111. Five hundred barrels of flour were sold for $4125, at 
 
 a profit of 10%. Find the cost per barrel. 
 
 112. An agent makes 20% by selling a book for 72 cts. 
 
 If he had sold it for $1, what per cent would he 
 have made ? 
 
 113. A merchant bought from a shoe dealer 12 cases of 
 
 shoes, each containing 60 pairs, at 87-^- cts. per 
 pair, and sold the whole for $756. Find his gain 
 per cent. 
 
 114. If 196 sq. rds. are 40% of the area of a field 30 rds. 
 
 in length, what is the width of the field ? 
 
 115. When brooms are $5.50 a dozen, what will be paid 
 
 for 18|- gross, if a discount of 10% is allowed on 
 the bill for cash ? 
 
 116. A grocer bought, at 60 cts. per gallon, 16 hhds. of 
 
 molasses of 63 gals, each, and sold it at a profit 
 of $120.96. What was his gain per cent? 
 
MISCELLANEOUS PROBLEMS. 305 
 
 117. A merchant in his first year of business increased his 
 
 capital 40%, and increased his capital the second 
 year 30%. He lost 33|-% of his capital the third 
 year, and had $18,200 left. What was his capital 
 at first? 
 
 118. A contractor engaged to build a railroad at $31,200 
 
 a mile. The work actually cost $90 per rod. 
 What was his gain per cent ? 
 
 119. A merchant sold goods at 25% discount and 4% off 
 
 from the selling price for cash. What was the 
 whole per cent discount? 
 
 120. At \\Jo commission an agent receives $97.29 for 
 
 selling goods. Find the amount of the sale. 
 
 121. A merchant sent $30,750 to his agent in New Orleans, 
 
 for the purchase of cotton. Find the sum spent for 
 cotton, if the agent charges 2J% commission for 
 buying. 
 
 122. Find the sum paid for insurance, at -%, on a house 
 
 worth $8000, and at f % on furniture worth $2000, 
 if the insurance is on - of the value of the property 
 insured. 
 
 123. A sea captain paid $345, at IY%, for insuring f of 
 
 the value of a ship. Find the value of the ship. 
 
 124. A town has to raise $192,000 for expenses. If 4% 
 
 is allowed for collecting, how much money must be 
 raised ? 
 
 125. A merchant sends $24,600 to his agent at St. Louis, 
 
 for the purchase of flour at $5 a barrel. How 
 many barrels can be bought if the agent charges 
 commission for buying? 
 
306 MISCELLANEOUS PROBLEMS. 
 
 126. A paper-mill worth $30,000 was insured for an 
 
 annual premium of If % on 90% of its value. In 
 the second year it was injured by fire to the amount 
 of $1780. How much did the mill owner save by 
 insuring ? 
 
 127. A city voted a tax of $74,500; the poll-tax was $1.25 
 
 on 2000 polls ; the assessed value of city property 
 was $6,000,000. What was the tax on $1000? 
 
 128 What insurance must be placed upon a store and its 
 contents, valued at $20,085, that the entire value 
 of the goods and store and of a premium of 2% 
 may be recovered in case of loss by fire ? 
 
 129. A premium of $88.14 is paid upon a cargo of wheat 
 
 insured at 2f % on of its value. Find the num- 
 ber of bushels shipped, if the average price is 80 
 cts. a bushel. 
 
 130. A 30% duty of $5594.40 was paid on 252 watches. 
 
 What was the invoice price of each watch? 
 
 131. Eleven and one-half yards of cloth 1-J yds. wide are 
 
 required for a dress. How many yards must be 
 bought if the shrinkage in sponging is 10% in 
 length and 8% in width? 
 
 132. If 30% of a merchant's sales is profit, what is his 
 
 gain per cent ? 
 
 133. A merchant insured a ship and cargo at 4|%. If 
 
 $ 158,650 cover both property and premium, what 
 is the value of the ship and cargo ? 
 
 134. How much money must be sent to purchase 10,000 
 
 bbls. of sugar, at $8.50 per barrel, if the commis- 
 sion for buying is 3%, and the sum prepaid for 
 freight is $315? 
 
MISCELLANEOUS PROBLEMS. 307 
 
 INTEKEST. 
 Find the interest of : 
 
 135. $1000 for 2 yrs. 7 mos. 18 dys., at 6%. 
 
 136. $1496 for 7 mos. 21 dys., at 6%. 
 
 137. $582 for 1 yr. 7 mos. 15 dys., at 6%. 
 
 138. $168 for 1 yr. 5 mos. 12 dys., at 2f %. 
 
 139. $548 for 7 mos. 18 dys., at 6|%. 
 
 140. $1272 from July 12, 1880, to Feb. 24, 1882, at 3J%. 
 
 141. $1975.30 for 60 dys., at 6%. 
 
 142. $1675 for 90 dys., at 6%. 
 
 143. $976 for 3 yrs. 6 mos., at 1% a month. 
 
 Find the rate per cent : 
 
 144. When the interest on $3000 for 3 yrs. is $630. 
 
 145. When the interest on $1500 for 2 yrs. is $172.50. 
 
 146. When the interest on $1278.50 for 3 yrs. 6 mos. is 
 
 $178.99. 
 
 147. When a sum of money is doubled in 8 yrs. 
 
 148. When $1758 amount to $1869.34 in 8 mos. 
 
 Find the time : 
 
 149. When the interest on $278.40, at 7-i%, is $100.92. 
 
 150. When $600, at 3%, amount to $660. 
 
 151. When the interest on $78, at \\/o a month, is 
 
 $28.08. 
 
 152. When the principal, at 5%, is doubled. 
 
308 MISCELLANEOUS PROBLEMS. 
 
 Find the principal that will : 
 
 153. Produce $424.94 interest in 3 yrs., at 
 
 154. Produce $285.60 interest, at 7%, in 1 yr. 8 mos. 12 
 
 dys. 
 
 155. Produce $81.37 interest, at 3f %, in 2 yrs. 9 mos. 
 
 18 dys. 
 
 What principal will amount to : 
 
 156. $88.80, at 6%, in 3 yrs. 4 mos. 
 
 157. $308.10, at 5f%, in 6 mos. 
 
 158. $570.475, at 6%, in 3 yrs. 4 mos. 6 dys. 
 
 159. $661.32, at \f a month, in 3 yrs. 6 mos. 
 
 160. Find the interest on $1825 from Jan. 1 to June 25, 
 
 at 5%, counting the exact number of days, and 
 allowing 365 dys. for a year. 
 
 BANK DISCOUNT 
 Find the proceeds of the following notes : 
 
 161. $300. SPRINGFIELD, 111., Aug. 12. 1884. 
 Sixty days after date I promise to pay Nicholas Welsh, 
 
 or order, $300, value received. 
 
 Discounted at 6%, Sept. 1. JOHN BRYCE. 
 
 162. $700. BOSTON, Nov. 13, 1880. 
 Ninety days after date I promise to pay to the order of 
 
 David Morrison seven hundred dollars, value received. 
 Discounted at 7%, Jan. 1, 1881. GEORGE BROWN. 
 
 163. $217.40. NEW YOEK, July 30, 1884. 
 Ninety days after date I promise to pay to the order of 
 
 Seth Jay two hundred seventeen and y 4 ^ dollars, value 
 received. 
 
 Discounted at 6%, Aug. 10, 1884. JAMES BENT, 
 
MISCELLANEOUS PROBLEMS. 309 
 
 164. $500. CHICAGO, July 9, 1883. 
 Ninety days from date, for value received, I promise to 
 
 pay to the order of John Hogan five hundred dollars, with 
 interest at 9%. 
 
 Discount at 6%, July 9, 1883. JOHN FOSTER. 
 
 165. $5897.50. TROY, June 24, 1881. 
 Four months from date, for value received, I promise to 
 
 pay to the order of Aaron Reed five thousand eight hun- 
 dred ninety-seven and y 5 ^ dollars, with interest at 6%. 
 Discounted at 5%, Aug. 15. JAMES CAREY. 
 
 Find the face of a note which : 
 
 166. Discounted at 6% for 90 dys. yields $344.57. 
 
 167. Discounted at 9% for 46 dys. yields $493.87. 
 
 168. Discounted for 6% for 3 mos. yields $984.50. 
 
 PARTIAL PAYMENTS. 
 
 169. A note for $680, dated June 15, 1884, payable on 
 
 demand, with interest at 6%, hears the following 
 endorsement: May 15, 1885, $425. What is due 
 June 15, 1885? 
 
 170. On a note of $1400, dated March 1, 1880, there was 
 
 received Oct. 19, 1880, $700; Jan. 1, 1881, $400. 
 What is due March 1, 1881, reckoning interest at 
 6%? 
 
 171. A note of $900,' dated Jan. 1, 1884, and bearing 
 
 interest at 5 f , has the following endorsements : 
 May 13, $240; Aug. 19, $300; Oct. 25, $180. 
 Kequired the balance due Jan. 1, 1885. 
 
310 MISCELLANEOUS PROBLEMS. 
 
 172. A note of $1800, dated Jan. 1, 1880, and bearing 
 
 interest at 5%, has the following endorsement: 
 June 1, 1881, $400. Find the balance due June 
 1, 1884. 
 
 173. A note of $600, dated Aug. 13, 1881, and bearing 
 
 interest at 6%, has the following endorsements: 
 Jan. 1, 1882, $200; April 1, 1882, $110. Find 
 the balance due Aug. 13, 1883. 
 
 174. A note of $1150, dated June 30, 1878, and bearing 
 
 interest at 6%, has the following endorsements: 
 Jan. 30, 1879, $15; April 30, 1880, $570; July 
 30, 1881, $420. Find the balance due Dec. 30, 
 1882. 
 
 COMPOUND INTEREST. 
 Find the compound interest of: 
 
 175. $300, at 6%, for 3 yrs. 4 mos. 18 dys. 
 
 176. $350, at 6%, for 3 yrs. 5 mos. 24 dys. 
 
 177. $840, at 8%, from June 13, 1880, to Aug. 1, 1881, 
 
 interest being payable quarterly. 
 
 178. $400, at 4|%, from Jan. 1, 1881, to Feb. 13, 1884. 
 
 179. $1100, at 6%, for 2 yrs. 7 mos. 6 dys., interest being 
 
 payable semi-annually. 
 
 180. $1000, at 8%, for 2 yrs. 3 mos. 18 dys., interest pay- 
 
 able quarterly. 
 
 STOCKS. 
 
 181. Find the cost of $2400 stock, at 97f 
 
 182. Find the cost of $2785 stock, at 105f 
 
 183. Find the cost of $5680 stock, at 103 J. 
 
MISCELLANEOUS PROBLEMS. 31 1 
 
 184. How much stock, at 85f , including brokerage, can 
 
 be bought for $23 76.84? 
 
 185. How much 6% stock will produce an income of 
 
 $840? 
 
 186. Find the price of stock, when $4647.50 will pay for 
 
 $5200 worth of stock. 
 
 187. How many hundred-dollar shares of 1% stock will 
 
 yield a yearly income of $686? 
 
 188. A gentleman gave his daughter $25,700 of \% 
 
 bonds. What yearly income from them will she 
 receive ? 
 
 189. What amount of 8% stock will yield a yearly income 
 
 of $8000? 
 
 190. What is the rate of dividend when the sum of $300 
 
 is received from $ 7500 stock ? 
 
 191. Find the rate of dividend when the sum of $1603.80 
 
 is received from $35,640. 
 
 192. How much stock, at 121|, can be bought for $6318? 
 
 193. How much stock, at 97|, can be bought for $1755? 
 
 194. Find the sum paid for $5600 stock, at 112|, and 
 
 brokerage \. 
 
 195. What income will be obtained from $5125, invested 
 
 in 6% stock, at 102? 
 
 196. Find the income from $8190, invested in 5% stock, 
 
 at 91. 
 
 197. Find the income on $1935, invested in 8% stock, at 
 
 198. Find the income from $6750 invested in 4 stock, at 
 75. 
 
312 MISCELLANEOUS PROBLEMS. 
 
 199. If $7656 be invested in stock, at 63$, and the stock 
 
 pays a dividend of 3^%, how much will be re- 
 ceived on the money invested ? 
 
 200. If $7000 be invested in stock, at 87, and the stock 
 
 pays a dividend of 7%, how much will be re- 
 ceived ? 
 
 201. If 9% stock is bought at 150, what rate of interest 
 
 will be received on the investment? 
 
 202. What rate of interest will be received on 5% stock 
 
 at 75? 
 
 203. What rate of interest will be received on 4% stock, 
 
 at62|? 
 
 204. How much money must be invested in 5% stock, at 
 
 80, to produce $400 income? 
 
 205. How much money must be invested in 4% stock, at 
 
 90, to produce $320 income? 
 
 206. How much money must be invested in 6% stock, at 
 
 75, to produce $200 income? 
 
 207. A man received $240 from his 6% dividend, on 
 
 stock bought at 105. How much money did he 
 have invested in the stock ? 
 
 208. What should be paid for a 4% stock, that 5% inter- 
 
 est may be realized on the investment ? 
 
 209. What should be paid for a 6% stock, that 8% inter- 
 
 est may be realized on the investment ? 
 
 210. If 4% stock, which produces an income of $180, is 
 
 sold at 90, what sum will be realized from the sale ? 
 
 211. What increase of income will there be, if $3600 of 
 
 4% stock is sold at 90, and the proceeds invested 
 in 7% stock, at 108? 
 
MISCELLANEOUS PROBLEMS. 313 
 
 212. Find the increase of income, if $3900 of 3% stock 
 
 is sold at 88, and the proceeds invested in 4-|% 
 stock, at par. 
 
 213. Find the increase of income, if $6000 of 8% stock 
 
 is sold at 120, and the proceeds invested in 6-^-% 
 stock, at 90. 
 
 SIMPLE PROPORTION. 
 
 214. If 16 bbls. of apples cost $28, what will 129 bbls. 
 
 cost? 
 
 215. If 15 workmen can do a piece of work in 25 dys., in 
 
 how many days can 25 men do the same work ? 
 
 216. If 8 horses eat a certain quantity of hay in 2 mos., 
 
 how long will the same quantity last 12 horses? 
 
 217. A meadow can be mowed by 40 men in 10 dys. 
 
 How many days will it take 30 men to mow it? 
 
 218. If 30 men can build a wall in 18 dys., how many 
 
 men will be required to build it in 12 dys. ? 
 
 219. A bankrupt owes $3000, and his assets amount to 
 
 $850. How much on a dollar will his creditors 
 receive ? 
 
 220. What does a bankrupt pay on a dollar, if his credit- 
 
 ors receive $376.275 on $2076? 
 
 221. A bankrupt's effects amounted to $2675.40, and his 
 
 debts to $3057.60. What did his creditors 
 receive on a dollar? 
 
 222. If 4 men reap 5 A. 159 sq. rds. in 1 week, how 
 
 many men at the same rate will reap 35 A. 154 
 sq. rds. ? 
 
314 MISCELLANEOUS PROBLEMS. 
 
 223. A wall whose height is 9.1875 ft. casts a shadow of 
 
 10.5 ft. Find the length of the shadow of a 
 steeple 93.8 ft. high. 
 
 224. A cistern can be filled in 54 min. by a pipe running 
 
 3J- gals, a minute. In how many minutes can it 
 be filled by another pipe, running 4^- gals, a minute? 
 
 225. A watch set on Saturday, at half-past eight in the 
 
 evening, loses 1 min. in 30 hrs. What time does 
 it show the next Thursday, at 4 o'clock in the 
 afternoon ? 
 
 226. When do the hour and minute-hands of a watch 
 
 coincide between 5 and 6 o'clock ? 
 
 NOTE. Since the hour-hand moves through 5 minute-spaces while 
 the minute-hand traverses 60, the minute-hand moves 12 times as 
 fast as the hour-hand. The minute-hand, therefore, in moving 
 through 12 minute-spaces, traverses 11 minute-spaces more than the 
 hour-hand. 
 
 When the hour-hand is at V., the minute-hand, being at XII., is 
 25 minute-spaces behind it. The question, therefore, is, if the 
 minute-hand, to gain 11 spaces, must move through 12 spaces, how 
 many spaces must it move through to gain 25 spaces ? 
 
 11 : 12 : : 25 : ? 
 
 227. When do the hour and minute-hands of a watch 
 
 coincide between 8 and 9 o'clock ? 
 
 228. When do the hour and minute-hands of a watch 
 
 coincide between 3 and 4 o'clock ? 
 
 229. When do the hour and minute-hands of a watch 
 
 coincide between 10 and 11 o'clock? 
 
 The true weight of a body weighed successively in the 
 scales of a false balance is the square root of the product of 
 the apparent weights. 
 
MISCELLANEOUS PROBLEMS 315 
 
 230. A body appears to weigh 5^g- Ibs. in one scale, and 
 
 5-$- Ibs. in the other scale, of a false balance. Find 
 its true weight. 
 
 The times in which bodies fall are proportional to the 
 square roots of the distances traversed. Since a body falls 
 16.1 ft. the first second, to find the time a body is falling, 
 divide the distance by 16.1, and extract the square root of 
 the quotient. 
 
 231. In how many seconds will a stone fall to the bottom 
 
 of a coal-pit 420 ft. deep? 
 
 COMPOUND PROPORTION. 
 
 232. If 60 bu. of corn feed 6 horses for 50 dys., in how 
 
 many days will 15 horses consume 75 bu. ? 
 
 233. If 20 cwt. are carried 50 miles for $5, how much will 
 
 be the cost of carrying 40 cwt. 40 miles? 
 
 234. If 20 men can perform a piece of work in 12 dys., 
 
 required the number of men who can perform an- 
 other piece of work three times as great in -J- of 
 the time. 
 
 235. If 12 horses, in 5 dys., draw 44 loads of stone, how 
 
 many horses will draw 132 loads the same dis- 
 tance in 18 dys. ? 
 
 236. If a footman travels 130 mi. in 3 dys., of 14 hrs. 
 
 each, in how many days, of 7 hrs. each, will he 
 travel 390 mi. ? 
 
 237. If 50 men dig a cellar in 7 dys., working 11 hrs. a 
 
 day, how many days will 24 men require, work- 
 ing 8 hrs. a day ? 
 
316 MISCELLANEOUS PROBLEMS. 
 
 238. A garrison of 1500 men has provisions for 12 wks., 
 
 at the rate of 20 oz. per day to each man. How 
 many men will the same provisions maintain for 
 20 wks., allowing each man only 8 oz. per day? 
 
 239. If 12 candles, of which 8 weigh a pound, serve 4 
 
 winter evenings, from five to eleven, how many can- 
 dles, of which 6 weigh a pound, will serve 3 spring 
 evenings, from seven to eleven ? 
 
 240. A contractor, having engaged to lay 10 mi. of rail- 
 
 way in 150 dys., finds that 90 men have finished 
 3 mi. in 80 dys. How many more men must he 
 engage to finish the work in the given time? 
 
 241. If 200 men in 12 dys., of 8 hrs. each, can dig a 
 
 trench 160 yds. long, 6 yds. wide, and 4 yds. deep, 
 in how many days, of 10 hrs. each, will 90 men 
 dig a trench 450 yds. long, 4 yds. wide, and 3 yds. 
 deep? 
 
 242. If 120 men make an embankment f of a mile long, 
 
 30 yds. wide, and 7 yds. high, in 42 dys., how 
 many men will it take to make an embankment 
 1000 yds. long, 36 yds. wide, and 22 ft. high, in 
 30 dys. ? 
 
 POWERS AND ROOTS. 
 Find the square root of: 
 
 243. 30976. 247. 2052.09. 
 
 244. 106929. 248. 4795.25731. 
 
 245. 622521. 249. 24674.1264. 
 
 246. 1234321. 250. 
 
MISCELLANEOUS PROBLEMS. 317 
 
 Find the cube root of : 
 
 251. 373248. 256. 52734.375. 
 
 252. 54872. 257. 7834.87438. 
 
 253. 389017. 258. 0.053157376. 
 
 254. 1092727. 259. f 
 
 255. 84604519. 260. 7. 
 
 MENSURATION. 
 
 261. What is the total surface of a cube, the edge of 
 
 which measures 4i in.? 
 
 262. How many planks, each 15 ft. long, and 10 in. wide, 
 
 will be required for the flooring of a room 30 ft. 
 in length and 22i ft. in width? 
 
 263. A square court, whose side is 42 yds., is paved with 
 
 28,224 square tiles. Find the dimensions of each 
 tile. 
 
 264. Find the area of a triangle whose base is 9 ft. 8 in., 
 
 and whose altitude is 5 ft. 3 in. 
 
 265. How many yards of carpeting, 1 yd. wide, will be 
 
 required for a room 27 ft. long, and 21 ft. 3 in. 
 wide, if the strips run across the room ? 
 
 266. How many yards in the side of a square field con- 
 
 taining 3 A. 44 sq. rds. 25 sq. yds. ? 
 
 267. The weight of a cubic inch of water is 253.17 grs. ; 
 
 that of a cubic inch of air, 0.31 grs. Find to three 
 places of decimals the number of cubic inches of 
 water equal in weight to 1 cu. ft. of air. 
 
 268. How many cubic feet in a piece of timber 18 ft. long, 
 
 15 in. wide, and 10 in. thick? 
 
318 MISCELLANEOUS PROBLEMS. 
 
 269. The sides of a triangular garden are 52.64, and 72 
 
 yds. respectively. Find the area. 
 
 270. Find the circumference and area of a circle whose 
 
 radius is 2 ft. 4 in. 
 
 271. The side of an equilateral triangle is 12 ft. Find its 
 
 area. 
 
 272. How many bricks, each 9 in. long, 4} in. wide, and 
 
 3 in. thick, will be required for a wall 175 yds. 
 long, 12 ft. high, and 1 ft. 10i in. thick? 
 
 273. The diameter of a circular shaft in a railway tunnel 
 
 is 5 ft. and its depth 30 fathoms. How many 
 cubic feet of earth were dug out in making it? 
 
 274. Find the cost of plastering a room 25 ft. 6 in. long, 
 
 17 ft. 3 in. wide, and 10 ft. 8 in. high. The price 
 for the walls is 21 cts., and for the ceiling 32 cts. 
 a square yard, and no allowance is made for doors 
 and windows. 
 
 275. A line 62 ft. long reaches from the top of a house 48 
 
 ft. high, to the bottom of a house on the opposite 
 side of the street. Find the width of the street. 
 
 276. Find the capacity in bushels of a round basket 20 in. 
 
 in diameter and 28 in. deep. 
 
 277. The diameter of a spherical balloon is 25 ft. How 
 
 many square yards of silk were required to make 
 it, and how many cubic feet of gas will be required 
 to fill it? 
 
 278. Find the weight of an ivory ball 2 in. in diameter, 
 
 the weight of ivory being 1825 oz. a cubic foot. 
 
 279v Hew many cubic feet in a stone roller 6 ft. 6 in. long 
 and 5 ft. 4 in. in circumference ? 
 
CHAPTER XVI. 
 
 METRIC MEASURES. 
 
 320. The Metric System is a system of weights and meas- 
 ures expressed in the decimal scale. 
 
 321. The standard meter, as defined by law, is the length 
 of a bar of very hard metal carefully preserved at Paris, 
 accurate copies of which are furnished to the governments 
 of all civilized nations. 
 
 322. The principal units of the metric system are : 
 
 The meter ( m ) for lengths ; 
 The square meter ( qm ) for surfaces ; 
 The cubic meter ( cbm ) for large volumes ; 
 The liter (*) (lee'-ter) for smaller volumes ; 
 The gram ( g ) for weights. 
 
 323. All these units are divided and multiplied deci- 
 mally, and the size of the measures thus produced is shown 
 by one of seven prefixes ; namely, deka, meaning 10 ; Jiekto, 
 meaning 100 ; kilo, meaning 1000 ; myria, meaning 10,000 ; 
 and deci, meaning 0.1 ; centi, meaning 0.01 ; milli, mean- 
 ing 0.001. 
 
 324. But, as in United States money we seldom speak 
 of anything else than dollars and cents, so in other meas- 
 ures it is only those printed in black letter in this chapter 
 that are in common use. 
 
 NOTE. A meter is a trifle more than 39.37 inches, and all the units 
 of the system are derived from the meter. All the compound names 
 are accented on the first syllable ; thus, millimeter. The teacher 
 should be supplied with a meter stick, a liter, and a cubic centimeter. 
 
320 METRIC MEASURES. 
 
 UNITS OF LENGTH. 
 
 325. A millimeter ( mm ) = 0.001 of a meter. 
 A centimeter ( cm ; == 0.01 
 
 A decimeter =0.1 " " 
 A meter ( m ). Principal Unit, 
 
 A dekameter = 10 meters. 
 
 A hektometer = 100 " 
 
 A kilometer ( km ) = 1,000 " 
 
 A myriameter = 10,000 " 
 
 326. A length given in any one of these meas- 
 ures may be expressed in terms of another measure 
 by simply moving the decimal point to the right 
 or left. 
 
 Thus, 17,856,342 mm may be written as kilo-me- 
 ters by observing that milli- meters are changed to 
 meters by moving the point three places to the 
 left ; and these meters into kilo-meters by carrying 5 
 it three places further, making, in all, six places, f 
 Therefore, 17,856,342"* = 17.856342 km . 
 
 Again, 4.876326 km may be written as centi- 
 meters, by observing that kilo-meters are changed I 
 to meters by moving the point three places to the I 
 right, and meters to centi-meters by moving it two 
 places further, making, in all, five places. There- 
 fore, 4.876326 km = 487,632.6 cm . 
 
 327. The rule, therefore, for this conversion is : 
 First change the point so as to convert the given 
 
 measures into terms of the principal unit ; then 
 change the point so as to convert the principal 
 into the required units. 
 
 328. Remember that, before adding or sub- 
 tracting, the quantities must be written in the 
 same units of measure. 
 
 r. 
 
 
 
 if 
 
 r 
 
METRIC MEASURES. 321 
 
 Ex. 165. (Oral.) 
 
 1. How many meters in a dekameter ? hectometer? kilo- 
 
 meter? How many dekameters in a hectometer? 
 kilometer ? 
 
 2. What part of a meter is a decimeter? centimeter? 
 
 millimeter ? What part of a centimeter is a milli- 
 meter ? 
 
 3. Read32.3 m ; 12.6 cm ; 15.4 km ; 59.8 mm . 
 
 4. Express 3256 m as kilometers ; as centimeters. 
 
 5. Express 5368 mm as centimeters ; as meters. 
 
 6. Express 12.4 km as meters; as centimeters. 
 
 Ex. 166. 
 
 Find the value of each of the following expressions in 
 meters : 
 
 1. 0.435 m + 852 cm + 4263 mra + 0.1595 km . 
 
 2. 0.927 kra 6495 cm ; 4.37 cra 42.87 ram . 
 
 3. 8 X 0.0457 km ; 3.04 X 60.93 cm ; 5.43 x 67.2 ram . 
 
 4. 38,019 mm ^ 0.097; 0.41 km -*- 25.625. 
 
 5. A book is 2.1 cm thick ; if the average thickness of the 
 
 leaves is 0.05 mm , find the number of pages in the book. 
 
 6. The expense of building a certain railroad is $25,000 
 
 on the average per kilometer. What is the whole 
 cost of the road, if its length is 72 km and 53 m . 
 
 7. The wheels of a locomotive that makes 45 km an hour 
 
 are 7.5 m in circumference. How many revolutions 
 will they make a minute ? 
 
 8. A locomotive runs 1284 m in 1} min. How many kilo- 
 
 meters will it go in 1 hr. 35 min. 15 sec. ? 
 
 9. The top of a monument is 143.9 m , and the base 67.19 m 
 
 above the level of the sea; the steps which lead 
 from the base to the top of the monument are each 
 19 cm high. How many steps are there ? 
 
322 METRIC MEASURES. 
 
 I I I I I I i~| 
 
 MEASURES OF SURFACE. 
 
 329. The principal unit of surface is a square meter ( qm ). 
 
 330. In square measure the multiplication and division 
 of units is by hundreds and hundredths, 
 
 instead of by tens and tenths' Suppose the 
 square in the margin to represent a square 
 meter, It is divided into ten equal horizon- 
 tal bands, and each band is one-tenth of the 
 square meter. Each band can be divided, 
 as the upper one is, into ten little squares measuring 
 one-tenth of a meter on a side. Each of these squares will 
 be 0.1 of the band, or 0.01 of the whole square. The square 
 meter, therefore, contains 10 X 10 or 100 square decimeters. 
 
 If the square meter were divided into 100 equal hori- 
 zontal bands, each band would be 0.01 of the square ; and 
 if each of the 100 bands were divided into 100 squares, that 
 is, into 100 square centimeters, the whole square would 
 contain 100 X 100 or 10,000 square centimeters. A square 
 meter, therefore, contains 10,000 square centimeters, 
 
 In like manner, a square meter contains 1,000,000 square 
 millimeters, 
 
 331. UNITS OF SURFACE. 
 
 A square millimeter ( qmm ) 0.000001 of a square meter. 
 
 A square centimeter ( qcm ) 0.0001 
 
 A square decimeter = 0.01 
 
 A square meter ( qm ) Principal Unit, 
 
 A square dekameter 100 square meters. 
 
 A square hektometer = 10,000 " " 
 
 A square kilometer ( qkm ) = 1,000,000 " 
 
 332. It will be observed that while centimeters are in 
 the second, a*nd millimeters in the third decimal place 
 from meters, square centimeters are in the fourth and square 
 millimeters in the sixth decimal place from square meters, 
 
METRIC MEASURES. 323 
 
 LAND MEASURE. 
 
 333. In measuring land the square meter is called a 
 centar ( ca ), the square dekameter is called an ar ( a ), and the 
 square hektometer a hektar ( ha ). 
 
 Ex. 167. (Oral.) 
 
 1. How many square meters in a square dekameter? 
 
 square hectometer ? square kilometer ? 
 
 2. How many centars in an ar ? in a hektar ? 
 
 3. What part of a hektar is an ar ? a centar ? 
 
 4. What part of a square meter is a square decimeter? 
 
 square millimeter ? 
 
 5. Read56.4 qra ; 2.05 qkm ; 531.6 qcm . 
 
 6. Read53 a ; 36.03 ha ; 56 ca ; 56 qm . 
 
 Ex 168. 
 
 1. Convert 1,854, 276 qm into hektars; into square kilo- 
 
 meters. 
 
 2. How many hektars in 2. 7856 qkm ? 
 
 3. Write 1.7431 qm as square centimeters; as square milli- 
 
 meters. 
 
 4. How many square kilometers in 17,467. 5 ha ? 
 
 5. How many square meters in 1.3614 qkm ? 
 
 6. How many square meters in 2.25 ha ? 
 
 7. How many square centimeters in 0.0137 of a square 
 
 meter ? 
 
 8. Write 3.571 qcm as square millimeters. 
 
 9. A man bought 3 ba of land at $200 per hektar, and sold 
 
 it for $2.50 per ar. How much did he gain ? 
 10. If 6 ha are divided into 64 equal lots, how many square 
 s meters will there be in each lot? 
 
324 
 
 METRIC MEASURES. 
 
 MEASURES OF VOLUME. 
 
 334. The principal unit of capacity or volume is a cubic 
 meter ( cbm ). 
 
 335. The cubic meter can be divided into 10 layers, 
 each a meter square and a deci- 
 meter thick. Each layer will, 
 
 therefore, be 0.1 of a cubic meter. 
 
 Again, each layer can be 
 divided into 10 equal parts. 
 Each part will, therefore, be 
 0.1 of the layer, or 0.01 of the 
 meter, and will be a decimeter 
 square and a meter long. 
 
 Also, each one of these parts 
 can be divided into 10 equal parts, each of which will be a 
 cubic decimeter, and will be 0.1 of 0.01, that is, 0.001 of 
 the cubic meter. 
 
 The cubic meter, therefore, contains 1000 cubic decimeters, 
 
 In like manner, each cubic decimeter can be divided into 
 1000 cubic centimeters, and each cubic centimeter into 1000 
 cubic millimeters. 
 
 336. UNITS OF VOLUME. 
 
 A cubic millimeter ( cmm ) = 0.000000001 of a cubic meter. 
 A cubic centimeter ( oom ) 0.000001 " " 
 
 A cubic decimeter = 0.001 " " 
 
 A cubic meter ( cbm ) Principal Unit, 
 
 337. It will be seen that cubic centimeters are in the sixth 
 and cubic millimeters are in the ninth decimal place from 
 cubic meters, 
 
 338. In measuring wood, the cubic meter is called a ster 
 ( st ) ; and in measuring liquids, grain, etc., the cubic deci- 
 meter is always called a liter, 
 
 When the liter is the unit, the numeral prefixes have the 
 bame value as in linear measure. 
 
METRIC MEASURES. 325 
 
 Ex. 169. (Oral.) 
 
 1. In a cubic meter, how many cubic decimeters? cubic 
 
 centimeters ? 
 
 2. What part of a cubic meter is a cubic decimeter? 
 
 3. What part of a cubic meter is a cubic centimeter ? 
 
 4. What part of a cubic decimeter is a cubic centimeter? 
 
 5. How many liters in a hektoliter ? 
 
 6. How many liters in a cubic meter? 
 
 7. How many cubic centimeters in a liter? 
 
 8. Express 7685. 25 1 as cubic meters ; as hektoliters. 
 
 9. If the water in the liter represented in the margin 
 
 stands 6 cm high, how many 
 cubic centimeters of water 
 are there in the measure? 
 How many will be required 
 to fill it? If the faucet be 
 turned, and the water allowed 
 to run out until the measure 
 Liter = cubic Decimeter. is only a quarter full, how 
 
 many cubic centimeters will run out? How many 
 
 will still remain? 
 
 Ex. 17O. 
 
 1. How many cubic meters in a rectangular box 125 cm 
 
 long, IIS wide, and 80 cm deep? how many liters? 
 
 2. How many cubic meters of earth must be removed to 
 
 dig a ditch 90 ra long, 85 cm wide, and 50 cm deep? 
 
 3. How deep must a cistern be to hold 6000 1 , if the bot- 
 
 tom is a square measuring 2.25 on a side ? 
 
 4. How many hektoliters in a bin 4 m long, 2 m wide, and 
 
 l m high? 
 
 6. How high must a box be to hold 30 1 , if it is 50 cm long 
 and 20 cm wide? 
 
326 METRIC MEASURES. 
 
 UNITS OF WEIGHT. 
 
 339. The units of weight are the weights of units of pure 
 water taken at its greatest density, 
 
 that is, a little above the freezing 
 point. 
 
 The principal unit is the gram, 
 which is the weight of a cubic centi- 
 
 Cubic Centimeter. Gram Weight 
 
 meter of water. 
 
 The numeral prefixes have the same value as in linear 
 measure. 
 
 340. A cubic centimeter of water weighs a gram. 
 A liter of water weighs a kilogram, 
 
 A cubic meter of water weighs a ton (1000 kg ). 
 
 Ex. 171. (Oral.) 
 
 1 . How many grams in a kilogram ? dekagram ? hektogram ? 
 
 2. What part of a gram is a centigram? decigram? milligram? 
 
 3. In a ton, how many kilograms? grams? 
 
 4. A hektoliter of water weighs how many kilograms? 
 
 what part of a ton ? 
 
 5. What is the weight in grams of 5 ccm of water? 
 
 6. Change 12,260 rag into grams. 
 
 7. Give the weight in kilograms of 0.275 cbm of water. 
 
 8. Change 0.546 kg to grams ; to milligrams. 
 
 9. Change 0.563 of a ton to kilograms. 
 
 Ex. 172. 
 
 1. What is the weight of water required to fill a vat 98 cm 
 
 long, 71 cm wide, and 38 cm deep ? 
 
 2. A mass of 21. 7 g is divided into 70 pills What is the 
 
 weight of each pill ? 
 
 3. At 2 cts. a kilogram, what will 2.25 tons of hay cost ? 
 
 4. At $6 a ton for coal, what will it cost to heat a building 
 
 30 days, if it takes 400 kg of coal a day ? 
 
METRIC MEASURES. 327 
 
 SPECIFIC GRAVITY. 
 
 341. The specific gravity of any substance is the number 
 found by dividing the weight of the substance by the weight 
 of an equal bulk of water. 
 
 342. Therefore the specific gravity of a substance is the 
 number that expresses the weight of a cubic centimeter of it 
 in grams ; or of a liter in kilograms ; or of a cubic meter in 
 tons, 
 
 343. The volume of a body is found by dividing its 
 weight by its specific gravity. 
 
 Ex. 173. 
 
 1. A bar of iron 50 cra long, 4 cm wide, l cm thick has a spe- 
 
 cific gravity of 7.8. Find its weight in kilograms. 
 
 2. A piece of iron weighing 117 kg is made into a bar 6 cm 
 
 wide and 2 cm thick. What is its length, if the spe- 
 cific gravity of the iron is 7.8 ? 
 
 3. A cubical vessel, 40 cm on an edge, is full of water. If 
 
 14 l are drawn off, and replaced by a liquid of which 
 the specific gravity is % of that of water, what is the 
 weight of the mixture ? 
 
 4. What will be the weight in air, in water, and in olive 
 
 oil, of which the specific gravity is 0.915, of a cube 
 of iron 6 cm on an edge, if the specific gravity of the 
 iron is 7.8? 
 
 NOTE. A body weighed in a liquid weighs less than in air 
 by the weight of the volume of the liquid which it displaces. 
 
 5. What is the specific gravity of a substance of which 
 
 7.3 ccm weighs 31.5 g ? 
 
 6. A block of granite 60 cm long, 50 cm wide, 15 cm thick 
 
 weighs 130. 5 kg . Find its specific gravity. 
 
328 METRIC MEASURES. 
 
 7. What is the weight of a load of 250 paving stones, 
 
 15 cm square and 8 cm thick, if the specific gravity of 
 the stone is 2.90? 
 
 8. A stick of timber 8 m long, 25 cm wide, and 23 cm thick is 
 
 carried by 4 men. What weight does each carry, if 
 the specific gravity of the timber is 0.53 of that of 
 water ? 
 
 9. Allowing that water in freezing is increased by y 1 ^ of 
 
 its volume, find the volume and the weight of a block 
 of ice SO " 1 long, 50 cm wide, 20 cm thick, and find how 
 much water it will give on melting. 
 
 10. A hall has a capacity of 450 cbm . Compute the weight 
 
 of oxygen in the air that fills the hall, knowing that 
 air contains 23% of its weight of oxygen, and weighs 
 y^ as much as water. 
 
 11. A bottle empty weighs 650 g ; full of oil, it weighs 
 
 1075 g . What part of a liter will the bottle hold, if 
 the specific gravity of the oil is 0.905 ? 
 
 12. A vessel full of water weighs 9.68 kg , and, full of oil, 
 
 of which the specific gravity is 0.91, weighs 9.266 kg . 
 Find its capacity and its weight when empty. 
 
 NOTE. To find the capacity, divide the difference of the two 
 weights by the difference of the weights of a liter of water 
 and a liter of oil. 
 
 13. In a decimeter cube full of water is placed a chain oi 
 
 iron, of which the specific gravity is 7.8. When the 
 chain is removed, the height of the water is just 5 cm . 
 Find the volume and weight of the chain. 
 
 14. If a body weighs 3.71 kg in air, and 2.38 kg in water, 
 
 what is its specific gravity ? 
 
 15. From 106. 25 g of rhubarb are made 125 powders. What 
 
 is the volume of water of which the weight is equal 
 to the weight of a powder ? 
 
METRIC MEASURES. 329 
 
 Ex. 174. 
 MISCELLANEOUS PEOBLEMS 
 
 1. A well is 18. 2 m deep and the wheel is 1.4 m round. 
 
 How many turns of the wheel will be required to 
 raise the bucket? 
 
 2. Find the number of liters in a vat 2 m by 75 cm by 50 cm . 
 
 3. Into how many pills of 325 mg each can a mass of 23. 4 g 
 
 be made ? 
 
 4. How many liters will a box hold which is 75 cra long, 
 
 15 cm wide, and 12 cra deep? 
 
 5. Add 3.473 m , SO 8 ", 83 mm , 4.5 m , and 16 cm . 
 
 6. There are 4 measuring lines: the first is 7.5 m , the 
 
 second is 3 m 75 cm , the third is 4 m 80 cm , and the 
 fourth is 8 m 6 cm . Express in meters the total length 
 of the four lines. 
 
 7. On the same railroad are four stations, between which 
 
 the consecutive distances are as follows : 7 km 249 m , 
 3 km 200, and 5.007 km . Find in kilometers the dis- 
 tance between the first and fourth stations. 
 
 8. A goldsmith has sold jewels of the following weights 
 
 respectively : 27 g 9 mg , 30 g 70 cg , 7 g 4 cg , and 19* 34 cg 
 7 mg . Find the total weight in grams. 
 
 9. From 17 km 6 m take 243 m 691 mm . 
 
 10. From a farm containing 340 ha 7 a there are sold 119 h8 
 
 29.03 a ; how many hektars are left? 
 
 11. A liter of mercury weighs 13 kg 598 g . Find the weight 
 
 in kilograms of 3.69 liters. 
 
330 METRIC MEASURES. 
 
 12. If 16.94 1 of olive oil weigh 15 kg 500*, find the weight 
 
 of one liter. 
 
 13. Into how many lots of 3.75 a may 8 ha 40 a be divided? 
 
 14. If 122. 6 g of chlorate of potassa yield 48* of oxygen, 
 
 what weight of oxygen may be obtained from l kg 
 of the chlorate ? 
 
 15. A field of wheat containing 8 ha furni.rhed 600 sheaves 
 
 per hektar ; 2 sheaves of wheat furnished a bundle 
 of straw weighing 5 kg . What will the whole straw 
 bring, at $15 a ton? 
 
 16. A farmer shears 620 sheep and 180 lambs ; the sheep 
 
 give on an average 4 kg of wool each, and the lambs 
 J of a kilogram each. The wool is sold for 50 cts. 
 a kilogram. How much money does the farmer 
 receive ? 
 
 17. A vessel full of water weighs 5.25 kg ; the weight of 
 
 the vessel when empty is 250 g . How many liters 
 will the vessel hold ? 
 
 18. Wheat weighs 80 kg a hektoliter. A field of 4.6 ba has 
 
 produced 9.2 t. of wheat. If a sheaf of the wheat 
 on the average gives 4 1 of grain, find the average 
 number of sheaves produced per hektar. 
 
 19. A piece of zinc weighs in the air 343 g , and in water 
 
 293* only. What is its volume ? 
 
 20. A piece of zinc weighs in the air 343 g , and in water 
 
 293*. What is its specific gravity ? 
 
 21. A jug empty weighs 1.02 kg ; full of water it weighs 
 
 3.8 kg . Find the capacity of the jug in liters. 
 
 22. The price of 8 casks of olive oil, containing each 9.05 hl , 
 
 is $ 1072. What will be the price of 20 1 ? 
 
ADVERTISEMENTS. 
 
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