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Dictation Problems and Reviews . . $ .50 Key to Common School Arithmetic . 1.00 Key to Elementary Algebra . . . . 1.25 Forwarded, postpaid, on receipt of the Price. Entered, according to Act of Congress, in the year 1871, by DANIEL B. HAGAR and HENRY B. MAGLATHLIN, In the Office of the L ibrarian of Congress, at Washington. Copyright, 1877, by DAN/EL B. HAGAR and HENRY B. MAGLATHLIN. Westcott & Thomson, E. Stanley Hart, Stcr-- 'ypers and Electrotypers, Philada. Printer, Philada. INTRODUCTION. The purpose of this manual is to facilitate the advance of young learners in the science of numbers by gradual steps. The lessons are intended to secure that normal develop- ment and discipline of the reasoning powers, and those correct habits of investigation, which alone form a sure foundation for progress in any branch of knowledge. The principles and rules have been carefully established by induction. The plan has been to make the reasons for each process entirely clear, and to enable the learner to state them in concise language. Mental and written exercises admitting substantially of the same solution have been combined, so as to render un- necessary the use of a separate mental arithmetic, and otherwise to abridge advantageously the ordinary course of arithmetical study. Pictorial illustrations, from original designs, have been freely introduced, with the view of making some parts of the subjects treated more easily understood, through the medium of the eye. It is hoped that this work, which is complete in itself, may satisfactorily meet the wants of intermediate classes in graded schools ; and also may prove useful in many dis- trict schools, in which the attendance is too limited to war- rant the use of a more extended treatise. 3 SUGGESTIONS TO TEACHEES. 1. Material objects should be used as illustrations far enough to make sure that the pupils clearly understand the value of numbers ; how numbers are composed in addition, how they are separated in subtraction, how multiplication and division are performed, and how the elementary tables are constructed. When these things are com- prehended, material objects should be dispensed with, and a thorough knowledge of the tables should be relied upon for the requisite results. 2. The tables should be made so familiar that when any two num- bers are named, the result of a desired operation upon them shall, by the power of repeated association, instantly flash upon the mind. 3. Care should be taken that the definitions are clearly understood before they are learned. 4. The attention of pupils should be directed to the successive steps taken in the solution of the problems first given under any sub- ject, and each pupil should be required to state the first step, the second step, and so on until all the steps are named and recorded on slates or blackboard. These steps should be combined, and thus the mode of building up a rule be made clear. The rule should be re- garded, not as a guide to the solution of problems, but as a concise statement of what the pupils have already learned to do. 5. In addition, pupils should usually avoid naming the numbers to be added, but should give only the successive results. They should have much practice in adding and subtracting by 2's, 3's, 4's, etc. In multiplication, they may profitably use either form of expression — 2 times 2, 3 times 2, 4 times 2, etc. ; or, two 2's, three 2's, four 2's, etc. Sometimes one of these expressions is preferable, sometimes the other. 6. The explanations given are not to be committed to memory. The definitions and principles, having been fully comprehended, ought to bb fixed in the memory. The rules may or may not be learned, as teacliers shall prefer. 7. Fractions should be amply illustrated by material objects, atten- tion being specially called to the number and the size of the parts into which a thing is divided. 8. Care should be taken that the explanations given by pupils are logical in order and accurate in expression. 4 CONTENTS. INTEGERS. SECTION PA« I. — Numbers 7 II. — Notation and Numeration 11 III. — Addition 16 IV.— Subtraction 27 V.— Review 37 VI. — Multiplication 40 VII.— Division 52 VIII.— Review 65 FACTORING. IX.— Factors 68 X.— Divisors 70 XL — Multiples 72 XII.— Cancellation 74 FRACTIONS. XIII.— Notation 77 XIV.— Reduction 82 XV.— Addition 92 XVI.— Subtraction 94 XVII. — Multiplication 97 XVIII.— Division 105 XIX.— Review Ill UNITED STATES MONEY. XX.— Notation 116 XXL— Reduction 119 XXIL— Computations 121 1* 5 6 CONTENTS. DENOMINATE NUMBERS. SECTION PAG-i XXIII. — Measures of Extension 13< XXIV. — Measures of Capacity Y6V XXV. — Measures of Weight 142 XXVI.— Circular Measures „. 146 XXVII. — Measures of Time 148 XXVIII.— Paper and Counting 150 XXIX.— Keview lt><> COMPOUND NUMBERS. XXX.— Keduction 1j>4 XXXI.— Addition 158 XXXII.— Subtraction 161 XXXI II.— Multiplication 165 XXXIV.— Division 168 XXXV.— Review 171 DECIMALS. XXXVI. — Notation and Numeration 174 XXXVII. — Addition and Subtraction 178 XXXVIIL— Multiplication 180 XXXIX.— Division 182 XL. — Reduction 184 XLI.— Review 187 PERCENTAGE. XLII— Notation 188 XLIII.— Cases 190 XLIV— Interest 1« XLV.— Review 19b APPENDIX. Rectangular Measurements 201 Miscellaneous Problems 209 Answers to Written Exercises 215 Elementary Arithmetic. SECTION I. NUMBERS. ARTICLE 1. — 1. Arthur has one white rabbit and one gray rabbit. How many rabbits has he ? 2. Arthur has two rabbits, and his sister has one. How many rabbits have both of them ? 3. Jane had two books, and her father gave her two more. How many books had she then ? 4. If you have three cents in one pocket, and two cents in another, how many have you in both ? 5. James gave four cents for an orange, and two cents for an apple. How many cents did he give for both ? 8 NUMBERS. 6. If you count the fingers and thumb on your right hand, how many will you find them to be ? 7. How many are six books and two books ? 8. Arthur has five lead pencils and four slate pencils. How many pencils has he ? 9. Six trees and three trees are how many trees ? 10. How many are one and one? Two and one? Three and one ? Four and one ? 11. How many are five and one? Six and one? Seven and one? Eight and one? Nine and one? 12. How many are two and two? Three and two? Four and two ? Five and two ? Six and two ? 13. How many are seven and two? Eight and two ? Three and two ? Three and three ? Four and three? 14. How many are five and three ? Six and three ? Seven and three ? Four and four ? 15. How many are five and four? Six and four? Five and five ? 16. Count from one to ten. Count the fingers and thumbs which you have on both hands. 17. If you had five books, and should have three more given you, how many books would you then have ? 18. How many are five and three ? Three and five ? 19. How many are six and two? Eight and two? Four and three ? Seven and three ? 20. If you have six books, and have four more given you, how many books will you then have ? NUMBERS. 9 2. — 1. How many ones in a collection of one one and one one ? Of two ones and one one ? 2. How many ones in a collection of nine ones and one one ? Of ten ones and one one ? 3. How many are nine and two ? Eight and three ? Nine and three ? Seven and four ? 4. How many dollars are six dollars and four dollars? Seven dollars and three dollars ? 5. How many are eight and four? Six and five? Six and six ? 6. How many are eight and six? Nine and six? Seven and seven? Eight and seven? 7. How many are eight and eight? Nine and eight? Nine and nine ? 8. How many are ten and five? Ten and four? Seven and four ? Nine and three ? 9. How many are five and five? Nine and five? Seven and six ? Eight and six ? 10. How many are seven and seven ? Ten and six ? Eight and eight? 11. How many are six and ten? Nine and six? Seven and eight ? Eight and nine ? 12. How many are eight and ten? Ten and nine? Nine and ten ? Ten and ten ? Count from one to twenty. 13. How many tens are one ten and one ten ? Two tens and three tens ? Count from one to thirty. 14. How many tens are four tens and three tens ? Six tens and four tens ? Count from one to one hundred. 1U NUMBERS. DEFINITIONS. 3. A Unit is one, or a single thing of any kind. 4. A Number is a unit, or a collection of units. 5. A Figure is a character used to express a number. Each of the first nine numbers is expressed by u single figure, thus — printed. 1, 2, 3, 4, 5, 6, 7, 8, 9. WR.TTEN. /234$6/8? NAMED. One, two, three, four, five, six, seven, eight, nine. The figure 0, which is called Zero, or Cipher, ex- presses the absence of number. 1 ten is named Ten. 6 tens are named Sixty. 2 tens are named Twenty. 7 tens are named Seventy. 3 tens are named Thirty. 8 tens are named Eighty. 4 tens are named Forty. 9 tens are named Ninety. 5 tens are named Fifty. 10 tens are named One hundred. When a number is expressed by two figures, side by side, the figure on the right, or in the first place, ex- presses Ones, and the figure on the left, or in the second place, expresses Tens. Thus, 57 expresses 5 tens 7 ones, or fifty-seven. Wit I TTEX EXJE It CIS ES. 6. Copy and name the number expressed — (1.) (2.) (3.) (4.) (5.) (6.) 19 17 16 62 75 93 91 70 74 53 42 37 23 62 57 77 57 82 36 91 80 66 90 28 41 82 11 46 28 69 55 73 85 64 82 99 NUMERATION AND NOTATION, 11 SECTION IX NUMERATION AND NOTATION. 7. — 1. By combining what two figures do we express ten, or 1 ten .0 ones ? Eleven, or 1 ten 1 one ? 2. By combining what two figures do we express twelve, or 1 ten 2 ones ? Thirteen, or 1 ten 3 ones ? 3. By combining what two figures do we express nineteen, or 1 ten 9 ones? Ninety-one, or 9 tens 1 one? 4. By combining what two figures do we express seventy-five, or 7 tens 5 ones ? Fifty-seven, or 5 tens 7 ones? 5. By combining what two figures do we express eighty, or 8 tens ones ? Ninety-three, or 9 tens 3 ones? DEFINITIONS. 8. Numeration is the method of naming numbers. 9. Notation is the method of writing numbers. 10. In Naming Numbers, ten ones are named one ten, ten tens are named one hundred, and so on. 11. A figure written alone, or at the left of a point (,\ called the Decimal Point, expresses ones, or Primary Units. 12. Orders of Units are expressed by the successive figures written side by side to express number. When a number is expressed by three figures, the first figure at the left of the decimal point expresses units of the First Order, or Ones; the second figure at the left of the point expresses units of the Second Order, or 12 NUMERATION AND NOTATION. Tens; the third figure at the left of the point expresses units of the Third Order, or Hundreds, and so on„ Thus, 365 expresses 5 units of the first order, 6 units of the second order, 3 units of the third order, or three hundred sixty- five. 13. In Reading Numbers, each of the three orders of figures at the left of the decimal point expresses a Class or Period of units, with a distinct name, having ones, tens and hundreds, as shown in the following NUMEEATION TABLE. PERIODS. Millions. Thousands. Units. 2 2 2 « ORDERS. which are 897 tons. Since he had 1367 tons and sold 897 tons. 470 tons. 897 tons, he then had left the difference between 1367 tons and 897 tons, which is 470 tons. 8. I had 3675 dollars, and gained 1320 dollars. If I should pay away 2720 dollars, how much would I have left? 9. How much is 3675 — 539 added to 363 + 73 ? 10. How much is the sum of 7867 and 1319, dimin- ished by 4261 ? 40 MULTIPLICATION. SECTION VI. MUL TIPLICA TION. 74. — 1. If John can remove 2 books at one time, how many can he remove at two times ? 2. How many are 2 books and 2 books, or 2 times 2 books ? 3. If you have 2 cherries, and James has 3 times as many, how many has James ? 4. How many are 2 and 2 and 2, or 3 times 2 ? 5. If you have 4 fingers on each hand, how many have you on both hands ? 6. John has 3 blocks, and his brother has 3 times as many. How many has his brother ? 7. If you should get 3 merit-marks each day, how many would you get in 4 days ? 8. How many are 4 times 3 ? 3 times 4 ? M UL Tl PLICA TIOX. 41 9. On one twig there are 4 cherries ; how many cherries are there on 2 similar twigs ? 10. How many cents are 3 times 4 cents ? 11. There are four pecks in one bushel ; how many pecks are there in 4 bushels ? 12. When 6 cents are paid for 1 quart of milk, how much must be paid for 2 quarts ? 13. How many are 2 times 6 ? How many cents are 2 times 6 cents ? 14. How many are 5 and 5 and 5, or 3 times 5 ? 15. At the rate of 5 dollars a week, how many dollars can be earned in 3 weeks ? 75. — 1. How many are once 1? Once 2? Once 3? Once 4? 2. How many are 2 times 1 ? 2 times 2 ? 2 t* 3? 2 times 4? 2 times 5? l 8 3. How many are 2 times 6 ? 2 times 7 ? 2 ti 8? 2 times 9? 2 times 10? 4. How many are 3 times 1 ? 3 times 2 ? 3 times 3? 3 times 4? 3 times 5? 5. How many are 3 times 6 ? 3 times 7 ? 3 times 8 ? 3 times 9 ? 3 times 10 ? 6. How many are 4 times 1 ? 4 times 2? 4 times 3 ? 4 times 4 ? 4 times 5 ? 7. How many are 4 times 6 ? 4 times 7 ? 4 times 8? 4 rimes 9? 4 times 10? 8. How many are 5 times 1 ? 3 ? 5 times 4 ? 5 times 5 ? 9. How many are 5 times 6 ? 8? 5 times 9? 5 times 10? 4 » 5 times 2 ? 5 times 5 times 7 ? 5 times 42 MUL TIPLICA TION. 10. How many are 2 times 2 ? 3 times 2 ? 4 times 2 ? ,5 times 2 ? 11. How many are 6 times 2 ? 7 times 2 ? 8 times 2?" 9 times 2? 10 times 2? 12. How many are 2 times 3 ? 3 times 3 ? 4 times 3? 5 times 3? 13. How many are 6 times 3 ? 7 times 3 ? 8 times 3? 9 times 3? 10 times 3? TABLES. Once 2 times 3 times 4 times 5 times 6 times 1 is 1 1 are 2 1 are 3 1 are 4 1 are 5 1 are 6 2 " 2 2 " 4 2 " 6 2 " 8 2 " 10 2 " 12 3 " 3 3 " 6 3 " 9 3 " 12 3 " 15 3 " 18 4 " 4 4 " 8 4 " 12 4 " 16 4 " 20 4 " 24 5 " 5 5 " 10 5 " 15 5 " 20 5 " 25 5 " 30 " 6 6 " 12 6 " 18 6 " 24 6 " 30 6 " 36 « 7 7 " 14 7 " 21 7 " 28 7 " 35 7 " 42 8 8 " 16 8 " 24 8 " 32 8 " 40 8 " 48 ' 9 9 " 18 9 " 27 9 " 36 9 " 45 9 " 54 74 « io 10 " 20 10 " 30 10 " 40 10 " 5o 10 " 60 111! "11 11 " 22 11 " 33 11 " 44 11 " 55 11 " 66 i2 " 12 12 " 24 12 " 36 12 " 48 12 " 60 12 " 72 7 times 8 times 9 times 10 times 11 times 12 times lare 7 lare 8 lare 9 1 are 10 lare 11 lare 12 2 " 14 2 " 16 2 " 18 2 " 20 2 " 22 2 " 24 3 " 21 3 " 24 3 " 27 3 " 30 3 " 33 3 " 36 4 " 28 4 " 32 4 " 36 4 " 40 4 " 44 4 " 48 5 " 35 5 " 40 5 " 45 5 " 50 5 " 55 5 " 60 6 " 42 6 " 48 6 " 54 6 " 60 6" 66 6 " 72 7 " 49 7 " 56 7 « 63 7 " 70 7 " 77 7 " 84 8 " 56 8 " 64 8 " 72 8 " 80 8 " 88 8 " 96 9 " 63 9 " 72 9 " 81 9 " 90 9 " 99 9 " 108 10 " 70 10 " 80 10 " 90 10 " 100 10 "110 10 " 120 11 " 77 M " 88 11 " 99 11 "110 11 "121 11 "132 12 " 84 12 " 96 12 "108 12 "120 12 "132 12 "144 MUL TIPLICA TION. 43 DEFINITIONS. 76. Multiplication is the process of taking one of t^o numbers as many times as there are units in the other. 77. The Multiplicand is the number to be multiplied. 78. The Multiplier is the number by which to multiply. 79. The Product is the result of the multiplication. The multiplicand and multiplier are Factors of the product. 80. The Sijrn of Multiplication is an oblique cross, X, mid is *ead times or multiplied by. Thus, 9 X 8 is read nine multiplied by eight, or eight times nine. MENTAL, EXERCISES. 81.— 1. What is the product of 9 by 7 ? Of 5 by 4 ? 2. What is the product of 7 by 5 ? Of 6 by 3 ? 3. If 1 orange cost 6 cents, how many cents will 8 oranges cost ? * Solution. — If 1 orange cost 6 cents, 8 oranges will cost 8 times 6 cents, which are 48 cents. 4. If you can earn 4 dollars in 1 week, how many dollars can you earn in 6 weeks ? 5. What will 8 pairs of boots cost at 7 dollars a pair ? 6. How many trees in 9 rows of 6 trees each ? 7. How many yards are 7 times 10 yards? 8. At 4 dollars each, what will 11 barrels of apples cost? 9. James had 12 chickens, and his brother 6 times as many. How many chickens had his brother ? 10. At 7 cents each, what will 9 oranges cost ? * 11. At 9 cents each, what will 9 pencils cost ? 44 M UL TIP LIC A TION. 12. If a horse eat 6 quarts of oats in 1 day, how many quarts will he eat in 6 days ? 13. At 10 cents each, what will 8 pencils cost ? 14. If 11 yards of calico be required for 1 dress, how many yards will be required for 3 dresses ? 15. If 12 cents must be paid for a quart of cherries, how many cents must be paid for 9 quarts ? WRITTEN EXERCISES. 82. Copy and multiply — (1.) Multiply 6 Solution.— Three times six are eighteen. By o Write 18 under the line at the bottom of the Product, 18 column ' (2.) (3.) (4.) Multiply 5 7 8 By ' _4 j* S Product, 20 64 (7.) Multiply 23 Begin with the ones, and multiply the ones g y g and the tens separately. Write the product of the ones by 3, which is Product, 69 9 ? U nder the column of ones. Write the product of the tens by 3, which is 6, under the column of tens. The result is 69, which is the product required. (5.) (6. 9 6 6 5 30 (8.) (9.) (10.) (11.) (12.) (18. Multiply 4.3 21 31 41 50 60 E, ' 2 A 6 5 7 8 Product, 86 186 350 MUL TIP LIC A TION. 45 MENTJLL EXERCISES. 83. — 1. What is the product of 9 and 8 multiplied together ? 2. What is the product of 3X5? Of 3X5X6? 3. What is the product of 4 X 3 X 2 ? Of 3 X 4 X2? Of 2X3X4? Of 3X2X4? 4. Is the product of the factors 4, 3 and 2 the same if multiplied together in different orders ? 5. In a certain garden there are 12 rows, with 11 trees in each row. How many trees are there in the garden ? 6. James is 12 years old, and his uncle is 5 times as old. What is the age of his uncle ? 7. If you earn 10 dollars in 1 month, how much can you earn in 10 months ? 8. How many cents must be paid for 12 yards of muslin at 12 cents a yard ? 9. How much must be paid for 9 quarts of cherries at 11 cents a quart ? Principle of Multiplication. 84. The product of factors is the same in whatever order they are multiplied. WRITTEN EXERCISES. 85.— 1. Multiply 87 by 4. Solution.— The product of the 7 ones Multiplicand, 87 b ? 4 is 28 ones > or 2 tens 8 ones * m» 7/.- ?: ~ / Write 8 under the line at the bottom Multiplier, A m , „ , .. ft of the column of ones, and reserve the 2 Product, 348 tens to unite with the tens of the next product. 46 MULTIPLICA TION. The product of the 8 tens by 4 is 32 tens, which, with the 2 tens that were reserved, make 34 tens. Write 34 under the line at the bottom of the column of tens. The result is 348, which is the product required. Write and solve in like manner — (2.) (3.) (4.) (5.) (6.) Multiply 93 68 75 34 92 By 5 6 8 7 9 Product, 465 600 828 (7.) (8.) (9.) (10.) Multiply 125 dollars. 84 bushels. 1 19 barrels. 137 'tons. By 3 7 2 5 Product, 375 dollars. 86.— 1. Multiply 25 by 10. By 30. 685 tons. Multiplicand, 25 Multiplier, 10 Product, 250 Multiplicand, 25 Multiplier, SO Product, 750 Solution. — To multiply by 10, simply annex a cipher to the multiplicand, be- cause the annexing of a cipher removes each figure one order to the left, and makes the value denoted tenfold as large as before. Solution. — Since 30 is 3 X 10, the product of 25 by 30 must be the product of 25 X 3 X 10. 3 times 25 is 75, and 10 times 75 is 750. Write and multiply in like manner — 5. 31 by 100. 2. 125 by 10. 3. 134hy40. 4. 133 by 20. 6. 62 by 500. 7. 124 by 700. M UL TI PLICA TION. 47 MENTAL EXERCISES. 87. — 1. If it take 6 men 8 days to do a piece of work, how many days will it take 1 man to do it ? Solution. — It will take 1 man 6 times as long as 6 men. If it take 6 men 8 days to do a piece of work, it will take 1 man 6 times 8 days, which are 48 days. 2. If 9 men can mow a field in 3 days, in what time can 1 man mow it ? 3. How many days must 1 man work to do a job which would require the labor of 7 men for 5 days ? 4. If a certain amount of provisions will last 9 per- sons 10 days, how long will it last 1 person ? 5. George is 8 years old, and his father is 4 years more than 3 times as old. What is the age of his father? 6. If 1 orange be worth 6 apples, how many apples must be given for 9 oranges ? 7. If you earn 12 dollars a month, and spend 7 dollars a month, how many dollars can you save in 12 months ? 8. When the price of apples is 3 cents each, and of lemons 5 cents each, what will 5 apples and 6 lemons cost? 9. If you can earn 40 cents a day, and you pay 30 cents a day for your board, how many cents can you save in 6 days ? 10. If it take 12 men to reap a field in 6 days, how many men will it take to reap it in 1 day ? 11. I bought 3 plows at 12 dollars each, and gave in payment 4 ten-dollar bills. How much change ought to be paid back to me ? 48 MUL TIP LIC A TION. 12. I bought 8 oranges at 5 cents each, and gave in payment a fifty-cent piece. How much change should be paid back to me ? 13. How many dollars must be paid for 5 tons of coal at 6 dollars a ton, and 4 cords of wood at 3 dollars a cord ? 14. Mary is 10 years old, and her mother is 4 years less than 4 times as old. What is the age of her mother ? 15. When blueberries are 9 cents a quart, and black- berries 12 cents, how much more will 8 quarts of black- berries cost than 8 quarts of blueberries ? WRITTEN EXERCISES. 88.— 1. Multiply 364 by 42. Solution. — 42 is equal to 4 tens and Multiplicand, 364 2 ones. 364 multiplied by 42 must Multiplier, 42 equal 2 times 364 -f 4 tens times 364. p . 7 ~. — V&q Write the multiplier under the mul- f /~o tiplicand, placing figures of the same Products, j 1456 or( ier in the same column. Product 15288 ^he P r °duct of 364 by the 2 ones is 728 ones, which is a partial product. 4 tens times 4 ones are 16 tens, or 1 hundred + 6 tens ; write 6 for the tens of a second partial product, and reserve the 1 hundred to unite with the product of the hundreds; 4 tens times 6 tens are 24 hundreds, which, with 1 hundred reserved, are 25 hundreds, or 2 thousands + 5 hundreds ; write 5 for the hun- dreds of the second partial product, and reserve the 2 thousands to unite with the product of the thousands ; 4 tens times 3 hun- dreds are 12 thousands, which, with 2 thousands, are 14 thou- sands ; write 14 for the thousands of the second partial product, which gives 1456 tens, or 14560. Find the required product by adding the two partial products, which gives 15288. Prove the Work by carefully reviewing it. MVL TIPLICA TION. 49 Multiply and prove in like manner — 2. 214 by 14. 3. 509 by 29. 4. 114 by 16. 5. 232 by ^i. 6. 444 by 55. 7. 420 by 17. 8. ^5 by 25, and prove by reversing the order of the factors. Multiplicand, Multiplier, Partial Products, Product, 43 25 \ 215 s, j 86 25 43 215 86 1075 Proof A 75 hoo \1075 Solution. — Reverse the order of the factors by taking 25 for the multiplicand and 43 for the multiplier. If the work is correct, the result must be the same by both processes, because the product of any set of factors must be the same in whatever order they may be multiplied. 9. 67 by 39, and prove by reversing the order of the factors. 89. Rule for Multiplication. — Write the multiplier under the multiplicand, placing figures of the same order in the same column. If the multiplier consists of but one order of units, multiply each order of the multiplicand in succes- sion, beginning at the right, by the multiplier, writing the right-hand figure of the result under the order multiplied, and uniting the units ex- pressed by the left-hand figure, if any, with the next result. If the multiplier consists of more than one order of units, multiply each order of the multiplicand by each order of the multiplier, write the right- hand figure of each partial product under the 50 M Vh TIPLICA TION. order of the multiplier, and add the partial prod- ucts. If either factor have one or more ciphers on the right, multi- ply without regard to the cipher or ciphers on the right of either factor, and annex to the result as many ciphers as there are on the right of the factors. Proof. — Review the work carefully, or reverse the order of the factors and multiply. The result should be the same by both methods. PKOBIjjEMS. 90. Find the product of — 1. 1251 dollars by 6. 2. 9 03 hogsheads by 4> 3. 1037 yards by 7. 4. 4009 bushels by 9. 91. — 1. What is the product of 642 multiplied by 403? Multiplicand, 642 ^ ' -it Omit to multiply by the tens, be- 1926 cause the product of any number by 2568 is 0. 5. 365 days by 27. 6. 58 miles by 43. 7. 731 tons by 1000. 8. 3140 acres by 120. Product, 258726 2. What is the product of 316 multiplied by 502 ? 3. What is the product of 1207 X 2001 ? 4. How many are 1005 times 1005? 5. What is the product of 56390 multiplied by 401 ? 6. What will 125 horses cost at 108 dollars each? 7. How many pounds of beef are there in 107 barrels, if each barrel contains 200 pounds ? M UL TIPLICA TION. 51 8. How many bricks in 203 loads, if each load con- tains 1037 bricks ? 92. — 1. What will a farm of 164 acres cost at 25 dollars per acre ? 2. How many soldiers in 16 regiments, if each regi- ment contains 819 men ? 3. If sound travel 1120 feet in 1 second, how many feet will it travel in 105 seconds ? 4. The multiplier is 68 and the multiplicand is 4320. What is the product ? 5. What is the product of one hundred thirteen mul- tiplied by itself? 6. How many cents will 3184 bushels of corn cost at 87 cents a bushel ? 7. What is the product of 555 X 44 X 6 ? 8. In a certain storehouse there are 21 apartments, and in each apartment 15 bins, each bin containing 490 bushels of wheat. How many bushels of wheat are there in all ? 9. A merchant shipped 11109 boxes of sugar, each weighing 198 pounds. What was the weight of the whole ? 10. If light travel at the rate of 192,000 miles in a second, how far will it travel in 5 minutes of 60 seconds each ? 93. Test Questions. — 1. What is multiplication? The multiplicand? The multiplier? The product? 2. What is the sign of multiplication? How is it read? How does it differ in form from the sign of addition ? 3. What are the factors of a product? Name a principle of multiplication. DIVISION. SECTION VII. DIVISION. 94. — 1. Two cows have 4 horns. How many horns has each cow? 2. How many times 2 horns are 4 horns ? 3. A farmer has 9 sheep. How many times 3 sheep has he ? 4. Nine sheep are how many times 3 sheep ? 5. A harrow has 12 teeth. How many times 3 teeth has it ? How many times 4 teeth has it ? 6. Twelve is how many times 4? How many times 3 ? 7. Eight fowls are how many times 2 fow T ls ? 8. A house has 16 windows. How many times 4 windows has it ? DIVISION. 53 95. — 1. How many times 1 in 2? In 3? In 4? In 5 ? In 6 ? 2. How many times 2 in 2 ? In 4 ? In 5 ? In 6 s ? In 8? In 10? 3. How many times 2 in 12 ? In 14 ? In 16 ? In 18? In 20? 4. How many times 3 in 3 ? In 6 ? In 9 ? In 12 5 In 15? 5. How many times 3 in 18 ? In 21 ? In 24 ? In 27? In 30? 6. How many times 4 in 4? In 8 ? In 12? In 16? In 20? 7. How many times 4 in 24 ? In 28 ? In 32 ? In 36? In 40? 8. How many times 5 in 5? In 10? In 15? In 20? In 25? 9. How many times 5 in 30 ? In 35 ? In 40 ? In 45? In 50? In 60? 10. What is one of the 2 equal parts of 6 ? Of 10? Of 16? 11. What is one of the 3 equal parts of 12 ? Of 15 ? Of 24? 12. What is one of the 4 equal parts of 8 ? Of 16 ? Of 40? 13. What is one of the 5 equal parts of 10 ? Of 20 ? Of 35 ? Of 45 ? 14. What is one of the 6 equal parts of 6 ? Of 12 ? Of 24? Of 30? 15. What is one of the 7 equal parts of 7 ? Of 21 ? Of 35? Of 42? 16. What is one of the 8 equal parts of 8 ? Of 32 ? 5* 54 DIVISION. TABLES. lin 2 in Sin 4 in 5 in 1, Once. 2, Once. 3, Once. 4, Once. 5, Once. 2iy Z times. 4, 2 times. 6, 2 times. 8, 2 times. 10, 2 times. 3, 3 " 6, 3 " 9, 3 " 12, 3 " 15, 3 " 4, 4 " 8, 4 " 12, 4 " 16, 4 " 20, 4 " 5, 5 " 10, 5 " 15, 5 " 20, 5 " 25, 5 " 6, 6 " 12, 6 " 18, 6 " 24, 6 " 30, 6 " 7, 7 " 14, 7 " 21, 7 " 28, 7 " 35, 7 " 8, 8 " 16, 8 " 24, 8 " 32, 8 " 40, 8 " 9, 9 " 18, 9 " 27, 9 " 36, 9 " 45, 9 " 10,10 " 20,10 " 30,10 " 40, 10 " 50,10 " 6 in 7 in 8 In 9 in 10 in 6, Once. v, Once. 8, Once. 9, Once. 10, Once. 12, 2 times. 14, 2 times. 16, 2 times. 18, 2 times. 2i\Jj Jtiines. 18, 3 " 21, 3 " 24, 3 " 27, 3 " 30,3 " 24, 4 " 28, 4 " 32, 4 " 36, 4 " 40,4 " 30, 5 " 35, 5 " 40, 5 " 45, 5 " 50, 5 " 36, 6 " 42, 6 " 48, 6 " 54, 6 " 60, 6 " 42, 7 " 49, 7 " 56, 7 " 63, 7 " 70, 7 « 48, 8 " 56, 8 " 64, 8 " 72, 8 " 80, 8 " 54, 9 " 63, 9 " 72, 9 " 81, 9 « 90, 9 " 60,10 " 70, 10 " 80,10 " 90, 10 " 100,10 " 96. — 1. If any number of things are separated into two equal parts, what is each part called ? One half of the number of things. 2. If Arthur has 6 apples, and should distribute them equally between 2 friends, how many apples would each friend receive? Solution. — If Arthur has 6 apples, and should distribute them equally between 2 friends, each friend would receive one half of 6 apples, which is 3 apples. 3. If you should wish to distribute 10 cents equally DIVISION. oa between 2 boys, what part of 10 cents would each boy receive ? 4. If you have 12 pears, and should give one half of them to your brother, how many would he receive ? 5. If any number of things be separated into three equal parts, what is each part called ? One third of the number of things. 6. If 3 boys should share 12 apples, equally dividing them, what part of 12 apples would each have? 7. What is one third of 18 dollars ? Of 24 gallons ? Of 30 bushels ? Of 36 days ? 8. When any number is separated into four equal parts, what is each part called ? One fourth of that number. 9. If 20 pears be shared equally among 4 boys, what part of the 20 pears will each boy have ? 10. How many is one fourth of 16 ? Of 24 ? 11. If any number be separated into five equal parts, what is each of the parts called ? One fifth of the number. 12. How many is one fifth of 35 ? Of 45 ? Of 50 ? 13. When any number is separated into six equal parts, into seven equal parts, etc., what is each of these parts called ? One sixth, one seventh, etc., of the number. 14. What is one sixth of 48 ? Of 60 ? One seventh of 21? Of 35? One eighth of 24? Of 40? Of 72? 15. If 6 hats cost 48 dollars, what will 1 hat cost? 16. What is one tenth of 30 ? Of 100 ? Of 120 ? 17. If 10 barrels of flour cost 100 dollars, what will 1 barrel cost ? 56 DIVISION. DEFINITIONS. 97. — 1. Division is the process of finding how many times one of two given numbers is contained in the other; or, Division is the process of separating one of two given numbers into as many equal parts as there are units in the other. 98. The Dividend is the number to be divided. 99. The Divisor is the number by which to divide. 100. The Quotient is the result of the division. 101. The Sign of Division is a short horizontal line, with a dot above and another below it, -^-, or a short upright curved line, ), and is read divided by. Thus, 40 -*- 8, or 8)40, is read, forty divided by eight. WRITTEN EXERCISES. 102. Copy and divide- r* • • oio n - -j j Solution.— 2 is contained in 8, Divisor, 2)8 Dividend. __ r . , , ,. .' 9 — 4 times. Write 4 under the divi- 4 Quotient. dend for the quotient. (2.) (3.) (4.) (5.) (6.) 3)18 5)15 4)12 6)30 8)J8 q)zq Solution. — 8 is contained in 50, 6 times, and 2 remain. Write 6 as the quotient, and 6, 2 Rem. t ^ e 2 as a remainder. (8.) (9.) (10.) (11.) (12.) 7)57 6)55 8)75 9)J7 7)58 Multiply the quotient by the divisor, and to this DIVISION. 57 product add the remainder, in each of the last five problems. If this result equals the dividend, the work is correct. MENTAL EXERCISES. 103. — 1. What do you understand by one half of a number ? One of the two equal parts into which the number is divided. 2. What do you understand by one third of a num- ber ? By one fourth of a number ? 3. What do you understand by two thirds of a num- ber ? By two fourths of a number ? 4. What part of 5 is 1 ? Is 2? Is 3? Is 4? Solution. — One is 1 fifth of 5 ; 2 is 2 times 1 fifth of 5, or 2 fifths of 5 ; 3 is 3 times 1 fifth of 5, or 3 fifths of 5 ; and 4 is 4 times 1 fifth of 5, or 4 fifths of 5. 5. What part of 6 is 1 ? Is 2? Is 3? Is 4? Is 5? 6. Seven is how many times 3 ? Solution. — Seven is 2 times 3, and 1 remains, which is 1 third of 3. Therefore, 7 is 2 and 1 third times 3. 7. Nine is how many times 2? 4? 3? 8? 8. Eight is how many times 6 ? 9. Nine is how many times 6 ? 7 ? 8 ? 10. At 8 dollars a yard, how many yards of velvet can be bought for 33 dollars ? 11. At 9 cents a pound, how many pounds of rice can be bought for 47 cents ? 12. 15 is how many times 4? Solution. — Fifteen is 3 times 4, with a remainder 3 ; or 3 and 3 fourths times 4. 58 DIVISION. 13. How many yards of braid, at 7 cents a yard, can be bought for 52 cents ? 14. Thirty-eight is how many times 7 ? 5 ? 15. Fifty-nine is how many times 5 ? 8 ? 16. If 9 pieces of cloth cost 95 dollars, what is the cost of each piece ? 17. Eighty-three is how many times 9 ? 7 ? 18. What is one fifth of 52 ? Of 63 ? Of 49 ? 19. What is one seventh of 31 ? Of 57 ? Of 68 ? 20. What is one eighth of 46 ? Of 39? Of 73 ? 21. If 75 persons are to cross a stream in a boat which will carry only 8 persons at a time, how many persons will remain after as many full boat-loads as pos- sible have crossed ? 22. If 10 bushels will fill a bin, how many bins can be filled from 87 bushels of wheat, and how many bushels will remain ? 23. How many plows, worth 11 dollars each, can be bought with 93 dollars, and how many dollars will be left? 24. In one hundred and thirty eggs are how many dozen, and how many remain ? DEFINITIONS. 104. A Remainder is a part of the dividend which may remain undivided. 105. An Integer is a number composed of entire units or ones. 106. A Fraction, as one half, two thirds, etc., is a number which represents one or more of the equal parts of a unit or one. DIVISION. 59 One half is written J, one third is written J, two thirds are written f , three fourths are written f, and so on. In such expressions the number written below the line denotes the number of parts into which the unit is divided, and the number written above the line denotes the number of parts taken. Principles of Division. 107. — 1. Division is the reverse of multiplication. 2. The dividend is equal to the product of the integer of the quotient multiplied by the divisor, plus the re- mainder. WRITTEN EXERCISES. 108. — 1. Divide 4315 by 4, or separate 4315 into 4 equal parts ; and prove the solution. Divisor, 4)4315 Dividend. Solution.— One of the 4 3 t equal parts of a number is 107 8^ Quotient one fourth of that uum ber. One fourth of 4 thousands is 1 thousand. Write 1 in the thousands' order in the quotient. One fourth of 3 hundreds is number of hundreds. Write in the hundreds' order in the quotient, and unite the 3 hun- dreds with the 1 ten, making 31 tens. One fourth of 31 tens is 7 tens, with 3 tens remaining. Write 7 in the tens' order in the quotient, and unite the 3 tens with the 5 ones, making 35 ones. One fourth of 35 ones is 8 ones, with 3 ones remaining,, Write the 8 in the ones' order in the quotient. Write the remainder, 3, over the divisor as a part of the quotient. The 3 written over 4, or f, may be regarded as indicating the division of 3 by 4. The result is 1078f , which is the quotient required. 60 DIVISION. 1078 J^ Proof. — To prove the work, multiply the integer T0T9 of the quotient by the divisor, and add to the product „ the remainder. If the work is correct, this result will equal the dividend. 4815 When only the divisor, dividend and quotient are written, the process is called Short Division. Divide, by short division, explain and prove — (2.) 4)3101 ~T75l (6.) 7)300 (3.) 5)5163 103 2 j (7.) 3)975 (4.) 7)814 (8.) 6)1992 (5.) 8)1137 (9.) 9)2889 10. How many is one fourth of 731 apples? 11. What is the quotient of 1363 days divided by 2? 12. What is the quotient of 1563 divided by 10 ? 10)1563 156.3 Or, 156f Solution. — To divide by 10, simply move the decimal point in the dividend one order to the left, because this changes each figure to an order next lower, and makes the value expressed only one tenth as large as before. The remainder is 3, and written at the right of the point, ex- presses 3 tenths, or written over the divisor, T \. The result is 156 T V, which is the quotient required. 13. What is the quotient of 715 divided by 100? 14. What is the quotient of 1634 divided by 10? 15. What is the quotient of 1783 divided by 1000? DIVISION. 61 MENTAL EXERCISES. 109. — 1. How many pencils, at 7 cents each, can be bought for 56 cents ? Solution. — Since one pencil can be bought for 7 cents, as many pencils can be bought for 5G cents as there are times 7 in 56, which are 8 times. Therefore, 8 pencils at 7 cents each can be bought for 56 cents. 2. How many barrels of flour, at 8 dollars each, can be bought for 64 dollars ? 3. How many pounds of sugar, at 9 cents a pound, can be bought for 72 cents ? 4. If you have 60 dollars, and spend it at the rate of 10 dollars a week, how many weeks will your money last you ? 5. If twelve eggs are 1 dozen, how many dozen are 72 eggs? 6. When pine-apples are 8 cents each, how many can be bought for 64 cents ? 7. When 96 dollars are paid for 12 barrels of flour, how much is paid for each barrel ? 8. How many melons can be exchanged for 110 peaches, at the rate of 11 peaches for 1 melon? 9. How many pounds of beef, at 11 cents a pound, can be bought for 7 9 "cents, and how many cents will remain ? 10. At 12 cents a pound, how many pounds of sugar, and what part of .a pound, can be bought for 109 dollars ? 11. How many are 12 times 9, plus 1 ? 9 times 12, plus 1 ? 12 John gained 24 dollars by purchasing wood at 5 dollars a cord and selling it at 8 dollars a cord. How many cords did he buy ? 6 62 DIVISION. WRITTEN EXERCISES. 110. — 1. Divide 672 by 4, and prove the solution. Solution. — 4 is contained in 6 hundred^ iY18 / T )^oV iaq * nun ^ re< ^ times - Write 1 in the order of 4yOi^\ loo hundreds in the quotient, which for con- 4 venience is placed on the right of the divi- 27 . dend. 2 J. Multiply 4, the divisor, by the 1 hundred, making 4 hundreds, which write under the 0/0 6 hundreds, and subtract from it, leaving 2 32 hundreds. Unite the 2 hundreds with the 7 tens, making 27 tens. 4 is contained in 27 tens, 6 tens times. Write the 6 in the order of tens in the quotient. Multiply the divisor by the 6 tens, making 24 tens, which write under the 27 tens, and subtract, leaving 3 tens. Unite the 3 tens with the 2 ones, making 32 ones. 4 is contained in 32 ones, 8 times. Write the 8 in the order of ones in the quotient. The result is 168, which is the quotient required. - LUO Proof. — To prove the work, multiply the quotient -4 by the divisor. If this product equals the dividend, 672 tne work is correct. When each step of the solution is written, the process is called Long Division. Divide and prove in like manner — 2. 540 by 5. a 896 by 2. 4. 822 hy 6. 5. 423 by 3. 6. 57^by^. 7. 655 by 5. 111. Rule for Division.— Write the divisor at the left of the dividend. Divide the least number of the left-hand orders 8. 936 by 8. 9. 796 by 2. 10. 504 by 7. DIVISION. 63 of the dividend thai will contain the divisor, and place the quotient at the right of the dividend, in long division, and beneath the dividend in short division. Multiply the divisor by this quotient; subtract the result from that part of the dividend which was used; to the remainder annex the next order of the dividend, and divide the number thus formed. Proceed in the same manner until all the orders of the dividend have been used. Should there be at last a remainder, write it, with the divisor under it, as a part of the quotient. When the divisor is 1, with one or more ciphers on the right, move the decimal point in the dividend as many orders to the left as there are ciphers on the right of the divisor. The orders on the left of the point will be the integer of the quotient, and the orders on the right the remainder expressed as a fractional part of the quotient. Proof. — Multiply the integer of the quotient by the divisor, and add to the product the remainder, if any. If the work is correct, this result will equal the dividend. PMOBZEMS. 112. Divide, explain and prove — (1.) (2.) (3.) (4.) 2)1363 8)4602 7)703 4 )2060 (5.) (6.) (7.) (8.) 14)4651 15)3910 21)443 12)6702 64 DIVISION. 113. How many is — 1. One fourth of 731 apples? 2. One third of 563 cents? 3. One sixth of 802 feet? 4. One seventh of 415 rods? 5. One ninth of 629 dollars ? What is the quotient of—, 6. 6363 days -=- 11 ? 7. 2741 pounds --13? 8. 1790 dollars --7? 9. 8000 cents -=- 25 ? 10. 4350 yards ~ 9 ? 114.— 1. What is one tenth of 3587? 2. If 15 acres of land cost 6090 dollars, how many dollars will one acre cost ? 3. How many hours will it take a train of cars to move 1225 miles, at the rate of 25 miles per hour? 4. What number is equal to 1488 -- 24 ? 5. What number is equal to 4141 -=- 101 ? 6. How many times may a 31 -gallon cask be filled from a vat containing 1929 gallons, and how many gallons will remain? 7. Into how many lots of 10 acres each can 365 acres of land be divided, and how many acres will remain un- divided ? 8. If 100 feet of boards will make a box, how many similar boxes can be made from 5767 feet of boards, and how many feet will remain ? 115. Test Questions.— 1. What is division ? What is the dividend ? The divisor ? The quotient ? 2. What is the sign of division? Show how it is used? What does it mean ? 3. What is a remainder? An integer? A fraction? 4. What is short division? Long division ? What principles of division are given ? ME VIEW. 65 SECTION VIII. REVIEW. 116. — 1. By what process do you find the product of two or more numbers ? 2. By what process do you find the quotient of one number divided by another ? 3. How do multiplication and division differ ? 4. If the multiplicand is dollars, what will be the de- nomination of the product? 5. If the multiplicand is 6 and the multiplier 4, how many times the multiplicand is the product ? 6. What does the product express ? 7. If the divisor is 4 dollars and the dividend 24 dollars, what does the quotient express ? 8. When the divisor is 4 and the dividend 24 dollars, what part of 24 dollars is the quotient ? 9. To what term in multiplication does the dividend in division correspond ? 10. To what terms in division do the factors in multi- plication correspond ? 11. How many are 7 times 12, plus 8 ? 12. How many are 9 times 11, minus 7 ? 13. When 5 tons of coal cost 35 dollars, what will 7 tons cost ? Solution. — If 5 tons cost 35 dollars, 1 ton will cost one-fifth of 35 dollars, or 7 dollars, and 7 tons will cost 7 times 7 dollars, which are 49 dollars. 14. When 8 barrels of flour cost 88 dollars, what will 5 barrels cost ? 06 REVIEW. 15. James bought 12 bags of meal for 24 dollars, and Smith bought 11 bags at the same rate. How many dollars did Smith pay for his meal ? 16. If you can earn 56 dollars in 8 weeks, how much can you earn in 3 weeks ? 17. If a horse can trot at the rate of 24 miles in 3 hours, how far can he trot in 2 hours ? 18. If 5 men can do a piece of work in 6 days, in how many days can 3 men do it ? Solution. — If 5 men Can do a piece of work in 6 days, 1 man will require 5 times 6 days, or 30 days, to do it, and 3 men will require one third of 30 days, or 10 days. 19. If 6 men can mow a field in 8 days, in how many days can 12 men mow it? 20. When 8 bushels of wheat will pay for 4 yards of cloth at 4 dollars a yard, what is the value of a bushel of wheat? 21. When 4 yards of cloth can be bought for 16 dol- lars, how many yards of cloth can be bought for 40 dollars ? Solution.— If 4 yards can be bought for 16 dollars, 1 yard can be bought for one fourth of 16 dollars, or 4 dollars. Since 1 yard can be bought for 4 dollars, as many yards can be bought for 40 dollars as there are times 4 in 40, or 10. 22. If 5 horses will consume 60 bushels of oats in a certain time, how many horses will consume 72 bushels in the same time ? 23. When 99 dollars will pay for 11 barrels of flour, how many barrels can be bought for 63 dollars ? 24. If you can earn 108 dollars in 12 months, how many dollars can you earn in 8 months ? MB VIEW. 67 WRITTEN EXERCISES. 117. — 1. A merchant bought 12 hogsheads of molasses it 65 dollars a hogshead, and sold them for 805 dollars. How much did he gain by the transaction ? 2. From 6304 X 15, subtract 372 -- 3. 3. If you sell 506 hats, which cost you 4 dollars each, for 6 dollars each, how much will you gain ? 4. From 1035 X 6, take 1060 X 5. 5. I bought 8 horses at 225 dollars each, and 17 yoke of oxen at 150 dollars a yoke. What is the cost of the whole? ■ 6. If a man's salary is 5000 dollars a year, and his expenses are 225 dollars a month, how much can he save each month ? 7. A farmer has 131 bushels of corn, twice as much rye, and three times as much wheat. What quantity of rye and of wheat has he ? 8. When 19 acres of land cost 760 dollars, how many acres can be bought for 2180 dollars? 9. If a quantity of provisions will last 355 men 25 days, how long will it last 5 men ? 10. When 755 dollars are paid for 5 horses, how much must be paid for 31 horses ? 11. I bought a farm containing 120 acres for 7200 dollars, and sold it for 75 dollars an acre. How much did I gain by the transaction ? 12. How much is 255 X 12 divided by 51 X 8 ? 13. How many casks, holding 31 gallons each, can be filled from 17 hogsheads of molasses, containing each 93 gallons ? 68 FACTORING. SECTION IX. FACTORS. 118,-1. How many ones in 2 ? In 3 ? In 10 ? 2. What two integers, then, multiplied together, will produce 2 ? 3 ? 10 ? What 15? 21 ? 3. What integers, other than itself and 1, multiplied together, will produce 14 ? 22 ? 39 ? 4. What integers, other than itself and 1, will divide 14 without a remainder ? 22 ? 39 ? 5. What integers, multiplied together, will produce 9 ? 10? 16? 18? 6. W T hat are all the integers which will divide 9 without a remainder ? 10? 16? 18? 7. Name some number which is not the product of any integers except itself and 1. 8. Name some number which is the product of other integers than itself and 1. DEFINITIONS. 119. The Factors of a number are the integers which, being multiplied together, will produce that number. 120. A Prime Number is an integer that has no factor except itself and 1. 121. A Composite Number is an integer that has other factors besides itself and 1. 122. The Prime Factors of a number are the prime numbers that are factors of that number. Since 1 is a factor of all numbers, in naming the prime fac- tors of numbers it need not be given. FACTORING. 69 123, Factoring is the process of finding the factors of composite numbers. 124—1. What are the prime factors of 12? 14? 16? 2. What are the prime factors of 15 ? 18 ? 20 ? 3. What are the prime factors of 21 ? 28 ? 30 ? 4. What are the prime factors of 33 ? 45 ? 49 ? 5. What are the prime factors of 27 ? 42 ? 44 ? 6. What are the prime factors of 36 ? 40 ? 54 ? 7. Name the prime numbers from 1 to 19. From 19 to 29. 8. Name the prime numbers from 31 to 41. From 41 to 83. 9. Name the composite numbers from 4 to 20. From 24 to 36. From 36 to 48. From 48 to 60. 125. Principle. — Every composite number is equal to the product of all its prime factors. WRITTEN EXERCISES. 126. — 1. What are the prime factors of 90? 2)90 Solution.— Dividing 90 by the n jyz prime number 2, we have as factors ^ 2 and the composite number 45. 3)15 Dividing 45 by the prime num- 5 bar 3, we have as factors 3 and 15. on— 4>V*V*V* dividing again by 3, we have as VU - ^AJAJAO factors 3 and 5 which are prime numbers. Hence, 90 = 2X3X3X5, and the prime factors of 90 are 2, 3, 3 and 5. 2. What are the prime factors of 84 ? 3. What are the prime factors of 75 ? 4. What are the prime factors of 96 ? 70 FACTORING. 127, Rule for finding the Prime Factors of a Number.— Divide the given number by any prime number greater than 1 that will divide it without a remainder and the quotient, if composite, in the same man- ner; and so proceed until a quotient is obtained which is a prime number. The last quotient and the several divisors are the prime factors. PROBLEMS. 128. "What are the prime factors of— 1. 95? 4. 122? 7. 108? 10. 148? 2. 63? 5. 116? 8. 200? 11. 210? 3. 92? 6. 184? 9. 728? 12. 735? SECTION X. DIVISORS. 129. — 1. What integers will divide 15 without a re- mainder? 21? 35? 2. What integers will divide 16 without a remainder? 27 ? 42 ? 3. What integers will divide 44 without a remainder ? 77? 81? 4. What factors have 6 and 24 in common? 15 and 20 \ 5. What factors have 22 and 25 in common ? 6. What prime factors have 12 and 18 in common? 7. W r hat is the greatest factor common to 12 and 18 ? 8. What is the product of the prime factors common to 12 and 18? FACTORING. 71 9. What is the product of the prime factors common to 8 and 24 ? 10. What is the greatest factor common to 3 and 24 ? DEFINITIONS. 130. A Divisor, or Measure, of a number is any factor of that number. 131. A Common Divisor of two or more numbers is any factor common to those numbers. 132. The Greatest Common Divisor of two or more num- bers is the greatest factor common to those numbers. 133. Principle. — The greatest common factor, or divi- sor, of two or more numbers is equal to the product of all the common prime factors of those numbers. WRITTEN EXERCISES. 134. — 1. What is the greatest common factor or di- visor of 330 and 550 ? 330 = 2 X 3 X 5 X 11 Solution. — Find the prime QfiO = 2 X 5 X 5 X 11 Actors of the numbers. ^y ry 1 1 = 1 in ^ ne P r i me factors common to the numbers are 2, 5 and 11, and their product is 110. Hence, the greatest common factor or divisor is 110. 2. Find the greatest common factor of 27 and 36. 3. Find the greatest common factor of 42 and 35. 4. Find the greatest common divisor of 14, 35 and 63. 135. Rule for finding the Greatest Common Divisor.— Find the prime factors common to the given numbers, anal the product of those factors will be the greatest co /union divisor of the numbers. 72 FACT0R1XG. PROBLEMS. 136. What is the greatest common factor or divisor of—* 1. 26 and 39? 14. 18 and 96? ! 7. 4, 6 and 18? 2. 17 and 51? \ 5. 45 and 300? 3. 27 and 63? 6. 21 and 105? 8. 12, 30 and 84? 9. 18, 54 and 72? 10. What is the length of the longest pole which will exactly measure 130, 150 or 170 feet? SECTION XI. MULTIPLES. 137. — 1. Name the numbers from 3 to 15 that will contain 3 without a remainder. 2. What are some of the numbers that are an exact number of times 3 ? 3. What are some of the numbers that are an exact number of times 5 ? Of times 8 ? Of times 9 ? 4. Of what integers is 12 an exact number of times? 5. Name some number that will contain either 3 or 4 an exact number of times. 6. Name some dividend that will exactly contain 2 and 6. 5 and 7. 6 and 9. 7. What is the least dividend that will contain both 10 and 15 an exact number of times? 8. What are all the prime factors of 10 and 15 ? 9. What are prime factors of the least number that will exactly contain both 10 and 15? FACTORING. 73 DEFINITIONS. 138. A Multiple of a number is any number which it will divide without a remainder. 139. A Common Multiple of two or more numbers is any number which each of those numbers will divide without a remainder. 140. The Least Common Multiple of two or more num- bers is the least number that each of those numbers will divide without a remainder. 141. Principle. — The least common multiple of two or more numbers is the least number that contains all the prime factors of those numbers. WRITTEN EXERCISES. 142. — 1. What is the least common multiple of 4, 12 and 30? / — 2 X 2. Solution. — Find the prime fac- 1Q = Q V I? X ? * ors °^ ^ ne gi ven numbers. to* ^y ^y - The multiple of 4 must contain "\ ' the prime factors 2 and 2 ; the % X 2 X 3 X o = 60. mu itiple of 12, the additional prime factor 3 ; and the multiple of 30, the additional prime factor 5. These factors are 2, 2, 3 and 5, and their product is 60. There- fore, the least common multiple of 4, 12 and 30 is 60. 2. Find the least common multiple of 8 and 12. 3. Find the least common multiple of 4, 6 and 20. 143. Rule for finding the Least Common Multiple.— First find the prime factors of the given numbers. The prod- uct of these different prime factors, each factor being taken the greatest number of times it occurs in any of the numbers, will be their least common } multiple. 7 FACTORING. PROBLEMS. 144. What is the least common multiple of — 7. 6, 8 and 10? 8. 12, 15 and 16 \ 9. 16, 35 and 70? 1. 16 and 18? 4. 14 and 21 ? 2. 36 and 108 ? 5. 42 and 63 ? 3. 24 and 84? 6. 19 and 57? 10. What is the smallest sum of money for which you can purchase a number of pears at either 5 cents, 10 cents or 12 cents each? SECTION XII. CANCELLATION. 145. — 1. What is the quotient of 60 divided by 15 ? 2. What is the quotient of one-third of 60 divided by one-third of 15? 3. What is the quotient of one-fifth of 60 divided by one-fifth of 15? 4. What common factors have 60 and 15 ? 5. If you take out from 60 the factors common to 15, what factor will remain ? 6. What is the quotient of 2 X 7 X 3 divided by 2X7? 7. If you strike out from 42 and 14 all common factors, and divide the remaining factors, what is the quotient ? DEFINITION. 146. Cancellation is the process of shortening compu- tations by striking out equal factors from the dividend •and divisor, and using only the remaining factors. FACTORING. 75 147. Principle. — Striking out equal factors from both dividend and divisor, or dividing them by the same num- ber j does not change their quotient WRITTEN EXERCISES. 148—1. Divide 11 X 3 X 2 by 11 X 2. 1 1 Solution.— Write the iHr X ^ X i 1 1X3X1 dividend over the divisor. ~r/T~77 , v — 3 Divide both dividend 11 1 X 7 and divisor by 11 and 2, ^ by canceling or striking out those common factors in both, leaving 1X3X1-5-1X1 = 3. When a factor is canceled, 1 remains, and if not written is understood. 2. Divide 150 by 30. Solution. — Since the dividend and divisor 8 j 0)15\ have 10 as a common factor, cancel it by striking g off the cipher from the right of both, leaving 3)15, which equals 5. 3. Divide 13 X 11 X 7 by 13 X 3. 1001 by 13 X 3. 4. Divide 2500 by 500. 360 by 10 X 9. 149. Rule for Cancellation.— Cancel in the dividend and divisor factors common, to both, and then divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor. PROBLEMS. 150. Divide — 1. 48X3x5by8X7X5. 2. 50X5X3 by 15 X 10. 3. 75XllX2byllX5X2. 4. 31 X 25 by 7 X 5 X 5. 5. 4X13X3 by 39X2. 6. 81X8X7 by 27 X 28. 7. How many tons of coal at 9 dc liars a ton can be exchanged for 81 barrels of apples at 3 dollars a barrel? 76 FACTORING. 8. How many tons of hay at 24 dollars a ton can lx exchanged for 8 thousand feet of boards at 15 dollars a thousand ? 9. Divide 11461 by 1400, using the factors 100 and 14. 141 00)114) 61(8^ Solution.— Strike off two orders of j jp figures from the right of the dividend and divisor, which divides each by 261 100, and gives for a new dividend 114 hundreds, and 61 remaining, and for a new divisor, 14 hundreds. The new dividend 114, divided by the new divisor 14, gives a quotient 8 and a remainder 2, which is 2 hundreds. Annexing to this remainder the first remainder, 61, gives 261, the entire remainder, and the entire quotient is 8 T 2 ? Vo- 10. Divide 450 by 70, using the factors 10 and 7. 11. Divide 11400 by 600, using the factors 100 and 6. 12. If you should earn 1350 dollars in 300 days, how much would you earn each day ? 151. Test Questions. — 1. What are the factors of a num- ber? What is a prime number? A composite number? What are the prime factors of a number? What is factoring? A principle of factoring? 2. What is a divisor, or measure, of a number? A common divisor of two or more numbers? The greatest common divisor of two or more numbers? A principle of the greatest common factor or divisor? 3. When is a number a multiple of another? A common multiple of two or more numbers? The least common multiple of two or more numbers? What is a principle of the least common multiple of two or more numbers? 4. What is cancellation ? What is a principle in cancellation ? FRACTIONS. 77 SECTION XIII. NOTATION OF FRACTIONS. 152. — 1. When a cake is divided into two equal parts, what is each of the parts called ? 2. Into how many halves can a cake be cut ? 3. When an apple is divided into three equal parts, what is each of the parts ? What are two of the parts ? How many thirds in an apple ? 4. When an orange is cut into four equal parts, what is each of the parts? What are two of the parts ? Three of the parts ? How many fourths in an orange ? 5. If a cake be divided into five equal parts, what is each of the parts? What are two of the parts? Three of the parts ? Four of the parts ? 7* One half is written J. One third " J. One fourth " \. One fifth " 78 FRA CTIONS. 6. Into how many halves can any thing be divided ? Into how many thirds ? Fourths ? Fifths ? Sixths ? 7. What is meant by one half of any thing ? By one third of any thing ? By two thirds ? By one fourth ? By two fourths ? By three fourths ? 8. Which are the smaller parts of any thing — halves or thirds ? Halves or fourths ? Thirds or fourths ? Halves or sixths ? Thirds or sixths ? Fourths or sixths ? 9. Halves, thirds, fourths, etc., are expressed by figures, as follows — Two thirds are written f . Three fourths u f . Four fifths " 4 . 5. Six sevenths " f . One twelfth " T V. j Nine eighths " f . One fifteenth " T V I Eleven sixteenths " yg-. One twentieth " -^o • | Three twenty-fourths" ^V 10. How many halves of 1 does \ express? How many thirds of 1 does f express ? How many fourths of 1 does f express ? 1 1 . What does the figure 4 under the dividing line in the expression f denote ? The number of equal parts into which the unit 1 is divided. 12. What does the figure 3 above the dividing line in the expression f denote ? The number of equal parts of the unit 1 which are taken. 13. How is the expression 3^ read? 14. In the expression Z\ y what expresses the integer ? What the fraction ? 15. In the expression 7f, what expresses the integer? What the fraction ? FRACTIONS. 79 DEFINITIONS. 153. A Fraction is a number which represents one or more of the equal parts into which a unit is divided. 154. The Denominator of a fraction is the number which shows into how many equal parts the unit is divided. 155. The Numerator of a fraction is the number which shows how many equal parts are taken. 156. The Terms of a fraction are its numerator and denominator. 157. A Mixed Number is a number expressed by an integer and a fraction. WRITTEN EXERCISES. 158. Write and read — 1 6 I. y. 7 10 ' • 19' 13. V. 19. 4|. 9 - 8 - 4. ii- 8. ti- 14. V- 20. 7f o l 8 »• ft' Q 1A V. i6 . 15. V 4 - 21. 21*. A 2 4 - TV- 10. T v 16. W- 22. 7Ji. r 1 6 0. 2T- 11. If. 1 ' • i i- .Zo. t/t/y"Q". a 1 2 0. yy. 12. T V 18. if* 24. 106 T 8 7 . 159. Write in figures — 1. Three fifths. 9. Two twenty-firsts. 2, Four sevenths. 10. Five thirty-seconds. 3. Five eighths. 11. Nineteen eighths. 4. Six tenths. 12. Thirty-one ninths. 5. Nine fourths. 13. Six and six elevenths. 6. Five sixths. 14. Twelve and four fifths. 7. Four ninths. 15. Twenty-one and one fourth. 8. Two elev enths. 16. Nineteen anc 1 one thirteenth 80 FRACTIONS. VALUE OF A FRACTION. 160. — 1. If you should cut 1 apple into halves, how many halves would there be ? 2. If you should cut 5 apples into halves, how many halves would there be ? One half of 5 apples is how many halves of 1 apple ? .3. How many apples are one half of 5 apples? 5 halves are what part of 5 apples ? 4. Is | greater or less than 1 ? Is f greater or less than 1 ? What is the value of f in ones ? Of \° ? 5. Is f greater or less than 1 ? What is the value of | in ones ? Of f ? 6. Considered as an expression of division, what is the value of f? Of V? 7. Considered as an expression of division, what is the value of V ? Of V ? FRACTIONS. 81 DEFINITIONS. 161. A Proper Fraction is a fraction whose numerator is less than its denominator. 162. An Improper Fraction is a fraction whose nume- rator is not less than its denominator. 163. Reduction of Fractions is the process of changing their form or denomination without changing their value. 164. Principle. — The value of a fraction is the quo- tient arising from the division of the numerator by the denominator. WRITTEN EXERCISES. 165. Write and name the following fractions — » 1. f. 4. %. 7 11 10. ,V, 2.1. 5. T V 8. V s - 11. ¥■ or 2. 2. What is the sum of ft, if- and if? 3. What is the sum of f^, f f and $$ ? 4. What is the sum of ^, f and f ? Solution. — Reducing the fractions to their least common denominator, we have £$ + f| + H = tt, °r 1H, which is the sum required. 5. What is the sum of f , $ and \\ ? 6. What is the sum of f, & and ft ? 7. What is the sum of 3f, 5£ and 11 ? ? * Solution. — Reducing the fractions ^4 ~ ° 8 to their least common denominator r 1 ~4 and adding their numerators, we have * " 8 as their sum \° = If. Write the § 11 =11 under the fractions, and add the 1 «,,«, ~9r7— 9D 1 With the inte S ers ' S ivin S 2 °t = 2 °i' awn, ^ — 40- the sum required> i_ «0 f_ Af 8 _ 15 g ■* - 40' } J "^T s ~~ z 40 20 *0 . 32 + 15 40- 67 40 8. What is the sum of 11 J, 15| and 5^ _9 2 ' 94 FRACTIONS. 197. Rule for Addition of Fractions.— Reduce the frac- tions to equivalent fractions having a common de- nominator; add their numerators , and write und ,r the sum the common denominator. If there be mixed numbers or integers, add the fractions and integers separately, and uniii the results. PROBLEMS. 8. What is the sum of- 1. 2. 3. 4. MandH? . i, T 3 ¥ and tV ? 9 16. „ n A 11 9 2 5' 2 5 dIlu 2 5 • Y % ^andfV? 5. 6. 7. 8. |, land 5? 8, T 8 T and2 7 V? 31 4| and 33? H.^andl^? 9. What is the sum of -J + fV + I + H ? 10. What is the sum of 3J + 2£ + f + $ ? 11. A farmer has in one bin 31f bushels of wheat, in a second bin, 15f bushels, and in a third, 16^ bushels. How many bushels has he in the three bins ? 12. If you should spend 6-^5- hours in study, 3 2V hours in play, and 11^ hours in sleep, how many hours would you spend in all ? SECTION XVI. SUBTRACTION OF FRACTIONS. 199. — 1. John and his brother have f of an apple. If John has f , what part of the apple has his brother ? 2. Jane has | of an orange ; if she should give Susan |, what part of the orange would she have left ? FRACTIONS. 95 3. How many fifths are | less f ? V l ess i ? 4. How, then, do you subtract one fraction from another when both have a common denominator ? 5. If f of an apple be taken from it, what part will remain ? 6. How much is 1 less \ ? 2 less \ ? 3 less f ? 7. If a cord of wood cost f of a dollar less than 6 dollars, what is its cost ? 8. How much is \ less -f- ? • Solution. — \ equals T 7 ¥ , and } equals T 4 ¥ ; and T 7 ¥ less T 4 ¥ is T \. 9. A man worked T 9 7 of a day, and his son f of a day. How much longer did the man work than his son ? 10. How much is \ less \ ? \ less -§ ? |- less f ? 11. How do you prepare fractions having different denominators for subtraction ? 12. If a man owning -ff of a mill should sell f of the mill, what part of it would he have left ? 13. How many are 2£ less f ? 5f less 2£? 14. What number must be added to 3| to make 9 ? 200. Principle. — Fractions must express like parts, or have a common denominator, before they can be subtracted. WRITTEN EXERCISES. 201.— 1. What is the difference between f and f ? t -9 6_ 2 6-2 4 Solution. — Reducing the 8 9~9 9~ 9 ~9 fractions to their least com- mon denominator and finding the difference between their numerators, we have f, the difference required. 2. Find the difference between -fa and f . 3. Find the difference between T 4 ir and J. 96 FRACTIONS. 4. Find the difference between 8$ and 5|. Solution. — Eeduce the fractions to ol _ - qA_ — - y~ fractions having the least common denomi- 4 12 12 na tor. Since A cannot be taken from A, 5~ == 5 — = 5~— we ta ke 1 one, or }$, from the 8 ones, — — leaving 7 ones, and adding the \\ to the Difference, 2^ T 3 2, have \\ ; 5 T 8 2 from 7^| leaves 2 T V, the difference required. 5. Find the difference between 14$ and 6f . 6. Find the difference between 28f- and 9y\. 202. Rule for Subtraction of Fractions.— Reduce the frac- tions to equivalent fractions having a common de- nominator, and write the difference of the nume- rators over the common denominator. If there be mixed numbers, subtract first the fractional part of the subtrahend, and then the integral part, and unite the results. PROBLEMS. 203. Subtract— 1 $f from f 2. A 3 4. T \ from $$. 5. A from T 5 r . 7. 13$ from 16$. 8. llj from 19}. 9. 33$f from 100. 6. f$ from $f 10. If a farmer should sell 24^ acres from a lot con taining 66f acres, how many acres would be left? 11. What is the difference between 85$ and 83$? 204. Test Questions.— 1. What is the principle in addition of fractions ? Plow do you add when all the numbers are frac- tions ? When there are fractions and mixed numbers or integers ? 2. What is the principle in subtraction of fractions? How do you subtract when the minuend and subtrahend are frac- tions? When the minuend and subtrahend are mixed numbers? FRACTIONS. 97 SECTION XVII. MULTIPLICATION OF FRACTIONS. CASE I. Fractions Multiplied by Integers. 205. — 1. If you should give \ of a melon to each of 5 boys, how many sixths of a melon would they all have? 2. If you should give ^ of a cake to each of 7 boys, how many thirds of a cake would be requir- ed? How many cakes ? 3. 7 times \ are how many times 1 ? 4. If 1 yard of cloth cost | of a dol- lar, how many dollars will 8 yards cost? Solution. — Since 1 yard of cloth cost f of a dollar, 8 yards must cost 8 times | of a dollar, which are V, or y>, equal to 6f dollars. 5. At ■£$ of a dollar each, how many dollars will 9 hats cost ? 6. If 1 pound of tea cost f of a dollar, how many dollars will 7 pounds cost ? 7. 7 times £ are how many times 1 ? 8. If a horse eat \\ of a bushel of grain in 1 week, how many bushels will he eat in 10 weeks ? 9. How many are 6 times f ? 7 times f ? 9 98 FRACTIONS. 10. How many are 8 times yy ? 5 times yy ? 11. If ^ of an acre will pasture a cow, how many acres will pasture 12 cows? 12. How many are 11 times |? 12 times f ? 13. If f of a yard of cloth is y of what is required for a suit of clothes, how many yards will be required for the suit ? 14. If you can earn ^ of a dollar in 1 day, how many dollars can you earn in 1 1 days ? 15. Multiplying the numerator of ^ by 11 gives ff, or |; dividing the denominator of ^ by 11 gives f ; in each case the fraction is multiplied by 11. How, then, may a fraction be multiplied by an integer ? 16. Multiply Y 5 ¥ by 8. A by 7. yV by 5. 17. How many are 5 times 6J? Solution. — 5 times 6 are 30, and 5 times \ are }, or 2J, which added to 30 gives 32J, the answer required. 18. If 1 ton of coal cost 5f dollars, how many dollars will 7 tons cost ? 19. If you can gather 6f quarts of berries in 1 day, how many quarts can you gather in 6 days ? 20. At the rate of 3| miles in 1 hour, how far can you walk in 8 hours ? 21. How much must be given for 11 pounds of tea, at If dollars a pound ? 22. What will 5 yards of cloth cost, at 3f dollars a yard? 206. Principle. — Multiplying the numerator, or di- viding the denominator, by any number, multiplies the fraction by that number. FRACTIONS. 99 WRITTEN EXERCISES, 207.— 1. Multiply H by 8. 1L V Q - UX8 — 88 _r 8 __ r>l 16 X *-~16~~ 16 -°16 ~ b 2 Or, n. 11 - y R — - ±- = — = £- 16 ^ 16 + 8 2 2 Solution.— 8 times 11 sixteenths are f f, which, reduced, is 5 t 8 q, or 5|, the result required. Or, since dividing the denominator of a fraction multiplies the fraction, 8 times \\ = V, or 5 J, the same result. 2. Multiply llf by 7. 4 4 11X7 = 77 jX7=f= 834 Solution.— Since llf equals 11-f-f, the product of llf by 7 is the same as 7 times 11 plus 7 times f. 7 times 11 are 77, and 7 times | are V, or b\. 77 + 5 J = 82J, the product required. 3. Multiply H by 9. ffo by 17. «HJ by 15. 4. Find the product of 25J by 16. 208. Rules for Multiplying a Fraction by an Integer.— Mul- tiply the numerator, or divide the denominator, by the integer. If the multiplicand be a mixed number, multi- ply the integer and fraction separately, and add the products. PROBLEMS. 209. Multiply- 1. t^ bv 9. 5. tf by 15. 9. 3fbyl7. 2. H by 20. 6. if by 9. 10. 1\ by 14. 3. i by 25. 7. A by 8. 11. 4^by31 4. & by 13. 8. if by 18. 12. 19 j by 63 100 FRACTIONS. CASE II. Integers Multiplied by Fractions. 210. — 1. \ of 3 inches is what part of 1 inch ? Solution. — \ of 3 inches must be 3 times \ of 1 inch, or | of linen. 2. \ of 3 apples is how many apples ? 3. i of 8 bushels is how many bushels ? 4. Arthur has 5 dollars, and James has § as many. How many dollars has James ? Solution. — James has f of 5 dollars. Since J of 5 dollars is f of 1 dollar, § of 5 dollars must be 2 times f, which are l °. or 3J dollars. Therefore James has 3J dollars. 5. How much will f of a yard of cloth cost, at the rate of 7 dollars a yard ? 6. f of 7 is what number? 7. If a man can do a piece of work in 45 days, in how many days can he do f of it ? 8. f of 45 is what number ? 9. If a barrel of flour is worth 16 dollars, how much is ^ of a barrel worth ? 10. -f- of 42 is what number? 11. Multiply 9 by f. 5 by f 8 by f 12. At 6 dollars a ton, what will 5f tons of coal cost? Solution. — Since 1 ton costs 6 dollars, 5f tons will cost 5§ times 6 dollars ; 5 times 6 dollars are 30 dollars, and f of 6 dol- lar are V, or 4| dollars; 30 dollars and 4J dollars are 34| dollars, the cost required. 13. At 5 dollars a yard, what will sost? 14. Multiply 8 by 4|. 7 by 3$. FRACTIONS. 101 15. If 6 men can do a piece of work in 4| days, how many men will it take to do it in 1 day ? 16. Multiply 4 by &ft_. 8 by 8|. 5 by 7 T V 211. Principle. — A number is multiplied by a fraction by obtaining such a part of the number as the fraction indicates. WRITTEN EXERCISES. 212.— 1. Multiply 35 by f, or find f of 35. Solution.— Since f == \ of 5, f times 35 must equal \ of 5 times 35, or 35 7 x -, which, by can- celing, or § ~ I > or 25. Or, since f = 5 times \ t we find f of 35 by taking 5 times \ of 35 ; \ of 35 is 5, and 5 times 5 are 25. 2. Multiply m by ^ or find T 4 T of 66. 3. Multiply 75 by ft or find \\ of 75. 213. Rules for Multiplying an Integer by a Fraction.— Mul- tiply the integer by the numerator of the fraction, and divide the product by the denominator. Or, Divide the integer by the denominator of the fraction, and multiply the quotient by the nume- rator. 35x1 = -^ il — 25 Or, 35xf = i 5 ■35- ' 9- \ X5 = 25 PROBLEMS. 214. Multiply— 1. 72 by A- 2. 96 by f . 3. 105 by -&. 4. 112 by f. 5. 215 by fV. 6. 360 by $. 7. 327 by A- 8. 516 by f. 9. -819 by A- 102 FRACTIONS. CA.S1S III. Fractions Multiplied by Fractions. 215. — 1. If ^ of a pear be separated into two equal parts, what part of the pear will one of those parts be ? 2. J of \ is what part of 1 ? 3. If | of an orange be equally shared by two boys, what part of the orange will each receive? 4. \ of | is what part of 1 ? Solution.— \ of f is equal to 2 times \ of \ ; 2 times \ are §, or J, the part required. 5. What is § of f? J of |? i of |? i of i is k and 6. What is *of *? lof #? ioff? 7. A man owning J of a ship sold f of his share. What part of the ship did he sell ? Solution.— He sold f of ] of the ship ; f of J is equal to 3 times \ of J ; \ of | is equal to ¥ V, and 3 times ^ are § |. Therefore, he sold JJ of the ship. 8. When a man had traveled y 9 ^ of a mile he had still to travel a distance equal to f of that gone over. How far had he still to travel, and how far did he travel in all? 9. 10. What is f of 4? | of £? What is | of T 2 T ? I of*? A of I? fof|? FRACTIONS. 103 11. What will f of a bushel of corn cost, at ^ of a dollar a bushel ? 12. At 2\ dollars a yard, what is the cost of f of a yard of cloth ? Solution.— Since 1 yard costs 2| dollars, | of a yard will cost f of 2| dollars. 2-| dollars are equal to | of a dollar ; f of | are f f , or 2 T V Hence, £ of a yard will cost 2 T 1 1 dollars. 13. If a cord of wood cost 3£ dollars, how much will f of a cord cost ? 14. What is f of 5i? fof3^? fof7i? 15. What is f of 2i? f.of 7£? f of 2£ ? 16. When fractions are connected by the word of what does the of denote ? Multiplication. WRITTEN EXERCISES. 216.— 1. Multiply f by f, or find f of f. & a , Solution. — f of | is the same as 3 o v, o 41/. 8^ 9^7~63~~Tl times \ of f ; f of | is g^, or &, and 3 Or, times -$$ is ~ G 1^, or §f, which, reduced, £ 3 8X3- __ 8 is 2 8 t» tae product required. Or, indi- 9 7 ~~ 5-x 7 ~ 21 eating the multiplication and cancel- 3 ing, we have ^ 8 T , as before. 2. What is the product of 8| by 4£ ? 2 26 1 21 Solution. — Reducing the mixed S' 3 = J"" ; 4j ~ J~* numbers to equivalent fractions, we 7 have 8| X 4| = y X V- Canceling ^X^ = 1S2 = 36- and multi P l y in g> we have if*, which, *" 5 J 5 reduced, is 36f, the product required. 3. Multiply iV by £-. or find f of &. 4. Multiply 13f by 2|. 29^ by 6£. 104 FRACTIONS. 217. Rules for Multiplying a Fraction by a Fraction.— Multi- ply the nuinerators together for the numerator, and the denominators for the denominator of the pro- duet. If there be mixed numbers, reduce them to frac- tions and then multiply. 1. 2. 3. 13. AVhat is the value of £ of if ? 14. What is the value of f of \ of f f ? 15. John owns f of a boat, and his brother owns -^ as much of it. What part of the boat does his brother own ? 16. What will \ of a yard of cloth cost, at 3^ dollars a yard ? PROBLEMS. 218. Multiply- &byf. 5. m by t 5 t- 9. tt by 7|, Abyf. 6. 3^ by I 10. 1* by ft. A byf 7. 16| by f. 11. 12 by If. t 2 oV by f 8. U by 3*. 12. 17| by 5A- 219. Test Questions.— 1. In what two ways may a fraction be multiplied? Show that multiplying the numerator multi- plies the fraction. That dividing the denominator multiplies the fraction. 2. How is an integer multiplied by a fraction? Show how you can take of a number the part denoted by a fraction. In what two ways may an integer be multiplied by a fraction ? 3. How do you multiply a fraction by a fraction ? ' How do you proceed if there are mixed numbers? 4. When fractions are connected by the word of, what does the of denote ? FRACTIONS. 105 SECTION XVIII. DIVISION OF FRACTIONS. CASE I. Fractions Divided by Integers. 220, — 1. A boy having f of an orange wished to di- vide it equally among 3 boys. What part of the orange could he give to each ? 2. \ of f is what part of 1 ? 3. If 5 books can be bought for f of a dollar, what will 1 book cost? 4. If 4 yards of cloth can be bought for £ of a dollar, what will 1 yard cost? Solution. — If 4 yards of cloth cost f of a dollar, 1 yard will cost I of | of a dollar, or ^ of a dollar. 5. If 1 man can do a piece of work in f of a day, in what time can 5 men do it ? 6. i of | is what part of 1 ? 7. When 6 books of equal value cost V 2 of a. dollar, what is the cost of 1 book ? 8. I divided by 6, or | of f, is what number? 9. Divide i by 2. f by 3. f by 5. 10. Divide f by 3. £ by 4. jf by 5. 11. Dividing the numerator of \j by 5 gives -fa ; multiplying the denominator of yy by 5 gives £§-, which •equals fa ; in each case the fraction is divided by 5. How, then, may a fraction be divided by an integer ? 12. Divide V 6 by 8. f by 6. if by 7. 13. Divide M by 3. \° by 5. V 8 by 9. 106 FRA cnoxs. 14. If 1 man can do a piece of work in 8| days, in what time can 3 men do it ? Solution. — If 1 man can do a piece of work in 8|- days, 3 men can do it in J of 8| days, or in 2\\ days. 15. If 4 bushels of corn cost 5| dollars, what is the cost of 1 bushel ? 16. Divide 12J by 7. 9| by 6. 10} by 9. 221. Principle. — Dividing the numerator or multiplying the denominator by any number divides the fraction by that number. WRITTEN EXERCISES. 222.— 1. What is the quotient of jf divided by 6 ? 12 12^-6 2 Solution. — Since dividing the Js ^ ls~ ~ 13 numerator of a fraction divides the fraction, \\ divided by 6 gives T 2 3, 2 the quotient required. 19 -19 2 u '.'^ 6 — m~~^-— 'if O r > since multiplying the de- nominator of a fraction divides the fraction, Jf divided by 6 = Jzxe ~ "h* tne same result. 2. What is the quotient of 31^ divided by 5 ? «5 Solution.— Since 31 \ is ±f£, -_®L = (}1 we may divide 31 J by 5 by di- viding its equivalent, *£*, which gives 2 j 5 , or 6}, the quotient re- £ quired. /?A Or, we may divide the mixed ^ number without first reducing it to an improper fraction. One fifth of 31 is 6, with a remainder 1, which is equal to f; this added to the J gives {. One fifth of J is I, which added to 6 gives 6\, the same result. 3. Divide & by 3. & by 34. 23 j by 7. ° r > i i 5)3lh FRACTIONS. 107 223. Rules for the Division of a Fraction by an Integer.— Divide the numerator, or multiply the denomin- ator, by the integer. If the dividend is a mixed number, reduee it to an improper fraction before dividing ; or, divide the integer and fraction separately , and unite the results. PROBLEMS. 224. Divide— 1. I by 6. 2. | by 9. 3. H by 22. 10. If 11 4. f by 27. If by 15. 5. 6. H by 7. boys should have 90|f dollars divided 7. 16* by 7. 8. 18f by 8. 9. 25| by 20. equally among them, how much would each receive ? CASE II. Integers Divided by Fractions. 225. — 1. How many thirds of a cake in 1 cake? In 2 cakes ? 2. How many times f of a cake in 2 cakes ? 3. How many pears, at | of a cent each, can be bought for 2 cents ? 4. When tea is f of a dollar a pound, how many pounds can be bought for 6 dollars ? Solution. — If $ of a dollar will purchase 1 pound of tea, 6 dollars will purchase as many pounds as | of a dollar is con- tained times in 6 dollars. 6 dollars equal 3 T ° of a dollar ; 3 7 ° -*- f = 30 -s- 4 = 7| oi 7£. Hence, 7| pounds can be bought. 108 FRACTIONS. 5. At | of a dollar a yard, how many yards of cloth can you buy for 3 dollars ? 6. 1 is how many times f ? -f-? 7. 2 are how many times f ' v I WX ^ AAW If J.AA14.XA T VAIAAVU ^1 • i are how many times f ? t? 1? i*-i=f+5=*o Or 15^1 ^i ^20 WRITTEN EXERCISES. 226. — 1. What is the quotient of 15 divided by f ? Solution. — 15 is equal to 6 ¥ °. 60 fourths divided by 3 fourths gives 20, the quotient required. Or, since 15 -f- 1 is 15, 15 -=- \ must be 4 times 15, and 15 -+- 1 must be J of 4 times 15, which is 20, the same result. 2. What is the quotient of 64 divided by $? 3. Divide 25 by f . 112 by |. 98 by |. 227. Rules for Division of an Integer by a Fraction —Rednei the integer to a fraction having the same denomi- nator as the divisor, and divide the numerator of the dividend by the numerator of the divisor. Or. Multiply the integer by the denominator of the divisor, and divide the result by the numerator. 228. Divide— 1. 15 by |. 2. 61 by | . 3. 21 by f. 4. 40 by | PROBLEMS. 9. 10. 11. 24 by f 17 by 36 by ^ 5. 51 by &. 6. 43 by f 7. 65 by A- 8. 90 by A. 13. What is the quotient of 40 divided by 3^? Solution. a=f, and jQ-m 320+25 A- 12. 110 by f£. 12 **= 12 * FRACTIONS. 109 14. Divide 17 by 2f. 28 by If 42 by 6§. 15. At -J of a dollar a yard, how many yards of cloth can be purchased for 8f dollars ? CASE III. Fractions Divided by Fractions. 229. — 1. At \ of a dollar a yard, how much cloth can be bought for £ of a dollar ? 2. At ^ of a dollar a bushel, how many bushels of apples can be bought for f of a dollar ? 3. How many times is f contained in f ? Solution. — f-^-f is equivalent to |$ divided by ^, and 8 twentieths are contained in 15 twentieths 1J times, which is the result required. 4. Divide i by f . § by f . $ by f f by |. 5. At jq of a dollar a pound, how many pounds of sugar can be bought for % of a dollar? 6. At j-Q of a dollar each, how many books can be bought for 2| dollars ? Solution. — 2f is equal to \ % ; at T 3 o of a dollar each, as many books can be bought for \ % of a dollar as 3 tenths are con- tained times in 24 tenths, or 8, which is the number required. 7. When butter is \ of a dollar a pound, how many pounds can you buy for 2| dollars ? DEFINITION. 230. A Complex Fraction is one that has a fraction in one or both of its terms. 3 Thus, - is a complex fraction, and indicates the division of I by f. in 110 FRACTIONS. WRITTEN EXERCISES. 231. — 1. What is the quotient of f divided by f ? A _j_ I — *t _=_ ^ — *? -a J? — #£ Solution.— f and § 5 5 ^5 45 10 5 5 reduced to fractions hav- ing a common denominator are f f and |§ ; we can then divide i by f, by finding the quotient of 36 forty-fifths divided by 10 forty-fifths, which is 3f , the quotient required. Or, . tX9 _18_ 8 Since f -*-l is f, $ -4- \ must be 5 .9 5X# 5 ^5 9 times |, or ^~- , and f -*- f must be ^ of 9 times f , or -f-^-f . which, reduced, is 3f , the same result. 2. What is the quotient of || divided by f ? 3. What is the quotient of I divided by 21 ? 232. Rules for Division of a Fraction by a Fraction.— ife- £Zw,c£ the fractions, if necessary , to a common de- nominator, and divide the numerator of the divi- dend by the numerator of the divisor. Or, Multiply the dividend by the denominator of the divisor, and divide the result by the numerator. 233. Divid 1. f by &. 2. H by f . 3. H by f . 4 tfby*. PROBLEMS. 5. f byf 6- A by |. 7. f by |. 8- H by 5V 13. What is the quotient of 3£ divided by 1£? Solution. 9. A by A- 10. A by &. 11. ,&bj J. 12. &bj &• 2>i 25 , -,1 5 ?-- y ,and 7-^ J; 5 -=^ REVIEW. Ill 14. How many barrels of potatoes, at 2| dollars a barrel, can be bought for 16^ dollars? 15. How many yards of calico, at T 3 g of a dollar a yard, can be bought lor % of a dollar ? 16. What is the value of A ? Of 2 f ? 16 3 17. How many tons of coal, at 3f dollars a ton, can be bought for 13 -J- dollars ? 234:. Test Questions. — 1. In what two ways can a fraction be divided by an integer? If the dividend be a mixed number, how do you divide ? 2. How do you divide a fraction by an integer? An integer by a fraction ? 3. What is a complex fraction? Express in the fractional form, the division of some fraction by a fraction. How do you divide a fraction by a fraction? SECTION XIX. REVIEW OF FRACTIONS. 235—1. How many fifths in 4|? In 7f ? In 9|? 2. How many ones in V ? In V ? In V ? 3. What is the value of V ? Of V ? 4. What improper fraction is equal to 6| ? 5. Express f in its lowest terms. 6. Express f in higher terms. 7. Reduce f and f each to eighteenths. 8. Reduce f and | to fractions having a common de- nominator. 112 BE VIEW. 9. A sum of money, diminished by f of a dollar, is equal to 22 7 o dollars. What is the sum ? 10. If I sell an article for f of its cost, what frac- tional part of the cost do I lose ? 11. If I sell an article for J part more than its cost, for how many fourths of the cost do I sell it ? 12. If I pay 37^ cents for a knife, and sell it for | of its cost, how many cents do I lose ? 13. A fanner had ll-£ bushels of wheat stolen, which was | of all he had. How much had he ? 14. 11^ is ^ of what number? 15. If a slate cost ^V of a dollar, how many slates can be bought for If dollars ? WllTTTEX EXERCISES. 236.— 1. Reduce 105f to eighths. 2. Reduce j\ and -^ to fractions having the least common denominator. 3. Reduce |-f and T V§- each to its lowest terms. 4. Having lost f of a dollar, I find I have left 13f dollars. What sum had I at first ? 5. A merchant owned ff of a ship, and sold f of the ship. What part of the ship had he left? 6. 5^ is \ of what number ? ^ of what number ? 7. if is \ of what number? \ of what number? 8. \i is i of what number? T V of what number? 9. If a horse will eat, in a given time, ^i of a ton of hay ; a cow, f of a ton ; and an ox, t 9 q of a ton, what quantity will they all eat in the same time? 10. A lady gave lOf dollars for a dress, 4^- dollars for a shawl, and I of a dollar for a handkerchief. How much did they all cost her ? REVIEW. 113 11. James gave -^ of his money for a coat, and y2 °f it for a library. What part of it had he left ? 12. What will 19 pounds of tea cost, at If dollars a pound ? lo. The multiplicand is 12| and the multiplier 3 J. What is the product ? 14. Smith owns f of 3^ of a ship, and Collins owns f as much as Smith. What part does Collins own ? MENTA L EX EH CIS ES. 237.— 1. 20 is f- of what number? Solution. — If 20 is f of some number, \ of that number must be \ of 20, or 10, and f, or the whole, of the number, must be 7 times 10, which is 70, the number required. 2. 40 is f of what number ? § of what number ? 3. 30 is f of what number ? f of what number ? 4. 16 is f of what number? f of what number? 5. 20 is f of what number ? \ of what number ? 6. If f of an acre of land be worth 40 dollars, what is an acre worth ? 7. A farmer sold a cow at a gain of 20 dollars, which was y of her cost. What was the cost ? 8. Jane is 16 years old, and is % as old as her sister. What is the age of her sister ? 9. A pole stands y of its length in the mud, \ in the w r ater, and the remainder, which is 5 feet, above water. What is the length of the pole ? 10. When f of a dollar is f of the price of a pound of tea, what is the price of a pound ? Solution. — If | of a dollar is § of the price o£ a pound, \ of the price must be \ of f, or |, of a dollar, and f, or the whole price, must be 3 times f, or J, equal to \\ dollars. 111 * 114 REVIEW. 11. f is | of what number? f of what number? 12. If | of a rod is £ of the width of a walk, what is the width of the walk ? 13. f is f of what number? f of what number? 14. How many times f is \ ? How many times f- is f ? 15. If 3 men can do a piece of work in 4^ days, in how many days can 5 men do it ? 16. A\ is f of what number? f of what number? 17. How many yards of cloth, 3 quarters of a yard wide, are equal to 7 yards 5 quarters wide ? 18. 7 is f of what number? f of what number? 19. 12 is f of how many times 2 ? 20. 30 is | of how many times 12 ? 21. f of 10 is § of what number? 22. f of 12 is f of how many times 4 ? 23. If | of a barrel of beef cost 16 dollars, how many cords of wood at 5 dollars a cord will pay for a barrel of beef? 24. If f of a ton of coal cost 4 dollars, what will | of a ton cost ? 25. If f of a yard of cloth cost 3 dollars, what will f of a yard cost ? WRITTEN EXERCISES. 238. — 1. At | of a dollar a yard, how many yards of cloth can be bought for 16^ dollars? 2. Divide W by 18. H by 32. 3£ by f. 3. I paid 95^ dollars for flour, at the rate of 8^ dollars per barrel. How many barrels did I buy ? 4. If £ of an acre of land cost 96 dollars, what will | of an acre cost ? REVIEW. 115 5. If the divisor is y and the quotient 5f, what is the dividend ? 6. If | of a ton of hay cost 18 dollars, what will Jf of a ton cost ? 7. If the product of two factors is 15f, and one of the factors is 3^, what is the other factor ? 8. If the divisor is ^ and the dividend is -gfa, what is the quotient ? 9. I own \% of a ship, and my brother owns ^. How many times as much as he, do I own ? 10. John sold 320 acres of land ; he then bought f as many as he sold, and found that number to be f as many as he had at first. How many had he at first ? 239. Test Questions.— 1. What is an integer ? A fraction? A mixed number? 2. What are the terms of a fraction? Which term is the numerator? Which the denominator? How is a fraction ex- pressed by figures ? 3. What is a proper fraction? An improper fraction? A complex fraction ? 4. What is the value of a fraction? Is the value of a fraction changed by multiplying both terms by the same number? By dividing both terms by the same number? 5. What is reduction of fractions? How may fractions be changed to fractions with higher terms? To fractions having lower terms ? 6. By what means may fractions be added or subtracted? How are fractions with different denominators prepared for addition or subtraction? 7. What is a principle in addition of fractions? In subtrac- tion of fractions? In multiplication of fractions ? In division of fractions? 116 UNITED STATES MONEY. SECTION XX. NOTATION OF UNITED STATES MONET. 240. United States Money ^^^^ s reckoned in dollars, cents and mills. %? & ^ & i , W CENT/ 10 mills (m.) are 1 cent. . .c. or et. 10 cents " 1 dime d. 10 dimes " 1 dollar $. $l=10d~100ct,=1000m. 241. United States Money consists of Coins and Paper Money. 242. Coins are pieces of metal stamped for use as noney. The coins of the United States are now made of gold, silver, nckel or bronze. TABLE. UNITED STATES MONEY. 117 TABLE OF COINS. Names. Values. Double-eagle....$20. Eagle 10. Half-eagle 5. 3-dollar piece... 3. Quarter-eagle... 21. Dollar 1. Bronze. \ Cent lc. Gold. Silver. Nickel. Names. Values. Dollar 100c. Half-dollar 50c. Quarter-dollar... 25c. 20-cent piece .... 20c. „ 10-cent piece.... 10c. 5-cent piece 5c. [ 3-cent piece 3c. In addition to the above, a Trade-Dollar is issued for the con- venience of foreign trade. 243. Paper Money consists of Notes, issued by banks and by the Treasury of the United States, as substi- tutes for coin. When Treasury Notes are of less face- value than $1, they are called Fractional Currency. 244. In business transactions dimes are regarded as a number of cents. Thus, 5 dimes are regarded as 50 cents. In denoting by figures sums of United States money, the sign $ is placed before dollars, and the decimal point (.) is placed before cents. Thus, $5.90 expresses five dollars and ninety cents. $.075 expresses seven cents and five mills. 245. — 1. How many mills in 1 cent ? In 3 cents ? In 9 cents ? 2. How many cents in 10 mills ? In 30 mills ? In 90 mills? 3. How many cents in 1 dime ? In 5 dimes ? In 8 dimes ? 4. How many dimes in 10 cents? In 50 cents? In 80 cents ? L18 UNITED STATES MONEY. 5. How many dimes in 1 dollar ? In 5 dollars ? 6. How many dollars in 10 dimes? In 50 dimes? 7. How many cents in 10 dimes ? In 1 dollar ? Ir I dollars ? In 5 dollars ? 8. How many dimes, and how many cents over, in 25 ;ents ? In 42 cents ? In 85 cents ? 9. How many dollars, and how many cents over, in L25 cents ? In 250 cents ? In 375 cents ? 10. How many cents, and how many mills over, in 15 nills ? In 45 mills ? In 67 mills ? 11. Since 10 mills are 1 cent, what part of a cent is 1 nill? Is 2 mills? Is 7 mills? 12. Since 10 cents are 1 dime, what part of a dime is L cent ? Is 3 cents ? Is 9 cents ? 13. Since 10 dimes are 1 dollar, what part of a dol- ar is 1 dime ? Is 2 dimes ? Is 7 dimes ? 14. Since 100 cents are 1 dollar, what part of a dol- ar is 1 cent? Is 3 cents? Is 17 cents? 15. Since 1000 mills are 1 dollar, what part of a dol- ar is 1 mill ? Is 7 mills ? Is 117 mills ? WRITTEN EXERCISES. 246. Copy and read — 1. $15.25. 4. $143.41. 7. $3,057. 2. $7.08. 5. $205.75. 8. $1,305. 3. $97,375. 6. $97,334. 9. $41,065. 247. Write in figures — 1. Ten dollars sixteen cents. 2. Twenty-seven dollars five cents. 3. Sixty-three dollars thirty-one cents. 4. Five hundred seventeen dollars. 10. $16.15. 11. $11.31. 12. $97,005, UNITED STATES MONEY. 119 SECTION XXI. REDUCTION OF UNITED STATES MONEY, 248. — 1. Change $7 to cents and to mills. 7 = No. of dollars. Solution.— Since $1 is equal to 100 100 cents, $7 must equal 7 times 100 YOO = No. of cents. cents > or 700 cents - 1n Since 1 cent is equal to 10 mills, 700 cents must equal 700 times 10 7000 = No. of mills. mills, or 7000 mills. 2. Change $15 to cents and to mills. 3. Change $15.67 to cents. $15.67 = Solution. — $15.67 may be changed to cents 1567 cents *>y removing the dollar-sign and decimal point, which gives 1567 cents, because $15 = 1500 cents, and 1500 cents plus 67 cents are 1567 cents. 4. Change $37.03 to cents. 5. Change $43,444 to mills. $43,444= Solution:— $43,444 may be changed to A3 AAA mills. m iU s by removing the dollar-sign and deci- mal point, which gives 43444 mills ; because $43 = 43000 mills, 44 cents = 440 mills, and 43000 mills, plus 440 mills, plus 4 mills, are 43444 mills. 6. Change $6,305 to mills. 7. Change 7000 mills to cents and to dollars. Solution. — Since there must be 7000 = No. of mills. one ten ^h as many cents as there are >vnn xt r x mills, 7000 mills mav be changed to 700= No. of cents. centg by dividing by * 10) or by remov . 7 = No. of dollars, ing one cipher from the right, which gives 700 as the number of cents. 120 UNITED STATES MONET. Since there must be one hundredth as many dollars as there are cents, 700 cents may be changed to dollars by dividing by 100, or by removing two ciphers from the right, which gives 7 a* the number of dollars. 8. Change 93000 mills to cents and to dollars. 249. Rules for Reduction of United States Money.— lb re- duce dollars to cents, remove the dollar-sign and annex two ciphers ; to reduce dollars to mills, an- nex three ciphers ; to reduce cents to mills, annex one cipher. To reduce dollars and cents to cents, or dollars, cents and mills to mills, remove the dollar- sign and the decimal point. To reduce cents to dollars, point off two orders from the right and prefix the dollar-sign / to re- duce mills to dollars, point off three orders from the right and prefix the dollar-sign; and to re- duce mills to cents, point off one order from the right and prefix the dollar-sign. PROBLEMS. 250. Reduce— 4. 5700 cents to dollars. 1. $57 to cents. 2. $53 to mills. 3. 98 cents to mills. 5. 53000 mills to dollars. 6. 980 mills to cents. 251. Test Questions.— 1. In what is United States money reckoned ? Of what does it consist ? Recite the table. 2. What are coins ? Of what are the coins of United States money made? Name the coins made of gold. Of silver. Of nickel. Of bronze. What is Paper Money? What is Frac- tional Currency ? 3. How are dimes regarded in business? In denoting sums of money by figures, what sign denotes dollars? What point is placed before cents? V NIT ED STATES MONEY. 121 SECTION XXII. COMPUTATIONS IN UNITED STATES MONEY. ADDITION. 252. — 1. If you should pay 30 cents for a slate, 25 cents for a writing-book and 10 cents for a pencil, how much would you pay for all ? 2. How much is 30 cents + 25 cents + 10 cents ? 3. Susan gave 50 cents for a collar, 40 cents for a thimble and 12 cents for some needles. How much did she give for the whole ? 4. If you give $9.50 for a coat and $3.25 for a vest, how much do you give for both ? WRITTEN EXERCISES. 253.— 1. What is the sum of $95.60, $19 and $4,375? $95.60 Solution. —Write the numbers and add, as 19. required by the rule for addition, and separate 4-375 the dollars from the cents in the sum by a deci- 118.975 mal point. Write and add- (2.) (3.) (4.) (5.) $4-5.13 $2,375 $11.14 $3.72 5.07 6.25 63.15 ■ 144 17. 7.625 7.99 5.138 $67.20 $9,002 6. $144.56 + $17.18 + $100.63 =- what amount? 121 UNITEP STATES MONET. 7. Johnson paid for a farm $6500, and for improve- ments on it $365.50. He then sold it for $150 more than the whole cost ; for how much did he sell it ? SUBTRACTION. 254. — 1. Arthur had 95 cents, and gave 50 cents for a knife. How much had he left ? 2. 85 cents less 17 cents are how many cents? 3. If you had $1.25, and should give 75 cents for an arithmetic, how much would you have left? 4. What sum added to 75 cents will make $1.25? What sum added to 50 cents will make $1.25 ? 5. If I have $9.25, how much more must I get to pay for a coat worth $10 ? 6. I have 7 dimes; how much more must I get to pay for a book worth $1.10? WRITTEN EXERCISES. 255. — 1. What is the difference between $106 and $43.50? t QQ Solution. — Write the numbers and subtract, as AS 50 squired by the rule for subtraction, and separate — -r '— the dollars from the cents in the difference by a $62.50 decimal point. Write and subtract — (2.) (3.) (4.) (5.) $164-15 $115,000 $31.15 $347.00 87.09 3 7.085 4.17 243.19 6. How much less is $1867.25 than $5555.43? 7. If you purchase goods at a cost of $316.50, and Bell them for $400, how much do you gain? UNITED STATES MONEY. MULTIPLICATION. 123 56. — 1. At 12 cents each, what will 12 writing- books cost ? 2. At 25 cents each, what will 5 spelling-books cost ? 3. James paid $1.25 for a hat, and Edward paid 3 times as much. How much did Edward pay ? 4. If flour is $10 a barrel, what will 10 barrels cost? 5. What will 5 bushels of wheat cost, at $2.10 a bushel ? WRITTEN EXERCISES. 257,-1. What is the product of $109.50 multiplied by 9? $109.50 Solution. — Write the numbers and multiply, as q required by the rule for multiplication, and sepa- -r rate the dollars from the cents in the product by a $985.50 decima i pomt . Multiply— 2. $61.34 by 8. 3. $40.65 by 11. 4. $3,125 by 25. 5. $19.06 by 12. 6. $20,013 by 13. 7. $93.56 by 100. 8. $1005 by 13. 9. $1056 by 171. 10. $34,055 by 1000. 11. $103.03 by 100. 12. $13.06 X 10 X 3 = what amount? 13. What will it cost to build 7 cottages, at $2500.50 each ? 14. If a man earns $125.87 each month, how much does he earn in 12 months? 15. How much will 640 acres of land cost, at $120 per acre ? 124 UNITED STATES MOSEY. DIVISION. 258. — 1. How many writing-books, at 12 cents each, can be bought for $1.44 ? 2. At 25 cents each, how many spelling-books can be bought for $1.25? 3. James paid $3.75 for 3 hats, how much were they apiece ? 4. At 3 dimes a yard, how many yards of cloth can be bought for $2.10 ? WRITTEN EXERCISES. 259.— 1. What is the quotient of $985.50 divided by 9? Solution. — Write the numbers and divide, 9 J $ 985.50 as required by the rule in division, and separate $109.50 the dollars from the cents in the quotient by a decimal point. 2. What is the cost of a barrel of flour when 14 bar- rels cost #Qi 9 Solution. — Continue the division after dividing the dollars, supplying the orders of cents in the dividend by ciphers. W$91.00($6.50 84 70 70 Divide — 3. $325.20 by 8. 4. $67.10 by 11. 5. $626.50 by 25. 6. $73.44 by 12. 7. $9356 by 100. 8. $13065 by 13. .56 by 4. 10. $634055 by 1000. 11. $1135 by 20. 12. $19.65 by 15. 13. $114.24 by 6. 14. $111144 by 100. UNITED STATES MONEY. 125 15. If 11 tons of hay cost $184.25, what is the cost of 1 ton ? 16. At $6.50 a barrel, how many barrels of flour can be bought for $91 ? 650)9 100( 14 Solution.— Prepare the numbers for di- ^' 9 _r_ viding by reducing both to cents. $6.50 — 2600 650 cents; and $91 -=9100 cents; 9100 -h- 2600 ^50 = 14, the result required. 17. How many tons of coal, at $5.25 a ton, can be bought for $105 ? 18. When 20 tons of coal can be bought for $105, what is the cost of 1 ton ? 19. At 80 cents a bushel, how many bushels of corn can be bought for $68 ? 20. Jones paid for his farm $10564, and Smith bought some land at one-fourth of that sum. What was the cost of Smith's land ? 21. One half of Jones's farm is 320 acres; if the whole cost him $7680, what was the price per acre ? ALIQUOT PARTS. 260.— 1. What part of a dollar is 50 cents? 2. What part of a dollar is 25 cents ? 3. How many cents in half a dollar? In a fourth of a dollar ? 4. What part of a dollar is obtained by dividing it by 2 ? By dividing it by 4 ? 5. What part of a dollar is 20 cents? Is 10 cents? 6. How many cents in a fifth of a dollar? In a tenth of a dollar? n * 126 UNITED STATES MONEY. 7. What part of a dollar is obtained by dividing it by 5 ? By dividing it by 10 ? 8. What part of a dollar is 33^ cents? 16-| cents? 8| cents ? 9. How many cents in one third of a dollar ? In one sixth of a dollar ? In one seventh of a dollar ? In one twelfth of a dollar ? 10o What part of a dollar is obtained by dividing it by 3 ? By dividing it by 6 ? By dividing it by 8 ? 11. What part of a dollar is 12£ cents? Is 37^ cents? Is 621 C ents ? Is 87£ cents ? 12. How many cents in one eighth of a dollar? In five eighths ? In seven eighths ? 13. What part of a dollar is 66f cents? Is 75 cents f Is 83^ cents ? 14. What will 25 yards of cloth cost at 12 \ cents a yard? Solution.— If 1 yard cost 12J cents, or \ of a dollar, 2o yards will cost 25 times \ of a dollar, or 2 f of a dollar, which are $3J, or $3.12i. 15. What will 60 pairs of hose cost at 25 cents a pair? 16. What will 37 bushels of apples cost at 33^ cents a bushel ? 17. How much must be paid for 30 bushels of corn at 66% cents a bushel ? Solution.— If 66f cents, or f of a dollar, must be paid for 1 bushel, there must be paid for 30 bushels, 30 times f of a dollar, or %° of a dollar, which are $20. 18. How much must be paid for 16 baskets of peaches at 87^ cents a basket? 19. At 75 cents each, what will 48 arithmetics cost? UNITED STATES M02^FT 127 DEFINITION. 261. An Aliquot Part of a number is any exact frac- tional part of that number. In business, frequent use is made of the convenient aliquot parts of a dollar, given in the following — TABLE. 10 cents are — of $1. 20 " " j of $1. 25 « « 7 of 01. 4 50 « >> I of$l. 8- cents are — of . o 12 mi 3 33± jof$l. lof$l. Uf$i. WRITTEN EXERCISES. 262. — 1. What is the cost of 49 yards of cloth at $1.75 a yard? Solution. At $1.00 per yard, the cost of 49 yards is $49. a .50 << « « " " ~2 of $49 == 24-50 " .25 " " " " " 7 °f $4-9 ^^ 12.25 " $1.75 « « " " " $85.75 2. How much must be paid for 54 bushels of wheat at $1.66| per bushel? 3. What will 72 hats cost at 87£ cents each? 4. What will 96 pounds of rice cost at 8£ cents a pound ? 263. Rule for Finding the Cost of Articles by Aliquot Parts.- First find the cost at $1, and then take the aliquot parts of this amount. 128 UNITED STATES MONEY. PROBLEMS. 264. — 1. What is the value of 758 yards of carpeting at $1.25 per yard? 2. How much must be paid for 178 yards of gingham at 37^ cents per yard ? 3. What will 1840 tons of coal cost at $4.62* per ton? 4. At 33^ cents a pair, what is the cost of 96 pairs of gloves ? BILLS. 265. A Bill of Goods is a written statement of articles sold, the quantity and price of each article, and the en- tire cost of the whole. 266. A Bill of Services is a written statement of labor performed, the time, kind and value of such services. 267. A Debtor is the party who owes a bill, and a Creditor is the party to whom the bill is owed. 268. A bill is Receipted when the creditor, or some one acting for him, acknowledges its payment in writing. 269. Test Questions.— 1. How do you write United States money for adding ? For subtracting ? 2. How do you write the numbers and multiply in United States money? How do you write the numbers and divide? 3. In dividing dollais, when there is a remainder, how may you continue the division? When the divisor expresses cents and the dividend dollars, how do you prepare the numbers for dividing? 4. What are aliquot parts of a number? How do you com- pute the cost of articles by aliquot parts? 5. What is a bill of goods? A bill of services? When is a bill receipted? UNITED STATES MONET. 129 WRITTEN EXERCISES. 270. Copy the following bills, and compute the amount due — Bill Unreceipted. Philadelphia, Jan. 4, 1871. John W. Brewster, Bought of William Collins & Co. 18 yards of Muslin, @ $.16l $ 3 00 28 yards of Cambric, @ .25 7 00 12 dozen Napkins, @ 2.12\ 25 50 6 dozen Towels, @ $5.25 31 50 $67 00 The character @ signifies at. Thus, 18 yards of muslin @ $.16| means 18 yards of muslin at $.16f per yard. Bill Receipted. Portland, April 3, 1871. Henry C. Warren, Bought of Hamstead Brothers. 10 pounds Oolong Tea, @ $1.20 20 pounds Rio Coffee, @ .37 ~ 15 pounds Sugar, @ . 12- Received payment, Hamstead Brothers. 130 UNITED STATES MONEY. Bill Receipted by Clerk. Hartford. March 11, 1871. Col. Ambrose Chase, To William T. Stone, Dr. 1871. Feb. 3 For Labor on wall, 9 days, @ $3.50 $31 50 tt March 10 5 " Excavating eellar, " Labor, laying stone, 8 days, 15 50 @ $3.25, 26 00 $73 00 Received payment, William T. Stone, per M. T. Sneider. Bill tvith Credit Items. Providence, April 27, 1871. Mr. Walter Bowen, To John Burgess & Sons, Dr. 1871. Jan. April 17 8 25 17 24 To 72 tons Coal, @ $8.62 j " 15 cords Oak Wood, @ 7.25 " 18 cords Pine Wood,® 6.33J Or. By Merchandise, as by his bill, $550.50 By Cash, 200.00 $621 108 114 00 75 00 Feb. U $843 750 75 50 Balance due J. B. & Sons $ 93 25 DENOMINATE NUMBERS. 131 SECTION XXIII. MEASURES OF EXTENSION. LINEAR MEASURES. 271. Linear or Long Measures are those used in ascertaining dis- tances and the dimen- sions of things. The units of length are an inch, a yard, a rod and a mile. One Inch. TABLE. 12 inches (in J are 1 foot ft. 3 feet " 1 yard yd,. 5% yards " 1 rod rd. 320 rods " 1 mile. . . . mi. 1 mi. = 320 rd. = 1760 yd. = 5280 ft. = 63360 in. Also, 4 inches are 1 hand, used in measuring the height of horses; 3 feet are 1 pace; and 6 feet are 1 fathom, used in measuring depth of water. 272. In Cloth Measure the yard is divided into halves, quarters, eighths and sixteenths. 273. The Surveyor's Chain, called Gunters chain, used 132 DENOMINATE NUMBERS. in measuring roads and boundaries of land, is 4 rods in length, and is divided into 100 links. Thus, 7 T 9 o% inches are 1 link ; 100 links, or 4 rods, are 1 chain; and 80 chains are 1 mile. 274. — 1. How many inches are 2 feet? Are 6 feet? 2. How many inches is 1 yard ? Are 2 yards ? 3. How many feet are 6 yards ? Are 8 yards ? 4. How many feet in 24 inches? In 72 inches? 5. How many yards in 36 inches ? How many feet in 72 inches ? How many yards in 72 inches ? 6. How many yards in 24 feet? In 36 feet? 7. How many yards in 3 rods ? In 5 rods ? 8. In 1 mile how many rods ? In \ of a mile how many rods ? Solution. — Since in 1 mile there are 320 rods, in \ of a mile there must be one half of 320 rods, or 160 rods. 9. How many rods in \ of a mile? In i of a mile? 10. How many inches in 3 quarters of a yard ? 11. How many eighths in 3 quarters of a yard? 12. How many yards in 33 feet? In 45 feet? 13. How many rods in 33 feet? In 66 feet? 14. What part of a foot is 4 inches? Is 8 inches? 15. At 5 cents a foot, what will 10^ yards of wire cost? 16. When ribbon is worth 20 cents a yard, what are 3 yards and 3 quarters worth ? 17. What part of a yard is 9 inches? Is 45 inches ? 18. At \ of a dollar a rod, how much will it cost to construct a path \ of a mile long ? DENOMINATE NUMBERS. 133 DEFINITIONS. 275. Denomination is the name of the unit expressing a measure or number. Of two denominations, the higher is that which expresses the greater value, and the lower is that which expresses the less value. 276. A Denominate Number is a number expressed in one or more denominations. WRITTEN EXERCISES. 277. — 1. How many feet are 46 rods ? Solution. — Since 1 rod is 5J yards, there must be 5J times as many yards as rods ; hence, in 46 rods there must be 5J times 46 yards, or 253 yards. Since 1 yard is 3 feet, there must be 3 times as many feet as yards ; hence, in 253 yards, there are 3 times 253 feet, or 759 feet, which is the answer re- quired. 759 ft 2. How many rods are 759 feet ? 3 ft ) 759 ft Solution. — Since 3 feet are one 2 yard, 759 feet must be as many yards &J yd. ) 253 yd. as there are times 3 feet in 759 feetj 2_ _2_ which are 253 times. Hence, 759 1 1 hf. yd.) 506 hf. yd. feet = 253 y ards - ~~7r j Since 5J yards are 1 rod, 253 yards 4- r > must be as many rods as there are times 5 \ yards in 253 yards, or times 11 half yards in 506 half yards, which are 46 times. Hence, 253 yards, or 759 feet = 46 rods 6 3. How many feet are 80 rods? Are 15 miles? 4. How many rods are 1320 feet? Are 4700 feet? 5. How many miles are 79200 feet ? Are 7920 chains ? 12 46 rd. 230 23 253 yd. 3 134 DENOMINATE NUMBERS. SURFACE MEASURES. 278. A Surface is that which has length, and breadth or width, without thickness. 279. A Square is a figure having four equal straight sides, and four equal corners or angles. 280. A Square Iuch is a square having each of its sides 1 inch in length. 281. A Square Foot is a square having each of its sides 1 foot in length. 282. A Square Rod is a square i square inch. having each of its sides 1 rod in length. 283. A Square Mile is a square having each of its sides 1 mile in length. 281. Surface Measures are those used in ascertaining extent of surface. The units of surface are a square inch, a square foot, a square yard, a square rod, an acre and a square mile. TABLE. 144 square inches (sq. in.) are 1 square foot . ..sq. ft. 9 square feet " 1 square yard . sq. yd, 30^ square yards " 1 sq. rod or perch. . P, 160 sq. rods or perches " 1 acre , A GJfO acres " 1 square mile . . sq. n% a 1 A = 160 P. = 4840 sq. yd. = 43560 sq. ft.= 6272640 sq. in. 285. In the Measurement of Land, 16 square rods are 1 square chain (sq. ch.), and 10 square chains are 1 acre. DENOMINATE NUMBERS. 135 286. — 1. How many square inches in 2 square feet? 2. How many square feet in 4 square yards ? In 8 square yards ? In 2 square rods ? In 3 square rods ? 3. How many square yards in 36 square feet ? In 72 square feet ? 4. How many square rods in 1 acre ? In |- of an acre ? 5. What part of an acre is 80 square rods ? Is 40 square rods? 6. How many square rods are 27 square feet? 7. How many square rods are 60| square yards ? 8. What part of an acre is 5 square chains ? 9. What will it cost to pave 108 square feet of a walk, at 50 cents a square yard ? WMITTEN EXERCISES. 287.— 1. How many 6quare feet in 25 acres ? Solution. 25 A. 160 1500 25 4000 P. soj 120000 1000 121000 sq. yd. 9 1089000 sq.ft. 2. How many acres in 1089000 square feet ? Solution. 9 sq.ft.) 1089000 sq.ft. 30J sq. yd. J 121000 sq. yd. 121 fourths ) 484000 fourths sq. yd. sq. yd. 160 P. ) 4000 P. 25 A. 3. How many square inches in 15 square rods ? 4. How many square rods in 588060 square inches ? 136 DENOMINATE NUMBERS. 5. In 100 square chains how many square rods ? 6. In 1600 square rods how many square chains ? 7. In 36 square miles how many square rods ? 8. In 23040 square rods how many square miles? 9. In 120 square yards how many square inches? 10. How many acres are there in a lot of land 80 rods long and 72 rods wide ? 11. How many acres of land are there in a road 6^ miles long and 5 rods wide ? CUBIC MEASURES. 288. A Solid, or Volume, is that which has length, breadth and thickness, or depth. 289. A Cube is a solid bounded by six equal squares, called faces. 290. A Cubic Inch is a cube whose faces are each 1 inch square. 291. A Cubic Foot is a cube whose faces are each 1 foot square. 292. A Cubic Yard is a cube whose faces are each 1 yard square. 293. Cubic Measures are those used in meas- uring things that have length, breadth and depth, or thickness. The unite are a cubic inch, a cubic foot and a cubic yard; also, a cord foot and a cord. A Cubic Inch. DENOMINATE NUMBERS. 137 TABLE. 1728 cubic inches (cu. in.) are 1 cubic foot. . . cu. ft. 27 cubic feet " 1 cubic yard. . cu. yd. Also, 16 cubic feet are 1 cord foot cd. ft. 8 cord feet f or ) ^ 1 CQrd cd 128 cubic feet, ) 1 cu. yd. - 27 cu. ft. = 46656 cu. in. 294. Wood as usually cut for the market is 4 feet long, and is piled in ranges 4 feet high. Of such ranges, a part that is 1 foot of the length of the range is 1 cord foot, or 1 foot of wood; and a part that is 8 feet of the length of the range is 1 cord of wood. 295. — 1 . How many cubic inches in 2 cubic feet ? 2. How many cubic feet in 2 cubic yards ? 3. How many cubic feet in 2 cord feet ? In 3 cord feet ? In 4 cord feet ? In | of a cord ? In | of a cord ? 4. How many cords in 48 cord feet? In 88 cord feet? 12* 138 DENOMINATE NUMBERS. WRITTEN EXERCISES. 296. — 1. How many cubic inches in 306 cubic yards ? Solution. 2. How many cubic yards in 14276736 cubic inches? 306 cu. yd. 27 2142 612 8262 cu. ft. 1728 66096 16524 57834 8262 14276736 cu. in. Solution. 1728)14276736(8262 cu.fi. 13824 4527 3456 10713 10368 3456 3456 27)8262(306 cu. yd. 81 162 162 3. How many cubic feet in 365 cords ? 4. How many cords in 46720 cubic feet? 297. Test Questions. — 1. For what is linear or long measure used ? What are its units ? Recite the table. 2. How is the yard divided in cloth measure? How many inches in 1 hand? How many feet in 1 pace? In 1 fathom? What is the length of Gunter's chain ? 3. What is a denomination ? Of two denominations, which is the higher? The lower? What is a denominate number? 4. What is a surface ? A square ? A square inch ? A square foot? A square rod? A square mile? Recite the table. 5. What is a volume or solid? A cube? A cubic inch? A cubic foot ? A cubic yard ? For what are cubic measures used ? What are the units of cubic measures ? Recite the table. How is wood usually cut and ranged ? DENOMINATE NUMBERS. 139 SECTION XXIV. MEASURES OF CAPACITY. LIQUID MEASURES. 298. Liquid Measures are used in measuring liquids. The units are a gill, a pint, a quart and a gallon. TABLE. J^ gills (gi.) are 1 pint pt. 2 pints " 1 quart. . . . qt. Jj, quarts " 1 gallon . . . gal. 1 gal. =4qi. = 8 pt. = 32 gi. A Barrel, regarded as a measure of cisterns, vats, etc., is 31 \ gal., and a Hogshead is 63 gallons. A Gallon, liquid measure, contains 231 cubic inches. 299. — 1. How many gills in 8 pints? In 12 pints? 2. How many pints in 32 gills? In 48 gills? 3. How many pints in 8 quarts ? In 1 1 quarts ? 4. How many quarts in 1 6 pints ? In 24 pints ? 5. How many quarts in 6 gallons? In 10 gallons? 6. How many gallons in 24 quarts ? In 48 quarts ? 7. What part of a quart is 2 gills ? Is 6 gills ? 8. What part of a gallon is 1 quart? Is 3 quarts? 9. At 40 cents a gallon, what will 3 quarts of wine cost ? 10. How many pint-and-a-half bottles can be filled from a gallon and a half ? 11. How many barrels can be filled from fifty hogs- heads ? 12. How many quarts are there in a hogshead of molasses? 140 DEXOMIXA TE NUMBERS. WRITTEN EXERCISES. 300. — 1. How many gills in 84 gallons ? Solution. 84 gal. 4 336 qt. 2 672 pt 4 2. How many gallons in 2688 gills ? Solution. 4) 2688 gi. 2)672 pt 4)336 qt 84 gal. 2688 gi. 3. How many pints in 116 hogsheads? 4. How many hogsheads in 928 pints ? 5. At 3 cents a gill, what will a barrel of liquor cost ? 6. At 6 cents a quart, how much must be paid for 40 gallons of milk ? DRY MEASURES. 301. Dry Measures are those used in measuring grain, fruit, coal, salt, and similar articles. The units are a pint, a quart, a peck and a bushel. TABLE. 2 pints (pt.) are 1 quart. . . qt. 8 quarts " 1 peck .... ph. 4 peeks " 1 bushel . . bu. lbu = 4pk.= 32 qt. = 64 pt. A Gallon, or 4 quarts, dry measure, contains 268| cubic inches; and a bushel, 2150^^ cubic inches. DENOMINATE NUMBERS. Ill 302. — 1. How many pints in 6 quarts ? In 12 quarts ? 2. How many quarts in 8 pecks? In 11 pecks ? 3. How many quarts in 12 pints? In 24 pints ? 4. How many pecks in 64 quarts? In 88 quarts? 5. How many pecks in 9 bushels? In 12 bushels? 6. How many bushels in 36 pecks? In 60 pecks ? 7. What will 2\ quarts of chestnuts cost, at 5 cents a pint ? 8. What will 3f gallons of milk cost, at 4 cents a quart ? 9. What will a peck and a half of berries cost, at 16f cents, or ^ of a dollar, a quart ? WRITTEN EXERCISES. 303. — 1. In 310 bushels, how many quarts? 2. In 9920 quarts, how many bushels? 3. How many pints in 64 pecks ? 4. How many bushels in 40960 pints ? 5. At 8 cents a quart, how much are 6 bushels of chestnuts worth? 6. At 8 cents a quart, how many bushels of chestnuts can be bought for $15.36 ? 304. Test Questions. — 1. What are liquid measures? Name the units of liquid measures. Kecite the table. 2. How many gallons in a barrel when it is regarded as a measure? A hogshead? How many cubic inches does a gallon contain ? 3. What are dry measures? Name the units of dry measures. Recite the table. 4. How many cubic inches does a gallon contain? How many does a bushel contain ? 142 DENOMINATE NUMBERS. SECTION XXV. MEASURES OF WEIGHT. WOIRDUPOIS WEIGHTS. 305. Avoirdupois Weights are those used in weighing produce, groceries, coal, iron, and similar articles. The units are an ounce, a pound, a hundred-weight and a ton. TABLE. 16 ounces (oz.) are 1 pound lb. 100 pounds " 1 hundred-weight . . cwt. 20 hundred-weight " 1 ton T. IT =20 cwt. = 2000 lb. = 32000 oz. Also, 56 pounds of corn or rye, 60 pounds of wheat or pota- toes, or 32 pounds of oats, are 1 bushel; 100 pounds of dry- fish are 1 quintal ; 100 pounds of grain are 1 cental; and 196 pounds of flour, or 200 pounds of beef or pork, are 1 barrel. 306. At the Custom Houses, in collecting duties on English goods, and in the wholesale and freighting of coal, 28 pounds are 1 quarter, 4 quarters, or 112 pounds, are 1 hundred-weight, and 20 hundred-weight, or 2240 pounds, are 1 ton, called the long ton. DENOMINATE NUMBERS. 143 307. — 1. How many ounces in 2 pounds? In 3 pounds ? In 5 pounds ? 2. How many pounds in 32 ounces ? In 48 ounces ? 3. How many pounds in 6 hundred- weight? In 9 hundred-weight ? 4. How many hundred- weight in 500 pound?? In 700 pounds ? 5. How many tons in 50 hundred- weight ? In 100 hundred-weight ? 6. What part of a pound is 12 ounces? 7. What part of a hundred- weight is 25 pounds ? Is 75 pounds ? 8. How many pounds in 1 quarter of a hundred- weight ? In 3 quarters of a hundred-weight ? 9. At $2 a hundred-weight, what will 1^ tons of iron cost? 10. At 10 cents an ounce, what will 1^ pounds of rhubarb cost ? 11. At $1.50 per bushel, how much must be paid for a bag of wheat weighing 90 pounds ? WRITTEN EXERCISES. 308. — 1. How many ounces in 316 pounds? 2. How many pounds in 65 tons ? 3. How many pounds in 5056 ounces ? How many tons in 13000 pounds ? 4. How many bushels in a load of wheat weighing 4590 pounds? 5. What is the weight of 76| bushels of wheat ? 6. At H cents per pound, what will be the cost of freighting 16 tons of goods? 144 DENOMINATE NUMBERS. TROY WEIGHTS. 309. Troy Weights are those used in weighing gold, sil- ver and gems. The units are a grain, a pennyweight, an ounce and a pound. TABLE. 2Jf grains (gr.) are 1 pennyweight .... pwt. 20 pennyweights " 1 ounce oz. 12 ounces " 1 pound lb. 1 lb. = 12 oz. = 240 pwt. = 5760 gr. A Pound Avoirdupois is equal to 7000 Troy grains. 310. Apothecaries, in compounding medicines and in putting up prescriptions, either use only the denomina- tions of grains, ounces and pounds, or subdivide the Troy pound into grains, scruples, drams and ounces* Thus, 20 grains (gr.) are 1 scruple .... 9. S scruples " 1 dram 3. 8 drams " 1 ounce §. 12 ounces " 1 pound lb. 311. — 1. How many grains in 1 pennyweight? In 2 pennyweights ? 2. How many pennyweights in 48 grains ? DENOMINATE NUMBERS. 145 3. How many pennyweights in 2 ounces? In 5 ounces ? 4. How many ounces in 40 pennyweights ? In 100 pennyweights? 5. How many ounces in 3 pounds ? In 10 pounds? 6. How many pounds in 48 ounces ? In 144 ounces ? 7. At $3.25 a pennyweight, what is the value of a jewel weighing 12 pennyweights? 8. At 3 cents a scruple, what is the value of a drug weighing 1 ounce? 9. How many doses, of 5 grains each, are there in 1 dram of medicine ? WRITTEN EXERCISES. 312. — 1. How many grains in 65 ounces ? 2. How many ounces in 31200 grains? 3. How many pennyweights in 51^ pounds ? 4. How many pounds in 12360 pennyweights? 5. At 25 cents a dram, what will 80 pounds of drugs cost? 6. At 25 cents a dram, how many pounds of drugs may be bought for $1920 ? 7. If the gold coin of the United States is composed of 9 parts of pure gold and 1 part of alloy, how many pennyweights of alloy are there in 20 pounds of coin ? 313. Test Questions.— 1. What are avoirdupois weights? Name the units of avoirdupois weights. Recite the table. 2. How many pounds of corn or rye are 1 bushel ? How many pounds of wheat or potatoes ? How many of oats? 3. How many pounds of fish are 1 quintal? How many 13 146 DENOMINATE NUMBERS. pounds of grain are 1 cental ? How many pounds of flour are 1 barrel ? How many pounds of beef or pork are 1 barrel ? 4. How many pounds are a quarter in the wholesale and freighting of coal? How many pounds are a hundred-weight? How many pounds in a ton ? 5. What is Troy weight? What are the units of Troy weight? Recite the table. Recite the table of apothecaries' weights. 6. How many Troy grains are 1 pound Troy ? How many Troy grains are equal to 1 pound avoirdupois ? Which is the heavier, a pound Troy or a pound avoirdupois ? SECTION XXVI. CIRCULAR MEASURES. 314. A Circle is a plane surface bounded by a line, all parts of which are equally distant from a point within called the center. 315. A Circumference is the line that bounds a circle. 316. An Arc is any part of the circumference ; as AD, or DB. 317. A Degree is one of the 360 equal parts of a circumference. 318. An Angle is the differ- ence of direction of two lines which meet at a point. Thus, the lines AB and A C, which A^- —c meet at A, form the angle BAC. 319. The Measure of an angle whose sides meet at DENOMINATE NUMBERS. 147 the center of a circle is that part of the circumference between the sides. Thus, the arc AD is the measure of the angle A CD which is formed in the circle, (page 146). 320. Circular Measures are those used in meas- uring the arcs of circles, angles and the difference of directions. The units are a second, a minute, a degree and a circumference. TABLE. 60 seconds (") are 1 minute '. 60 minutes " 1 degree °. 360 degrees " 1 circumference . . C. 1C. = 360° = 21600' = 1296000". 321. A Minute of the circumference of the earth, or a geographic mile, is about \\ common miles. 322. — 1. How many seconds in 2 minutes? In 3 minutes ? In 5 minutes ? 2. How many minutes in 120 seconds? In 180 seconds ? In 300 seconds ? 3. How many minutes in 3 degrees? In 4 degrees? 4. How many degrees in 120 minutes? In 240 minutes ? In 360 minutes ? 5. What part of a circumference is 30 degrees ? 6. How many degrees in \ of a circumference ? 148 DENOMINATE NUMBERS. WRITTEN EXERCISES. 323. — 1. How many seconds in 240 degrees? 2. How many degrees in 864000 seconds ? 3. How many minutes in f of a circumference? 4. How many degrees in 14200 minutes? 5. How many seconds in § of an hour ? 6. If a vessel sail 2| degrees of the circumference of the earth in one day, in how many days will it sail 180 degrees ? 7. In sailing 180 degrees of the circumference of the earth, how many common miles does a vessel pass over? SECTION XXVII. MEASURES OF TIME. 324. Measures of Time are those used in measuring time or duration. The units are a second, a minute, an hour, a day and a year. TABLE. 60 seconds (s.) are 1 minute m. 60 minutes " 1 hour h. @4 hours " 1 day d. 365 days " 1 common year . c. y. 866 days 1 leap year l*y* lc.y. = 865d. = 8760h.=52B600m. = 31586000s. Also, 7 days are 1 week, 12 months are 1 year, and 100 years are 1 century. DENOMINATE NUMBERS. 149 325. The Names of the Months, and the number of days in each, are — Days. Days. January, 1st month, 31. July, 7th month, 31. February, 2d " 28 or 29. August, 8th " 31. March, 3d " 31. September, 9th " 30. April, 4th " 30. October, 10th " 31. May, 5th " 31. November, 11th " 30. June, 6th " 30. December, 12th " 31. The exact length of the year is about 365^ days ; hence, every fourth year February has 29 days, and the year has 366 days. In general, every year whose number can be divided by 4 without a remainder is leap year. Thus, the year 1872 is a leap year. 326. — 1. How many seconds in 1 minute? In one half a minute ? In one third of a minute ? 2. How many minutes in 120 seconds? 3. How many minutes in 1 hour ? In \ of an hour ? 4. How many hours in 120 minutes? In 180 minutes? In 240 minutes? 5. How many hours in 2 days ? In 3 days ? 6. What part of a day is 6 hours? Is 18 hours? 7. How many days in 48 hours? What part of a day are 6 hours ? 8. How many months in 5 years ? In 9 years ? 9. How many weeks in 49 days ? In 63 days ? 10. How many days has a leap year? A common year? 11. What months have 30 days each ? In what years does February have 29 days ? 12. How many days are there in 7 weeks? In 9 weeks ? 13* 150 DENOMINATE NUMBERS. WRITTEN EXERCISES. 327. — 1 How many minutes in 18 days? 2. How many days in 25920 minutes ? 3. How many hours in a common year ? 4. How many seconds in 24 hours ? 5. How many hours in 86400 seconds ? 6. How many years in 8760 hours? 7. How many seconds in a common year ? 8. How many days in 31622400 seconds? 9. If you are 11 years old, how many minutes have you lived, allowing 365^ days to a year ? SECTION XXVIII. PAPER AND COUNTING. 328. Paper is bought and sold by the sheet, quire, ream, bundle or bale. TABLE. 24 sheets are 1 quire. 20 quires " 1 ream. 2 reams " 1 bundle. 5 bundles " lbale. 329. In counting cer- tain articles, the units dozen, gross and great gross are used. DENOMINATE NUMBERS. 151 f TABLE. 12 things are 1 dozen doz. 12 dozen " 1 gross gro. 12 gross " 1 great gross. . grt. gro. Two things are a, pair, and 20 things are a score. 330. — 1. How many sheets of paper in 2 quires? 2. How many quires in 1 bundle ? In 2 bundles ? 3. How many quires in 48 sheets ? 4. How many reams in 80 sheets ? In 100 sheets ? 5. How many things in 5 dozen ? How many pens in a gross ? 6. How many dozen in a great gross ? In half of a great gross ? 7. How many pairs are 40 ? How many scores are 40? Are 80? Are 120? WRITTEN EXERCISES. 331. — 1. How many sheets are 9| reams? 2. How many reams are 4560 sheets ? 3. How much will a gross of pens cost, at 16 cents a dozen ? 4. If you buy 5 T 3 o reams of paper, how many sheets will you obtain ? 5. In some boxes there are 2064 eggs ; what is their value at 30 cents a dozen ? 6. How many boxes will be required to pack 2592 screws, if each box will hold 9 dozen ? 7. How much will 25 gross of pens cost, at 2 cents per pen ? 8. What will 75 bundles of paper cost, at 15 cents per quire ? 152 BE VIEW. 332. Test Questions. — 1. What is a circle? A circumfer- ence? An arc? A degree? 2. What is an angle? The measure of an angle? 3. What are circular measures? The units of circular meas- ures? Kecite the table. 4. How many miles is a minute of circumference? 5. What are measures of time? The units of measures of time? Recite the table. 6. How many days are 1 week? How many months are 1 year? How many years are 1 century? 7. Name the months, and give the number of days in each. 8. What is the exact length of a year? How often has Feb- ruary 29 days ? When February has 29 days, how many days has the year ? In general, what years are leap years ? 9. By what is paper bought and sold? Recite the table. What units are used in counting certain articles ? Recite the table. How many things are a pair ? How many things are a score ? SECTION XXIX. REVIEW OF DENOMINATE NUMBERS. 333. — 1 . How many inches in f of a yard ? Solution. — 1 yard is 3 feet; f of 3 feet are 2 feet; and since 1 foot is 12 inches, 2 feet are 2 times 12 inches, or 24 inches. Hence, f of a yard are 24 inches. 2. How many pints in f of a bushel ? 3. At f of a dollar a cord foot, what will i of a cord of wood cost ? 4. At $.10 per ounce, what will 1£ pounds of spice cost? 5. What part of a bushel is 3 pecks ? REVIEW. 153 6. At 2 dollars a bushel, what will 3 pecks of wheat cost? 7. James is 108 months old, and his age is f of his brother's. What is his brother's age ? 8. At 16 cents a yard, what will 9 yards of cloth cost? 9. At 12 cents a peck, how many bushels of oats can be bought for 96 cents ? 10. If 40 perches of land cost 9 dollars, what will an acre cost? 11. If i^ of a ton of hay be worth 3 dollars, what will 30 hundred-weight be worth ? WRITTEN EXERCISES. 334. — 1 . How many minutes in a leap year ? 2. A walk is 40 rods long ; what will it cost to pave it, at 25 cents per linear foot ? 3. Into how many lots, of 20 square rods each, can an acre and a half be divided ? 4. How much must be paid for 25 barrels of beef, at 1 1 cents a pound ? 5. At 11 cents a pound, how many pounds of beef can be bought for $550 ? 6. For how much will a hogshead of liquor sell, if retailed at 10 cents a half gill ? 7. A certain town is 36 square miles in extent ; how many acres does it contain ? 8. How many bushels in a load of corn, which, at 86 cents a bushel, costs $34.40 ? 9. How many hogsheads of molasses, of 63 gallons each, worth 55 cents a gallon, can be bought for $1386 ? 154 COMPOUND NUMBERS. SECTION XXX. REDUCTION OF COMPOUND NUMBERS. 335. — 1. How many inches are 6 feet 5 inches ? Solution. — Since 1 foot is 12 inches, 6 feet must be 6 times 12 inches, or 72 inches ; and 72 inches plus 5 inches are 77 inches. Hence, 6 feet 5 inches are 77 inches. 2. How many cord feet in 5 cords 6 cord feet ? 3. Reduce 1 peck 3 quarts 1 pint to pints. 4. Reduce 2 tons 9 hundred-weight to hundred-weights. 5. How many feet in 77 inches ? Solution. — Since 12 inches are 1 foot, 77 inches must be as many feet as 12 inches are contained times in 77 inches, which are 6 times, with a remainder of 5 inches. Hence, 77 inches are 6 feet 5 inches. 6. How many cords in 46 cord feet? 7. Reduce 37 gills to quarts. 8. How many pecks in 33 pints? 9. How many tons are 49 hundred-weight? 10. What will 2 gallons 3 quarts of vinegar cost, at 5 cents a pint? 11. A wall is 4 yards 6 inches long ; what is its length in inches ? DEFINITIONS. 336. A Simple Denominate Number is a number ex- pressed in units of only one denomination. Thus, 5 yards is a simple denominate number. 337. A Compound Denominate Number is a number ex- pressed in units of more than one denomination. Thus, 6 yards 7 inches is a compound denominate number. COMPOUND NUMBERS. 155 338. Reduction of Denominate Numbers is the process of changing them from one denomination to another without changing their value. 339. Principle. — Denominate numbers are reduced tc lower denominations by multiplication, and to higher de- nominations by division, REDUCTION DESCENDING. 340. Reduction Descending is the process of changing a number to an equivalent number expressed in units of a lower denomination. WRITTEN EXERCISES. 341. — 1. How many pints are 3gal. 2qt. lpt. ? Solution. — Since 1 gallon is 4 quarts, J gal. J qt. 1 pt. 3 ga j lons must be 3 t i mes 4 quartS) r 12 •^ quarts ; and 12 quarts plus 2 quarts are 14 qt. 14 quarts. 2 Since 1 quart is 2 pints, 14 quarts — — must be 14 times 2 pints, or 28 pints; ~J PL and 28 pints plus 1 pint are 29 pints. Hence, 3gal. 2qt, lpt. are 29pt. 2. Reduce 4yd. 2ft. 7in. to inches. 3. Reduce 5 sq. yd. 7 sq. ft. 27 sq. in. to square inches. 342. Rule for Reduction Descending.— Multiply the num- ber of the highest denomination given, by that ** umber of the next lower which equals one of the higher, and to the product add the number, if any, of the next lower denomination. Reduce this result in lihe manner, and so proceed until the given number is reduced to the required denomination. 156 COMPOUND NUMBERS. PROBLEMS. 343.— 1. Reduce 3° 4' 16" to seconds. 2. Reduce 3 A. 140P. to square yards. 3. Reduce lmi. 138rd. 4yd. to feet. 4. Reduce 100 cu. yd. 20 cu. ft. 100 cu. in. to cubic inches. 5. Reduce 7bu. 3pk. 5qt. to quarts. 6. Reduce 9oz. llpwt. 17gr. to grains. 7. Reduce 4T. 9cwt. 91b. to ounces. 8. Reduce lhhd. 20gal. 3qt. to quarts. 9. How much must be paid for 1 A. 40 sq. rd. of land, at $1.50 per square rod? 10. What will it cost to make lmi. lOOrd. of fence, at $1.20 a rod? 11. How much must be paid for 13 63 29 of med- icine, at 5 cents a grain ? 12. How many hours has a man lived who is 50y. 125d. old, allowing 365| days to the year? REDUCTION ASCENDING. 344. Reduction Ascending is the process of changing a number to an equivalent number, expressed in units of a higher denomination. WRITTEN EXERCISES. 345. — 1. How many gallons are 29 pints? Solution. — Since 2 pints are 2 pt. )29 pt. 1 quart, 29 pints must be as A at )1A at. 1 vt many quarts as 2 pints are con- — r — r — tained times in 29 pints, which 3 gal. 4 qt. are 14 timeSj wit j 1 a rem ainder 89 pt.^3 gal. 2 qt. 1 pt. of 1 pint. Hence, 29 pints are 14 quarts 1 pint. COMPOUND NUMBERS. 157 Since 4 quarts are 1 gallon, 14 quarts must be as many gallons as 4 quarts are contained times in 14 quarts, which are 3 times, with a remainder of 2 quarts. Hence, 14 quarts are 3 gallons 2 quarts ; and 29 pints are 3 gallons 2 quarts 1 pint. 2. Reduce 175 inches to yards. 3. Reduce 7515 square inches to square yards. 4. Reduce 37038 seconds to hours. 346. Rule for Reduction Ascending.— Divide the given number by that number of its denomination, whieh equals one of the next higher, and write the remainder, if any. Divide the quotient in like manner, and so con- tinue until the given number is changed to the re- quired denomination. The last quotient, with the remainders, if any, written in their order from the highest to the lowest, will be the required number. Reductions Descending and Ascending, being per- formed by opposite processes, are proofs of each other. PROBLEMS. 347. — 1. Reduce 11056 seconds to degrees. 2. Reduce 18755 square yards to acres. 3. Reduce 7569 feet to miles. 4. Reduce 4700260 cubic inches to cubic yards. I 5. Reduce 275 quarts to bushels. 6. Reduce 4601 grains to ounces. 7. Reduce 142494 ounces to tons. 8. Reduce 335 quarts to hogsheads. 9. How many acres of land will $300 purchase, at $1.50 per square rod ? 14 158 COMPOUND NUMBERS. 10. How many miles of fence, at $1.20 per rod, can be built for $504 ? 11. How many ounces of medicine, at 5 cents a grain, can be bought for $44 ? 12. How many years has a man lived who is 438300 hours old, allowing 365^ days to a year ? 348. Test Questions.— 1. What is a denominate number? A simple denominate number? A compound denominate num- ber? 2. What is reduction ? The principle of reduction ? 3. What is reduction descending ? How is a number reduced from a higher to a lower denomination ? 4. What is reduction ascending? How is a number reduced from a lower to a higher denomination ? 5. Why are reductions descending and ascending proofs of each other? SECTION XXXI. ADDITION OF COMPOUND NUMBERS. 349. — 1. What is the sum of 6 yards 1 foot, and 4 yards 1 foot? 2. Add 6 cords 3 cord feet, and 5 cords 6 cord feet. Solution. — 6 cords 3 cord feet, and 6 cord feet, are 6 cords 9 cord feet, or 7 cords 1 cord foot ; 7 cords 1 cord foot, and 5 cords, are 12 cords and 1 cord foot. 3. Add 10 bushels 2 pecks, and 4 bushels 3 pecks. 4. If you should buy in one month, 4 gallons 3 quarts of kerosene, and the next month, 3 gallons 3 quarts, how much would you buy in all ? COMPOUND NUMBERS. 169 5. Mary is 9 years 7 months old, and Alice is 8 years 8 months old. What is the sum of their ages ? 6. A farmer sold two turkeys ; one weighed 10 pounds 11 ounces, and the other 11 pounds 7 ounces. What was the weight of both ? DEFINITION. 350. Addition of Compound Numbers is the process of uniting two or more compound numbers of the same kind to find their sum. WRITTEN EXERCISES. 351_1. What is the sum of 151b. 5oz. 15pwt.; 71b. 9oz. 12pwt. ; and 6oz. 4pwt. ? Solution. — Write the numbers so 15 lb. 5 oz. 15 pwt. that figures expressing units of the Y g 12 same denomination shall be in the s> v same columns. Begin with the pennyweights, and 23 lb. U oz. 11 pwt. add the pennyweights, ounces and pounds separately. The sum of the pennyweights is 31 pennyweights, or 1 ounce 11 pennyweights. Write the 11 pennyweights under the column of pennyweights, and reserve the 1 ounce to add with the ounces. The sum of the ounces is 21 ounces, or 1 pound 9 ounces. Write the 9 ounces under the column of ounces, and reserve the 1 pound to add with the pounds. The sum of the pounds is 23 pounds, which write under the column of pounds. The entire sum is 23 pounds 9 ounces 11 pennyweights. 2. What is the sum of 6mi. 120rd. 3yd. ; 7mi. 160rd. 5yd. ; and 55rd. 3yd. ? 3. What is the sum of 15gal. 3qt. lpt. ; 7gal. qt. lpt. ; and 2qt. Opt. ? 160 COMPOUND NUMBERS. 352. Rule for Addition of Compound Numbers.— Write the numbers so that units of the same denomination shall stand in the same column. Begin with the lowest denomination, and add the numbers of each denomination separately . If the sum is less than one of the next higher denomi- nation, write it as a part of the required result. If the sum is equal to or exceeds one of the nexv higher denomination, reduce it to that denomi- nation, wi'ite the remainder, if any, as a part of the required result, and add the units of the higher denomination with the column of that denomi- nation. r B OK L EMS. 353. Write and add— (1.) (2.) 15cwt. 75 lb. 7 oz. 10 A. 146 sq.rd. 3 5 73 15 25 9 3 75 (3.) (4.) 5 CM. .yd. 17 cu.ft. 703 eu.in. 9 oz. 18 pwt. 11 gr. 11 10 835 1 19 23 11 106 4 20 3 9 112 6 (5.) (6.) 100 bu. 3 ph. 25 rd. 3 yd. 2 ft. 76 2 16 £ 1 13 3 11 3 3 4 2 COMPOUND NUMBERS. 161 7. What is the sum of 2d. 7h. 6m. ; 5d. 13h. 25m. ; llh. 11m.; and 7d. 15h. 55m.? 8. Add 1° 30' 15" 19° 45' 17" ; and 31° 40' 16". 9. A boy gathered in one day, lpk. 3qt. lpt. of ber- ries; another day, lpk. 5qt. lpt. ; and a third day, 2pk. 2qt. lpt. How many berries did he gather in all ? 10. In one field there are 17 A. 120 sq. rd. ; and in another 15A. 140 sq. rd. How much land is there in both fields? 11. In one car there are 16T. llcwt. 751b. of coal; in another, 15T. 19cwt. 311b. ; and in a third, 14T. 17cwt. 501b. How much coal is there in the three cars? SECTION XXXII. SUBTRACTION OF COMPOUND NUMBERS. 354. — 1. In one sack there are 2bu. 3pk. of wheat, and in another, lbu. 2pk. How much more is there in one sack than in the other ? 2. If 5ft. lOin. be cut from a line 7ft. 9in. long, how much of the line will be left ? Solution. — There will be left of the line the difference be- tween 7 feet 9 inches and 5 feet 10 inches. 7 feet 9 inches les9 10 inches are 6 feet 11 inches ; and 6 feet 11 inches less 5 feet are 1 foot 11 inches. Hence, there will be left 1 foot 11 inches. 3. If you should have in a cask llgal. 3qt. of molasses, and should sell 8gal. 3qt. of it, how much would there be left? 4. From lOh. 30m. subtract 8h. 40m. 162 COMPOUND NUMBERS. 5. In one firkin there are 121b. 8oz. of butter, and in another 91b. 12oz. How much more is there in the one than in the other ? 6. In one load there are 2cd. 1 cd. ft„ of wood, and in another load led. 7 cd. ft. How much is required to make the smaller load equal the greater ? DEFINITION. 355. Subtraction of Compound Numbers is the process of finding the difference between two compound numbers of the same kind. WRITTEN EXERCISES. 356.— 1. From 17bu. 2'pk. 6qt. take 8bu. 3pk. 4qt. 17 h 2 k ft t Solution. — Write the subtrahend Q . under the minuend, so that units of the M 2 *^ same kind shall stand in the same 8 bu. 3pk 2 ql columns. Begin at the right and subtract the units of each denomination of the subtrahend from those of the same denomination in the minuend. 4 quarts from 6 quarts leaves 2 quarts, which is the difference of the quarts. Since 3 pecks cannot be taken from 2 pecks, take 1 bushel from the 17 bushels, leaving 16 bushels, and add it, reduced, to the 2 pecks, thus obtaining 6 pecks ; then, 3 pecks from 6 pecks leaves 3 pecks, which is the difference of the pecks. 8 bushels from 16 bushels leaves 8 bushels, which is the dif- ference of the bushels. The entire difference is 8 bushels 3 pecks 2 quarts. 2. From 131b. 5oz. 16pwt. subtract 91b. 7oz. 14pwt. 3. From 6gal. 2qt. Opt. subtract 4gal. 3qt. lpt. 4. From 22yd. 2qr. 2na. subtract 13yd. 3qr. 3na. COMPOUND NUMBERS. 163 357. Rule for Subtraction of Compound Numbers.— Write the subtrahend under the minuend, so that units of the same denomination shall be in the same column. Begin with the lowest denomination, and subtract the units of each denomination of the subtrahend from those of the same denomination in the minu- end, if possible, and write the difference beneath. If the number of any denomination of the sub- trahend is greater than that of the same denomi- nation in the minuend, increase the number in the minuend, by adding to it as many units as make one of the next higher denomination, and subtract; then, regarding the number of the next higher de- nomination of the minuend as one less, proceed as before. PROBLEMS. 358. Write and subtract — (1.) (2.) From 25 mi. 100 rd. 7 yd. 9 A. 120 P. 24 sq. yd. Take 16 120 6 6 125 19 From Take (3.) 15 wk. d. 10 h. 12 9 11 From Take (5.) 11 1 S 14 3 7 2 16 (4.) 44 T. 75 lb. 10 oz. 21 80 15 (6.) 3° 7' 0" 1 15 . 45 7. A man who has started on a journey of 63mi. 160rd. has traveled 44mi. 125rd. 3yd. How much farther has he to travel to complete his journey ? 164 COMPOUND NUMBERS. 8. James is 13 years 9 months old, and Henry is 11 years 11 months old. What is the difference in their i? 9. From a cask which contained 75 gallons of vinegar there has been drawn 46gal. 3qt. lpt. How much re- mains in the cask ? 10. The longitude of Boston is 71° 3' 30" West, and of Chicago, 87° 37' 47" West. What is the difference of their longitudes ? 11. What is the time from May 17, 1869, to June 16, 1871? Solution.— The time from May 1871 y. 6 mo. 16 d. 17> 1869j to May 17> m ^ ig 2 ^ . from 1869 5 17 May .17, 1871, to June 16, 1871, is 2 y. mo. 30 d. 30d. The entire difference of time is 2y. 30d. 12. A man was born May 16, 1819; how old was he July 4, 1871 ? 359. Test Questions. — 1. What is addition of compound numbers ? How do you write the numbers for adding ? 2. How do you add the units of the several denominations ? How do you proceed if the sum of the units of any denomina- tion is less than a unit of the next higher denomination? How do you proceed if their sum is equal to or greater than a unit of the next higher denomination? 3. What is subtraction of compound numbers? How do you write the numbers for subtracting ? 4. How do you subtract? How do you proceed if the num- ber of any denomination of the subtrahend is greater than that above it ? COMPOUND NUMBERS. 165 SECTION XXXIII. MULTIPLICATION OF COMPOUND NUMBERS. 360.— 1. If a box contain 2 hundred-weight 20 pounds of sugar, how much will 2 similar boxes contain ? 2. How much wood is there in 3 ranges, each con- taining 3 cords 5 cord feet ? Solution. — 3 times 5 cord feet are 15 cord feet, or 1 cord 7 cord feet, and 3 times 3 cords are 9 cords ; 9 cords and 1 cord 7 cord feet are 10 cords 7 cord feet, the answer required. 3. John is 9 years 4 months old, and his father is 4 times as old. How old is his father ? 4. A boy gathered 1 peck 3 quarts of berries every day for 6 days. How many did he gather in all ? 5. If it take 1 ounce 3 drams of medicine for a pre- scription, how much will it take for 6 similar prescrip- tions ? 6. If 4 yards 3 quarters of cloth are required for a suit of clothes, how many yards will be required for 7 suits? 7. How much wood is there in two ranges, each con- taining 2 cords 6 cord feet? 8. If a team can plow 1 acre 40 square rods in 1 day, how much can it plow in 4 days ? DEFINITIONS. 361. Compound Multiplication is the process of taking a compound number as many times as there are units in the multiplier. 166 COMPOUND NUMBERS. WRITTEN EXERCISES. 362.— 1. Multiply 6gal. 3qt. lpt. by 5. 6 gal 3 qt. 1 pL Solution. — Write the multiplier urn q der the lowest denomination of the ■— — multiplicand. 34 gal 1 qt. 1 pt. Begin at the right and mu itipl y the number of each denomination in the order of the denominations. 5 times 1 pint are 5 pints, or 2 quarts 1 pint ; write the 1 pint as the number of that denomination in the product, and reserve the 2 quarts to be added to the product of the quarts. 5 times 3 quarts are 15 quarts; 15 quarts and the 2 quarts re- served are 17 quarts, or 4 gallons 1 quart. Write the 1 quart as the number of that denomination in the product, and reserve the 4 gallons to be added to the product of the gallons. 5 times 6 gallons are 30 gallons ; 30 gallons and the 4 gallons reserved are 34 gallons, which write as the gallons of the product. The entire product is 34 gallons 1 quart 1 pint. 2. Multiply 6rd. 2yd. 1ft. by 5. 3. Multiply 101b. 3pwt. 5gr. by 8. 363. Rule for Multiplication of Compound Numbers.— Write the multiplier under the lowest denomination of the multiplicand. Begin with the lowest denomination, and multi- ply the number of each denomination in its order. If the product is less than one of the next higher denomination, write it as a part of the required product. If the product is equal to or exceeds one of the next higher denomination , reduce it to that denom- ination, write the remainder , if any, as a part of the required, product, and add the units of the higher denomination to the product of that denom- ination. COMPOUND NUMBERS. 167 PROBLEMS. 364. Copy and multiply — (1.) (2.) 1 hhd. 10 gal 3 qt. 50 eu.ft. 1121 m. in. 7 8 (3.) (4.) U A. 63 P. 1 sq. yd. 4 bu. 3 pk. 2 qt. 4 9 (5.) (6.) 6 wk. 3 d. 10 h. 13° a U" 10 8 7. A farmer sold 6 loads of wood, each containing 2cd. 5 cd. ft. How much did he sell in all ? 8. What is the weight of 7 boxes of sugar, if each weighs lcwt. 311b. 8oz.? 9. Jane is 8y. 3mo. old ; how old is her aunt, who is 5 times as old ? 10. If a train of cars move 27mi. 120rd. 2yd. in an hour, how far will it move at the same rate in 11 hours? 11. Flint has in his farm 40 A. 35P., and my farm is 7 times as large. What is the extent of my farm ? 12. How much molasses is there in 12 casks, each containing 62gal. 3qt. lpt. ? 13. If it require loz. lOpwt. 15gr. of silver to make 1 spoon, how much will be required to make 18 spoons? 14. In a certain to.wn there are 3 farms. Each farm is divided into 9 equal lots, and each lot contains 12 A. 72P. What is the extent of the 3 farms ? 168 COMPOUND NUMBERS, SECTION XXXIV. DIVISION OF COMPOUND NUMBERS. 365. — 1. If 6 pounds 12 ounces of tea be divided equally among 3 persons, how much will each receive ? 2. If 8 bushels 3 pecks of chestnuts be divided equally among 5 boys, how many will each boy receive? Solution. — Each boy will receive one fifth of 8 bushels 3 pecks. One fifth of 8 bushels is 1 bushel, with a remainder of 3 bushels ; 3 bushels are 12 pecks, and 12 pecks plus 3 pecks are 15 pecks ; one fifth of 15 pecks is 3 pecks. Hence, each boy will receive 1 bushel 3 pecks. 3. Johnson is 37 years 4 months old, and his age is 4 times that of his son. What is the age of his son ? 4. If it require 33 yards 1 quarter to make 7 suits, how much will it require to make 1 suit ? 5. In 3 equal ranges of wood there are 10 cords 7 cord feet. How much is there in each range ? DEFINITION. 366. Division of Compound Numbers is the process of finding how many times one denominate number is con- tained in another of a similar kind ; or, it is the process of separating a compound number into equal parts. WRITTEN EXERCISES. 367.-1. Divide 34gal. lqt. lpt. by 5. 5)34 gal 1 g t. 1 pt. Solution.— Write the divisor at ~ z~z " ^T the left of the dividend. gal. d qt. 1 pt. Begin with the highest denomina- tion, and divide the number of each denomination in its order. One fifth of 34 gallons is 6 gallons, with a remainder of 4 COMPOUND NUMBERS. 169 gallons. Write the 6 gallons as the gallons of the quotient ; the 4 gallons are 16 quarts, which, added to the 1 quart, give 17 quarts. One fifth of 17 quarts is 3 quarts, with a remainder of 2 quarts. Write the 3 quarts as the quarts of the quotient ; the 2 quarts are 4 pints, which, added to the 1 pint, give 5 pints. One fifth of 5 pints is 1 pint, which write as a part of the quotient. The entire quotient is 6 gallons 3 quarts 1 pint. 2. Divide 32rd. Oyd. 1ft. by 5. 3. Divide 811b. 5pwt. 16gr. by 8. 368. Rule for Division of Compound Numbers.— Begin with the highest denomination; divide the number of eaeh denomination in its order, and write the several quotients as the parts of the same denom- inations of the required quotient. When there is a remainder reduce it to the next lower denomination, and add the result to the number of that denomination before dividing. When divisor and dividend are both compound numbers, they must be reduced to simple denom- inate numbers of the same denomination prepar- atory to dividing. Multiplication of compound numbers and division of compound numbers, being performed by opposite pro- cesses, are proofs of each other. PROBLEMS. 369. Copy and divide — (1.) (2.) 7 )7 hhd. 4.5 gal. 1 qt . 8) 4-05 cu. ft. 32 8 cu. in, (3.) (4.) 4 )U A. 63 P. lsq.yd. 9)37 bu^J ph. 2 qt 15 170 COMPOUND NUMBERS. 5. If 6 persons should share equally 19cd. 6 cd. ft. of wood, how much would each person receive ? 6. Mary's mother is 40y. 3mo. old, and Mary is one fifth as old. How old is Mary ? 7. If 7 equal boxes of sugar weigh 7cwt. 271b. 8oz., what is the weight of one box ? 8. If a train of cars move, at a uniform rate, 308m. 44rd. in 11 hours, how far does it move in 1 hour? 9. My farm contains 281A. 85P., and Flint's farm is one seventh as large. How large is Flint's farm ? 10. When 22oz. 19pwt. lOgr. of silver are required to make 18 spoons, what is the weight of 1 spoon? 11. A farmer has 65bu. 2pk. 4qt. of grain, which he wishes to put in bags, each containing lbu. 3pk. 4qt. How many bags will be required ? 12. In a certain town there are 3 farms, and each is divided into 9 equal lots. The entire extent of the farms is 37 A. 56P. ; what is the size of each lot ? 13. In what time can a man saw 1 cord of wood, if he can saw 12 cords in 93h. lOmin.? 370. Test Questions. — 1. What is multiplication of com- pound numbers? How do you write the numbers for multiply- ing ? How do you multiply ? 2. How do you proceed if any product is less than one of the next higher denomination ? If any product is equal to one or more units of the next higher denomination? 3. What is division of compound numbers? How do you divide? How do you proceed if there is a partial remainder? 4. Why are the multiplication of compound numbers and the division of compound numbers proofs of each other? REVIEW. Ill SECTION XXXV. BE VIEW OF COMPOUND NUMBERS. 371. — 1. When oats are 10 cents a peck, what will 1 bushel 2 pecks 4 quarts of oats cost ? 2. Reduce f of a yard to units of lower denom- inations. Solution. — Since 1 yard is 3 feet, f- of a yard is f of 3 feet, which is V 5 of a * oot > or 2 ^ eet anc * §> or h °f a foot. Since 1 foot is 12 inches, J of a foot, is \ of 12 inches, or 6 inches. Hence, £ of a yard, reduced to units of lower denominations, is 2 feet 6 inches. 3. Reduce f of an acre to units of lower denominations. 4. Reduce f of a ton to units of lower denominations. 5. 2 gallons 3 quarts are how many pints ? 6. Reduce 31 pints to units of higher denominations. Solution. — 31 pints are 15 quarts 1 pint, and 15 quarts are 3 gallons 3 quarts. Hence, 31 pints in units of higher denomina- tions are 3 gallons 3 quarts 1 pint. 7. Reduce 39 inches to units of higher denominations. 8. Reduce 46 ounces Troy to units of higher denom- inations. 9. How many hours from 9 o'clock A. m. to 2 o'clock P. M. ? 10. How many minutes from 15 minutes before 10 o'clock to 15 minutes past 11 o'clock? 11. James has lib. 8oz. of candy, and Edwin has f as much. How much has Edwin ? Solution. — 1 pound 8 ounces is 24 ounces. Hence, James has 24 ounces, and since Edwin has f as much, he has f of 24 ounces, or 18 ounces, which is 1 pound 2 ounces. 172 REVIEW. 12. If you have 1 pound 9 ounces of gold, and should sell f of it, how much would you have left ? 13. At $2 a hundred- weight, what will | of 2 tons 5 hundred- weight of hay cost ? 14. If you should gather in one day 3 quarts 1 pint of berries, and the next day 5 quarts 1 pint, how much would they be worth, at 12 cents a quart? 15. How many months old is a boy who has lived 11 years less 8 months ? 16. Jason has 1 pound 6 ounces of candy, and his brother has 3 times as much. How much more than Jason has his brother ? 17. My granary has two bins; one will hold 36 bushels 1 peck, and the other one fifth as much. How much less does one contain than the other ? WRITTEN EXERCISES. 372. — 1. How many steps, of 3 feet each, must you take in walking 1 mile 6 rods ? 2. How many miles has a man travelled who has gone 1793 steps of 3 feet each? 3. I have 2 plots of ground, each containing 1A. 120P. Into how many lots, of 20 square rods each, can the plots be cut ? 4. A planet has moved in a given time 17° 30'. How many seconds has it moved ? 5. At 90 cents a cord foot, how many cords of wood can be bought for $16.20? 6. I have 2 pounds of silver. How many spoons, weighing loz. 9pwt. each, can be made from it, and how many pennyweights will remain ? REVIEW. 173 7. How many days, of 8 hours each, do you waste in a term of 12 weeks, each week containing 5 school days, if you are idle 2 hours each school day ? 8. For how much will 4bu. 2pk. of strawberries sell, if put into quart boxes and retailed at 30 cents a box ? 9. What will 4 casks of molasses, each containing 84gal. 3qt., sell for, at 15 cents a quart? 10. How many loads, each containing led. 2 cd. ft., can be taken from a range of wood containing 25cd. 6cd. ft. ? 11. If a train of cars can move in 1 hour, 25mi. 80rcL, in what time can it move 250mi. 160rd. ? 12. How many kegs, each containing 5gal. 3qt. lpt., can be filled from a cask containing 58gal. 3qt. ? 373. Test Questions.— 1. What is a denomination? What are the denominations of linear measure? Of surface measure? Of cubic measure? 2. What are the measures of lines, surfaces or solids called ? Measures of extension. Recite the table of linear measures. The table of surface measures. The table of cubic measures. 3. What are the denominations of liquid measure? Of dry measure? What are the measures of liquids, or of grain, fruits and like articles, called? Measures of capacity. Recite the table of liquid measures. Of dry measures. 4. What are the denominations of Avoirdupois Weight ? Of Troy Weight? What are these called? Measures of weight Recite the table of Avoirdupois Weight. Of Troy Weight. 5. What are the denominations of circular measure ? Recite the table. What are the denominations of the measure of time? Recite the table. 6. How does reduction descending differ from reduction ascending? Compound addition from compound subtraction? Compound multiplication from compound division? 15* 174 DECIMALS. SECTION XXXVI. NOTATION AND NUMERATION OF DECIMALS. 374.— 1. If a block of wood be divided into ten equal parts, what is 1 of the parts called ? What are 2 of the parts called ? 3 of the parts ? 7 of the parts ? 2. If one tenth of a block of wood be divided into ten equal parts, what is 1 of the parts called ? What are 2 of the parts called ? 3 of the parts ? 7 of the parts ? 3. What part of 1 is A of A? A of A? A of A? A of A? 4. What part of one tenth is one hundredth ? How many hundredths is one tenth ? DECIMALS. 175 5. If one hundredth of a block of wood be divided into ten equal parts, what is 1 of the parts called ? What are 2 of the parts called ? 3 of the parts ? 7 of the parts ? 6. What part of 1 is T V of ^ of T V ? fV of T V of -^ ? 7. What part of one hundredth is one thousandth ? How many thousandths is one hundredth ? DEFINITIONS. 375. A Decimal Fraction, or Decimal, is a number of tenths, hundredths, thousandths, etc., expressed by placing a point (.) before the numerator and omitting the de- nominator. Thus, .3 expresses 3 tenths, or { q ; and .03 expresses 3 hun- dredths, or t §q. 376. The point is called the Decimal Point, and in a numerical expression separates the decimal from the order of ones or unite. Thus, 3.003 expresses 3 ones and 3 thousandths, or 3 T o 3 oq. 377. The first order at the right of the decimal point is tenths, the second hundredths, the third thousandths? etc., as shown in the following TABLE. I •a -a I i • I I 654321 .23456 7 INTEGERS. DECIMALS. 176 DECIMALS. A Mixed Decimal consists of an integer and a decimal. Thus, 15.165, which is read fifteen ones and one hundred sixty- five thousandths, or fifteen and one hundred sixty-fice thousandths, is a mixed decimal. 378. Principles. — 1. Ten of any decimal order are always equal to one of the next higher order. 2. The denominator of a decimal is 1, with as many ciphers annexed as there are orders in the decimal. WRITTEN EXERCISES. 379—1. Write and read .308. Solution. — .308 = 3 tenths hundredths 8 thousandths; or, .3 = 30 hundredths a* 300 thousandths ; and 300 thousandths -f- 8 thousandths = 308 thousandths. Hence, .308 is read S08 thousandths. 2. Write and read 61.25. Solution. — 61.25 = 61 ones and decimal .25 ; .25 = 2 tenths 5 hundredths ; or, since .2 = 20 hundredths, .25 = 20 hundreths -f 5 hundredths = 25 hundredths. Hence, 61.25 is read 61 ones and 25 hundredths ; or 61 and 25 hundredths. 3. Write and read .3165. 4. Write and read 515.006. 5. Write three hundred five thousandths. Solution. — 305 thousandths = 30 hundredths and 5 thou- sandths ; 30 hundredths = 3 tenths and hundredths. Hence, 305 thousandths == 3 tenths hundredths 5 thousandths = .305. 6. Write one thousand three hundred thirty-one ten- thousandths. 7. Write eleven thousand eleven, and eleven thou- sandths. DECIMALS. 177 380. Rules for Reading and Writing Decimals.- -Read a decimal as though it were an integer, giving it the name of its right-hand order. Write a decimal as though it were an integer, and place the decimal point so that each figure shall stand in its proper order, marking the ab- sence of units of any order by a cipher. PROBLEMS. 381. Write and read — 1. .347 4. .138 7. 14.005 2. .1405 5. .10065 8. 9.093 3. .0072 6. .000002 9. 1.6444 Write in the decimal form — io. T v 1Q 17 16. 14^. 11. ^r- 1/1 434 17 O 3 Lt * ^1000* 12. t&t- 15. tWoV- 1Q 1 6444 10 * J-ioooou- 19. Ninety-eight hundredths. 20. One hundred twenty-five thousandths. 21. One thousand eighteen ten-thousandths. 22. Six, and seventy-five hundredths. 23. Two thousand four hundred sixty-two, and seven tenths. 24. Four hundred, and forty-four thousandths. 25. Twenty-four, and twenty-four millionths. 26. Thirty-five thousand three hundred fifty-one, and twenty-one hundredths. 27. One million two hundred thousand thirty, and one millionth. 178 DECIMALS. SECTION XXXVII. ADDITION AND SUBTRACTION OF DECIMALS. 382.— 1. What is the sum of 3 tenths and 5 tenths? 2. What is the sum of 7 tenths and 8 tenths ? Solution. — 7 tenths and 8 tenths are 15 tenths ; and since 10 tenths are equal to 1, 15 tenths are 1 and 5 tenths, or 1.5. Hence, the sum of 7 tenths and 8 tenths is 1.5. 3. What is the sum of 12 hundredths and 11 hun- dredths? 4. If you pay 25 hundredths of a dollar for a slate, and 10 hundredths of a dollar for some paper, what part of a dollar do you pay for the whole ? 5. If you pay 25 hundredths of a dollar for a slate, and 10 hundredths of a dollar for some paper, how much more does the slate cost than the paper ? 6. If you have 75 hundredths of a bushel of chest- nuts, and your brother has 15 hundredths of a bushel, how many more have you than your brother ? 383. Principle. — Decimals which express like parts of a unit may be added or subtracted like integers. WRITTEN EXERCISES. 384.— 1. Add 13.634, 35.423 and 8.56. hi.b34 Solution. — Write the numbers so that figures fjo.Jf.2S of the same order shall stand in the same column. 8.50 Add as in addition of integers, which gives 57.617, 57 6 17 ^ e resu ^ required. 2. Add .315, 17.563 and 63.05. DECIMALS. 179 3. From 963.75 subtract 585.125. 963,750 Solution. — Write the numbers so that figures ron -igc of the same order shall stand in the same column. - Subtract as in subtraction of integers, which 378.625 gives 378,625, the result required. These numbers are prepared for subtracting by annexing a cipher to the minuend, which makes its decimal express thou- sandths, or like parts with the decimal of the subtrahend, with- out altering the value. 4. From 196.35 subtract 173.91. 5. From 73.007 subtract 68.75. 385. Rule for Addition and Subtraction of D3cimals.— Write the numbers so that figures of the same order shall stand in the same column. Add or subtract in the same manner as in in- tegers, and place the decimal point at the left of the order of tenths in the result. PROBLEMS. 386.— 1. What is the sum of .81, 3.65, 4.5 and 7.315? 2. What is the sum of .9, 14.501, 6.75 and 19? 3. What is the sum of .125 + .62 +.437 ? 4. What is the difference between 41.634 and 7.595? 5. What is the difference between 18.5 and 9.995? 6. What is the difference between .735 of a ton and .598 of a ton? 7. What is the difference between 98 and .98 ? 8. Wilson is 63.125 years old, and Johnson is 58.75 years old. What is the sum of their ages ? 9. Horatio has 125.675 thousand feet of box-boards, and his father 239.703 thousand feet. How many more has the one than the other? 180 DECIMALS. 10. A farmer received $69,875 for corn, $93.1875 for wheat, and $42.9375 for oats. How much did he re- ceive for the whole ? 387. Test Questions. — 1. What is a decimal? A mixed decimal ? 2. Kecite the decimal orders from tenths to millionths. What principles of decimals are given? 3. How do you read decimals? How do you write deci- mals ? 4. What is the principle for addition and subtraction of deci- mals ? How do you add or subtract decimals ? SECTION XXXVIII. MULTIPLICATION OF DECIMALS. 388. — 1. How many tenths are 3 times 1 tenth? 3 times 3 tenths ? 5 times 4 tenths ? 2. How many ones are 5 times 4 tenths ? 5 times 9 tenths ? Solution. — 5 times 9 tenths are 45 tenths; and since 10 tenths are 1, 45 tenths are 4 ones and 5 tenths, or 4.5. 3. How many hundredths are 3 times 1 hundredth ? 3 times 3 hundredths ? 5 times 4 hundredths ? 4. How many tenths are 5 times 4 hundredths ? 5. When tenths are multiplied by ones, what is the denominator of the product? When hundredths are multiplied by ones, what is the denominator ? 6. What is the product of 1 tenth by 1 tenth, or of rV hy ^ ? Of 3 tenths by 2 tenths, or T \ by ^ ? DECIMALS. 181 7. What is the product of 1 hundredth by 1 tenth, or ru (T by tV ? Of 3 hundredths by 2 tenths, or -j-fo by ^ ? 8. When tenths are multiplied by tenths, what is the denominator of the product? When hundredths are multiplied by tenths ? 9. How many decimal orders will there be in the product, decimally expressed, if you multiply rcr by 3 ? iVby T 3 or .01, may be written 1%, and read one percent. lob yOT - 06 > " " 6#, " six per cent 4 i ~J^~ y or . 04 2 ' " " 4 J % , " four and one half per cent 115 1 1C ■jm > or 1 . 1 5 f " •< H5 f c 9 « one hundred fifteen per cent. 190 PERCENTAGE. 4:12. The Base is the number or quantity of which the per cent, is computed. 413. Percentage is the process of computing by hun- dredths. The term Percentage is applied, also, to that part of the base which corresponds to the rate per cent. WRITTEN EXERCISES. 414:. Express decimally- 1. 3%. 5. 4f%. 9. 45%. 2. 4%. 6. Hfb. 10. 67%. 3. 7%. 7. 17*%. 11. 125%. 4. 9%. 8. 7i%- 12. 135%. SECTION XLIII. CASES IN PERCENTAGE. CA.SE I. Base and Rate Per Cent, given, to find the Percentage. 415. — 1. What part of a number is 4% of it? Solution. — 4% is T $ 7 , or ^; hence, 4% of a number is j% of the number. 2. What part of a number is 15% of it? 30% of it? 3. What part of a number is 20% of it? 12% of it? 50% of it? 4. What is the simplest form of 20% ? Solution.— 20% is &%, which in its simplest form is J. PERCENTAGE. 191 5. What is the simplest form of 50% ? Of 60% ? Of 80%? 6. What is the simplest form of 12£% ? Of 16f % ? Of 33^% ? 7. A farmer had 20 cows, and sold 10% of them. How many did he sell ? 8. If from a flock of sheep 5% of them should be taken, what per cent, would remain ? 9. A man bought a horse for $100, and sold him at a profit of 25%. For what sum did he sell him? 10. If you buy a horse for $100, and sell him at a loss of 20%, how much do you get for him? WRITTEN EXERCISES. 416.— 1. What is 7% of 9546 yards? 9546 yds. Solution.— Since 7% is .07, 7% of 9546 .07 yards is .07 of 9546 yards, which is 668.22 yards, 668.22 yds. the P ercenta g e required. 2. What is 5% of 860 pounds? 3. What is 12% of 750 men? 417. Rule for Finding any Per Cent, of a Number.— Multi- vly the given number by the rate per cent, expressed decimally. PROBLEMS. 418. How much is — 1. 4% of $61.50? 7. 19% of 562? 2. 3|% of $974? 8. 12% of 880? 3. 7±% of 160yd.? 9. 10% of $750? 4. 25% of 530rd.? 10. 8% of 450 men? 11. 15% of 560cwt.? 12. 60% of 7251b.? 5. lf% of 9160? 6. 3£% of 850? 192 PERCENTAGE. 13. If you earn in a year $300, and spend 45% of that sum, how much do you spend ? 14. What is a merchant's commission for selling goods to the amount of $7500, at 2£% ? 15. Of a cargo of corn, consisting of 2560 bushels, 15% was damaged. How much was not damaged? 16. If a town having 9540 inhabitants should in- crease in population 20%, what would be the number of its inhabitants ? CASE II. Base and Percentage given, to find the Rate Per Cent. 419. — 1. What per cent, of a number is | of it? 2. What part of 5 is 2 ? What per cent, of 1 is f ? 3. I spent \ of my money. What per cent, of it did I spend ? 4. I had $20, and spent $5. What per cent, of my money did I spend ? Solution. — I had $20, and spent $5; hence, I must have spent s 5 o, or i of my money; and \ equals T 2 o 5 o, or 25%. 5. What per cent, of $10 is $2 ? Of $45 is 30 ? 6. 8 days are what per cent, of 20 days ? Of 48 days ? 7. I received $3 for collecting $60. Wnat per cent, did I receive ? 8. A farmer bought a cart for $50, and sold it for $60. What per cent, did he gain ? 9. James bought a knife for $.60, and sold it for $.48. What per cent, was the loss ? 10. What per cent, does a merchant gain who buys flour at $8 per barrel, and sells it at the rate of 3 barrels for $30? PERCENTAGE. 198 WRITTEN EXERCISES. 420.— -1. What per cent, of 400 is 24 ? 400)2 A.00(. 06 = 6°Io Solution.— Since 24 is jfo of 2 1 go 400, it is .06, or 6? , of 400, which is the rate per cent, required. 2. What per cent, of 520 is 26 ? 3, What per cent, of 95 bushels are 3.6 bushels ? 421. Rule for finding the Rate Per Cent, that a given Percentage is of the Base— Divide the percentage by the base, ex- tending the quotient to hundredths. PROBLEMS. 422. What per cent, of— 1. 54 yards are 9 yards? 2. 125 men are 8 men? 3. $650 are $39.25 ? 4. 530rd. are 132ird. ? 5. 9160 is 160.3? 6. 400 is 28 ? 7. 560 is 106.4? 8. 725 is 435 ? 9. $756 are $75 ? 10. 450 men are 36 men ? 11. 560cwt. are84cwt.? 12. 7251b. are 4351b. ? 13. If you earn $300, and spend $135 of it, what rate per cent, of the sum earned is the sum spent ? 14. A house, which cost $5670, was sold for $283.50 above the cost. What was the rate per cent, of gain ? 15. Of an army of 45450 men, 12626 men were lost in battle. What was the rate per cent, of loss ? 16. If a horse, which cost $250, was sold for $100 above cost, what was the rate per cent, gained ? 17. A certain town, which had 9540 inhabitants, has increased by 1908 persons. What is the rate per cent, of increase? 17 194 PERCENTAGE. SECTION LIV. INTEREST. 423. — 1. When the allowance for the use of money is 6%, how many cents is it for $1, or 100 cents? How much is it for $100 ? 2. When the allowance for the use of $1 is 6 cents, what is the rate per cent. ? When it is 5 cents ? 3. When the allowance for the use of $100 for 1 year is $6, what is the yearly rate per cent. ? When it is $5, what is the yearly rate per cent. ? 4. When the yearly allowance for the use of money is at the rate of 6%, what is the allowance for the use of $5 for 3 years ? Solution.— The allowance for 1 year at 6% is t |q of the base; hence, the allowance for the use of $5 for ] year must be t^; of $5, or $.30, and for the use of $5 for 3 years must be 3 times $.30, or $.90. PERCENTAGE. 195 5. What is the allowance for the use of $8 for 4 years, at a yearly rate of 7 % ? 6. What is the allowance for the use of $25 for 5 years, at a yearly rate of 4% ? 7. What is the allowance for the use of $50 for 4 months, at a yearly rate of 6 % ? Solution. — The allowance for the use of $50 for 1 year at 6 per cent, is $3 ; hence, for 4 months, or £ of a year, it must be J of $3, or $1. 8. What is the allowance for the use of $40 for 3 months, at a yearly rate of 7 % ? 9. What is the allowance for the use of $90 for 2 months, at a yearly rate of 6% ? 10. In computing the allowance for the use of money, a month is commonly regarded as how many days ? Ans. 30 days. 11. What is the allowance for the use of $200 for 1 month 20 days, at a yearly rate of 6 % ? Solution. — The allowance for the use of $200 for 1- year is $12 ; hence, for 1 month, or T V of a year, it is T V of $12, or $1, and for 20 days, or f of a month, it is f of $1, or $.66f ; $1 and $.66f is $1.66f, the allowance required. 12. What is the allowance for the use of $100 for 2 months 15 days, at a yearly rate of 8% ? 13. What is the allowance for the use of $300 for 1 month 10 days, at a yearly rate of 5% ? 14. What is the allowance for the use of $120 for 1 year 4 months 6 days, at a yearly rate of 5% ? 15. What is the allowance for the use of $200 for 2 years 5 months 15 days, at a yearly rate of 12% ? At a yearly rate of 16|% ? 196 PERCENTAGE. DEFINITIONS. 424. Interest is an allowance for the use of money or its equivalent. 425. The Principal is the sum for the use of which in- terest is allowed. 426. The Rate of Interest is the rate per cent, of the principal allowed for its use for one year. 427. The Amonnt is the sum of the principal and in- terest. WRITTEN EXERCISES. 428.— 1. What is the interest of $345.50 for 3 years 3 months, at 6% ? Principal, $345.50 SoLUTiON.-The inter- d . qq est of $345.50 for 1 year „ is .06 of that sum, or Int. for ly., §20.7300 $ 2 0.73 . hence> for 3 years &_ the interest is 3 times Int. for 3y., $62.19 #20.73, or $62.19. , \ / , _, io l The interest of $345.50 Int. for Smo., or jy. 9 5.18- foY 1 year . g $2Q n . hence> Int. for 3y. 3mo. y $67. 37 j for 3 months > or * of a J * * year, it is \ of $20.73, or $5.18J. The interest for 3 years 3 months is $62.19 +$5.18|, or $67.37J. 2. What is the interest of $500 for 5 years 6 months,, at 7%? 3. What is the interest of $750 for 4 years 5 months^ at 8%? 4. What is the interest of $840 for 3 years 4 months, at 6%? 5. What is the interest of $920 for 4 years 2 months, at 5% ? PERCENTAGE. 197 6. What is the amount of $450 for 7 months 20 days, at 4%? Solution. — The inter- est of $450 for 1 year is .04 of the principal, or -°4 $18 ; hence, for 6mo., or } year, it is i of $18, or $9; for lmo., or \ of 6mo., it is \ of $9, or $1.50; and for 20d., or fmo., it is f of $1.50, or $1; hence, for 7mo. 20d. it is $9 + $1.50 + $1, or $11.50. The amount is the sum of the principal and in- terest, $450 + $11.50, which is $461.50. $18.00 $9.00 Principal, Hate, Int. for ly., Int. for Gmo.y or ~y., Int. for lmo., or j of 6mo., 1.50 Int. for gOd., or fmo. s 1.00 Int. for 7 mo. 20d., $11.50 Principal, Amount, 450 $461.50 7. What is the amount of $600 for 5 months 12 days, at7%? 8. What is the amount of $720 for 2 months 3 days, at6%? 9. What is the interest of $5000 for 25 days, at 8% ? 10. What is the interest of $4200 for 18 days, at 6% ? 429. Rules for Computing Interest.— Multiply the prin- cipal by the rate of interest expressed decimally, and that product by the time expressed in years. If there be months or days, find the interest for the number of months, by taking fractional parts of a year's interest; and for the number of days, by talcing fractional parts of a month's interest. To find the amount, add the principal and in- terest. 17* 198 PERCENTAGE. PMOBLEMS. 430. — What is the interest of — 1. $1800 for 3 years 9 months, at 7% ? 2. $34.50 for 1 year 6 months, at 6% ? 3. $75.50 for 6 years 11 months, at 8%? 4. $5600 for 4 years 5 months, at 5% ? 5. $8000 for 2 years 2 months, at 4% ? 6. $10000 for 15 years 1 month, at 3|%? 7. $240 for 10 months 6 days, at 9%? 8. $6450 for 3 months 3 days, at 10% ? 9. $65.40 for 1 month 24 days, at 6%? 10. What is the amount of $725 for 1 year 6 months 10 days, at 8%? 11. What is the amount of $830 for 2 years 3 months 27 days, at 7% ? 12. What is the amount of $300 for 18 days, at 6% ! 13. What is the interest of $407.60 from Jan. 1, 1869, to April 1, 1871, at 6%? 14. What is the amount of $1000 from May 15, 1870, to October 18, 1871, at 7%? 15. What is the amount of $1200 from February 1, 1871, to November 1, 1872, at 8% ? 431. Test Questions. — 1. What is any per cent, of a num ber? What is the rate per cent.? The base? What is per- centage ? 2. How do you find any per cent, of a number? The rate per cent, a given percentage is of a base? 3. What is interest? The principal? The rate of interest? The amount? How do you compute interest? REVIEW. 199 SECTION XLV. REVIEW OF PERCENTAGE. 432. — 1. What is a broker's charge, at \%, for buy- ing stocks amounting to $400 ? 2. An agent sells goods to the amount of $500, and reserves 2^% of the amount for his services. How much did he receive ? 3. I bought broadcloth for $5 per yard, and sold it at 20 % above cost. How much did I gain per yard ? 4. I bought cloth at $5 per yard, and sold it so as to gain $1 per yard. What rate per cent, was the gain ? 5. I bought a watch for $60, and sold it at a loss of 10%. How much did I get for it? 6. At what price must I sell a watch that cost me $50 to gain 20% ? 7. If I sell a watch which cost me $60, at a loss of 20%, how much do I get for it? 8. How many dollars are $20 less 5% of $20? 9. I sold a carriage which cost me $80 for $60. What was the rate per cent, of loss ? 10. If you sell what cost you $6 for $8, what is the rate per cent, of gain ? 11. What is the interest of $200 for 3 years 6 months, at 6%? 12. What is the interest of $400 for 15 days, at 6%? 13. What is the amount of $300 for 4 months at 7% ? 14. What is the amount of $50 for 6 months, at 8% ? 15. What is the interest of $600 for 5 years 9 months, at 6%? •200 REVIEW. WB I TT KX EXER CISE8. 433. — 1. How much must be paid for insuring a house for $1500, at 2^% ? 2. What is the interest of $524 for 3 months 12 days, at 4i% ? 3. If you buy wheat at $1.80 per bushel, and sell it so as to gain 5 % , what do you get per bushel for it ? 4. A school-house which cost $7500 is insured for 66f% of its cost. For how much is it insured? 5. What is the amount of $1004.45 for 1 year 6 months 6 days, at 6 % ? 6. If you give your note for $800, and pay it 2 years 8 months afterward, with interest at 7%, what sum will be required ? 7. If you should have a note for $1200, which is pay- able in 2 months 3 days, without interest, and in bor- rowing money upon it you should allow interest in advance, for the time, at 6%, how much would that interest be ? 8. I bought a village lot for $500, and sold it for $550. What rate per cent, was my gain ? 9. If I should ask $550 for a village lot, but should fall upon,the price 10%, what would then be my price? 10. I bought a horse for $250 ; at what price must I sell him to gain 20%? 11. What sum will be required to pay a note for $1540, which has been on interest at 6% for 3 years 4 months 24 days ? 12. I bought a farm, May 16, 1870, for $4670, and paid for it, October 22, 1871, with interest at 8%. How much did I pay ? APPENDIX. RECTANGULAR MEASUREMENTS. 434. A Line is that which lias only length. 435. A Straight. Line is a line a B that has only one direction. Tims, the line A B is a straight line. 436. An Angle is the difference of direction of two lines that meet. Thus, the lines AB and AC, meeting at A y form the angle CAB. 437. A Perpendicular Line is a straight line which meets another straight line so as to form two equal angles. Thus, (he line CD is a perpendicular line. 438. A Right Angle is an angle formed by two lines perpendicular to each other. Tims, the angles ACD and DCB are each right angles. 439. A Rectangle is any figure which has four straight sides and four equal angles. Thus, the figure in the margin, having four straight sides and four equal angles, is a rectangle. r 20 r 202 APPENDIX. \ 440. A Rectangular Solid is any volume bounded by six rectangular faces. Tims, the figure in the margin, having six rectangular faces, represents a rectangular solid. CASE I. Rectangular Surfaces. 441. — 1. How many square feet are in a rectangular surface which is 3 feet long and 1 foot broad? 2. How many square inches are in a rectangular strip of paper which is 6 inches long and 1 inch wide? 3. How much will a rectangular board 9 feet long and 1 foot broad cost, at 3 cents per square foot? DEFINITIONS. 442. The Dimensions of a rectangular surface are its length and breadth. 443. The Area of a rectangle is the extent of surface bounded by its two dimensions or sides. Thus, the rectangle in the margin will be seen to contain 12 square inches, if it be sup- posed to be 4 inches long and 3 inches wide. For, upon each inch of length (here may be conceived to be 1 square inch, making a row of 4 square inches ; and as there will be as many such rows as there are inches in the width, or 3 rows, the area of the rectangle must be 3 times 4 square inches, or 12 square inches. WRITTEN EXERCISES. 444. — 1. How many square feet of surface are there in a floor 18 feet long and 15J feet wide? APPESDIX. 203 18 sq.fi. x 151 = i>7 ' 9 *2j, number of hogsheads. 3. I have a bin 8 feet long, 6 feet broad, and 5 feet deep. How many tons of Franklin coal will it hold? 4. A bin will exactly contain 160 bushels of wheat. What is its capacity in cubic feet? 5. I have a cistern 6 feet long, 4 feet wide, and 5 feet deep. How many hogsheads, of 63 gallons each, is its capacity ? 6. A vat will hold exactly 3840 gallons. What is its capacity in cubic feet ? 457. Rules for Estimating Bins, Vats, or Cisterns.— 1. Mul- tiply the contents in cubic feet by .8 for bushels, or by 7* for gallons. 2. Divide the raparih/ in bushels by .8, or in gallons by 7j, for contents in cubic feet. 208 APPENDIX. PROBLEMS. 458. — 1. How many bushels of grain will fill a cubical box whose dimensions are each 5 feet? 2. A vat has 96 cubic feet of interior space. How many gallons of water will it hold ? 3. A tank is 6 feet long, 4 feet 6 inches broad, and 5 feet deep. How many pounds of water will it contain, the weight of a gallon of water being 8J pounds? 4. A chest will contain exactly 100 bushels of grain. What are its contents in cubic feet? 5. A wagon-body is 8 feet long, 3 feet 6 inches wide, and 2 feet deep. How many bushels will it contain? 6. A bin is 6 feet long and 4 feet wide. How deep must it be to contain exactly 2 tons of Lehigh coal ? 7. I have a bin which exactly holds 5 tons of Lack- awanna coal. It is 10 feet long and 4 feet deep. What is its width ? 8. How many bushels of wheat can be put into a bin that is 8 feet long, 6 feet 3 inches wide, and 4 feet 6 inches deep ? 9. I have a bin 6 feet long, 6 feet wide, and 6 feet deep, and two others each 3 feet long, 3 feet wide, and 3 feet deep. How many more bushels will the first hold than the other two ? 459. Test Questions.— 1. What is a line? A straight line? Kn angle? A perpendicular line? A right angle? A rectangle? A rectangular solid? 2. What are the dimensions of a rectangular surface? The area of a rectangle? What are the dimensions of a rectangular solid? The cubic contents of a rectangular solid? 3. How many cubic inches does the standard bushel contain ? The standard gallon ? APPENDIX. 209 MISCELLANEOUS PROBLEMS. 460. The Articles in parentheses denote the portions of the text for which the problems may be used as sup- plementary exercises. (Art. 73.) 1. Express by figures, fifty-nine thousand nine; seven thousand eighty; and fifty-one thousand one hundred three. 2. Write and read 134640; 60041 ; and 4602000. 3. What is the sum of two hundred fifty-two thou- sand six hundred six and one hundred seventy-two thousand nine hundred seventy-five? 4. How many are 96004 — 964? 83334 — 9453? 5. If you have 5675 dollars, and should spend 4987, how 7 much would you have left ? 6. How many are 7806 + 760 + 9376 + 97 ? 7. How many are 95631 — 777 added to 66406 -f 9972? 8. A mill had 6750 barrels of flour. To A it sold 2173 barrels, and to B 978 barrels. How much then had it unsold ? 9. How many are 152445 + 707050 less 93987? 10. The coach was first made in England in the year 1564, which was 1083 years after iron shoes were first made for horses. In what year were the iron shoes first made? 11. How many are 1685 + 75 + 832 + 9675 ? 12. How many are 59 + 641 + 9 + 8086 + 93015 ? 13. How many are 19678 — 3789, less 10000 — 9889? 210 APPENDIX. 14. A man had 50000 dollars, and paid from it for a house 10550 dollars, for land 18075, and for goods 15787. How much money had he then left? (Art. 117.) 1. How much will 15 horses cost at 225 dollars each? 2. Add 3605 to the product of 716 by 97. 3. Subtract 1148 from the product of 517 by 68. 4. A farmer bought 316 sheep at 3 dollars each, and sold 250 of them for 1 000 dollars, and the remainder at a loss of a dollar each. How much did he make? 5. From 1840 x 18 take 2045 x 13. 6. How much is 13465 divided by 47? 7. A planter has 64575 gallons of molasses. How many casks of 63 gallons each will be required to hold it? 8. How many are 225 x 36, divided by 810-*- 15? 9. When 64 acres of land can be bought for 1600 dollars, how many acres can be bought for 3225 dollars? 10. If the multiplier is 47 and the product is 13113, what is the multiplicand ? 11. If the divisor is 71 and the quotient 1002, what is the dividend ? 12. The quotient is 1365, the divisor 63, and the re- mainder 14. What is the dividend? 13. What is the quotient of 13675 divided by 18? By 63? By 37? 14. What is the quotient of 67350 divided by 100? By 31? By 60? 15. Bought 224 barrels of flour for 1568 dollars, and sold the same for 2128 dollars. What was the gain on each barrel ? APPENDIX. 211 (Art. 238.) 1. Reduce 14-j-^- to an improper fraction. 2. What is the Value of Vs 5 ? ° f "H"? 3. How many ones are there in ^-J 5 -? In 6 I e r 3 ? In 1119 tt • • 4. Change §-, -§, and T 7 ^ each to twenty-fourths. 5. Arrange |-, -^-, and -| in the order of their values. 6. John is llf years old, and William 13^. What is the sum of their ages? 7. A pole stands -f\ of its length in the mud, -^ of its length in the water, and the rest above water. How much of its length is above water ? 8. What is the difference between -| and | ? 9. W r hat is the difference between y and 5§ ? 10. How much greater is the sum of 3^ and 2| than their difference? 11. What is the difference between 18 and a fifth of fifteen-fifths? 12. At f of a dollar a bushel, what will 84 bushels of potatoes cost? 13. If a boy can earn \\ dollars in a week, how much can he earn in -| of a week ? 14. At 2-J- dollars a day, in how many days will a man earn 28f dollars? 15. How much is § of ff ? \ of -i-f 2 -? 16. What number added to -f will give \\\ ? 17. If 5 yards of cloth cost If dollars, what part of a dollar is it a yard ? 18. What fraction is equivalent to J of f of f? 19. A man owning f of a farm sells f of his share for 450 dollars. At this rate what is the value of the whole farm? 212 APPENDIX. 20. A certain estate is worth 24000 dollars, and 1 ^ of the value of the estate is j- of the value of the house upon it. What is the value of the house? 21. A farmer has 36f bushels of corn in a bin, which is just f of all he raised. How much did he raise? 22. Divide 2 by the sum of 2§ and f, and to the quotient add If — f. (Art. 384.) 1. How many dollars will 112 bushels of apples cost at 62 J cents a bushel? 2. Bought 2 bushels of chestnuts at $3.50 per bushel, and sold them at 8 cents a pint. How much was gained by the transaction ? 3. A man is 31 years old, allowing 365J days to a year. How many hours has he lived? 4. The distance between two places is exactly 15 \ miles. How many yards are they apart? 5. A lot of land containing exactly 47 square rods was sold at 2 cents a square foot. How much did it sell for? 6. How many acres are 4392 square rods? 7. How many hogsheads of vinegar of 63 gallons each, at 5 cents a quart, can be bought for $75.60? 8. When hay is $25 per ton, how much is it per hun- dred-weight? 9. How many ounces are there in 5 tons? 10. How many acres are 25090560 square inches? 11. How many pounds are 37440 troy grains? 12. How many chests of tea of 35 pounds each, at 45 cents a pound, can be bought for $63 ? APPENDIX. 213 (Art. 373.) 1. How many minutes are there in 1 week 4 days 15 hours ? 2. How much is the sum of 14 lb. 3 oz. 11 pwt. ; 5 lb. 7 oz. 13 pwt., and 11 oz. 17 pwt, 13 gr.? 3. From 59° 42' 15" take 11° 39' 47". 4. How many ounces are there in 13 cwt, 67 lb. 14 oz.? 5. How much must be paid for 183073 square yards of land at 50 cents a square rod? 6. A certain range of wood contains exactly 10 cords. If 2 loads, each containing 1 cord 3 cord feet, be taken from it, how much will be left? What will the remain- der be worth at 75 cents per cord foot? 7. What is the quotient of 34 cd. 6 cd. ft. 4 cu. ft. divided by 4 ? Of 118 bu. 1 pk. 5 qt. -*- 6 ? 8. What is the product of 10 reams 5 quires 13 sheets multiplied by 9? 13 A. 15 p. 3 sq. yd. x 10? 9. Three thieves carry off from a house 7 silver cups, each weighing 10 oz. ; 1 dozen and 9 silver forks, each weighing 2 oz. 8 pwt, ; and 13 silver spoons, each weighing 3 oz. Dividing the silver among the thieves, and giving to one of them a double share, how much would each have? 10. What is the value of 1 mi. 25 rd. 2 yd. 2 ft. + 5 mi. 150 rd. 3 yd. 2 ft,? 11. What is the value of 71 gal. 3 qt, 1 pt. 3 gi. x 8? Of 68 d. 18 h. 56 min.-4? 12. A dray-load is found to weigh 1 ton 4 cwt, 15 lb., and it consists of 21 boxes. What is the weight of each box? 214 APPENDIX. (Art, 407.) 1. Write one thousand, and one ten thousandth. 2. Write fifteen, and fifteen millionths. 3. Find the sum of eleven and six hundredths, added to thirteen, and one hundred six thousandths. 4. From 93 take .0093; from 1 take .000001. 5. What is the product of 1.04 multiplied by .104? 6. Express .275 as a common fraction in its simplest form. 7. Express T ^ as a decimal of 3 orders. 8. What is the value of 17 divided by 1.7? 9. What is the value of (- x '-z\ x 1.3? 10. Divide 1.56 by .005, and .0003 by .05, and find the sum. 11. Multiply 7 ten-thousandths by 15 thousandths, and divide the result by .25. 12. How many thousand feet of boards at $13.50 a thousand can be bought for $101.25? 13. A man traveled 5 days; the first day he went 16.05 miles, the second 35.16 miles, the third 21y^ 7 miles, the fourth 11.009 miles, and the fifth 31^^- miles. How far did he travel in all ? 14. I bought a cask of refined petroleum containing 48.5 gallons. How much of it can I sell and have left 13.125 gallons? 15. A person sold .15 of an estate to one person, and then J of the remainder to another person. What part of the estate did he still retain? r^j APPENDIX. 215 (Art, 433.) 1. How much is 5 J % of 8670 yards? 11 % of $125 ? 15 per cent, of 624 bushels ? 2. What % is made by selling goods at $76.56 which cost $63 ? 3. A horse which cost $250 was sold at $75 above cost. What was the rate % of gain ? 4. Bought goods for $1250 and sold them for $1206.25. What was the loss per cent,? 5. Bought coal at $7.50 per ton, and sold it so as to gain 15 %. At what price was it sold? 6. What is the interest of $245 for 3 years, at 7 % ? For 2 years 6 months, at 7 % ? 7. What is the amount of $370 for 8 months 18 days, at 9 %? 8. What is the interest of $1000 for 1 month 6 days, at 8 %? 9. What is the amount of $1250 for 2 years 2 months 15 days, at 6 % ? 10. What is the amount of $965 for 3 years 6 months 24 days, at 4 % ? 11. What is the interest of $50.50 from Jan. 1, 1877, to Oct, 10, 1878, at 7 %? ^ 12. What is the amount of $834.80 from Feb. 15, 1877, to Nov. 21, 1878, at 6 % ? 13. What is the interest of $500 from July 1, 1877, to Jan. 16, 1878, at 7 % ? 14. What is the amount of $1540.50 from April 1 I to Oct, 21, 1877, at 8 %? 15. What is the amount of $2000 from May 3, 1877, to June 13, 1878, at 5 %? ANSWERS. Art. 42. Art. 49. 6. 19497. 3. 164. 1. 9602. Art. 69. 4. 116. 2. 14152. 1. 42169. 5. 96. 3. 425435. 2. 5700. 6. 169. 4. 870 yards. 3. 4785. 9. 88. 5. 9514 pounds. 4. 1858. 10. 137. 6. 2296 men. 5. 3630. 11. 149. 7. 1834 cords. 6. 10375. 12. 88. 8. 9976. 7. 2195. Art. 43. 9. 4084. 8. 16911. 1. 208. 2. 168. 10. 25308. 11. 1597124. Art. 70. 1. 61. 3. 147. Art. 50. 2. 3110 dollars. 4. 209. 1. 1799. 3. 811. 5. 178. 2. 6125 dollars. 4. 3268 feet. 6. 220. 3. 1045. 5. 1785 dollars. 7. 159. 4. 10196. 6. 910; 989. 8. 179. 5. 5649. 7. 18981. 9. 194. 6. 573 dollars. 8. 127987. 10. 158. Art. 51. Art. 73. 11. 148. 1. 204. 1. 2340. 12. 178. 2. 281. 2. 3334. Art. 45. 3. 427. 3. 834. 2. 231. 4. 2068. 4. 4110. 3. 570. 5. 15155. 5. 11567. 4. 1041. 6. 157. 6. 7457. 5. 963. 7. 341. 8. 2275. 6. 488. 8. 544. 9. 3572. Art. 47. 1. 8660. 9. 2465. 10. 7606. 10. 4925. Art. 86. 2. 7886. Art. 66. 2. 1250. * 3. 3885. 2. 1716. 3. 5360. 4. 5942. 3. 4899. 4. 2660. 6. 2722. 4. 5594. 5. 3100. Art. 48. 6. 7099. 6. 31000. 1. 903 dollars. 7. 8902. 7. 86800. 2. 806 men. 8. 5200. Art. 88. 3. 881 miles. Art. 68. 2. 2996. 4. 921 bushels. 1. 918. 3. 14761. 5. 1301 feet. 2. 538. 4. 1824. 6. 7907 tons. 3. 18888. 5. 9512. 7. 9919 bales. 4. 81907. 6. 14652. 8. 1279 horses. 6. 1001. 7. 7140. 216 AN8 WEMS. 217 Art. 90. 1. 7506 dollars. 2. 3612 hogsheads. 3. 7259 yards. 4. 36081 bushels. 5. 9855 days. 6. 2494 miles. 7. 731000 tons. 8. 376800 acres. Art. 91. 2. 158632. 3. 2415207. 4. 1010025. 5. 22612390. 6. 13500 dollars. 7. 21400. 8. 210511. Art. 92. 1. 4100 dollars. 2. 13104. 3. 117600. 4. 293760. 5. 12769. 6. 277008. 7. 146520. 8. 154350. 9. 2199582. 10. 57600000. Art. 110. 2. 108. 3. 448. 4. 137. 5. 141. 6. 143. 7. 131. 8. 117. 9. 398. 10. 72. Art. 113. 1. 681J. 2. 57 5f. 3. lOOf. 4. 515. 5. 332^. 6. 260}$. 7. 21*. 8. 558 T V Art 1. 2. ,113. 182| apples. 187f cents. 133|feet. 59 f rods. 691 dollars. 4. 5. 6. 578 T 5 T days. 8. 9. 10. Art 1. 2. 3. 4. 5. 6. 8. 210}i pounds. 255 7 dollars. 320 cents. 483|. .114. 358 T V 406 dollars. 49. 62. 41. 62, and 7 gallons will remain. 36, and 5 acres will remain. 57, and 67 feet will remain. Art. 117. 1. 25 dollars. 2. 9443C, 3. 1012 dollars. 4. 910. 5. 4350 dollars. 6. 191 T 8 2 dollars. 7. 262 bu. of oats, 393 bu. of wheat. 8. 54J8-. 9. 1775 days. 10. 4681 dollars. 11. 1800 dollars. 12. 7f«f. 13. 51. Art. 126. 2. 2, 2, 3, 7. 3. 3, 5, 5. 4. 2, 2, 2, 2, 2, 3. Art. 128. 1. 5, 19. 2. 3, 3, 7. 3. 2, 2, 23. 4. 2, 61. 11. 12. 5. 2, 2, 29. 6. 2, 2, 2, 23. 7. 2, 2, 3, 3, 3. 8. 2, 2, 2, 5, 5. 9. 2, 2, 2, 91. 10. 2, 2, 37. 2, 3, 5, 7. 3, 5, 7, 7. Art. 134. 2. 9. 3. 7. 4. 7. Art. 144. 1. 144. 2. 108. 3. 168. 4. 42. 5. 126. 6. 57. 7. 120. 8. 240. 9. 560. 10. 60. Art. 148. 3. 25|. 4. 5, 4. Art. 150. 1. 2f 2. 5. 3. 15. 4. 4f. 5. 2. 6. 6. 7. 27. 8. 5. 10. 6f. 11. 19. 12. m . Art. 108. 2. V; ¥o°. 3. a$*;*H 5. V; W- 6. V; W- Art. 170. 1. y . 2 6* 218 ANS WEBS. 3. 1JL6 . Art. 180. 11. 63f. 4. if* Lf 12. 20J-J, 5. Vs 1 2. J. Art. 201. 6. W- 3. |. 9 - 7 - ^' 28' 7. H** 4. tV 3. i%. 8. w- 5. T V 5. 7,V 9. W- 6. f 6. 19A. 10. 4*, 7. i. Art. 303. 11 209 J.i. 15 . 8. J. 1 5 !• 2¥- 12. \% 6 . Art. 180. O 3 <*• T8- Art. 172. .Q 55. 5 4 °' 66 ) ^6* A 6.4. 4=. T8) T8) Q 9 «• 62* 4. it. 2. 91. 18* 3. 73. Art. 102. 5. tf j. 5. 13 T \. 1« T2) 12* o. 34 . 6. I7&. Art. 174. 1. 14. O 1 8 . _3_ . *■•• 3 0) 30) A- 7. S/q. O 48.20, «5» 8 8 ) 8 8 ) J. 24.16. *=' 3 6) 36) 14 8 8- 8. 8J. 9. 66^. 2. 26f. 3. 17. K 12. 20. °« 24) 24) A- 10. 42 T \. ft 16.35. «. ?0 ) 40 > li- 11. 2&. 4. 80J. rj 20.21. *• TO ) 70 ) ft- Art. 207. 5. 16|. 8. ?2) 42) 1 6 T2- 3. 2J; 7 T V T ; 5Jfe 6 25f. 7. 79. 8. 31. Q 32. 35. «*• ¥o ) ¥o ) if 4. 414. in s . 9 . ip_ J-U. 12-) T2 ) 12- 11 2 4.110. 153 L *- T'g'O ) 180 ) 18 0* Art. 200. 1. 6 T V 9. 24|f. lO 495. 180. 336 1<5. Y2 0) 72 0) 72 0* 2. 16 T V 10. 21. Art. 106. 3. 19*. 11. 304J. 2. 2& 4. li. 12. 72/ 7 . 3. 2& 5. 4f. 13. 113; 78 J yards. 5. 2Uf 6. 2f. Art. 178. 6. 2tf. 7. 2i O 42.12 *' 1~8> 6T- 8. 33J. 8. 3ff. 3. If; f*. Art. 108. 9. 60f. Art. 180. 1. 2^V. 10. 104. 1. f|. 2. t¥2. 11. 129^. 2. 4». 3. Hi. 12. 1215. q 45 4. |g. 4. Wo- Art. 214. 5. 6ft 3. 30#. 5. &. 6. ioa. 4. 49J. 6. T 3 oV 7. 40^. 5. 25 r 5 7 . Art. 184. 2. J; A- 8. 2|f. 9. 3§. 6. 280. 7. 102 T V 3. A; t V 10. 71. 8. 387. -4iVS VKEJJ& 9. 78. 5. 867. 5. 4f. Art. 316. 6. 49f. 6. 15f dollars. 3.1f. 7. 238|-. 7. 4|. 4. 36*; 181ft. 8. 3371. 8. T V Art. 318. 9. 84. 9. 14f. 1. ft. 10. 40*. 10. 640. 2- t¥i- 11. 81. Art. 253. 3. ft. 12. 200. 3. $16.25 4. |. 14. 6; 17 A; 6ft. 4. $82.28 5. ftft- 15. 10. 6. $262.37 6. 2}|f. Art. 331. 7. $7015.50 7. 10H- 2. 1ft. Art. 255. 8. 2ft 3. 81. 2. $77.06 9. lfH. Art. 333. 3. $77,915 io. H- 1. 4. 4. $26.98 11. 20. 2. 1ft. 5. $103.81 12. 89f, 3. 1ft. 6. $3688.18 13. m- 4. 1ft- 7. $83.50 14. J. 5. 2*. Art. 257. 15.1. 10. 5. 2. $490.72 16. 2| dollars. 11. If 3. $447.15 Art. 333. 12. }. 4. $78,125 Q 3 . 1 . Q 9 14. 6. 5. $228.72 Art. 334. 15. 4f. 6. $260,169 1. ft- 16. 4f ; 41. 7. $9356. 2. ft. 17. 31. 8. $13065. 3. ft. Art. 336. 9. $180576. 4. ft. l. H 3 -- 10. $34055. 5. ft. O 14.15 11. $10303. 6. ft. 3. 1; 1. 12. $391.80 7. 2ft. 4. 14| dollars. 13. $17503.50 8. 2H. 5.|*. 14. $1510.44 9. 11!- 6. 261; 311. 15. $76800. 10. 8ft dollars. 7- 2ft; 2|f Art. 259. Art. 336. 8. 5ft; 5}f. 3. $40.65 2. 73f. 9. 21 tons. 4. $6.10 3. 331; 134|; 2201. 10. 161 dollars. 5. $25.06 Art. 338. Art. 338. 6. $6.12 1. 26|. 1. 26 yards. 7. $93.56 2. 911. 2. If; *Hti 4. 8. $1005. 3. 52J. 3. 11. 9. $3.64 4. 64. 4. 126 dollars. 10. $634,055 219 220 ANSWIMS. 11. $56.75 8. $450. 12. $1.31 Art. 334. 13. $19.04 1. 527040. 14. $1111.44 2. $165. 15. $16.75 3. 12. 17. 20. 6. $403.20 18. $5.25 7. 23040. 19. 85. 8. 40. 20. $2641. 9. 40. 21. $12. Art. 341. Art. 262. 2. 175 in. 2. $90. 3. 7515 sq. in. 3. $63. 4. $8. Art. 264. 1. $947,50 2. $66.75 3. $8510. 4. $32. Art. 270. Art. 343. 1. 11056 // . 2. 18755 sq. yd. 3. 7569 ft. 4. 4700260 cu. in. 5. 253 qt. 6. 4601 gr. 7. 142544 oz. Bill receipted, $21.37J. 8. 335 qt. Art. 277. 9. $300. 5. 15; 99. 10. $504. Art. 287. 11. $44. 9. 155520. 12. 441300 h. 10. 36. Art. 345. 11. 65. 2. 4 yd. 2 ft. 7 in. Art. 300. 3. 5 sq. yd. 7 sq. ft. 27 sq. in. 5. $30.24 4. 10 h. 17 m. 18 sec. 6. $9.60 Art. 347. Art. 303. 1. 3° 4 / 16". 6. 6. 2. 3 A. 140 P. Art. 312. 3. 1 mi. 138 rd. 4 yd. 7. 480. 4. 100 cu. yd. 20 cu. ft. 10( Art. 323. 5. 2400. 6. 72. 7. 12600. cu.in. 5. 8 bu. 2 pk. 3 qt. 6. 9 oz. 11 pwt. 17 gr. 7. 4 T. 9 cwt. 5 lb. 14 oz. 8. 1 hhd. 20 gal. 3 qt. Art. 327. 9. 1A. 40 sq. rd. 8. 366. 10. lmi. 100 rd. 9. 5785560. 11. 12 63 23. Art. 331. 12. 50 y. 4. 2544. Art. 351. 5. $51.60 2. 14 mi. 17 rd. 6. 24. 3. 23 gal. 2 qt. 7. $72. ANSWERS. 221 Art. 353. 3. 10 lb. 1 oz. 10 pwt. 17 gr. 1. 19cwt. lib. 5oz. Art. 369. 2. 87 A. 76 P. 1. lhhd. 6 gal. 1 qt. 1 pt. 2f gi. 3. 20 cu. vd. 21 cu. ft. 28 cu. in. i 2. 50 cu. ft. 1121 cu. in. 4. lib. 9oz. 19pwt. 6gr. 3. 3 A. 95 P. 22 sq. yd. 8^ sq.ft. 5. 191 bu. 3pk. 4. 4 bu. pk. 4 qt. 1 pt. \\ gi. 6. 54 rd. 2 vd. 2 ft. 5. 3 cd. 2 cd. ft. 5 cu. ft. 576 7. 15 d. 23 h. 37 m. cu. in. 8. 52° 55' 48". 6. 8y. Omo. 18 d. 9. l'bu. lpk. 3qt. 1 pt. 7. 1 cwt. 3 lb. 14$ oz. 10. 33 A. lOOsq.rd. 8. 28 mi. 4 rd. 11. 47 T. 8cwt. 561b. 9. 40 A. 35 P. Art. 356. 10. 1 oz. 5 pwt. 12| gr. 2. 31b. 10 oz. 2pwt. 11. 35 bags. 3. 1 gal. 2 qt. 1 pt. 12. 1 A. 61 P. 10 sq. yd. 108 4. 8 yd. 2 qr. 3 na. sq. in. Art. 358. 13. 7 h. 45 min. 50 sec. 1. 8 m. 300 rd. 1 yd. Art. 372. 2. 2 A. 155 P. 5sq. yd. 1. 1793 steps. 3. 1 wk. 4 d. 23 h. 2. 1 mi. 6 rd. 4. 22 T. 19cwt. 941b. lloz. 3. 28 lots. 5. 3g 53 19. 4. 63000". 6. 1° 51' 15". 5. 2 \ cords. 7. 19 m. 34 rd. 2 yd. 1ft. 6 in. 6. 16 spoons; and 16 pwt. will 8. 1 y. 10 mo. remain. 9. 28 gal. 1 pt. 7. 15 days. 10. 16° 34' 17". 8. $43.20 12. 52 y. 1 mo. 18 d. 9. $203.40 Art. 362. 10. 20f loads. 2. 32 rd. yd. 2 ft. 11. 9 t % 3 t hours. 3. 811b. 5pwt. 16 gr. 12. 10. Art. 364. Art. 379. L 8hhd. 12 gal. 1 qt. 3. Three thousand one hun- 2. 405 cu. ft. 328 cu. in. dred sixty-five ten thou- 3. 57 A. 92 P. 4 sq. yd. sandths. 4. 43 bu. 1 pk. 2 qt. 4. Five hundred fifteen, and six 5. 64 wk. 6d. 4h. thousandths. 6. 104° V 52". 6. .1331 7. 15 cd. 6 cd. ft. 7. 11011.011 8. 9 cwt. 20 lb. 8 oz. Art. 381. 9. 41 y. 3 mo. 1. Three hundred forty-seven 10. 301 mi. 44 rd. thousandths. 11. 281 A. 85 P. 2. One thousand four hundred 12. 11 hhd. 61 gal. 2 qt. five ten-thousandths. 13. 21b. 3oz. 11 pwt. 6gr. 3. Seventy-two ten-thou- 14. 336 A. 24 P. sandths. Art. 367. 4. One hundred thirty-eight 2. 6rd. 2yd. Oft. 9f in. thousandth?. 222 ANSWERS. 5. Ten thousand sixty-five hun- dred thou- sandths. 6. Two millionths. 7. Fourteen, and five thou- sandths. 8. Nine, and nine- ty-three thou- sandths. 9. One, and six thousand four hundred forty- four ten-thou- sandths. 10. .5 11. .31 12. .04 13. .017 14. .434 15. .1111 16. 14.05 17% 9.003 18. 1. 06444 19. .98 20. .125 21. .1018 22. 6.75 23. 2462.7 24. 400.044 25. 24.000024 Art. 384. 2. 80.928 4. 22.44 5. 4.257 Art. 386. 1. 16.275 2. 41.151 3. 1.182 4. 34.039 5. 8.505 6. .137 of a ton. -7. 97.02 8. 121.875 9. 114.028 10. $206. Art. 300. 3. 2.865 4. .0450 Art. 303. 1. 17.5 2. 1.75 3. 17.5 4. 1.75 5. 3.01 6. .0301 7. 30.1 8. 30.1 9. 4.872 10. .4872 11. 4.872 12. 48.72 13. 7.5 14. 215042. 15. 3.65 16. $34.85 Art. 305. 3. 3.15 1.34 Art. 307. 1. 1.23 2. 43. 3. 3.3 4. 3.3833-f 5. .05 6. 100. 7. .01 8. 10. 9. .42 10. 75. 11. 10. 12. .123 13. .756 14. .03145 15. .0000905 17. .0609+ 18. 15.5 yd. Art. 300. 2. &. 3. tI* 4. h 5. f. Art. 401. i. B- 2. h 3. jV 4. tI*. 5. f. 6. i : n 12 »• T2T- 8. tfe. 9. l\h Art. 403. 2. .75 3. .08 Art. 405. 1. .85 2. .125 3. .025 4. .016 5. .625 6. .0625 7. .096 8. .008 9. 1.275 11. .3846+ Art. 407. 1. *. 2. .0017 3. i. 4. .368+ 5. $55.25 6. 31.856 7. 20.4 8. $507.78 9. $60,477 10. 5305. 11. 100. 12. 2146.85 pound* Art. 414. 1. .03 2. .04 3. .07 4. .09 5. .04J 6. .09£ 7. .17J A XS WEMS. 223 8. .07| 9. .45 10. .67 11. 1.25 12. 1.35 Art. 416. 2. 43 pounds. 3. 90 men. Art, 418. 1. $2.46 2. $34.09 3. 11.6 yd. 4. 132.5 rd. 5. 160.3 6. 27.625 7. 106.78 8. 105.6 9. $75. 10. 36 men. 11. 84cwt. 12. 435 lbs. 13. $135. 14. $187.50 15. 2176 bu. 16. 11448. Art. 420. 2. 5%. 3. m&. Art. 422. 1. 16|%. 2. 6f%. 3. 6A#. 4. 25%. 5. lfH%. 6. 7%. 7. 19%. 8. 60%. 9. 9ff%. 10. 8%. 11. 15%. 12. 60%. 13. 45%. 14. 5%. 15. 27|g|%. 16. 40%. 17. 20%. Art. 428. 2. $192.50 3. $265. 4. $168. 5. $191.67 7. $618.90 8. $727.56 9. $27,778 10. $12.60 Art. 430. 1. $472.50 2. $3,105 3. 41.777 4. $1236.667 5. $693,333 6. $5279.167 7. $18.36 8. $166,625 9. $.588 10. $813.61 11. $965.08 12. $300.90 13. $55,026 14. $1099.75 15. $1368. Art. 433. 1. $37.50 2. $6,681 3. $1.89 4. $5000. 5. $1095.85 6. $949,333 7. $12.60 8. 10%. 9. $495. 10. $300. 11. $1854.16 12. $5205.493 ANSWERS TO APPENDIX. Art. 444. Art. 452 3. 76 ft. 1. 216. 4. 45. 2. 1. Art. 446. 3. 1500. 1. 1400 sq. rd. 4. 5280. 2. 1849 sq. ft. 5. 2. 3. 43 rd. 6. $7.20 4. 15 rd. Art. 456, 5. 40 sq. yd. 3. 6. Art. 450. 4. 200. 4. 7. 5. 14?, 6. 512. Ir t. 458. 1. 100. 2. 720. 3. 84373. 4. 125. 5. 44.8 6. 3 ft. 7. 4 ft. 9 in 8. 180. 9. 129.6 224 ANSWERS. ANSWERS TO MISCELLANEOUS PROBLEMS. Art. 73. 1. 59009. 7080. 51103. 3. 425581. 4. 95040. 73881. 5. 688 dollars. 6. 18039. 7. 171232. 8. 3599 barrels. 9. 765508. 10. 481. 11. 12267. 11. 101810. 13. 15778. 14. 5588. Art. 117. 1. 3375 dollars. 2. 73057. 3. 34008. 4. 184 dollars. 5. 6535. 6. 286ff 7. 1025. 8. 150. 9. 129. 10. 279. 11. 71142. 12. 86009. f759tf. 13. ]217A. ( 369|f. 14. \ 2172U. I 1122JJ. 15. 2J dollars. Art. 238. 1. - 1 - 79 . 2*. 26JJ; 64 V. 3. 31; 63; 6° 4. 1 fi . 20 . 14' 6. 25 & years. v. ih 8. T V 9. X. 10. V. 11. 17f 12. $52.50 13. $3.00 14. 11 J. 1 K 21 . 91 16. L/o- 17. f. 18. I. 19. $1500. 20. $6300. 21. 81fJ bushels. 22. IJf. Art. 334. 1. $70. 2. $3.24 3. 271746. 4. 27280. 5. $255.9H 6. 27/„. 7. 6. " 8. 81.25 9. 160000. 10. 4. 11. 6i. 12. 4." Art. 372. 1. 16740. 2. 20 1b.lloz.lpwt. 13 gr. 3. 48° 2 / 28". 4. 21886. 5. 83026. 6. 843.50 f8 rd. 5 cd. ft. 9 7. i cu. ft.; 19bu.2 I pk. 7 qt. 1 pt. f 92r. 9q. 21 s. 8. \ 130 A. 150 p. I 30 vd. 3. 4. [One, 79 oz. 14 o pwt. ; each of J the others, 39 [ oz. 17 pwt. 10. 6 mi. 176 rd. vd. n ft. , , f 577 gal. 3 qt. 11 * 1 17 d. 4 h. 44 m. 12. 1 cwt. 15 lb. Art. 407. 1. 1000.0001 2. 15.000015 24.166 f 92.9907 \ .999999 5. .10816 6. U. 7. .176 8. 10. 9 1 ^ 10.' 312.006 11. .000042 12. 7*. 13. 114.3055 14. 35.375 gal. 15. .74375 Art. 433. 1. 476.85 vd. ; $13.75; 9.36 bu. 2. .27 + 3. 30. 4. 35. 5. $8.62* 6. $51.45; $42.87i 7. $393.86| 8. $8. 9. 81415.62| 10. 81102.67 11. 86.27 12. $923.29 13. $18.96 14. 81605.54 15. 82111.11 3 QAlO IfeTl «— COWPERTHWAIT & CO.'S EDUCATIONAL SERIES. Hagar's Mathematical Series. Primary Lessons in Numbers. Elementary Arithmetic. Common School Arithmetic. Elementary Algebra. Elementary Geometry. FOR TEACHERS. Dictation Problems in Arithmetic, Key to Com. 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