ARITHMETICS FUNDAMENTAL PROCESSES STATE NORMAL SCHOOL LQ8 WALSH-SUZZALLO ARITHMETICS BOOK ONE FUNDAMENTAL PROCESSES BT JOHN H. WALSH ASSOCIATE SUPERINTENDENT OF SCHOOLS THE CITY OF NEW YORK PROFESSOR OF THE PHILOSOPHY OF EDUCATION TEACHERS COLLEGE, COLUMBIA UNIVERSITY D. C. HEATH & CO., PUBLISHERS BOSTON NEW YORK CHICAGO 3 COPYRIGHT, 1914, BY D. C. HEATH & Co. IBS PREFACE THESE books are designed to give an elementary school child all the arithmetical power necessary to cope with the situations of common everyday life. A social survey of the usual responsibilities of men has determined omis- sions and additions. The methods of learning and teach- ing employed have been indicated by an analysis of thousands of typical arithmetical errors collected in ordi- nary schools. Because a standard of social efficiency has been applied throughout, it is expected that the immediate competency of the pupil leaving school will be greatly increased. Because a standard of pedagogical economy has likewise been rigidly observed, it is also expected that the pupils will develop this competency one to two years earlier. In consequence, this series has been so arranged that a child may acquire an easy and accurate command over all fundamental processes, both simple and complex, by the end of the sixth year. The seventh and eighth school years are thus left free for a study of those business insti- tutions and practices, the understanding of which is vital to an extended use of arithmetic in practical life. This particular book, the first one in the series, covers iv PREFACE the fundamental elements of arithmetical manipulation and thought. Thus a child who goes no farther than the fourth or fifth grade will be so familiar with essentials that he can readily learn more arithmetic by himself. No attempt has been made to project the pupil into unfamiliar situations. His experience is already greater than his power of mathematical interpretation. The teacher of primary arithmetic has accomplished enough when she has taught the child to solve his own problems. Hence, a special effort has been made to keep both exam- ples and problems within the reach of the child. While abstract work is the chief difficulty of the first school grades, it must ever be remembered that it is always a concrete and personally vital problem which pro- vokes the need to compute. In all development of new technique, the teacher should start with an easy problem actually or orally expressed. Once the child has decided from the concrete situation what he is to do, he should be permitted to do it symbolically or abstractly. Mere drill on combinations, processes, and manipulations may well be abstract, provided the pupil feels the need of such perfection and knows that in the end his facility is to be used in the solution of real problems. In this text, every new process is introduced by a prob- lem that calls for its use. Similarly, the last drill upon a process is given in the form of oral and written problems. This is to remind the teacher that all abstract work, how- ever important, is merely an intermediate stage in the PREFACE v effective teaching of arithmetic. Ample provision must be made for concrete work at the beginning and end of all practice upon the mechanics of arithmetic. The use of more than one way of working an example has been carefully avoided. We should be content with one concrete method of computation, particularly in the primary grades. To attempt to develop two different habits of response to the same situation leads to confusion and waste. Short methods of operation are given only in the higher grades. Special attention is called to the device used to give the child a clear understanding of the manner in which formal processes are performed. Whenever a process is expressed in figures, each stage of the mental process accompanying it is also expressed in simple language. These parallel language statements take the form of simple directions given in the most economical and efficient order. If the child does not clearly see how the process has been con- ducted in the course of the teacher's demonstration, refer- ence to the text makes it clear. In establishing a habit it is advantageous to follow an accurate, permanent, and consistent form. It is quite impossible for the teacher to give a great amount of attention to each individual in our large classes, and it is wise to develop in the child the power to understand and practice arithmetical manipu- lations by himself. In line with this effort to give the child power to care for himself, the child is constantly taught to test or check his work. iv PREFACE the fundamental elements of arithmetical manipulation and thought. Thus a child who goes no farther than the fourth or fifth grade will be so familiar with essentials that he can readily learn more arithmetic by himself. No attempt has been made to project the pupil into unfamiliar situations. His experience is already greater than his power of mathematical interpretation. The teacher of primary arithmetic has accomplished enough when she has taught the child to solve his own problems. Hence, a special effort has been made to keep both exam- ples and problems within the reach of the child. While abstract work is the chief difficulty of the first school grades, it must ever be remembered that it is always a concrete and personally vital problem which pro- vokes the need to compute. In all development of new technique, the teacher should start with an easy problem actually or orally expressed. Once the child has decided from the concrete situation what he is to do, he should be permitted to do it symbolically or abstractly. Mere drill on combinations, processes, and manipulations may well be abstract, provided the pupil feels the need of such perfection and knows that in the end his facility is to be used in the solution of real problems. In this text, every new process is introduced by a prob- lem that calls for its use. Similarly, the last drill upon a process is given in the form of oral and written problems. This is to remind the teacher that all abstract work, how- ever important, is merely an intermediate stage in the PREFACE v effective teaching of arithmetic. Ample provision must be made for concrete work at the beginning and end of all practice upon the mechanics of arithmetic. The use of more than one way of working an example has been carefully avoided. We should be content with one concrete method of computation, particularly in the primary grades. To attempt to develop two different habits of response to the same situation leads to confusion and waste. Short methods of operation are given only in the higher grades. Special attention is called to the device used to give the child a clear understanding of the manner in which formal processes are performed. Whenever a process is expressed in figures, each stage of the mental process accompanying it is also expressed in simple language. These parallel language statements take the form of simple directions given in the most economical and efficient order. If the child does not clearly see how the process has been con- ducted in the course of the teacher's demonstration, refer- ence to the text makes it clear. In establishing a habit it is advantageous to follow an accurate, permanent, and consistent form. It is quite impossible for the teacher to give a great amount of attention to each individual in our large classes, and it is wise to develop in the child the power to understand and practice arithmetical manipu- lations by himself. In line with this effort to give the child power to care for himself, the child is constantly taught to test or check his work. vi PREFACE These texts are not a substitute for a good teacher; they are an important and necessary aid. They aim to be efficient and economical guides to both teacher and pupil. The topics, problems, and processes are representative of the arithmetical facts, power, and skill that will be most useful to the average citizen. The methods of treat- ing them have been garnered from the most successful teaching practice. CONTENTS SECTION ONE PRELIMINARY REVIEW I'AOK 1 2 2 3 4 COUNTING ADDING BY ONES . . READING NUMBERS . . ADDING BY Twos . . TREASURING READING AND WRITING NUMBERS 6 ADDING BY THREES ... 6 ADDING BY FOURS ... 7 ADDING BY FIVES ... 8 MEASURING 9 WRITTEN ADDITION ... 11 MAKING CHANGE .... 13 SUBTRACTION 14 NOTATION AND NUMERA- TION 20 ADDING BY 6's. BY 7's . 21 ADDING BY 8's. BY 9's . 22 ADDITION, WITH CARRY- ING 24 TESTING RESULTS ... 25 ONE HALF. ONE QUARTER 26 ROMAN NUMBERS ... 27 TELLING TIME 27 EQUAL PARTS 30 QUART AND PECK ... 30 SUBTRACTION, WITH CAR- RYING 31 MULTIPLYING BY 2 DIVIDING BY 2 MULTIPLYING BY 3 DIVIDING BY 3 MULTIPLYING BY 4 DIVIDING BY 4 MULTIPLYING BY 5 SECTION TWO . . 35 DIVIDING BY 5 .... 53 . . 40 LIQUID MEASURE. ... 54 . . 44 RECTANGLES 55 . . 45 TIME 58 . . 48 ROMAN NUMBERS ... 58 . . 49 NOTATION AND NUMEUA- 52 TION 59 Vlll CONTENTS PAOB ROMAN NUMBERS ... 61 DOLLARS AND CENTS . . 62 ADDITION OF MONEY . . 64 SUBTRACTION OF MONEY . 66 FUNDAMENTAL OPERATIONS 67 MULTIPLICATION TABLE . 70 MULTIPLYING BY 6 ... 72 DIVIDING BY 6 .... 73 REMAINDERS IN DIVISION 75 MULTIPLYING BY 7 ... 78 DIVIDING BY 7 .... 79 MULTIPLYING BY 8. BY 9 82 DIVIDING BY 8. BY 9 . 83 PROBLEMS WITH Two OPERATIONS 84 MULTIPLYING BY 10 . . 88 DIVIDING BY 10 .... 88 MULTIPLYING BY A MUL- TIPLE OF 10 89 DIVIDING BY A MULTIPLE OF 10 91 INEXACT DIVISION ... 92 AREAS OF RECTANGLES . 93 FRACTIONAL PARTS ... 94 DENOMINATE NUMBERS . 96 RECTANGULAR SOLIDS . . 98 SECTION THREE READING AND WRITING NUMBERS ....'.. 99 MULTIPLIERS OF Two FIG- URES 100 LONG DIVISION .... 104 PROBLEMS WITH MORE THAN ONE OPERATION . 106 LONG DIVISION .... 108 ADDITION AND SUBTRAC- TION 109 ADDING AND SUBTRACTING DOLLARS AND CENTS . 112 CENTS AS DIVISORS . . . 114 URING RECTANGLES . 115 FRACTIONAL PARTS . 120 MIXED NUMBERS CONTAIN- ING HALVES 122 ADDING AND SUBTRACTING MIXED NUMBERS . . . 123 HALVES AND FOURTHS . . 125 THIRDS AND SIXTHS . . 129 RATIO 132 FIFTHS 133 MULTIPLYING AND DIVID- ING BY 11 145 MULTIPLYING 'AND DIVID- ING BY 12 146 MULTIPLIERS CONTAINING CIPHERS 148 LONG DIVISION .... 151 CONTENTS IX PAGE MULTIPLES AND FACTORS 158 REDUCTION OF FRAC- TIONS 159 ADDING AND SUBTRACTING FRACTIONS 160 ADDING MIXED NUMBERS 161 SUBTRACTING MIXED NUM- BERS 162 FRACTIONAL PARTS OF NUMBERS 165 MULTIPLYING BY A MIXED NUMBER 167 LONG MEASURE .... 174"^ AREAS SQUARE MEASURE . . . CUBICAL CONTENTS . 179 SECTION FOUR NOTATION AND NUMERA- TION 183 ADDITION AND SUBTRAC- TION 185 MULTIPLICATION AND DI- VISION 188 WRITTEN MULTIPLICATION 190 LONG DIVISION .... 191 ALIQUOT PARTS OP A DOLLAR 196 CANCELLATION 198 FRACTIONS 201 PROPER AND IMPROPER FRACTIONS 205 LOWEST TERMS. HIGHER TERMS 206 COMMON DENOMINATORS . 207 ADDITION OF FRACTIONS . 208 ADDITION OF MIXED NUM- BERS 209 SUBTRACTION OF MIXED NUMBERS 210 PRODUCT OF A MIXED NUMBER AND A WHOLE NUMBER 214 MULTIPLICATION OF FRAC- TIONS 218 DIVISION OF FRACTIONS . 220 FRACTIONAL RELATIONS . 222 DECIMALS 225 DECIMAL NOTATION AND NUMERATION .... 226 READING AND WRITING DECIMALS 227 REDUCTION OF DECIMALS. 227 MULTIPLYING A DECIMAL BY A WHOLE NUMBER . 229 DIVIDING A DECIMAL BY A WHOLE NUMBER 229 x CONTENTS PAGE PACE ADDITION AMD SUBTRAO BILLS 236 TION OF DECIMALS . . 230 LETTERS 240 COMPOUND NUMBERS . . 233 CHECKS 241 MEASUREMENTS .... 234 NOTES 242 VOLUMES 235 RECEIPTS 242 BUSINESS FORMS . , 236 ACCOUNTS 244 ARITHMETIC FIRST BOOK SECTION I INTRODUCTORY REVIEW Counting 1. How many men are there in the picture on the left-hand page ? 2. How many children ? 3. How many horses ? 4. How many cows ? 5. How many sheep ? 6. How many hens ? \ 7. How many ducks ? >, 8. How many pigeons are there in the picture ? 9. How many doors has our class room ? 10. How many windows ? 11. How many panes of glass are there in each window ? 12 How many rows of desks are there ? 13. How many desks are there in each row ? SECTION ONE Adding by Ones 1. Sarah has two girl dolls and one boy doll. How many dolls has she ? 2. William has three pears in his basket and one pear in his hand. How many pears has he ? 3. Of a flock of geese, four are in the pond and one is on the bank. How many geese are there in the flock ? 4. Jane spent five cents for a pencil and one cent for a pen. How many cents did she spend for both ? 5. John has six cents. If his father gives him one cent, how many cents will he have ? Reading Numbers One Two Three Four Five Six Seven Eight Nine Ten 123 4567 89 10 a. Sight Drills Give sums rapidly : 111111111 123456789 INTRODUCTORY REVIEW Adding by Twos 1. Arthur has two large rockets and two small ones. How many rockets has he ? 2. How many are two and two ? 3. Ann rode three miles on the cars and two miles in the stage. How many miles did she ride ? 4. How many are three and two ? 2 and 3? 5. Fred has four cents in his bank. How many cents will be in the bank after he puts into it two more cents? 6. How many are four and two ? 2 and 4 ? 7. There are five pears on one plate and two on another. How many pears are there on both plates ? s. How many are five and two ? 2 and 5 ? 9. Helen found six eggs in one nest and two in another. How many eggs did she find ? 10. How many are six and two ? 2 and 6 ? 11. Harry is seven years old and his sister Mary is two years older. How old is Mary ? 12. How many are seven and two ? 2 and 7 ? 13. There are eight children now in the room. How many will there be when two more come ? 14. How many are eight and two ? 2 and 8 ? 4 SECTION ONE Measuring 1. How many steps across the class room ? 2. How many steps are there in the length of the class room ? 3. How many glassfuls will the pitcher hold ? 4. How many pitcherf uls will the pail hold ? 5. How many glassfuls will the pint hold ? 6. How many pints will the quart hold ? 7. How many quarts will the pitcher hold ? 8. How many quarts will the pail hold ? Playing Store 9. Use cents to pay for articles costing : (a) Four cents ; (6) six cents ; (c) three cents ; (d) Two cents ; (e) five cents ; (/) seven cents. 10. How many cents are there in a nickel ? 11. Using a nickel and cents, pay for articles cost- ing : (a) Six cents ; (6) eight cents ; (c) ten cents ; (d) nine cents ; (e) seven cents. 12. How many cents are there in a dime ? is. How many nickels are there in a dime ? 14. How many cents are there in two nickels ? is. If you have three cents and a nickel in your pocket how many cents have you ? 16. If you have one cent and a nickel ? 17. If you have four cents and a nickel ? is. If you have two cents and a nickel ? INTRODUCTORY REVIEW 5 Reading and Writing Numbers Ten is written 10. Eleven is written 11. Twelve is written 12. Thirteen is written 13. Write : Fourteen ; fifteen ; sixteen ; seventeen ; eighteen ; nineteen. Read the following : 16, 18, 13, 19, 15, 12, 10, 14. i. How many cents in a dime ? In two dimes ? In three dimes ? In four dimes ? In five dimes ? In six dimes? In seven dimes? In eight dimes? In nine dimes ? Twenty is written 20. Thirty is written 30. Forty is written 40. Fifty is written 50. Write : Sixty ; seventy ; eighty ; ninety. Twenty-one is written 21. Fifty-four is written 54. Forty-three is written 43. Thirty-two is written 32. Read the following: 22, 33, 44, 51, 65, 78, 86, 92. The right-hand figure of a number is called the ones' figure. Drill Exercises Add: a. 2 6 2 3 2 4 2 10 8 2 9 2 5 2 7 2 1). 7 2 5 2 8 2 9 2 2 6 2 4 2 3 2 10 6 SECTION ONE Adding by Threes 1. There are three roses on one bush and three on another bush. How many roses are there on both bushes ? 2. How many are three and three ? 3. Mr. Smith sold four pigs to one man and three to another. How many pigs did he sell ? 4. How many are four and three ? 3 and 4 ? 5. A girl spent five cents for ice cream and three cents for cake. How much did she spend ? 6. How many are five and three ? 3 and 5 ? 7. There are six quarts of milk in one can and three quarts in another. How many quarts are there in both cans ? 8. How many are six and three ? 3 and 6 ? 9. Frank has seven hens and three ducks. How many has he of both ? 10. How many are seven and three? 3 and 7 ? 11. How many fish are in a jar if there are eight gold fish and three silver fish in it ? 12. How many are eight and three ? 3 and 8 ? is. Ned has nine marbles in a bag and three in his pocket. How many marbles has he ? 14. How many are nine and three? 3 and 9 ? INTRODUCTORY REVIEW 7 Adding by Fours 1. There are four boys and four girls in a car. How many children are there in the car ? 2. How many are four and four ? 3. A girl has a five-cent piece and four pennies in her bank. How much money is there in the bank ? 4. How many are five and four ? 4 and 5 ? 5. Along the walk there are six maple trees and four elms. How many trees are there ? 6. How many are six and four ? 4 and 6 ? 7. James has seven pounds of flour in his basket. The basket weighs four pounds. How many pounds do both weigh ? s. How many are seven and four ? 4 and 7 ? 9. Mary is eight years old. Her brother is four years older. How old is her brother ? 10. How many are eight and four ? 4 and 8 ? 11. If a book cost nine cents and a pad cost four cents, how much will both cost ? 12. How many are nine and 4 ? 4 and 9 ? Counting 1. Count by 2's to twenty. By 3's to eighteen. By 4's to twenty. By 5's to ninety. By 10's to ninety. 2. Count by 2's to eight, then by 4's to twenty. 3. Count by 3's to fifteen, then by 5's to forty. 8 SECTION ONE Adding by Fives 1. A boy has five marbles in one hand and five in the other. How many has he in both hands ? 2. How many are five and five ? 3. One hen has six chickens, and another hen has five. How many chickens have both hens ? 4. How many are six and five ? 5 and 6 ? 5. It is now seven o'clock. What time will it be in five hours? 6. How many are seven and five ? 5 and 7 ? 7. John picks eight quarts of berries and Thomas picks five quarts. How many quarts do both boys pick? 8. How many are eight and five? 5 and 8 ? 9. My peach tree is nine feet high and my pear tree is five feet higher. How high is the pear tree ? 10. How many are nine and five? 5 and 9 ? 11. John took five steps to the right and then four steps straight ahead. How many steps did he take ? 12. How many are four and five ? 5 and four ? INTRODUCTORY REVIEW 9 Sight Drills Give answers rapidly : A c D E F G H / a. 1 2 3 4 5 6 7 8 9 4 4 4 4 4 4 4 4 4 b. 1 2 3 4 5 6 7 8 9 5 5 5 5 5 5 5 5 5 c. 4 4 4 4 4 4 4 4 4 2 5 3 6 1 7 9 8 d. 5 5 5 5 5 5 5 5 5 7 1 6 8 2 4 9 3 Measuring 1. Measure the length of a book in inches. The width. 2. Measure the length and the width of a pane of glass. 3. Find the length of the class room in feet. Its breadth in feet. 4. Find the length and the breadth of the school yard in feet. 5. Find the dimensions of the blackboard. Of the door. Of a window. 6. How many inches are there in a foot ? 7. Find the number of inches in one half foot. 10 SECTION ONE Sight Drills Give sums. Add upward. E F G 444 54 64 74, 666 62 72 82 88 8 51 61 71 777 72 82 92 432 61 71 81 676 72 82 92 215 93 63 33 443 154 54 60 71 63 72 83 245 421 A B c z> a. 4 4 4 4 14 24 34 44 b. 6 6 6 6 22 32 42 52 c. 8 8 8 8 11 21 31 41 d. 7 7 7 7 32 42 52 62 e. 8 7 6 5 21 31 41 51 / 2 3 4 5 32 42 52 62 9- 4 5 6 3 23 .53 43 73 Say 15, 16; 25, 29; etc. h. 1 4 3 1 2 1 2 2 13 24 33 44 i. 21 36 44 57 7 2 1 1 1 1 2 1 INTRODUCTORY REVIEW 11 Addition Written Exercises Add: Before working these written exercises, give orally ,*he sum of each column in an example, beginning with the ones' column. a. b. c. d. c. A B c D E F 24 62 71 43 25 47 23 62 36 26 34 52 22 64 61 43 18 35 63 54 47 27 36 61 54 22 33 32 74 43 28 64 53 35 65 15 45 71 35 45 24 33 11 95 63 50 32 32 23 15 2 31 14 32 23 24 32 1 5 35 32 32 32 32 21 41 21 51 22 32 22 32 5 34 3 32 32 4 23 13 4 12 42 23 ""*" " ~ ~ 12 31 41 42 23 53 32 52 41 33 16 4 3 33 4 27 5 11 31 41 32 23 12 14 4 12 62 42 23 12 3 12 3 4 3 23 12 21 61 62 3 24 12 SECTION ONE Written Problems i. A boy earned thirty cents, twenty-five cents, and three cents. How much did he earn ? PROCESS 30c. Write the numbers under each other, placing 25 the ones' figures of each in a row. Write under 3 each column its sum. Ans 58c 2. Add forty cents, thirteen cents, and six cents. To save unnecessary repetition, write c. but once in the example and again in the answer. f c. stands for cents. Ans. 59 c. The sign of addition is + , called plus. 3. 31 + 42 + 3 + 1. 4. 22 + 33 + 14. 5. Twenty-five boys, plus twelve boys, plus two boys. 6. Ten cows, plus twenty-six cows, plus three cows. 7. A farmer has twenty-four cows in one field and fourteen in another field. How many cows has he in both fields ? 8. Mr. Smith sold eighteen pies Monday and eleven Tuesday. How many did he sell in the two days ? 9. A boy paid twenty-five cents for a ball and twelve cents for a bat. How much did he pay for both? INTRODUCTORY REVIEW 13 Making Change Let one pupil represent a storekeeper and another a cus- tomer. The latter announces the total of his purchase, giving money to the former, who then hands over the change. The customer satisfies himself of the correctness of the latter, stat- ing its amount. He then becomes the storekeeper and a third pupil becomes a customer. Make change in the business way. If a half-dollar is given for a thirty -eight cent purchase, the storekeeper hands two cents, saying "thirty-eight, forty" ; then a dime, saying "fifty." Make change for the following purchases : 1. Amount of purchase, 6 cents. Sum given, 25 cents. 2. Amount of purchase, 43 cents. Sum given, 50 cents. 3. Amount of purchase, 11 cents. Sum given, 25 cents. 4. Amount of purchase, 29 cents. Sum given, 35 cents. 5. Amount of purchase, 68 cents. Sum given, 75 cents. 6. Amount of purchase, 8 cents. Sum given, 50 cents. 7. Amount of purchase, 17 cents. Sum given, 25 cents. s. Amount of purchase, 3 cents. Sum given, 25 cents. 9. Amount of purchase, 27 cents. Sum given, 30 cents. 14 SECTION ONE Subtraction. Preparatory Exercises 1. If John needs 10 cents for a ball and he al- ready has 5 cents, how many more cents does he re- quire ? 2. What number added to 7 will make 14 ? 3. A boy placed 9 marbles in two piles, one of which contained 5 marbles. How many were there in the other pile ? 4. In how many years will a 5-year old girl be 8 years old? 'Is. A man is making a trip of 6 miles. If he has gone 2 miles already, how many miles has he still to go? 6. Mabel and Samuel together picked 8 quarts of berries. If Samuel picked 3 quarts, how many quarts did Mabel pick? 7. Nine boys are required for a baseball team. Only six have been chosen. How many more must they have ? a. If I buy 7 cents' worth of candy and give the storekeeper a dime, how much change should I receive ? 9. There were 10 trees in a grove. After 3 were cut down, how many were left standing ? i 10. William rode 12 miles in an automobile. How many more miles did he ride than Edwin, who rode only 8 miles ? INTRODUCTORY REVIEW 15 Drill Exercises Tell the number that must be added to the lower one to make the upper one : ABODE F G H I a. 9 8 2 9 3 7 4 9 8 _1 _7 _i _8 -1 -6 -2 -6 -1 The sign of subtraction is , called minus. Subtraction 1. From 86 take 32. I. 793856954 _1 _7 _2 -6 -4 -3 -2 -1 -3 c. 976847689 -3 -5 -1 -5 -1 -2 -5 -2 -5 d. 578675689 -2 -3 -4 -2 -4 -3 -4 -3 -4 PROCESS Write 32 under 86, placing. the ones' figures of both in the same column. Begin 32 at the ones' column and think 2 and 4 (writing 4) are 6, 3 and 5 (writing 5) are 8. 16 SECTION ONE Written Exercises The answer in subtraction mainder. is called the re- Find remainders : A 5 C D E F a. 88 59 78 49 76 63 -41 -34 15 28 -33 -41 &. 46 55 67 88 89 54 -15 - 94 .I/T: 36 75 -46 -32 c. 79 58 62 85 93 87 -23 -32 31 41 -52 -63 d. 45 86 66 67 64 77 -44 -21 45 33 -22 -32 e. 56 47 83 78 99 65 -32 -41 32 51 -34 -44 / 63 75 96 48 84 97 -42 -31 43 25 -52 -23 Addition and Subtraction Sight Exercises A B C D E F G a. 17 21 21 32 37 37 20 -1-4 -4 -17 + 5 -32 -5 -6 b. 12 19 19 11 19 19 19 + 7 -7 -12 4-8 -8 -11 + 6 INTRODUCTORY REVIEW 17 Oral Problems 1. If John spends 5 cents out of a dime, how much change does he get ? 2. There were 14 pears on a branch. How many were there after Mary picked off 7 ? 3. A boy had 9 marbles. How many had he after losing 4 of them ? 4. A girl is 8 years old. How old was she 3 years ago? Oral Drills The sign of equality is = , called equals. 13 + 4=17 means 13 plus 4 read " 13 and 4 are 17." 17 4 = 13 means 17 minus be read " 17 less 4 are 13." Give answers : . It may be 13. It may A B c D a. 14 + 4 o 18- 14 = ? 418- 4 = ? 24 + 4 = ? b. 13 + 6 ft 19- 13 = ? 19- 6 = ? 43 + 6 = ? c. 11 + 3 A 14- 11 = ? 14- 3 = ? 61 + 3 = ? d. 12 + 5 = ? 17- 12 = ? 17- 5 = ? 32 + 5 = ? e. 15 + 2 = ? 18- 16 = ? 18- 2 = ? 75 + 2 = ? f. 10 + 9 = ? 19- 10 = ? 19- 9 = ? 50 + 9 = ? ff- 14 + 3 = ? 17- 14 = ? 17- 3 = ? 84 + 3 = ? A. 16 + 2 = ? 18- 16 = ? 18- 29 . 26 + 2 = ? I 12 + 7 = ? 19- 12 = ? 19- 7 = ? 92 + 7 = ? 18 SECTION ONE Addition and Subtraction Oral Problems 1. In an orchard there are 1 1 trees in the first row and 10 trees in the second. How many are there in both rows ? 2. After plowing 36 acres how many acres must Mr. Jones plow to finish a 40-acre field ? 3. I bought goods amounting to 42 cents and received 8 cents in change. How much did I hand the storekeeper ? 4. A girl has 44 cents. How many more cents does she need in order to buy a 50-cent doll ? 5. There are 18 yards of cloth in one roll and 10 yards in another. How many yards are there in both rolls ? 6. After selling 22 dozen eggs, a grocer has 7 dozen left. How many dozen did he have at first ? 7. Edward has 24 marbles and Arthur has 6 more than Edward. How many has Arthur ? ^/s. Sarah has 26 cherries. How many will she have if she gives 10 to Jane ? ^. Frank is 16 years old. In how many years will he be 21 ? 10. A boy picked 27 quarts of berries. How many quarts will he have after selling 19 ? 11. Mr. Jones is 25 years old. How old was he 9 years ago ? INTRODUCTORY REVIEW 19 Written Problems 1. In an orchard there are 26 trees in the first row and 31 trees in the second row. How many are there in both rows ? 2. After cutting 24 acres of wheat, how many more acres will Mr. Wise have to cut to finish a 45-acre field? 3. I bought goods amounting to 52 cents and received 23 cents in change. How much did I hand the storekeeper? 4. A boy has 62 cents. How many more cents does he require in order to buy a 75-cent baseball ? 5. There are 23 yards of silk in one roll and 25 yards in another. How many yards are there in both rolls? 6. After selling 22 dozen eggs, a grocer has 16 dozen left. How many dozen did he have at first ? 7. Fred has 24 marbles, and George has 1 5 more than Fred. How many has George ? 8. Minnie has 26 cherries. How many will she have if she gives 13 to her brother ? 9. Robert is 11 years old. In how many years will he be 27 ? 10. A girl picked 36 quarts of berries. How many quarts will she have after selling 25 ? 11. Mr. Smith is 38 years old. How old was he 17 years ago ? 20 SECTION ONE Notation and Numeration One hundred is written 100. Two hundred is written 200. Three hundred, 300. 1. Write in figures (a) four hundred. (6) Five hundred, (c) Six hundred, (d) Seven hundred. One hundred one is written 101. 2. Write in figures (a) one hundred three. (&) One hundred four, (c) One hundred five, (d) One hun- dred six. (e) One hundred seven. One hundred ten is written 110. Two hundred twenty-five is written 225. 3. Write in figures: a. One hundred twenty. Five hundred thirty-one. 6. Three hundred one. Seven hundred eighty-four. 4. Read the following: a. Ill 202 317 496 500 6. 630 782 867 999 840 In writing hundreds, three figures are used. The right-hand figure is called the ones' figure, the next is called the tens' figure, and the left-hand one is called the hundreds' figure. 5. Write in figures: a. One hundred ninety. Five hundred twenty-two. 6. Two hundred seventy. Six hundred seventy-three. INTRODUCTORY REVIEW 21 Adding by 6's. By 7's. 1. There are 6 houses on one side of the street and 6 houses on the other. How many houses are there on both sides? 2. How many are 6 and 6 ? 3. After spending 7 cents, a girl has 6 cents left. How many cents did she have at first ? 4. How many are 6 and 7? 5. Charles has saved 8 cents. He needs 6 cents more to buy a book. How much does the book cost? 6. How many are 6 and 8 ? 7. If I pick 9 roses, there will be 6 still on the bush. How many are there on the bush? s. How many are 6 and 9? 9. In a class there are 7 boys and 7 girls. How many pupils are in the class? 10. How many are 7 and 7? 11. A pole was cut into two pieces, one 8 feet long and the other 7 feet long. How long was the pole at first? 12. How many are 7 and 8 ? 13. After giving 9 cherries to his sister, a boy has 7 left. How many cherries had he ? 14. How many are 7 and 9 ? 22 SECTION ONE Adding by 8's. By 9's 1. How many pies are 8 pies and 8 pies ? 2. There are 9 cars in one train and 8 in another. How many cars are there in both ? 3. How many are 8 and 9 ? 4. In a game of ball there are 9 boys on one side and 9 on the other. How many are there on both sides ? Sight Drills Give sums : A B c D E F G H a. 6 7 8 9 7 8 9 8 6 6 6 6 7 7 7 7 &. 6 6 6 7 7 8 9 9 7 8 9 8 9 9 8 9 Sight Exercises Add the following : ABODE a. 20 25 20 25 20 20 10 1L5 20 30 6. 35 30 30 40 20 10 15 20 20 40 c. 15 40 15 25 15 40 15 20 40 30 d. 30 40 50 60 70 30 40 50 60 70 INTRODUCTORY REVIEW 23 Written Exercises i. A rope was cut into three pieces 42, 63, and 24 ft. long. What was the length of the original rope ? * PROCESS Begin at the bottom. Think 7 (4 + 3), 9 ; write 9. Thii 8 (2 + 6), 12 ; write 12. 42ft. 63 > k 24 Ans. 129 ft. 2. Add: A B c D E a. 33 40 70 92 75 20 63 80 80 3 54 25 _3 16 61 b. 64 27 33 40 3' 14 91 4 73 93 80 _1 81 25 _93 c. 102 122 306 5 620 203 122 50 203 113 10 122 401 30 204 d. 620 90 140 30 120 61 305 200 304 302 3 402 24 43 416 e. 234 82 311 232 13 40 105 25 32 143 121 12 123 232 143 24 SECTION ONE Addition with Carrying 3. Add 53, 136, 78, and 684. PROCESS Begin with the ones' column. Think 12 (4 + 8), 18, 21 ; write 1 and carry 2. Think 10 (2 + 8), 17, 20, 25 ; * write 5 and carry 2. Think 8 (2 + 6), 9 ; write 9. Ans. 53 136 78 684 951 4. Find sums : a. c. d. e. A B C D E 64 20 23 16 5 33 15 64 5 23 127 357 227 117 215 26 14 7 119 8 25 2 321 103 342 140 27 33 25 32 56 18 33 15 359 4 9 234 8 80 123 234 345 456 203 47 58 207 345 88 6 9 163 20 8 449 909 175 473 219 81 44 184 250 62 314 30 400 26 105 5 16 37 105 30 INTRODUCTORY REVIEW Testing Results *. Add 278, 166, 37, and 69. 25 TEST After obtaining the sum by adding the col- umns upward, test the result by adding the 278 166 columns downward, thus: 14, 21, 30; 10,16, 37 19, 25 ; 4, 5. 69 Ans . 550 6. Find sums . Test : A B c D E .a. 35 160 313 464 7 323 48 46 203 322 150 204 65 45 48 &. 106 220 27 423 152 38 417 341 240 236 23 75 35 86 324 400 130 305 100 88 c. 606 108 22 40 200 70 75 123 38 106 147 6 34 50 . 84 1 201 5 527 40 32 2 580 44 35 d. 26 + 26 + 26 + 26. h. 34 + 605 + 21 + 83. e. 45 + 45 + 45 + 25. i. 210 + 35 + 406 + 54. /. 34 + 26 + 101 + 5. 3- 25 + 27 + 320 + 105. g. 4 + 89 + 25 + 103. k. 84 + 106 + 50 + 19. 26 SECTION ONE One Half. One Quarter H < One half is written |. One quarter is written J. Fold circles, squares, etc., to show that a half is one of the two equal parts of a thing, and that a quarter is one of its four equal parts. Oral Examples 1. How many halves in a pie ? How many quarters? 2. If a boy eats one quarter of a pie, how many quarters are left? 3. How many quarters of a pie are there in half a pie ? 4. If a half of a pie costs 10 cents, what is the cost of the whole pie ? 5. Find the cost of a quarter of a pie when a half pie costs 10 cents. 6. When milk costs 6 cents a quart, what does a pint cost ? 7. There are 12 eggs in a dozen ; how many eggs are there in a dozen and a half ? INTRODUCTORY REVIEW Roman Numbers 27 Letters are sometimes used to express numbers, especially in clocks and watches, also in chapters of books, etc. I represents 1 VII represents 7 II represents 2 VIII represents 8 III represents 3 IX represents 9 IV represents 4 X represents 10 V represents 5 XI represents 11 VI represents 6 XII represents 12 On the face of a clock IIII is used to represent 4. Telling Time 1. If the minute hand goes 12 spaces in an hour, how many spaces does it go in one half hour ? 2. What time is it when the minute hand points to XII and the hour hand to IIII ? 3. Tell the time when the minute hand is at VI and the hour hand is halfway between II and III. 4. When it is a quarter-past six, where is the minute hand ? The hour hand ? 5. When it is a quarter to one, where is the minute hand ? The hour hand ? e. What is J of 12? Jof 12? 28 SECTION ONE Addition and Subtraction Oral Problems 1. A man had 50 bushels of tomatoes. He sold all but 4 bushels. How many bushels did he sell ? 2. A girl had 50 cents. How many cents had she after spending 41 cents ? 3. A cheese weighed 45 pounds and the box weighed 10 pounds. How many pounds did both weigh ? 4. A boy had 36 firecrackers after he had shot off 9. How many had he at first ? 5. I gave the grocer 75 cents and he gave me 9 cents change. What did the goods cost? 6. A telegraph pole is 24 feet above the ground and 6 feet in the ground. How long is the pole ? 7. There are 15 flowers on one bush and 8 on another. How many are there on both ? 8. How many days are there from December 15 to December 25? 9. A farmer has 42 sheep. How many more must he buy in order to have 50 ? 10. Find the total cost of 28 cents' worth of cake and 9 cents' worth of candy. 11. One day a boy rode to a town 25 miles distant and back again. How far did he ride ? INTRODUCTORY REVIEW 29 Written Problems i/ A man had 75 bushels i ofpotatoes. He used 21 bush- els and sold the rest. How many bushels did he sell ? 2. A boy had 75 cents. How many cents had he after spending 33 cents ? 3. A cheese weighed 48 pounds and the box weighed 12 pounds. What was the weight of both ? 4. A girl had 36 cherries after giving 36 to her brother. How many did she have at first? 5. I gave the baker 25 cents and he gave me 11 cents change. What did the goods cost? 6. A flag pole is 38 feet above the ground and 12 feet in the ground. How long is it ? 7. There are 37 roses on one bush and 29 on another. How many are there on both ? 8. How many days are there from March 13 to March 29? 9. A farmer has 24 sheep. How many does he need in order to have 37 ? 10. What is the total cost of 28 cents' worth of coffee and 22 cents' worth of sugar? 11. A city is 48 miles away. How many miles does a person ride in going and returning ? 30 SECTION ONE Equal Parts 1. Fold a paper circle into two equal parts. What is each part called ? One half is written ^-. 2. Fold a paper circle into four equal parts. What is each part called ? One fourth is written J. 3. Fold a paper square into three equal parts. Each part is called one third. One third is written ^. 4. Fold a paper square into six equal parts. Each part is called one sixth. One sixth is written J. 5. How many thirds are there in a pie? How many sixths ? 6. How many sixths in one half a pie ? In one third of a pie ? Quart and Peck NOTE. The pupil should discover by measuring that there are 8 quarts in a peck. 1. How many quarts are there in a peck ? 2. Change 2 pecks to quarts. 3. Change 16 quarts to pecks. 4. If I feed a quart of grain every day to my chickens, how long will a peck last ? 5. If I feed them 2 quarts of grain every day how long will a peck last ? INTRODUCTORY REVIEW 31 Subtraction Written Exercises 1. Find the difference between 90 and 74. PROCESS 90 Write the larger number, 90, above the _ JA smaller. Begin at the ones' column. . Since 4 is greater than 0, think 4 and 6 (writing 6) are 10. Carry 1 to 7 which makes 8. Think 8 and 1 (writing 1) are 9. Test by adding 16 and 74 upward: thinking 6 and 4, 2 (carrying 1) and 7. 2. Subtract. Test : A B C D E a. 31 46 92 84 63 -19 -27 -79 -69 -48 b. 50 92 83 30 63 -25 -75 -26 -15 -35 c. 63 92 84 72 50 -24 -65 -15 -36 -26 d. 61 66 90 81 60 -19 -27 -19 -69 -15 e. 81 76 56 70 82 -29 -27 -28 -25 -36 /. 93 75 82 61 54 -46 -38 -26 -38 -27 32 SECTION ONE 3. Fron 318 take 276. PROCESS 318 Think 6 and 2 (writing 2) are 8. Think 27 f) 1 an ^ 4 (writing 4) are 11. Carry 1 to 2, which makes 3. As this is the same as the A*) A^yis ' upper number, do not write anything under the hundreds' column. 4. Subtract : ABODE a. 239 327 479 569 648 -156 -246 -382 -481 -560 &. 725 878 915 879 712 c. -654 -792 -830 -693 -651 625 568 449 316 229 -571 -475 -365 -294 -195 s. Subtract 92 from 500. PROCESS 500 -92 Ans. Think 2 and 1 to 9, which (writing 0) are (writing 4) are 8 (writing 8) are 10. makes 10. Think 10 10. Carry 1. Think 1 5. Carry and and 4 408 A B C D E a. 900 800 700 600 500 b. -75 -63 -37 -44 -88 311 311 100 100 100 -91 -291 -50 -95 -15 INTRODUCTORY REVIEW 33 Review Exercises $ In addition and in subtraction the numbers have names, as follows : ' Addition Subtraction 256 Addend 512 Minuend + 256 Addend -256 Subtrahend Ans. 512 Sum Ans. 256 Difference In subtraction the difference is also called the remainder. 1. Find sums. Test : a. 198 + 73 + 206 + 5 + 382. 6. 46 + 514+103 + 8 + 29. c. 184 + 291 + 332 + 45 + 9. d. 55 + 326 + 84 + 173 + 270. e. 321 + 78 + 97 + 84 + 259. 2. Find differences. Test : a. 321-259. 6. 291-184. c. 915-876. d. 840-326. e. 665-486. /. 824-632. g. 270-73. h. 533-486. i. 733-543. j. 514-29. *. 910-707. I 642-438. m. 382-206. n. 623-341. o. 551-275. 34 SECTION ONE Written Problems 1. A girl has 58 cents. She needs 17 cents more to buy a doll. What is the price of the doll ? 2. A farmer raised 75 loads of hay. He used 17 loads and sold the rest. How many loads did he sell ? 3. A boy had 75 cents. How much had he after spending 50 cents? 4. When a boy saves 16 cents more, he can buy a 75-cent pair of gloves. How much has he saved already ? 5. How much older is a man 50 years old than his son who is 16 ? 6. How many days from May 14 to Decoration Day, May 30 ? 7. Mr. Smith is 18 years younger than Mr. Brown, who is 41 years old. How old is Mr. Smith ? 8. Jacob caught 17 fish and Horace caught 17 more than Jacob. How many did Horace catch ? 9. Mrs. Jones is 27 years old. How old will she be 16 years from now ? 10. Mr. Jones is 36 years old. How old was he 18 years ago ? 11. A girl has 50 cents; how much will she have if she spends 26 cents ? SECTION II Preliminary Exercises 1. Add the following : 123456789 4-1 +2 +3 + 4 +5 +6 +7 +8 +9 Instead of writing 1, 2, 3, 4, etc., twice, as examples in addition, they may be written as multiplication examples thus : 123456789 x2 x2 x2 x2 x2 x2 x2 x2 x2 These are read 2 times 1, 2 times 2, 2 times 3, etc. 2. How many are 2 times 1 ? 2 times 2 ? 2 times 3? 2 times 4? 2 times 5? 2 times 6? 2 times 7? 2 times 8? 2 times 9? The sign of multiplication is x , called times. 3. Find the cost of 2 baseballs at 5 cents each. If 1 baseball costs 5 cents 2 baseballs will cost 2 times 5 cents. Am. 10 cents. The answer in multiplication is called the product. 35 36 SECTION TWO Oral Problems 1. How many feet have 2 cows ? 2. If I pay 9 cents for 1 bat, how much must I pay for 2 bats ? 3. How many desks are there in 2 rows, if there are 8 desks in each row ? * 4. Mary has 2 nickels. How many cents has she ? 5^5. A boy has 2 pockets and 3 marbles in each. How many marbles has he in both pockets ? y- 6. How many feet have 2 birds ? s. John has 36 cherries, James has 4 more than John. How many has James ? 6. A farmer has 40 cows. How many will he have if he sells 4 ? 7. How many sheep at $ 4 each can be bought for $ 36 ? 8. What will 4 yards of 9-cent calico cost ? 9. There are 36 trees in 4 equal rows. How many are there in each row ? 10. If one quarter pound of tea costs 10 cents, what is the cost of one half pound? 11. How many tomato plants will be needed for 9 rows if 10 plants are needed for each row ? 12. How many nickels make 20 cents ? 13. A girl has in her bank a quarter, a dime, and a nickel. How many cents are there in her bank ? 14. How many pints of milk will make 12 half- pint glasses of milk ? REVIEWS 57 Written Problems 1. If I pay 15 cents for a quarter of a pound of tea, what is the price of a pound ? 2. There are 144 pens in a gross. How many pens are there in a quarter of a gross ? 3. How many gallons are there in 100 quarts ? 4. At 18 cents per quart, what will a gallon of paint cost ? 5. Thomas has 36 cents ; Martha has 24 cents more than Thomas. How much money has. Martha ? 6. A farmer has 60 sheep. How many will he have if he sells 36 ? 7. How many sheep at $4 each can be bought for $ 64 ? 8. What is the cost of 4 yards of dress goods at 16 cents a yard ? 9. There are 96 trees in 4 equal rows. How many are there in each row ? 10. If one quarter yard of silk costs 36 cents, what is the cost of one half yard ? 11. How many cabbage plants are required for 4 rows if there are 23 plants in each row ? 12. How many nickels make 70 cents ? 13. How much money are a half dollar, a quarter, a dime, a nickel, and 6 cents ? 14. How many pints of lemonade will make 36 half-pint glasses of lemonade ? 58 SECTION TWO Time 60 seconds (sec.) 1 minute (min.) 60 minutes 1 hour (hr.) 24 hours 1 day (da.) 7 days 1 week (wk.) 1. How many minutes in ^ hour? 2. The minute hand takes 5 minutes to go from XII to I. How many minutes does it take to go from XII to VI ? 3. "When the minute hand is at I, it is 5 minutes after the hour. How many minutes is it after the hour when the minute hand points to V ? To II ? To III ? 4. When the minute hand is at VII it is 25 minutes to the next hour. How many minutes to the next hour is it when the minute hand points to IX ? To XI? To VIII? ToX? 5. Where is the minute hand at 10 minutes after 3 ? At 20 minutes to 5 ? 6. What part of an hour space does the hour hand go in 30 minutes? In 15 minutes? In 20 minutes? In 10 minutes ? Roman Numbers 13 14 15 16 17 18 19 20 XIII XIV XV XVI XVII XVIII XIX XX NUMBERS OF FOUR FIGURES 59 Notation and Numeration 1. One thousand is written 1,000. Two thousand is written 2,000. 2. Read: 3,000; 5,000; 7,000; 9,000. 3. Write in figures : four thousand ; six thousand ; eight thousand. 4. One thousand one is written 1,001. Two thou- sand six is written 2,006. Three thousand twenty is written 3,020. Six thousand five hundred thirty-six is written 6,536. 5. Write in figures : four thousand three hundred seventy-four ; six thousand seven hundred ; eight thousand ninety ; one thousand sixty-two ; seven thousand eight; two thousand five; nine thousand five hundred ninety ; one thousand three ; five thou- sand seventy-five ; three thousand six hundred seventy- three. 6. Read the following numbers : A B C D 1,834 9,703 8,046 3,000 4,039 5,000 3,604 4,500 607 750 707 2,908 6,300 2,030 98 5,660 8,270 572 2,345 789 509 1,111 670 1,050 1,062 7,008 2,010 9,590 7. Find the sum of each column. 60 SECTION TWO 8. Write in figures : Six hundred ninety-nine. Eight hundred five. Three thousand seven hundred thirty-eight. Nine thousand eight hundred four. Two thousand seven hundred twenty-six. Seven thousand. 9. Add the foregoing. 10. Write the following, and add : Three thousand sixty-four. Seven hundred nineteen. Eighty-two. Five hundred sixty. Two thousand three hundred five. Four hundred. In a number of four figures the right-hand fig- ure is called the ones' figure, the next is called the tens' figure, the next is called the hundreds' figure, and the left-hand figure is called the thousands' figure. A comma is generally used to separate the thou- sands' figure from the hundreds' figure, except in numbers representing dates, such as Jan. 1, 1914. Numbers of four figures are frequently read as hun- dreds : 1,800 being read as eighteen hundred; 1,912, as nineteen hundred twelve. ROMAN NUMBERS 61 Roman Numbers 20 is written XX. 60 is written LX. 30 is written XXX. 70 is written LXX. 40 is written XL. 80 is written LXXX. 50 is written L. 90 is written XC 100 is written C. A number between 20 and 29, 30 and 39, 40 and 49, etc., is expressed by placing the letter or letters representing the ones after the letter or letters repre- senting the tens. 28 is written XXVIII. 34 is written XXXIV. 49 is written XLIX. 63 is written LXIII. 57 is written LVII. 72 is written LXXII. 96 is written XCVI. 85 is written LXXXV. Express in Roman numbers : ABODE 22 66 82 49 18 33 77 75 31 27 44 88 64 26 32 55 99 59 81 46 Read : A B c D XXIII XLVIII LXVI XLV XXIV LXVII XXV LXIX LVII XXXVI XLII XXXV XLIX XCIX XXIX XLVII NOTE. Owing to their limited application, but little attention should be given to Roman numbers. 62 SECTION TWO Dollars and Cents Twenty-three dollars five cents is written $ 23.05. A period, called a decimal point, is placed before the cents. Read the following : A B c D $4.76 $1.25 $43.55 $805.07 $9.83 $3.75 $27.09 $914.30 Write : 1. Five dollars and seventy-nine cents. 2. Eight dollars and sixty-four cents. 3. Forty-two dollars and eight cents. 4. Sixty dollars and sixty cents. Seventy-five cents is written $ .75. Sometimes a cipher is written before the decimal point, seventy-five cents being written $ 0.75 ; and two ciphers may be written after the decimal point when there are no cents, twenty-seven dollars being written $ 27.00. Read the following: ABODE $.84 $0.16 $.47 $0.05 $.15 $.09 $0.25 $.63 $0.13 $.94 Write: 1. Four cents. Twenty cents. Twenty-six cents. 2. Ten cents. Fifty cents. Eighty-five cents. 3. Three cents. Sixty cents. Ninety-nine cents. 4. Seven cents. Eleven cents. Seventy-three cents. FEDERAL MONEY 63 Sight Exercises Give a. b. c. d. e. f- ff- h. i. j- answers : A $1.50 + .25 B $0.38 + 0.40 $0.30 + 0.30 c $.60 + .35 D $3.47 + 1.20 $2.40 + 2.40 $.50 + .80 $5.60 + 1.15 $2.40 x2 $0.30 x3 $.20 x4 $1.10 x5 $1.50 -.25 $1.00 -.05 $0.40 + .38 $.75 -.45 $3.47 -1.20 $2.00 -.10 $.50 -.25 $4.00 -.30 $2.40 + .60 $3.50 -3.25 $.60 + .50 $4.00 + 3.70 $4.80 $0.60 $.80 $4.40 2 $1.50 x2 3 $0.25 x4 40 $.50 x3 4 $1.12 x4 $3.00 2 $2.50 + .60 $1.00 $.66 $1.60 5 $2.50 -.60 22 $.60 + .45 4 $1.60 -.45 64 SECTION TWO Addition Written Exercises i. A grocer paid out the following amounts : $2.84, $0.47, $0.09, $16.56, $3.25. How much did he pay? $2.84 Arrange the numbers so that the deci- ._ mal points will be in the same column, OQ then add. Place a decimal point in the Q ( -I f\ 1st column, 9, (ignoring 0), 15, 20 (write 0). 2d column, (carrying 2), 9, 18, 26, 32, 41 (write 1). 3d column, (carrying 4 and ignoring 0), 12, 19, 25 (ignoring 0) (write 5). 4th column, 7, 8, 9 (write 9). ADDITION 65 3. Add and test : a. $49.62 b. $.12.13 c. $36.81 d. $4.23 6.38 8.75 .66 51.75 4.79 24.30 3.75 9.67 .24 3.98 .10 .82 .07 .70 20.99 9.56 e. $40.09 / $0.98 g. $15.18 h. $ 1.43 27.59 62.44 7.29 77.48 1.35 4.85 38.54 4.69 6.47 9.00 .85 5.40 .36 .84 8.00 .41 i. $9.10 j. $48.57 k. $8.88 I $56.89 7.45 4.84 17.77 9.99 45.85 6.95 9.00 .48 27.58 5.63 6.38 7.21 6.24 17.77 37.36 20.75 m. $ 21.27 + $ 7.54 + $ 5.68 + $ 17.85 + $ 0.97. n. $ 26.79 + $ 2.15 + $ 7.17 + $ 0.98 + $ 12.49. 4. Find the sum of three dollars nineteen cents, forty-three dollars eight cents, seven dollars, five dollars, five dollars seventy-three cents, and eight cents. 5. Add nine dollars ninety-nine cents, forty-eight dollars sixty-seven cents, eighty-eight cents, seventy- five cents, and nineteen dollars nine cents. 66 SECTION TWO Subtraction Written Exercises To test the result, cover the minuend and $ 40.33 then add the remainder to the subtrahend, _957Q adding upwards : 4 + 9 ; (carrying 1) 6 + 7 ; (carrying 1) 5 + 5; (carrying 1) 2 + 2. This $ 14.54 sum should be the same as the minuend covered. 1. Find remainders : A B a. $18.01 -$9.43 $27.44- $19.39 6. $48.63 -$8.77 $82.19- $78.90 c. $90.50 -$9.86 $35.04 -$16.66 d. $35.52 -$5.83 $63.15- $35.87 e. $53.60 -$8.75 $70.06- $42.19 2. From ninety dollars take ninety cents. 3. Find the difference between sixty dollars sixteen cents and forty-three dollars three cents. 4. From the sum of $3.50 and $2.40 take the difference between $7.50 and $6.20. 5. From $10 take the sum of $3.30, $2.60, and $1.50. 6. How many times can 15 cents be subtracted from 75 cents? 7. How many times must 25 cents be increased by 15 cents to make a total of 70 cents ? REVIEWS 67 Sight Drills 1. Add: A B C D E F G a. 50 40 20 40 32 20 14 + 30 +50 +70 +20 +30 +42 +80 I. 60 61 43 54 63 47 58 + 24 +24 + 22 + 25 + 36 + 20 + 31 2. Subtract . a. 80 80 62 62 90 94 96 -50 - -30 -40 -22 -70 -60 -36 b. 84 95 84 95 76 67 59 -53 - -72 -31 -23 -30 -44 -29 3. Multiply . a. 40 30 20 41 32 21 11 x2 x3 x4 x2 x3 x4 x9 5. 12 13 14 22 23 24 33 x4 x3 x2 x4 x3 x2 x3 c. 20 30 40 50 60 70 80 x9 x8 x7 x6 x5 x4 x3 4. Divide : a. 80 80 60 60 60 60 60 2 40 20 30 2 3 10 b. 90 90 84 84 84 96 88 30 3 42 21 4 3 4 68 SECTION TWO Oral Problems i. How many nickels are there in a half dollar ? SOLUTION The number of nickels in 50 cents is the number of times 5 cents is contained in 50 cents, which is 10. Ans. 10 nickels. 2. How many quarts in 5 gallons? SOLUTIOX Since there are 4 quarts in 1 gallon, in 5 gallons there are 5 times 4 quarts, or 20 quarts. Ans. 20 quarts. 3. Change 4 feet to inches. 4. Change 32 pecks to bushels. 5. At 5 cents per pound, how many pounds of sugar can be bought for 45 cents ? 6. If a train goes 45 miles in one hour, and 40 miles the next hour, how many miles does it go in two hours ? 7. A girl is taking a 90-mile trip in an automobile. How many miles has she yet to go after having traveled 60 miles ? 8. How many pounds of 5-cent sugar can be bought for a quarter dollar ? A half dollar ? REVIEWS 69 Written Problems i. How many nickels are there in 85 cents? SOLUTION To find the number of 5-cent 5 cents) 85 cents pieces in 85 cents, divide 85 cents AnsTvi (nickels) by 5 cents, which, gives 17. The answer is 17 nickels. 2. How many quarts in 23 gallons? SOLUTION Since 1 gallon is equal to 4 quarts, 23 gal- 4 qt. Ions are 23 times 4 quarts, or 92 quarts. x 03 NOTE. Although 23 is written as the A n n f multiplier, the product is obtained by mul- tiplying the larger number, 23, by the smaller number, 4. 3. Change 7 feet to inches. 4. Change 96 pecks to bushels. 5. At 5 cents per pound, how many pounds can I buy for 95 cents? 6. If a boat sails 47 miles in the morning, and 45 miles in the afternoon, how far has it sailed ? 7. A boy is taking a 90-mile trip. How many miles has he to go after he has gone 47 miles ? a. How many pounds of 5-cent sugar can be bought for a dollar and a quarter ? 70 SECTION TWO Multiplication Table 1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 3 6 9 12 15 18 21 4 8 12 16 20 24 5 10 15 20 25 6 12 18 24 7 14 21 8 16 9 \ Copy the foregoing table, and fill the vacant spaces. Write 2 times 9 in the second row and 9 times 2 in the second column, 3 times 8 and 3 times 9 in the third row, 8 times 3 and 9 times 3 in the third column, etc. Oral Exercises 1. Count by 6's to 60. 3. Count by 8's to 80. 2. Count by 7's to 70. 4. Count by 9's to 90. 5. How many are (a) six 6's ? (6) Seven 6's ? (c) Eight 6's ? (d) Nine 6's ? 6. How many are (a) Seven 7's ? (6) Eight 7's ? (c) Nine 7's? 7. How many are (a) eight 8's ? (&) Nine 8's ? 8. How many are nine 9's ? REVIEWS 71 Oral Problems 1. A farmer has 65 tons of hay. How many tons will he have after selling 43 tons ? 2. I paid 84 cents for 4 pounds of coffee. What was the price per pound ? 3. How many cars are there in two freight trains, one having 32 cars and the other having 30 cars ? 4. What is the cost of 3 yards of ribbon at 32 cents per yard ? 5.' After giving away one half of his firecrackers, Tom has 42 left. How many did he have at first ? 6. When tea is 60 cents a pound, what part of a pound can I buy for 30 cents ? 7. What part of a girl's dollar is left after she spends 25 cents ? 8. How many trees are there in 4 rows each con- taining 22 trees ? 9. How many pints are there in 44 quarts ? How many gallons? 10. How many feet in 48 inches ? 11. A boy gave his sister 12 cents, which was one half of his money. How many cents had he left ? 12. A man is 37 years old. In how many years will he be 40 ? is. How many days are there in November and December ? 14. How many days are there in 6 weeks ? 72 SECTION TWO Multiplying by 6 Drill Exercises Give products : 6x6 7x6 8x6 9x6 6x7 7x5 8x5 9x5 6x8 7x4 8x4 9x4 6x9 7x3 8x3 9x3 Written Exercises i. Multiply A B c D E a. 666 777 888 999 666 x6 x6 x6 x6 x7 I. 666 666 678 456 567 x8 x9 x6 x6 x6 c. 345 789 456 345 234 x6 x6 x8 x8 x9 d. 345 264 360 299 647 x7 x6 x6 x6 x6 2. Multiply by 6 : A B c D E a. 344 1 ,448 1,454 1,583 1,628 &. 676 1,056 1,536 1,295 1,468 c. 458 1 ,374 1,665 1,378 1,357 d. 597 1 ,570 1,498 1,467 1,555 DIVISION 73 Dividing by 6 Drill Exercises Give quotients : ABODE a. 6)36 6)42 8)32 7)42 6)30 b. 8)40 7)35 9)54 8)48- 7)21 c. 9)36 6)48 8)24 7)14 8)16 d. 6)54 7)28 9)18 9)27 9)45 e. 5)45 6)24 5)30 4)24 5)20 /. 4)32 5)35 6)12 3)24 4)32 Written Exercises i. If one tent shelters 6 soldiers, how many tents will be required to shelter 5,436 soldiers ? When the first quotient figure is written in its proper place, each remaining figure in the WO4ou tents dividend must have a figure in the quotient. Alls. 906 tents DIVISION TEST 6)6,120 6)4,806 6)6,024 Quotient 1,020 Ans. 801 Ans. 1,004 Ans. Divisor x 6 x 6 x 6 Dividend 6,120 4,806 6,024 To test the result, cover the dividend with a piece of paper and write on the latter the product of the quotient by the divisor without rewriting the latter. 74 SECTION TWO Written Exercises Divide. Test answers: ABC a. 6)1,056 6)1,572 6)2,286 D 6)2,160 I. 6)2,670 6)3,966 6)3,588 6)2,370 c. 6)4,068 6)4,416 6)5,244 6)5,658 d. 6)6,816 6)6,432 6)6,468 6)7,344 e. 6)7,740 6)8,670 6)8,832 6)8,634 / 6)9,360 6)9,546 6)8,910 6)5,040 g. 6)2,358 6)3,456 6)4,560 6)6,780 h 6)7,890 6)8,910 6)9,870 6)8,766 i. 6)3,996 7)1,638 8)2,600 9)2,376 j. 6)4,662 7)2,345 8)1,800 9)4,995 k. 6)5,328 7)2,268 8)1,888 9)3,996 DIVISION 75 Remainders in Division Sight Exercises 1. If I desire to divide 7 apples between 2 boys, how many whole apples can I give each ? How can I divide the remaining apple between them? What is each part called ? 2. What is the quotient of 7 -* 2 ? When the dividend does not contain the divisor an exact number of times, write the remainder, then draw a line, and underneath write the divisor. 2)9 3)20 4)23 Ans. 4^- Am. 6f Ans. 5f ARC a. =? J^-=? *.= ! I. 15-^2=? 19-3 = ? 21-4=? 3. Give answers : A B C D E F a. 3}22 8)41 6)37 9)25 5)51 6)51 &. 2)25 3)38 4)51 8)25 7)51 6)25 n ' {. i . i 7 ' A B c D a. of 9 JoflO iof 13 Jof 31 b. Jofll iof 13 |of 15 |of 52 c. Jof 25 of 19 jof 37 Jof 52 d. Jof 22 Jof 25 J of 45 ^of 38 76 SECTION TWO Oral Problems 1. How many hours are there in a day and a half ? 2. A farmer had 80 acres of land. He sold 30 acres. How many acres were left ? 3. If one man takes 60 days to do a piece of work, how long would it take 3 men to do it ? 4. What is the cost of 40 newspapers at ^ cent each? 5. There are 36 pupils in one room and 20 in an- other. How many are there in both rooms ? 6. How many cars in 3 trains, each containing 20 cars ? 7. How many weeks are there in 35 days ? 8. A girl is 8 years old and her mother is 32 years old. What is the difference in their ages ? 9. How many books are on two shelves one con- taining 16 books and. the other containing 12 books ? 10. How many ounces in a pound and a quarter ? 11. How many minutes in 3 hours ? 12. How many horses can be shod with 24 shoes ? is. Three feet make 1 yard. If a rail is 10 yards long, how many feet long is it ? 14. Two boys together weigh 150 pounds. If one weighs 80 pounds, what is the weight of the other ? is. How many months are there in three years ? is. How many inches in 2 feet ? REVIEWS 77 Written Problems 1. How many hours are there in 4 days ? 2. A farmer has 80 acres of land. How much land would he have if he sold 36 acres ? 3. If one man takes 65 days to do a piece of work, how long would it take 5 men to do it ? 4. Find the cost of 96 newspapers at ^ cent each. 5. Mr. Jones has 36 pupils in his class and Mr. Brown has 38 pupils. How many in both classes ? 6. How many cars in 4 trains each containing 24 cars? 7. How many weeks are there in 98 days ? s. A boy is 18 years old and his father is 46 years old. What is the difference in their ages ? 9. There are 26 books on one shelf and 24 on another. How many books are there on both shelves ? 10. How many ounces are there in 5 pounds ? 11. How many minutes are there in 9 hours ? 12. How many horses can be shod with 96 shoes ? 13. If a fence is 16 yards long, how many feet long is it ? 14. Two boys weigh together 150 pounds. If one weighs 83 pounds, what is the weight of the other ? is. How many mq|^hs are there in 8 years ? 16. How many inchl? in 5 feet ? 78 SECTION TWO a. 7x7 8x5 9x5 b. 7x8 8x6 9x6 c. 7x9 8x7 9x7 Written Exercises i. Multiply A B C a. 777 888 999 x7 x7 x7 b. 777 567 765 x9 x7 x7 c. 876 789 987 x7 *1 x7 d. 878 567 765 x7 x8 x8 e. 657 576 576 x9 x8 x9 2. Multiply by 7: A B C a. 434 1,234 1,265 b. 766 1,067 1,347 c. 854 1,285 1,409 Multiplying by 7 Drill Exercises BCD 8x4 7x5 7x6 D 111 x_8 678 x7 897 x7 657 x8 756 x8 D 1,394 1,406 1,389 MULTIPLICATION AND DIVISION Dividing by 7 Written Exercises i. Divide and test : 79 A 5 c D - a. 7)1,057 7)1,554 7)2,261 7)2,394 b. 7)2,660 7)2,156 7)3,493 7)2,835 c. 7)4,361 7)8,099 7)5,075 7)3,059 d. 7)6,048 7)4,263 7)2,569 7)9,982' e. 7)3,920 7)7,329 7)9,156 7)7,049 / 7)2,583 7)7,098 7)9,212 7)4,760 2. Divide by 7. Test: a. 9,842 8,953 9,569 8,638 6. 3,038 9,863 7,469 9,723 c. 8,932 6,699 5,978 8,995 Multiplication and Division i. Multiply by 7: A B C D a. 957 1,380 1,368 1,278 6. 826 1,279 1,276 1,367 2. Divide by 7. Test: a. 9,576 8,946 9,660 9,758 6. 5,782 5,362 8,855 9,429 80 SECTION TWO Oral Problems 1. A man's farm is divided into 4 fields, each containing 8 acres. How many acres are there in the farm ? 2. If 8 panes of glass are required for each window, how many windows will require 48 panes ? 3. If a man works 8 hours a day, how many hours does he work in 10 days ? 4. In an orchard of 80 trees all but 20 are peach trees. How many peach trees are there in the orchard ? s. How many players are needed in 2 games of football if each game requires 22 players ? 6. A boy requires 12 days to do one fourth of a piece of work. How many days does he require to do the whole work ? 1915 JANUARY 1915 Sun. Mon Tue. Wed. Thu. K-i. Sat. 1 2. 3 4 5 6 7 8 9 10 11 12 IS 14 15 16 17 18 19 20 21 22 23 2 >lfi 25 26 27 28 29 30 1915 FEBRUARY 1915 Sun. Mon. Tue. Wed. Thu. Fri. Sat. 1 2 3 4 5 6 7 8 9 10 11 12 15 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 7. How many days are there in January and February, 1915? 8. How many yards of wire are needed to make a fence 20 yards long, if the fence is 4 wires high ? 9. How many chestnuts in 10 burs containing 2 nuts each and 10 containing 3 nuts each ? REVIEWS 81 Written Problems 1. A man has a farm consisting of 7 fields, each containing 14 acres. How many acres are there in the farm? 2. If 8 panes of glass are required for each win- dow, how many windows will require 96 panes ? 3. If a farmer works 13 hours a day, how many hours does he work in 7 days ? 4. In an orchard of 80 trees all but 28 are apple trees. How many are apple trees ? 5. How many players are there in 4 games of baseball, if each game has 18 players ? 6. A man requires 28 hours to do one third of a piece of work. How many hours does he require to do the whole work ? 1916 FEBRUARY 1916 Sun. Mon. Tue. Wed. Thu. Fri. Sat. 1 2 5 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 1916 MARCH 1916 Sun. Mon. Tue. Wed. Thur Fri. Sat. 1 2 3 4 5 6 7 8 9 10 11 12 15 14 15 16 17 18 19 20 21 22 23 24 25 2b 27 28 29 30 31 7. How many days are there in February and March, 1916? 8. How many yards of wire are required to make a fence 27 yards long if the fence is 3 wires high ? 9. How many chestnuts in 26 burs containing 2 chestnuts each and 13 containing 3 chestnuts each ? } SECTION TWO Multiplying by 8. By 9 Written Exercises i. Find products : ABC a. 888x8 999x8 888x9 987x8 I. 887x8 787x9 887x9 878x9 c. 678x8 789x8 897x8 678x9 d. 788x8 788x9 878x8 787x9 e. 999x9 989x9 899x9 998x9 / 567x8 467x9 576x8 576x9 2. Multiply by 8: A B c D a. 1,056 1,167 1,234 963 I. 1,067 1,178 1,245 954 c. 1,088 1,189 1,225 947 d. 1,079 1,198 1,229 986 e. 1,097 1,187 1,239 927 / 1,099 1,188 1,236 976 3. Multiply by 9: A B C D a. 1,023 1,078 1,099 929 6. 1,034 1,089 1,088 938 c. 1,045 1,093 1,077 947 rf. 1,056 1,095 1,066 956 e. 1,067 1,097 1,055 965 /. 1,079 1,068 1,044 987 )991|1 off over the 70. When the division is performed, write before the 1, the 4 tens remaining, making the remainder 41. 141ft Ans. . 40)2,316 d. 50)3,742 g. 30)6,418 j. 20)5,437 6. 70)5,864 e. 60)3,915 k. 50)7,218 k. 40)6,351 c. 80)6,873 / 90)7,777 i. 70)8,326 I 60)4,987 MEASUREMENTS 93 Areas of Rectangles Oral Exercises 1. How many postage stamps in a sheet 7 stamps long, 4 stamps wide? 2. How many square yards are there in a floor .10 yards long, 8 yards wide ? 3. How many square feet in the surface of a board 16 feet long, -| foot wide ? 4. How many boards \ foot wide will cover a floor 12 feet wide? 5. Find the area of a fence 20 feet long, 6 feet high. 6. How many square feet of cloth will be required to cover the top of a table 4 feet long, 2^ feet wide ? 7. The top of a paving brick is 8 inches long, 4 inches wide. How many square inches will it cover ? 8. The side of a paving brick is 8 inches long, 2J inches high. How many square inches does it contain ? 9. How many square inches are there in the end of a brick whose dimensions are 4 inches by 2J inches ? 10. How many square yards are there in a play- ground 24 yards long, 20 yards wide ? 11. How many square yards of matting will cover a hall 10 yards long, 2 \ yards wide ? 94 SECTION TWO Fractional Parts To divide a thing into halves, it is divided into two equal parts ; to divide a thing into quarters, it is divided into four equal parts; to divide a thing into thirds, it is divided into three equal parts. i. Draw a rectangle, 4 inches long and 3 inches wide. How many square inches does it contain ? 2. How many square inches are there in each horizontal row of squares ? In each vertical row? 3. What part of the whole rectangle is each verti- cal row of squares ? What part are two vertical rows ? Three vertical rows ? 4. What part of the whole rectangle is one horizon- tal row of squares ? Two horizontal rows ? Three fourths is written Two thirds is written . 5. One third of 12 squares is 4 squares ; how many squares are 2 thirds of 12 squares? One fourth of 12 squares is 3 squares; how many squares are 3 fourths of 12 squares? FRACTIONAL PARTS 95 Oral Exercises 1. Give answers : A B C D a. I of 12 J- of 15 f of 15 J of 16 b. | of 16 J of 15 f of 15 f of 15 c. f of 15 of 18 i of 18 | of 18 d. $ of 16 f of 16 J- of 20 | of 20 2. How many inches (a) in \ foot ? (6) J foot ? (c)lfoot? (d)Jfoot? (e) | foot? 3. How many ounces (a) in J pound? (&) J- pound ? (c) j- pound ? (d) -f pound ? (e) % pound ? 4. How many quarts (a) in \ peck ? (6) \ peck ? (c) J peck ? 5. What part of a foot is (a) 6 inches? (6) 4 inches ? (c) 3 inches ? (d) 9 inches ? (e) 8 inches ? e. What part of a pound is (a) 8 ounces ? (6) 4 ounces? (c) 12 ounces? 7. What part of a peck is (a) 2 quarts ? (b) 4 quarts? (c) 6 quarts? 8. If a bushel of oats weighs 32 pounds, how many pounds does (a) a peck weigh ? (b) A quart ? 9. If a gallon of water weighs 8 pounds, what does (a) a pint weigh ? (b) A quart ? 10. How many pounds in J bushel of potatoes if a bushel weighs 60 pounds ? 11. If 2 pecks of wheat weigh 30 pounds, what is the weight of a bushel ? 96 SECTION TWO Denominate Numbers Oral Exercises 1. How many hours (a) in a day and a half? (6) In 1 day 12 hours? 2. How many pints in a gallon ? 3. How many quarts (a) in 3 gallons? (6) In 3 gallons 2 quarts ? 4. How many quarts (a) in 2 pecks ? (6) In 3 pecks ? (c) In 4 pecks ? (d) In a bushel ? 5. How many eggs (a) in 3 dozen? (6) In 4 dozen ? 6. How many days (a) in 7 weeks 1 day ? (6) In 1 1 weeks ? 7. How many weeks (d) in February, 1915 ? (b) How many weeks in January, and how many days over? 8. Change 4 feet 2 inches to inches. 9. Change 1 pound 6 ounces to ounces. 10. How many inches (a) in 2 feet? (6) In 3 feet ? (c) In a yard ? 11. How many minutes (a) in J hour ? (6) In f- hour ? 12. What part of a day is (a) 6 hours? (6) 12 hours ? (c) 18 hours ? is. Change 160 ounces to pounds. 14. Change 169 ounces to pounds and ounces. 15. Change 32 feet to yards and feet. DENOMINATE NUMBERS 97 Written Exercises 1. How many hours (a) in 2J days? (6) In 2 days 12 hours? 2. (a) How many quarts in 13 gallons? (6) How many pints ? 3. How many quarts (a) in 16 gallons? (6) In 16 gallons 2 quarts? 4. How many quarts (a) in 12 pecks? (b) In 3 bushels? 5. How many eggs (a) in 8 dozen? (b) In 7 dozen and a half ? 6. How many days (a) in 14 weeks ? (6) In 14 weeks 2 days ? 7. (a) How many weeks in a year of 365 days, and how many days over ? (b) How many days over in leap year, 366 days ? 8. Change 8 feet 3 inches to inches. 9. Change 3 pounds 2 ounces to ounces. 10. How many inches (a) in 2 yards? (b) In ^ yard? (c) In 2J yards? 11. How many minutes (a) in ^ hour? (6) In f hour? (c) In f hour? (d) In hour? 12. What part of an hour is (a) 12 minutes? (6) 24 minutes ? (c) 36 minutes ? (d) 48 minutes ? 13. How many bushels in 64 pecks ? 14. (a) How many gallons in 64 quarts ? (b) How many pints? 98 SECTIOX TWO Rectangular Solids Oral Exercises A cube each of whose edges measures 1 inch con- tains 1 cubic inch. One Cubic Inch i. How many faces has a cube ? How many edges ? What is the length of each edge of a one-inch cube ? How long is each face? How wide? 2. How many one-inch cubes will it take to make a cube 3 inches long, 3 inches wide, and 3 inches high? How many one-inch cubes in the bottom layer? In the top layer? 3. How many one-inch cubes will it take to cover the bottom of a box 4 inches long, 3 inches wide ? If the box is 2 inches high, how many one-inch cubes will it take to fill the box ? 4. How many cubes of loaf sugar will fill a box that will hold 6 layers, each layer containing 10 rows of 8 cubes each ? 5. A pile of bricks is 10 bricks long. It contains 6 rows to a layer, and is 4 layers high. How many bricks are there in the pile ? SECTION III Reading and Writing Numbers Read the following : Oral Exercises A B c D E a. 10,000 50,590 10,008 23,406 35,600 6. 11,000 61,725 11,017 44,444 52,983 c. 20,000 70,354 22,034 53,007 72,500 d. 31,000 83,976 36,090 85,021 93,811 e. 54,000 92,417 40,003 11,642 64,300 Written Exercises Write in figures : 1. Sixty-one thousand. 2. Fifty-three thousand seven hundred. 3. Forty thousand sixty. 4. Thirty-two thousand nine. 5. Twelve thousand three hundred forty-five. 6. Seventy thousand seven. 7. Eighty-eight thousand eight hundred eighty- eight. 8. Ninety-one thousand seven hundred twenty- seven. 90 100 SECTION THREE Multipliers of Two Figures i. Each of 467 pieces of copper wire weighs 21 ounces. What is their total weight ? PROCESS Multiply by 1, placing the right-hand 467 OZ. figure of the product under the 1. Multiply x 2 1 by 2 tens, placing the right-hand figure of the product under the 2. Combine the par- tial products. Am. 9,807 oz. Written Exercises 2. Find products : Test by performing the multiplication a second time without referring to the first operation until the product is again obtained. v* . A B C D a. 444 x 22 184 x 53 132 x 75 121 x82 6. 468 x 21 186 x 52 134 X 74 122 x81 c. 222 x 33 195 X 51 136 X 73 101 x99 d. 308 x 32 143 X 66 138 X 72 102 x98 e. 316x 31 151 X 65 139 X 71 103 x97 f. 225 x 44 156 X 64 111 X 88 104 x96 g- 231 x 43 158 X 63 112 X 87 105 x95 h. 222 x 42 161 X 62 113 X 86 106 x94 i. 237 x 41 163 X 61 116 X 85 107 x93 j- 123 x 55 128 X 77 118 X 84 108 x92 k. 172 x 54 131 X 76 119 X 83 109 x91 REVIEWS Sight Drills 101 1 . Multiply : ^ B c D E F G H a. 90 80 70 60 50 40 30 20 x2 x3 x4 x5 x6 x7 x8 x9 f &. 84 93 72 61 41 51 31 21 x2 x3 x4 x5 x6 x7 x8 x9 c. 94 83 62 71 31 21 41 51 x2 x3 x4 x5 x6 x7 x8 x9 2. , Divide : 180 240 360 350 360 280 320 20 30 40 50 60 70 80 e. 126 279 246 357 305 497 324 21 31 41 51 61 71 81 / 630 819 243 568 183 255 164 90 91 81 71 61 51 41 3. Give quotients and remainders : 0- 190 248 186 248 369 357 460 21 41 61 81 91 71 51 A. 250 250 460 330 360 150 290 31 61 91 81 51 21 71 i 100 200 300 400 400 400 400 99 99 99 99 98 97 96 102 SECTION THREE Sight Problems 1. At 32 to a class, how many pupils are there in 4 classes ? 2. At the rate of 21 miles per hour, how long would a trolley car require to go 147 miles ? 3. Find the cost of 6 acres of land at $61 per acre. 4. How many suits of clothes at $21 each can be bought for $168? 5. What is the total weight of 6 tubs of butter if each weighs 51 pounds? 6. How many barrels will hold 369 gallons of oil if each holds 41 gallons ? 7. How many yards around the 4 sides of a square each side of which measures 92 yards ? 8. If a conductor receives $ 4.05 from 81 passen- gers, what does each pay ? 9. At 51 nails to the pound, how many pounds will 459 nails weigh ? 10. How many nails in 8 pounds if there are 61 nails to the pound ? 11. A piece of rope 126 yards long is divided into 42 pieces. How many yards in each piece ? 12. How many quarts of milk are there in 72 gallon-cans ? 13. At 32 pounds to the bushel, how many bushels of oats will weigh 128 pounds ? REVIEWS 103 14. How many 21 -acre fields are there in a farm of 189 acres ? is. How far does a train travel in. 4 hours at the rate of 42 miles per hour ? 16. At 41 bushels to the acre, how many acres will produce 328 bushels of corn ? 17. If a family uses 3 quarts of milk a day, how many quarts are used in July ? is. How many hours would a train require to go 128 miles at the rate of 32 miles per hour? 19. If 5 men require 41 hours to do a piece of work, how long would it take 1 man to do it ? 20. What part of 276 is 92 ? 21. Find the cost of 72 2-cent postage stamps. 22. At 99 cents each, how many baseballs can be bought for $ 10, and how much change will there be ? 23. If a hogshead contains 63 gallons, how many hogsheads are there in 189 gallons? 24. If 1 fourth of a number is 40, what is 3 fourths of the number? 120 25. If 3 fourths of a number is 120, what is 1 fourth of the number ? What is the number ? 26. What part of 120 is 40 ? What part of 120 is 30 ? What part of 120 is 90 ? 104 SECTION THREE Long Division i. A pile of copper wire pieces weighs 9,807 ounces. If each piece weighs 21 ounces, how many pieces in the pile ? PROCESS Ans". 467 (pieces) Take 98 as tlie first partial dividend and 21^9807 divide it by 21. Write 4, the first quotient ft . figure, above the 8. Multiply 21 by 4, and place the product 84 under 98. Subtract 84 from 98, and to the remainder 14 annex 0, the 126 next figure of the dividend, which makes 140, 147 the second partial dividend. Divide 140 by 21, and write the quotient 6 above the 0. Multiply 21 by 6 and place the product 126 under 140. Subtract 126 from 140, and to the remainder 14 annex 7, the remaining figure of the dividend, which makes 147 the next partial dividend. Divide 147 by 21 and write the quotient 7 above the 7. Multiply 21 by 7 and place the product 147 under 147. As this is the same as the last partial dividend, there is no remainder. The quotient is, therefore, 467. Write pieces in parenthesis. Ans. 286 Ans. 153 Ans. 137 Ans. 123 34)9724 53)8109 72)9864 81)9963 68 53 72 81 292 280 266 272 265 216 204 159 504 204 159 504 Write each figure in the quotient directly over the last figure of the partial dividend that produces it. LONG DIVISION 105 Testing Quotients Test Quotient 286 Divisor x 34 153 x53 137 x72 123 x81 1144 459 274 123 858 765 959 984 Dividend 9724 8109 9864 9963 2. Find quotients : A B c D a. 462-21 714-5-51 1,701-81 3,936 - 32 b. 483-21 867-5-51 2,592-81 8,946- -42 c. 672-21 612-5-51 3,483-81 9,828 - -42 d. 693-^-21 918-5-51 4,374-81 6,604 - -52 e. 882-*- 21 816-5-51 5,278-91 1,820- -52 / 651-5-31 6,832-5-61 1,092-91 6,552 - -52 9- 961-31 1,403-5-61 2,002-91 1,352- -52 h. 682-5-31 8,174-61 1,183-91 2,232 - -62 i. 992-*- 31 2,745-5-61 3,003-91 1,550- -62 j- 713-31 9,516-5-61 4,004-91 2,914-62 k. 943-5-41 7,881-71 7,612-22 1,944-72 I 533-5-41 1,562-71 3,476-5-22 5,184-72 m. 984-5-41 9,514-5-71 5,588-22 1,476-5-82 n. 902-5-41 1,775-5-71 4,352-32 6,806-5-82 0. 861-5-41 9,656-*- 71 7,488-5-32 6,900-5-92 P- 561-*-51 l,704--71 7,104-5-32 2,208-5-92 3- 612^51 9,372-5-71 4,224-5-32 4,416-5-92 o 106 SECTION THREE More than One Operation Sight Problems 1. John has 12 marbles; James has 5 more than John. How many marbles have both ? 2. If a girl saves 3 cents per day, how much less than a dollar does she save in October ? 3. If a boy saves 2 cents per day, how much more than a half dollar does he save in October ? 4. How many days in the first three months of 1916? (Leap year.) s. How many days in April, May, and June ? 6. From a 40-quart can of milk there have been sold 8 gallons. How many quarts remain ? 7. What number divided by 21 gives a quotient of 4 and a remainder of 6 ? 8. A man has 39 cows in three fields. In one field he has 12 cows; in another field he has 14 cows. How many cows in the third field ? 9. If a traveler's expenses are $3 per day, how much does he spend in three weeks ? 10. Mary has as much more than 25 cents as Jane has less than 25 cents. If Mary has 32 cents, how much has Jane ? 11. What must be added to 21 to obtain the dif- ference between 15 and 58 ? 12. If 8 pounds of sugar cost 48 cents, how many pounds can be bought for 60 cents ? REVIEWS 107 Written Problems 1. William has 25 marbles; Thomas has 16 more than William. How many marbles have both ? 2. If a girl saves 3 cents a day, how much more than 50 cents does she save in February, 1916 ? 3. If a boy saves 4 cents a day in February, 1915? how much less than $ 2 does he save ? 4. How many days in the first six months of 1914 ? 5. How many days in the last six months of 1915 ? 6. From a 41 -gallon barrel of oil there have been sold 48 quarts. How many gallons remain ? 7. What number divided by 21 gives a quotient of 3 and a remainder of 17 ? 8. A man has 60 cows in three fields. In one field he has 17 cows, in another he has 27 cows. How many has he in the third field ? 9. If a traveler's expenses are $ 4 per day, how much does he spend in 7 weeks ? 10. Sarah has as much more than 75 cents as Laura has less than 75 cents. If Laura has 58 cents, how much has Sarah ? 11. What must be added to 39 to obtain the differ- ence between 17 and 91 ? 12. If 3 pounds of coffee cost 72 cents, how many pounds can be bought for 96 cents ? 13. When a 16 pound bag of sugar is worth 96 cents, what is the cost of 13 pounds? 108 SECTION THREE Long Division i. A machine turns out 8,649 feet of wire in 83 minutes. What is the output per minute ? What is the production per minute when 9,499 feet are de- livered in 73 minutes ? PROCESS Am. 104J ft. Place the first quotient Ans. 130 T 9 ft. 83)8649 ft. figure over the right-hand 73)9499 f t . 00 figure of the first partial nrt oo , j ,. (6 product, and a quotient 040 O1 Q figure over each of the remaining figures of the 17 dividend, writing a cipher 9 when a partial dividend does not contain the divisor. TEST 104 Quotient Test the result 130 Quotient x 83 Divisor bv addin s the re ' x 73 Divisor Q mainder to the prod- . Qa > I _ p ,1 , OiJU uct of the quotient 832 by the divisor. The 910 8632 sum will equal the 9490 + 17 Remainder dividend if the work +9 Remainder Dividend is correct 9 499 Dividend 2. Divide : a. 7,656 + 25 c. 7,000 + 66 e. 2,000-*- 19 6. 7,358 + 35 d. 9,400 + 77 /. 7,009 + 28 REVIEWS 109 Addition and Subtraction Sight Drills For daily drills a chart similar to the following may be prepared : ABODE F G H a. b. G. d. e. / 9- h. 2 13 25 39 48 57 64 76 3 5 15 29 38 47 54 66 72 19 28 37 44 56 62 73 9 18 27 34 46 52 63 75 8 17 24 36 42 53 65 79 7 4 14 26 32 43 55 69 78 16 22 33 45 59 68 77 6 12 23 35 49 58 67 74 1. Add to each of the numbers in the table : ABCDEFGH 50 41 32 23 14 25 36 47 60 51 42 33 24 35 46 57 To the order, " Add 50 by lines," successive pupils answer rapidly : 52, 63, 75, etc. ; 53, 65, 79, etc. To the order, " Add 47 by columns," the replies are : 49, 50, 52, etc. ; 60, 62, 66, etc. 2. Subtract each of the numbers in the table from 99 91 95 89 96 87 98 85 92 97 90 94 88 93 86 84 The order here is : " Take from 99 by lines " or " Take from 91 by columns." 110 SECTION THREE Oral Exercises NOTE. Combine with the entire first number the tens of the second number, then the ones. Thus 86 + 53 = 86 + 50(136) + 3 = 139. Also 86 - 53 = 86 - 50(36) - 3 = 33. i. Add: A B c D E a. 30 + 80 70 + 65 42 + 93 73 + 66 54 + 72 6. 40 + 90 85 + 90 71 + 65 81+98 65 + 33 c. 70 + 60 60 + 54 82 + 93 63 + 54 76 + 82 d. 80 + 90 37 + 80 63 + 53 35 + 82 82 + 21 2. Subtract : e. 90-30 85-35 49-23 99-19 /. 80-40 74-40 68-34 86-21 g. 70-20 63-23 87-45 70-23 h. 60-30 72-30 56-22 66-23 3. Add: A i. 40^ + 70^ k. 60^ + 60^ B C 50^ + 85^ 31^ + 63^ 65^ + 60^ 42^ + 52^ 70^ + 35^ 53^ + 46^ 85^ + 40^ 64^ + 33^ 4. Subtract : A m. $1.10-$ .30 n. $1.80 -$1.50 o. $1.20-$ .70 p. $1.80 -$1.60 34-13 55-22 94-33 86-25 $1.05 -$.50 $1.06- $.86 $1.07- $.70 $1.08 -$.48 C 94^-32^ 87^-63^ 69^-46^ REVIEWS 111 Oral Problems 1. There are 35 boys and 30 girls in a school. How many pupils are there ? 2. In a school of 86 pupils, 42 are girls. How many are boys ? 3. When a boy saves 30 cents more, he will have a dollar. How much has he now ? 4. A girl has read 35 pages in her reader and she has 60 more to read. How many pages are there in the book ? 5. If I pay 5 hundred dollars for one lot and 4 hundred dollars for another, what do I pay for both ? 6. Of a farm of 96 acres, 40 acres are in wheat and the rest in corn. How many acres are there in corn? 7. A conductor collected 47 fares going out, and coming back, 40 fares. How many did he collect on the round trip? 8. There are 42 telephone poles on one side of the road and 36 telegraph poles on the other side. How many poles are there on both sides ? 9. A farmer has 42 cows. How many must he buy in order to have 65 ? 10. How many are 23 and 23 ? 11. A grocer mixed 40 pounds of black tea with 25 pounds of green tea. How many pounds of mixed tea were there ? 112 SECTION THREE Dollars and Cents Written Exercises i. Add. Test results : ^ B c D E a. $36.85 $ 7.44 $ 25.82 $31.25 $ 8.47 6.95 8.82 5.33 9.45 20.00 18.26 18.25 20.31 10.24 5.78 5.37 4.76 2.59 7.13 .29 7.44 37.53 .45 8.88 48.86 &. $11.47 $35.79 $18.27 $5.10 $ 48.74 10.83 3.46 3.42 43.83 .09 45.09 11.89 20.35 7.65 6.85 7.67 6.34 2.87 17.25 .77 8.54 9.95 28.92 5.74 13.24 .98 8.87 4.73 18.29 7.65 c. $29.99 $ 12.25 $25.26 $10.79 $ 3.89 14.89 8.24 8.43 3.04 22.50 2.46 5.83 .24 13.40 6.18 .25 16.79 53.10 7.44 20.06 25.57 9.86 1.75 8.26 .88 3.21 43.50 .89 .29 9.47 d. $36.46 $9.32 $ 13.08 $ 12.46 $17.27 5.16 8.56 3.15 13.81 6.66 8.17 42.97 11.48 11.36 .75 9.10 2.73 5.27 1.88 24.00 29.13 9.37 5.21 .75 4.44 6.48 10.66 40.69 29.95 .98 FEDERAL MONEY 113 2. Subtract. Test : A B c D E a. $90 .00 $ 87.50 $75, .98 $56.03 $47 .96 -8 .47 - 29 .43 -47. 99 27. 06 -8 .89 6. $81 .62 $53 .70 $45, ,76 $ 47.57 $30 .13 -51 .77 -9 .84 -17 .89 30. 75 -25 .08 c. $ 36.80 $ 52.23 $93, ,2o f> 85. 64 $92 .73 -19 .95 - 27 .26 -47 .65 - 43.76 -50 .95 3. Multiply. Test: Point off two places in the answer for cents A B c D E a. $14 .09 $ 12 .48 $11. ,59 1 ;io .83 $9 .84 x7 x8 x7 x9 X 10 I. $7 .65 $6 .45 $ 8. 90 $4.32 $3 .25 X 10 X 11 X 21 x31 X 41 4. Divide. Test : Place the decimal point in the quotient when it is reached in the dividend. A B CD a. 4) $98.76 5) $95.75 6) $96.84 7)$ 90.09 75.12 9)$ 96.75 11)$ 80.08 12)$ 15.12 c. 21)$ 60.06 21)190.09 21)$ 48.93 21)$ 68.88 114 SECTION THREE Cents as Divisors Written Exercises i. At 5 cents per yard, how many yards can be bought for $1.95? PROCESS cents cents Write the divisor and the dividend as cents. 195 cents contains 5 cents 39 times. 5)195 Write yd. in a parenthesis. A.ns. 39 (yd.) 2. At 13 cents per pound, how many pounds of pork can be bought for $ 1.69 ? (169 cents divided by 13 cents gives the number of pounds.) 3. Divide the following : a. $3.64n-$.07 e. $10.50-5-25* 6. $ 4.84 -*-$. 11 /. $10.50-*- 42*' c. $4.84-5-$.44 g. $40.50 + 50* d. $4.41-*- $.21 h. $17.22-*- 14* 4. At 25 cents each, how many penknives can be bought for $12.75? 5. At 13 cents per quart, how many quarts of berries will $ 4.03 pay for ? 6. At 30 cents per quart, how many quarts of ice cream can be bought for $ 10.80 ? 7. At 5 cents for each fare, how many fares amount to $3.45? 8. At 25 cents each, how many dolls can be bought for $ 6.25 ? MEASUREMENTS 115 Rectangles A rectangle is a figure of four sides having square corners. The floor of the classroom is a rectangle ; so is the ceiling, each wall, the door, the window. The opposite sides of a rectangle are equal in length. When a rectangle has four equal sides, it is called a square. To find the area of a rectangle means to find the number of square inches, square feet, square yards, etc., it contains. Preparatory Exercises 1. A square having each side 1 foot long contains a square foot. How many 1-foot squares can be cut from a piece of paper 3 feet long, 2 feet wide ? 2. A square having each side 1 yard long contains a square yard. How many 1-yard squares can be cut from a piece of sheeting 3 yards long, 2 yards wide ? To find the number of square inches, square feet, square yards, etc., in the area of a rectangle mul- tiply the number of inches, feet, yards, etc., in its length by the number of the same denomination in its width. 116 SECTION THREE Sight Exercises i. Give the area of each of the following rectangles : a. 10 in. by 11 in. /. 10 yd. by 8 yd. &. 14 ft. by 10 ft. g. 20 in. by 12 in. c. 20 yd. by 13 yd. h. 21 ft. by 10 ft. d. 16 in. by 10 in. i. 12 yd. by 10 yd. e. 9 ft. by 8 ft. j. 11 in. by 9 in. The perimeter of a figure is the distance around it. 2. The number of yards in the perimeter of a field 20 yards wide, 24 yards long is 20 yd. + 24 yd. + 20 yd. + 24 yd., or 88 yd. 3. Give the perimeter of each of the following rectangles : a. 12 in. by 10 in. /. 13 yd. by 30 yd. &. 21 ft. by 21 ft. g. 20 in. by 12 in. c. 20 yd. by 14 yd. h. 22 ft. by 22 ft. d. 12 in. by 21 in. i. 21 yd. by 13 yd. e. 20 ft. by 30 ft. > 30 in. by 40 in. 4. What is the length of a room 12 feet wide that contains 240 square feet ? 5. Give the perimeter of a square whose area is 100 square inches. 6. How many square yards of carpet are needed to cover a floor 4 yards wide, 5 yards long ? MEASUREMENTS 117 Oral Problems i. Find the perimeter of a so yd- field 30 yards long, 20 yards wide. 2. A rug contains 140 square feet. If its length is 14 feet, so yd. what is its width ? 3. A field 30 yards long, 20 yards wide is inclosed by a wire fence. How many yards of wire are needed for each wire in the fence ? 4. How many square inches of gilt paper are re- quired to cover each of the six square faces of a cube 8 inches long ? 5. A room is 20 feet long, 15 feet wide, 10 feet high. a. How many square feet are there in each of the two side walls ? b. How many square feet are there in each of the two end walls ? c. How many square feet are there in the ceiling ? d. How many square feet of plastering are required ? e. How many square feet of carpet are needed to cover the floor ? 6. How many square feet are there in a fence 6 feet high, 30 feet long ? 118 SECTION THREE Oral Problems 1. Find the cost of 4 pounds of tea at 40 cents a pound. 2. I paid $ 1.20 for 6 pineapples. What was the price of each ? 3. Andrew has 90 cents; Bernard has 60 cents more than Andrew. How much has Bernard ? 4. A $2 bill is given in payment for goods amounting to $ 1.20. How much change is received ? 5. What is the cost of J yard of silk at the rate of $1.60 a yard? 6. If a man receives $2.40 a day of 8 hours, how much is he paid an hour ? 7. When goods cost $1.60 a yard, what part of a yard can be bought for 40 cents ? a. After spending 80 cents I have 80 cents left, how much had I at first ? 9. Find the total cost of 25 cents' worth of cake and 3 pounds of candy at 20 cents per pound. 10. A girl pays 30 cents for 3 half-pound boxes of candy. What does the candy cost a pound ? 11. If 2 feet of wire rope cost 20 cents, what would be the cost of a yard ? 12. Find the cost of a yard of ribbon at the rate of 20 cents for two thirds of a yard. 13. A farmer has 80 sheep. How many has he after he sells three quarters of them ? REVIEWS 119 Written Problems 1. Find the cost of 6 pounds of tea at 45 cents per pound. 2. I paid $ 1.05 for 7 quarts of berries. What was the price of a quart ? 3. Harry has 84 cents ; Fred has 26 cents more than Harry. How much money has Fred ? 4. I buy goods amounting to $ 3.85 and I give the grocer a $5 bill. How much change do I get? 5. What is the cost of ^ yard of velvet at $3.50 per yard? 6. If a man receives $ 2.80 per day of 8 hours, how much is he paid by the hour? 7. When goods cost $ 2.80 per yard, what part of a yard can be bought for 35 cents? 8. After giving his sister 55 cents, a boy still has $ 1.35. How much money had he at first ? 9. What is the cost of 3 pounds of candy at 35 cents per pound and 45 cents' worth of ice cream? 10. A boy pays $1.05 for 3 half-pound packages of tea. What does the tea cost a pound? 11. If 2 feet of wire cable cost 38 cents, what is the cost of a yard? 12. Find the cost of a yard of silk at the rate of 72 cents for two thirds of a yard. 13. A farmer has 144 sheep. How many would he have if he sold three fourths of them? 120 SECTION THREE Fractional Parts Oral Exercises i. Draw a rectangle 5 inches long and 3 inches wide, and divide it into 1-inch squares. 2. (a) How many squares are there in each horizontal row ? (&) How many hori- zontal rows are there? (c) How many square inches in the rectangle? 3. (a) How many squares are there in each verti- cal column? (6) How many vertical columns are there? (c) How many square inches are 5 times 3 square inches? (d) How many square inches are 3 times 5 square inches? 4. (a) What part of the rectangle is one horizontal row? (b) Two horizontal rows? 5. (a) What part of the rectangle is one vertical column ? (&) Two vertical columns ? (c) Three ver- tical columns ? (d) Four vertical columns ? 6. (a) 3 is what part of 15? (6) 5 is what part? (c) 6 is what part ? (d) 9 is what part ? (e) 10 is what part? (/) 12 is what part? 7. a. of 36, 60, 66, 90. e. } of 15, 25, 35, 45. 6. | of 24, 36, 80, 48. / f of 5, 15, 25, 35. c. f of 12, 18, 24, 30. g. f of 20, 30, 40, 50. d. I of 16, 20, 32, 40. h. J of 15, 20, 35, 40. FRACTIONAL PARTS 121 Written Exercises A B c D E F 1. Find i of 276, 480, 178, 360, 696, 720. 2. Find 1 of 360, 297, 165, 576, 480, 720. 3. Find 4 of 484, 360, 576, 720, 244, 676. 4. Find i of 510, 650, 360, 720, 480, 975. 5. Find * of 270, 444, 720, 516, 888, 978. Find of 408. METHOD 2 P by 136 of 4P = 272 dividing 408 , Ans. by 3, Find 1 third of 408, which is 136, and cancel these two numbers. Multiply 136 by 2. 6. Find 1 of 342, 444, 396, 165, 576, 720. 7. Find of 480, 360, 576, 720, 244, 676. 8. Find if of 510, 650, 360, 720, 480, 975. 9. Find fl 7T of 510, 650, 360, 720, 480, 975. 10. Find of 510, 650, 360, 720, 480, 975. 11. Find B (> of 360, 480, 720, 516, 888, 978. 12. (a) Find * of 216. 0) Find i of 216. (c) Compare the two results. 13. (a) Find of 216. (b) Find of 216. (c) Compare the results, (d) Find -J of 216. 14. (a) Find f of 216. (6) Find of 216. (c) Compare the results. is. Find the sum of J- of 216 and J of 216. 122 SECTION THREE Mixed Numbers Containing Halves . Preliminary Exercises *__ i. I buy one half dollar's worth of tea _,_ , and one half dollar's 2 2 worth of sugar. What , T < i ,1 o do 1 pay lor both : 2. How many half dollars (a) in $ 1 ? (6) In $2? (c) In $11? (d) In $21? , 2J, etc., are called mixed numbers. 3. Change to a whole number or to a mixed num- ber: (a) |. (6) f. (c) f. (d) f. 4. A man sold a pie and a half to a boy, and a half pie to a girl. How many did he sell to both ? 5. A girl baked 3 pies. How many of them are left after \ pie is eaten ? 6. A man earned $ 3 per day. He spent $ 2^ per day. How much did he save each day ? 7. Give sums at sight : a. 3 6. 31 c. 3 d. 3 e. 3J + 31 +2 +11 +3 + 8. Give remainders at sight : a. 6| 6. 6 c. 61 d. 6 e. 6 -31 -i -4 -11 -4J ADDITION OF MIXED NUMBERS 123 Written Exercises f cloth contain yards. How many yards in all ? i. Three lots of cloth contained 13J, 6J, and 29J PROCESS First add the fractions. Their sura is f , which is equal to 1^. Write the frac- tion , and carry 1 to the column of ones, pt"/> 131 yd. 291 -4ns. 491 yd. 2. Add the following : *v~4- A B c D E a. 201 201 371 25 25J 37 371 61 301 301 &. 481 181 37J 231 17 371 m 561 151 181 c. 631 551 27 1 23 8J 20 6i 171 A <$ 94 9J 17^ 48 8J 64 d. 20 37J 3| 25J 17 364 20J 18 26^ 6J 64 30 50 /I 1L * > 27 18 e. 214 15i 1 si- 30J 20J 214 15| is. V 20| 20J 21* 15J IS' 104 204 214 \4_ 18|- 5 I 204 124 SECTION THREE Subtraction 3. Find answers : A a. 4* B 41 c 41 t D 31* E 20* 50* b. ? + 36 50* ? + 36* 50 ? + 36* 50 ? + 30* 50 ? + 19* 501 5Ql 50 50 50 4. Subtract 191 from 100. PROCESS Say ^ and \ (writing 1) are 1. Carry- ing 1 to 9, say 10 and (writing 0) are 10. Carrying 1 to 1, say 2 and 8 (writing 8) are 10. Test by adding 80^ and 19f 100 19| 801 Am. 5. Subtract : A B c D E a. b. c. d. 80 -19* 75* -48* 93* -57 62 -13* 48* -29 -281 471 * 2 -19 -S* 44* -16* 63 -14* -22 62 73* -25* 84* -46 72* -261 95 -59* 861 -38* 77* -49 68 -19* 50J -23 REDUCTIONS 125 Halves and Fourths Preparatory Exercises i. How many quar- . ters in 1 quarter and 1 \ quarter? -i 2. What single piece ^ of money has the same L value as 2 quarter dol- / * lars? ^ 3. What fraction has the same value as |-? NOTE. When f is changed to ^-, it is expressed in lowest terms. 4. How many fourths are 3 fourths and 1 fourth ? 5. Give the sum of -|- + 1 as a whole number. Improper Fractions. Mixed Numbers ib f 9 f 9 % 9 are ca ll e d proper fractions, since each is less than 1. |, f? f > V~> are called improper fractions, since each is 1 or more. 2^-, 3^, ?-, are called mixed numbers, since each consists of a whole number and a fraction. 6. Change each of the following improper fractions to a whole number or to a mixed number : a. f . 6. |. c. J. rf. f. e. f / f. jr. f . h. |. 7. How many dollars are (a) 4 quarter-dollars ? (6) 4 half-dollars ? (c) 6 half-dollars ? 126 SECTION THREE Addition 8. Find the sum (a) of 38J- and 271. (6) Of 38J and 271 (c) Of 38f and 271. (d) Of 38f and 27f . a. 38J i, 381 c . 38| d. 38f + 271 +27i -h27i + 27f Ans. 65J Ans. 65f Ans. 66J Ans. 66j In (a) the sum of the fractions is f. Change |- to ^ and write \ under the fractions. Add the whole numbers. In (6) change \ to { and add to J. Write the sum, |-, under the fractions ; etc. In (c) change \ to \ and add to f, making |-, which is equal to 1. Write ^ under the fractions and carry 1 to the whole numbers. In (d) change the sum of the fractions |- to 1|^, which is equal to 1J. Write J under the fractions and carry 1 to the whole numbers. Sight Exercises NOTE. Change | in a result to \. Give sums : A B C D E a. 31 31 4f If 3J + 31 +3J- +51 +1| +6j & li 2i 2J 3} 31 li 3f 21 2 31 li 4 2J If 31 SUBTRACTION OF MIXED NUMBERS 127 Written Exercises Add the following : A B CD a. 131 6J 20} 7J 3} 16J- 8J 29 9J 25 261 a 501 231 b. 53} 271 53} 271 651 48| 291 48} 291 Sight Give answers : Exercises A B c D E a. 16J -4 24} -M 33} -It 501 b. 291 38} 1 ~* 1?" 42} 35 46 c. 28 '-I* 36 47 xrcl OO^ 631 Subtract : Written Exercises A 5 C D ' E a. 38} -19J 62} 95} -331 -27} 65} -16J 46 -17J b. 73 -25 84 -36J 50 29-2- 57} -28J 128 SECTION THREE _ Oral Problems 1. If I buy goods amounting to $ 1 J, how much change should I get out of a $ 10 bill ? 2. A grocer sold 1J pounds of black tea and f pound of green tea. How many pounds did he sell of both? 3. At $ 1^- each, what is the cost of 3 baseball gloves ? 4. If I have 6 yards of calico and I use three fourths of a yard, how much have I left ? 5. A baker has 3J apple pies and 4J- peach pies. How many has he of both kinds ? 6. One cow gave 15 \ quarts of milk and another gave 16^ quarts. How many quarts did both cows give? Written Problems 1. If I buy goods amounting to $ 19J, how much change should I receive out of a $ 50 bill ? 2. A grocer sold in a day 24 \ pounds of black tea and 18|- pounds of green tea. How many pounds did he sell of both? 3. At $ 17J each, what is the cost of 4 tables ? 4. If I have 60 yards of muslin, how many yards shall I have after using 13 J yards? 5. A dealer has 23 \ apple pies, 14J peach pies, and 3 plum pies. How many pies has he ? ADDITION OF MIXED NUMBERS 129 Thirds and Sixths Preparatory Exercises i. How many halves 1 in 1 ? How many thirds \ __ in 1 ? How many sixths -i JL in 1 ? / ^ \ / 1.11 _L \T 7*-* . 2. How many sixths in one half? How many /~ sixths in one third? How many sixths in two thirds? \I 3. How many sixths in 1 sixth + 1 sixth ? How many thirds? 4. How many sixths in 1 sixth + 1 third ? Give the sum of ^ 4- ^ in lowest terms. 5. How many sixths in 1 sixth + 1 half? Give the sum of J + \ in lowest terms. Sight Exercises Give sums. Change f in a result to J, f to J, J tof. A B C D E F a. IJ 21 '31 4f 51 6 !i 2J 3J- 4J 5J 6J 6. 71 8| 9f 8f 7f 6| "J- 8J 91 8f 7J 6J 130 SECTION THREE Written Exercises Add: A B C D a. 281 28| 28* 28* 8* 8f / 8* 8 - 191 19$ 1Q1 LJ G 19* 1}. 27* 381 4 39| 32 7 * 661 50 5* 43 25* 4* c. 16* 33* 11* 62| 22| 8* 38 351 39 14* 46f 1* d. 45J 24* 571 14* 181 58J -lO"o" 95 24* 7* 18| 49 e. 47* 26 36f 46| 37 7 5f 15| 8* 59f 28| 8| f. 29* 8* 651 43 35* 681 17 38* 14* 9 i 3| 5* '- g. 34f 28 641 28J 24f 41* 5* 8* 34 19| 25* 48 SUBTRACTION OF MIXED NUMBERS Sight Exercises 131 Subtract : A D /"? Jj L/ D F a. 2 a 9 ^1 3 11 11 6 6 H 41 6 4.1 re 2 ^F b. 7f li 11 ii 3 2 8 61 Subtract : Written Exercises A B C z> a. 75 94 87 93 13J b. 46} 26} 32} 18J 41} 27J 52J 38| 65 48J 57f 674 54| c. 70J 43J d. 33$ 16J 82| 901 37J 41} IZi 82 68J 29J 70 e. 53 26J 80i 18f 58J 'S 72} 49i 61| 132 SECTION THREE Ratio Oral Problems 1. What part of a dozen (a) is 2 ? (&) 3 ? (c) 4 ? (d) 6? 2. If a dozen oranges cost 18 cents, what is the cost (a) of a half dozen ? (&) Of J- dozen ? (c) Of dozen ? 3. At 3 peaches for 5 cents, (a) find the cost of 6 peaches. (6) Of 9 peaches, (c) Of a dozen peaches. 4. If 3 men can do a piece of work in 20 days, how long would it take 6 men to do the same work ? 5. How many men can do in 10 days a piece of work that requires 5 men for 20 days ? 6. At the rate of 20 pages in 3 days, how long would it take to finish 100 pages ? 7. If 2 fifths of a number is 40, what is 4 fifths vof the number? s. When bricks sell for $ 7 a thousand, what is the cost of 7,000 bricks ? 9. At 3 for 2 cents, find the cost of 60 news- papers. 10. When a 6-acre field produces 100 bushels of wheat, what should a 3 6-acre field produce at the same rate ? 11. At the rate of 40 miles in 3 hours, how long should it take a freight train to go 160 miles? ADDITION AND SUBTRACTION 133 Fifths Sight Exercises i. Give answers : A B C D E a. 4* 3f 5* 8f 9* 6f +2f + 3 6. 9* : ' 8 t fi2 fil U"F" \J"F" 5 5 -4J -2 5 e. 6 7 8 9 8 -If -If - 1* - 8f -7f 2. Add: Written Exercises A B C D a. 19f 20J I. 59f 28| 51 3 35f 18 ie| 36f 71 48f 3 59f 29* 3. Subtract : A B C D a. 40 51 62 73 -16* -25f -37f -58* b. 84* -37f 95f -69J 86f -57* 77* -59 134 SECTION THREE Oral Review Problems 1. Grace weighs 64 pounds, and her brother weighs 36 pounds more than Grace. How many pounds do both weigh together? 2. How much less than a dollar has a boy who has a half dollar, a quarter, a dime, a nickel, and a cent ? ' 3. How many pounds are there in 3 loads of hay weighing 21 hundred pounds each ? s >/ 4. How many seats are there in 8 classrooms containing 40 seats each? 5. How many pounds are there in 320 ounces ? 6. Change 2 feet 9 inches to inches. 7. In a school of 480 pupils, one sixth were absent on a stormy day. How many were present ? 8. How many feet of wire netting will inclose a plot 31 feet wide, 100 feet long? 9. At 10 cents each, how much fare is received from 64 passengers ? How much fare at 5 cents each ? 10. Mark has read 50 pages in a book and there are still 32 to read. How many pages are there in the book ? 11. An airship sailed 2 hours 45 minutes. How many minutes less than 3 hours was it in the air ? 12. How many weeks are there in 350 days ? How many weeks and days are there in 355 days ? is. How many half-pint cups are there in 2 quarts ? 14. How many days are there in the last two months of the year ? REVIEWS 135 is. At 1J bushels per acre, how many bushels of wheat are required to sow 40 acres ? is. When wool loses ^ of its weight in washing, what will a quantity of wool weigh after washing if it weighs 80 pounds before it is washed ? 17. When oats weigh 32 pounds per bushel, what is the weight of a peck of oats ? is. At the rate of 40 cents for \ yard, how many yards of silk can be bought for $ 2.40 ? 19. (a) How many feet of ribbon are required for 30 bows, when each bow takes 24 inches ? (b) How many yards ? 20. At ^ cent a pound, how many pounds of cotton must a man pick to earn (a) $ 1 ? (6) $ 2J ? 21. At 12 miles per hour, (a) how many hours does an automobile take to go 480 miles ? (&) How many days of 10 hours each ? 22. 0) What part of 63 is 21 ? 63 is 42 ? 23. A boy had a trip of 120 miles to make. How many miles remained after he had gone f of the distance ? 24. What is the perimeter of a triangle when each of the three equal sides measures 21 inches? 25. What is the fare for 124 miles at 2 cents per mile ? (&) What part of 21 in. 21 in. 21 in. 136 SECTION THREE 26. At 20 quarts of gasoline a day, how many gal- lons will a motor boat use in 10 days ? 27. A boy set out 360 plants, of which J died. How many remained ? 28. How many eggs are there in a box of 30 dozen ? 29. A man weighs 150 pounds and his son weighs 50 pounds less. How many pounds do both together weigh ? so. How much more than $ 15 is the cost of eight barrels of apples at $2|- per barrel ? 31. What day of the year is (a) Feb. 1 ? (&) March 1, 1915 ? (c) March 1, 1916 ? 32. How many passengers at 5 cents each will pay $ 1.50 in fares ? 33. On one trip a conductor collected 5 cents each from 8 passengers and 3 cents each from 10 passen- gers ; how much did he collect on that trip ? 34. A conductor collected on one trip $ 1.30 in fares,' 20 of them being 5-cent fares and the remainder 3- cent fares. How many were 3-cent fares ? 35. How many eggs are there in 3^ dozen ? 36. What is the cost of 4 horses at $200 each ? 37. How much less than $500 will be the cost of 8 cows at $50 each? 38. How much more than $500 will be the cost of 11 cows at $50 each? WRITTEN REVIEWS 137 Written Review Problems 1. A boy weighs 96 pounds, and his father weighs 75 pounds more than the son. What is the total weight of the two? 2. How much less than 1.0 dollars has a man who has a $ 5 bill, $ 2 bill, $ 1 bill, a silver dollar, a half dollar, a quarter, a dime, a nickel, and a cent ? 3. Find the weight of 2 loads of hay weighing 1,975 pounds each. 4. How many seats are there in 16 class rooms each containing 40 seats ? 5. How many pounds are there in 496 ounces ? 6. Change 15 feet 9 inches to inches. 7. In a school of 475 pupils, two fifths of them were unable to reach school on a very stormy day. How many were present? 8. How many feet of wire netting will inclose a plot 40 feet wide, 140 feet long ? 9. At 5 cents each how much fare is received from 97 passengers ? 10. Flora has read 147 pages in a book and has still 154 pages to read. How many pages are there in the book ? 11. An airship sailed 1 hour 37 minutes. How many minutes longer should it have sailed to remain in the air (a) 2 hours ? (b) 3 hours ? 12. How many weeks and days in 400 days? 138 SECTION THREE 13. How many quarter pint cups are there in 8 gallons ? 14. How many days are there in the last six months of the year? is. At 1J bushels per acre how many bushels of wheat are required to sow 48 acres ? 16. When raw silk loses of its weight in boiling, what will be the weight after boiling of a quantity that weighs 96 pounds before it is boiled ? 17. What is the weight of a half peck of corn at the rate of 56 pounds to the bushel ? is. If corn is bought at the rate of 15 cents a peck, how many bushels can be bought for $21.60 ? 19. How many yards of ribbon are required for 36 bows, if each bow takes 21 inches? 20. At \ cent per pound, how many pounds of cotton must a man pick to earn $ 13.86 \ 21. At 12 miles per hour, how many days of 8 hours each will it take an automobile to go 480 miles ? 22. (a) What part of 120 is 24 ? (6) What part of 120 is 72 ? (c) What part of 120 is 96 ? 23. A boy had a journey of 120 miles to make. How many miles remained after he had finished J of the distance ? 24. Find the perimeter of a square if each of the 4 equal sides measures 13 \ inches. 25. What is the fare for 124 miles at 2J^ per mile ? WRITTEN REVIEWS 139 26. At 25 quarts of gasoline per day how many gallons will a motor boat use in 32 days ? 27. A man set out 396 cabbage plants, -^j- of which died. How many remained ? 28. How many eggs are there in 6 boxes of 30 dozen each ? 29. A man weighs 153 pounds, and his son weighs 58 pounds less. What is the total weight of both ? so. How much more than $ 15 is the cost of 8 barrels of apples at $2.75 per barrel? 31. What day of the year is (a) Feb. 22? (6) March 4, 1915? (c) May 30, 1916? (d) July 4, 1917? 32. How many passengers at 5 cents each will pay $2.45 in fares? 33. On one trip a conductor received 5 cents each from 21 passengers and 3 cents each from 13 passen- gers ; what was the total amount received on that trip ? 34. A conductor collected altogether $ 1.50 in fares, 24 of them being 5-cent fares and the remainder 3-cent fares. How many 3-cent fares were there ? 35. How many eggs are there in 9J dozen ? 36. What is the cost of 5 horses at $ 175 each? 37. How much less than $ 500 will be the cost of 7 cows at $ 55 each ? 38. How much more than $ 500 will be the cost of 11 cows at $55 each? 140 SECTION THREE Addition and Subtraction Sight Drills 1. Add 75 to each of the following numbers : A B C D E F G H I J a. 26 74 48 59 99 18 36 69 .95 49 6. 83 19 15 46 96 74 28 66 15 56 c. 47 35 84 26 78 89 57 86 77 38 d. 65 58 37 45 39 55 29 68 92 87 e. 94 27 93 67 85 75 97 16 88 79 2. Subtract each of the foregoing numbers from 120. 3. Add 350 to each of the following numbers : A B C D E F a. 90 120 35 105 2300 900 b. 70 210 46 206 3400 800 c. 50 130 18 307 4500 700 d. 80 320 27 408 5600 640 e. 60 440 39 509 6200 530 4. Give answers rapidly : Think 121 + 50 + 6. Think 121 - 50 - 6. A B c a. 121 + 56 122-63 162 + 84 b. 121-56 140 + 51 162-84 c. 134 + 45 140-51 173 + 75 d. 134-45 151+62 173-75 e. 122 + 63 151-62 184 + 22 WRITTEN REVIEWS 141 Addition and Subtraction Written Exercises i. Add. Test A B c D a. 43,875 62,776 48,256 2,862 2,250 2,342 38,451 2,473 8,429 5,419 4,836 52,813 885 763 629 694 9,680 8,274 985 4,178 b. 13,336 12,643 84,195 47,386 4,236 656 464 7,562 48,512 5,472 7,358 3,625 4,087 58,688 969 10,533 6,472 4,865 4,272 8,524 c. 15,625 12,075 4,524 58,498 7,405 4,437 7,525 4,295 964 43,949 61,329 14,868 38,228 5,350 2,316 5,564 2,545 9,527 6,395 9,040 21,864 20,685 4,567 6,977 2. Subtract. Test by adding upward : a. 90,000 83,210 75,634 62,350 7,086 75,065 9,876 48,907 6. 90,090 80,000 70,700 60,000 83,956 1,234 36,895 5,080 142 SECTION THREE 3. Add: A B C a. $123.45 $907.48 $ 680.70 67.98 35.06 90.00 76.59 243.65 77.96 803.20 76.78 40.90 47.00 90.12 465.58 59.83 83.75 69.49 6. $316.25 $949.34 $469.00 57.42 73.44 74.05 54.58 750.27 526.51 602.72 46.81 57.36 59.50 245.54 425.83 35.00 382.28 35.10 c. $537.46 $456.56 $ 63.42 45.91 346.21 633.31 48.91 86.90 678.92 756.84 27.64 87.14 88.00 78.04 694.00 68.27 215.84 318.25 4. Subtract : A B c a. $385.34 $472.80 $316.25 198.76 73.86 158.87 6. $907.48 $ 680.70 $469.00 244.65 465.58 74.05 D 159.24 984.85 76.54 59.00 266.13 29.32 . 52.63 265.76 83.74 801.82 724.00 96.98 ; 37.42 862.00 375.89 92.66 348.15 438.56 D 123.45 67.89 801.82 724.98 i. DRILL EXERCISES Multiplication and Division Drill Exercises Give products : 143 ^4 B c D E a. 2x 20 3x20 4x20 5x20 2 x91 6. 2x 30 3x30 4x30 5x30 3 x81 c. 2x 40 3x40 4x40 5x40 4 x71 d. 2x 50 3x50 4x50 5x50 5 x61 e. 2x 60 3x60 4x60 5x60 6 x51 /. 2x 70 3x70 4x70 5x70 7 x41 F G H 7 j a. 6x 20 7x20 8x20 9x20 2 x21 I. 6x 30 7x30 8x30 9x30 3 x31 c. 6x 40 7x40 8x40 9x40 4 x41 d. 6x 50 7x50 8x50 9x50 5x51 e. 6x 60 7x60 8x60 9x60 6 x61 f- 6x 70 7x70 8x70 9x70 7 x71 2. Give quotients : A B C D E a. 30 )270 40)280 50)450 60)480 20 )160 6. 70 )350 80)480 90)810 20)180 30 )180 C. 80 )720 70)630 60)540 50)300 40 )360 A 40 )320 30)240 20)160 30)180 50)250 e. 50 )350 70)490 90)630 80)560 60 )360 / 60 )420 40)360 30)210 50)400 70 )280 144 SECTION THREE Oral Problems fy. / i. How many farms of 80 acres each can be made ' from 640 acres of land ? - / 2. How much would be received for 9 cows, if they were sold at $50 each? 3. A man bought a lot for 6 hundred dollars, and built a house costing 18 hundred dollars. How many hundred dollars did both cost ? 4. A cord of wood contains 128 cubic feet. How many cubic feet are there (a) in ^ cord? (6) In \ cord? 5. From a box containing 30 dozen eggs, 200 eggs were sold. How many remained ? 6. If a man works 309 days in a year, how much does he earn, at the rate of $ 3 per day ? 7. How many cubic inches are there in a block of stone 40 inches long, 30 inches wide, 4 inches thick ? 8. At 60 firecrackers to the pack, how many packs contain 540 firecrackers? 9. What is the cost of 480 pounds of butter at J dollar per pound ? 10. If 21 men require 32 days to do a piece of work, (a) what part of this time should it take 84 men to do the same work ? (6) How many days ? 11. A dealer received $32 each for 3 suits of clothes. How much less than $ 100 did he receive for them ? MULTIPLICATION AND DIVISION 145 Multiplying and Dividing by 11 Oral Exercises 1. Count by ll's from 11 to 132. 2. How many are (d) two ll's? (b) Eleven 2's? (c) Three ll's? (d) Eleven 3's? (e) Four ll's? (/) Eleven 4's (g} Five ll's? (h) Eleven 5's? (i) Six ll's? (j) Eleven 6's? (k) Seven ll's? (7) Eleven 7's? (m) Eight ll's? (n) Eleven 8's? (o) Nine ll's? (p) Eleven 9's? Written Exercises 1. Multiply by 11: A a. Ill b. 666 c. 765 d. 454 2. Divide by 11 : A B C D a. 1,210 1,342 1,595 1,859 b. 2,761 3,949 5,423 7,403 c. 9,119 8,712 7,260 5,808 3. Divide by 11. Find quotients and remainders: A B C D a. 3,681 8,937 9,432 9,004 b. 3,700 9,000 8,400 6,700 B c D E 222 333 444 555 777 888 909 876 654 543 432 321 565 787 898 579 146 SECTION THREE Multiplying and Dividing by 12 Learn the following table : 12 times 1 are 12 12 times 7 are 84 12 times 2 are 24 12 times 8 are 96 12 times 3 are 36 12 times 9 are 108 12 times 4 are 48 12 times 10 are 120 12 times 5 are 60 12 times 11 are 132 12 times 6 are 72 12 times 12 are 144 Oral Exercises 1. How many inches are there in 1 1 feet ? 2. At 12 cents per yard, what is paid for 7 yards of muslin ? 3. How many cents. are 12 nickels? 4. How many eggs are there in 8 dozen ? 5. Find the number of square inches in 12 rows, each containing 12 square inches. 6. How many cents are there in a dozen dimes ? 7. How many panes of glass are there in 6 windows, each containing 12 panes? 8. How many feet are there in 12 yards ? 9. How many years are there in 60 months ? 10. How many dimes are there in $ 1.20 ? 11. How many inches in 3 feet 6 inches ? 12. How many years and months in 40 months ? 13. How many square feet are there in a plot of ground 12 feet square? MULTIPLICATION AND DIVISION 147 Sight Exercises Give products : A B C D a. 9x12 12 x 8 20 x 12 12x30 b. 6x12 12x5 12x40 50x12 c. 7x12 12 x 4 60 x 12 12x70 d. 3x12 12x2 12x80 90x12 Written Exercises Multiply by 12 : A BCD E a. Ill 222 333 444 555 b. 666 777 818 729 645 c. 564 765 654 543 432 d. 321 454 565 787 809 Sight Exercises Give quotients i A D /"* Jj (_/ D a. 120 + 12 96+12 360 + 12 840 + 12 b. 144 + 12 48 + 12 600 + 12 960 + 120 c. 108+12 60 + 12 840 + 12 720 + 12 d. 132 + 12 84 + 12 480 + 12 240 + 120 Written Exercises Divide by 12 : A B C D a. 1,332 1,572 1,620 1,884 b. 2,772 3,948 5,424 7,416 c. 8,712 9,180 8,580 6,756 d. 5,088 3,384 4,500 6,336 148 SECTION THREE Multipliers containing Ciphers Sight Exercises Give products : A B C D E F a. 21 32 11 22 33 23 x400 x300 x50Q x 400 x 300 x 300 6. 2,100 3,200 1,100 2,200 3,200 2,200 x4 x3 x5 x4 x3 x3 c. 210 320 110 120 130 210 x40 x30 x50 x40 x30 x 30 Written Exercises i. Multiply 185 by 500. PROCESS The product by ones is 0; write in the ones' column. The product by tens rnrv * * is ; write in the tens' column. Place 92500 Ans. the right-hand figure of the product by 5 hundred in the hundreds' column. 2. Multiply: ABC a. 486x200 329x300 239x400 6. 184x500 157x600 139x700 c. 118x800 109x900 478x200 d. 327x300 248x400 177x500 MULTIPLICATION 149 2. Multiply 427 by 230. 497 PROCESS The product by ones is 0; write in the ones' column. Place the right-hand figure of the product by 3 under 3. 854 Place the right-hand figure of the product 98210 Ans. by 2 under 2. ABC a. 475x210 309x320 217x430 b. 175x520 146x63(0 128x740 c. 119x830 105x950 337x250 d. 209x360 208x470 170x580 3. Multiply (a) 408 by 205 ; (6) 205 by 408. (a) 408 205 PROCESS In (a) place the right-hand figure of the product by 5 in the ones' col- umn. Since the (b) 205 408 2040 1640 8200 83640 Ans. 83640 Ans. product by tens is 0, place a cipher in the tens' column. Place the right- hand figure of the product by 2 hundred in the hundreds' column of the same line. ABC a. 102x806 108x907 207x405 b. 203x409 305x306 506x108 c. 389x205 473x205 309x309 150 SECTION THREE 4. Multiply (a) 279 by 346; (6) 346 by 279. PROCESS (a) 279 In (a) place the right- (6) 346 346 hand figure of the 279 product by 6 under 6, ""^ xi &iiu -xxo,uu. iiguit, Of 9400 the product by 4 under 837 4 ? an d the right-hand 96534 Ans. figure of the product by 96534 Ans. 3 under 3. Test (a) by multiplying 346 by 279, as in (6). If the results agree, both products are probably correct. ABC a. 235x123 246x321 325x235 b. 437x214 562x173 684x142 c. 784x125 863x114 746x134 d. 527 x 184 408 x 238 336 x 260 e. 150x630 227x398 317x314 / 208x407 638x156 416x239 g. 356x249 256x349 583x158 h. 475 x 206 567 x 146 900 x 106 i. 136 x 634 678 x 135 760 x 130 j. 235 x 389 786 x 127 508 x 103 k. 480 x 108 890 x 1 12 326 x 275 I 340 x 290 120 x 780 156 x 570 w. 305x309 165x606 124x642 n. 208x404 234x367 205x350 LONG DIVISION 151 Long Division Written Exercises i. Divide 58,376 by 99 ; 43,721 by 975 ; 68,374 by 3,456. Ans. 589 a PROCESS O ^ /i i o yj /j_ o 2i /\ /yj c* 1 vl ~ T 1 Q -xi/i/d. rtT: qiy ? ^L/to* -L *^*i 4. g ' 99)58376 495 975)43721 3456)68374 3900 3456 887 792 4721 33814 3900 31104 956 821 2710 Write the first quotient figure over the 65 last figure of the first partial dividend. Write a quotient figure (or a cipher) over each remaining figure of the dividend. 2. Divide : A B C a. 8,058+102 54,324 + 108 86,785 + 1,021 6. 9,315 + 405 41,814 + 207 93,472 + 2,032 c. 9,178 + 706 65,178 + 306 96,736 + 3,023 d. 9,911+901 90,720 + 405 93,288 + 4,056 e. 7,308 + 812 88,704 + 504 90,126 + 5,007 /. 9,824 + 614 96,480 + 603 96,496 + 6,031 g. 9,545 + 415 98,982 + 702 98,308 + 7,022 h. 9,592 + 218 96,360 + 803 96,420 + 8,035 152 SECTION THREE Oral Exercises 1. At 20^ each, how many articles can be bought for ABODE a. 20^? GO**? 100^? HO**? ISO/? 6. 40/? 80/? 120^? 160/? 200 /? 2. At 19/ each, how many articles can be bought for a. 20^? 60/? 120^? 160^? 180X? 6. 40^? 80^? 140^? 180^? 200^? and how many cents remain in each case ? 3. Give the number of articles at 18 ^ each that can be bought for each of the following sums, and state the amount remaining in each : a. 40^ 20^ 100^ 160 J* 180^ 6. 80^ 60 X 140^ 120^ 200^ 4. Give quotients : A B CD a. 30)60 30)120 30)180 30)240 6. 30)90 30)150 30)210 30)270 5. Give quotients and remainders : a. 29)30 29)90 29)150 29)210 6. 29)60 29)120 29)180 29)240 c. 28)30 28)90 28)150 28)210 DRILL EXERCISES 153 Long Division Drill Exercises Give quotients : NOTE. Omit remainders, if any. A B CD a. 200)1,800 201)1,800 199)1,800 250)1,800 6. 300)2,400 302)2,400 299)2,400 360)2,400 c. 400)2,800 403)2,800 399)2,800 470)2,800 d. 500)2,500 504)2,500 499)2,500 580)2,500 e. 600)4,200 605)4,200 599)4,200 690)4,200 /. 700)6,300 704)6,300 699)6,300 790)6,300 g. 800)5,600 803)5,600 799)5,600 880)5,600 h. 900)7,200 902)7,200 899)7,200 999)7,200 t. 240)1,920 241)1,920 239)1,920 249)1,920 j. 320)2,240 321)2,240 319)2,240 320)2,239 k. 410)3,690 411)3,690 409)3,690 410)3,689 I 520)2,080 521)2,080 519)2,080 520)2,076 m. 630)3,780 631)3,780 629)3,780 630)3,774 n. 710)4,970 711)4,970 709)4,970 710)4,953 o. 820)4,100 821)4,100 819)4,100 820)4,096 p. 910)8,190 911)8,190 909)8,190 910)8,183 154 SECTION THREE Oral Problems 1. What is the total cost of a lot at 3 hundred dollars and a house at 21 hundred dollars? 2. A man had 31 hundred dollars in bank. How much remained after he drew out 8 hundred dollars ? 3. Give the cost of 21 horses at $300 each. 4. A driver sold 30 cows at $41 each. How much did he receive for them ? 5. At 4^ per pound what is the cost of a barrel of sugar containing 300 pounds? 6. How many 20-acre fields are there in a farm of 640 acres ? 7. At 20 bushels to the acre what would be the yield of 34 acres of wheat ? 8. How long would a train require to go 820 miles at the rate of 41 miles an hour ? 9. If a man has saved $991, how much more must be saved to have $ 1,000 ? 10. How 'many square yards are there in a plot of ground 32 yards long, 30 yards wide ? 11. Mr. Payne has two farms, one of 260 acres and the other of 120 acres. How many acres are there in both ? 12. If 3 men can do a piece of work in 21 days, how long would it require 1 man to do it ? 7 men ? 13. How much shall I have left of $ 1,000 after paying for 3 horses at $ 300 each ? WRITTEN REVIEWS 155 Written Problems 1. What is the total cost of a lot at $ 375 and a house at $ 2,475 ? 2. A man had $950 in bank. How much re- mained after he took out $ 875.75 ? 3. Find the cost of 21 horses at $375 each. 4. A drover sold 30 cows at $45 each. How much did he receive for them ? 5. At 4^ per pound what is the cost of a barrel of sugar containing 375 pounds ? 6. How many 25-acre plots can be made out of 600 acres of land ? 7. At 19 bushels to the acre what would be the yield of 34 acres of wheat ? 8. How long would a train require to go 975 miles at the rate of 30 miles per hour ? 9. A man requires $ 1,000 ; he has already saved $ 837.50. How much does he still need ? 10. What is the area of a plot of ground 32 yards long, 29 yards wide ? 11. Mr. Black has two farms, one of 269J acres and the other of 1 10 \ acres. How much land is there in both? 12. If 3 men can do a piece of work in 35 days, how long would it require 1 man to do it ? 7 men ? 13. How much change should I receive out of a $1,000 bill given in payment for 3 horses at $317 each? 156 SECTION THREE Written Exercises In dividing 969 by 19, find the first quotient figure by dividing 9 by 2, 19 being nearly 20. In dividing 975 by 39, get the first quotient figure by dividing 9 by 4. i. Divide : A B c a. 969-19 8,213-*- 191 61,280-*- 1,915 b. 928-*- 29 9,376-293 82,124-*- 2,933 c. 975-*- 39 9,480-*-395 86,922-*- 3,951 d. 784-49 7,455-*- 497 89,496-*- 4,972 e. 885-*- 59 8,386-*- 599 83,9 16-*- 5,994 / 897-*- 69 8,376-*- 698 90,5 84-*- 6,9 68 g. 869-4-79 6,368-*- 796 87,648-^-7,968 h. 979-*- 89 9,834-*- 894 7 1,592-*- 8,949 i. 891-*- 99 7,936-*- 992 69,489 -*- 9,927 j. 704-88 9,691-*- 881 52,8 90-*- 8,8 15 k. 936-*- 78 9,396-*- 783 70,497 -*- 7,833 I 952-*- 68 9,590-*- 685 82,212-6,851 m. 928-58 9,424-589 94,272 -*- 5,892 n. 864-*-48 9,723-463 82,348-4,844 o. 988-*- 38 9,650-*- 386 88,918-*- 3,866 p. 952-28 9,690-285 99,960-2,856 q. 936-18 9,752-184 95,836-1,843 r. 945-*-27 9,936-276 99,432 -*- 2,762 s. 999-37 9,672-*- 372 93,025-3,721 ORAL REVIEW 157 Oral Problems 1. How many ounces are there in 3 pounds ? 2. Change 50 ounces to pounds and ounces. 3. How many postage stamps are there in a sheet 15 stamps long, 3 stamps wide ? 4. What is the cost of 23 two-cent stamps? 5. At 14 cents a gallon, how many gallons of oil can be bought for 42 cents ? e. If 45 men require 30 days to do a piece of work, how long would it take 15 men to do it? 7. At 16 bushels to the acre, how many acres will produce 48 bushels ? 8. Seven times 8 are how many more than 50 ? 9. At the rate of 15 examples an hour, how long does it require a girl to work 45 examples ? 10. How many times is 99 contained in 100 ? what is the remainder ? How many times in 200 ? what is the remainder ? In 300 ? what is the remainder ? In 500 ? what is the remainder ? 11. How many times is 98 contained in 100 ? what is the remainder ? In 300 ? what is the remainder ? In 700 ? what is the remainder ? 12. At 99^ a yard, how many yards of silk can be bought for $ 6, and how many cents change are there ? 13. A girl bought 7 yards of goods at 99 ^ per yard and received 7^ change. How much did she give the storekeeper ? 158 SECTION THREE Multiples and Factors Since 14 is divisible by 2 and by 7, it is a multiple of these numbers ; 12 is a multiple of 2, of 3, of 4, and of 6. Since 14 is divisible by 2 and by 7, these two num- bers are factors of 14 ; 2, 3, 4, and 6 are factors of 12. 1 is not considered a factor of a number, neither is the number itself. Oral Exercises 1. What number will exactly divide 25 ? 49 ? 81 ? 121? 2. What two numbers will exactly divide 21 ? 26? 33? 34? 35? 39? 46? 51? 55? 57? Prime Numbers Since they have no factors, 1, 2, 3, 5, etc., are called prime numbers. Sight Exercises 1. Name a prime number between 6 and 9. 2. Name two prime numbers between 10 and 15. Two between 16 and 20. One between 21 and 25. One between 25 and 30. 3. Name two prime numbers between 30 and 40. Three between 40 and 50. Two between 50 and 60. Two between 60 and 70. Three between 70 and 80. Two between 80 and 90. One between 90 and 100. FRACTIONS 159 Reduction Oral Exercises When a thing is divided into 8 equal parts, each part is called an T eighth, written -J. <^ ~^||> i. How many eighths _L x are there in a half ? In / a quarter? In three 4 fourths ? 2. Change to 8ths a. 6. c. f . i i In the fractions J, J, the numbers 2, 4, 8, are called the denominators. In the fractions J, f, -|, , the numbers 1, 3, 5, 7, are called the numerators. The numerator and the denominator are called the terms of the fraction. When f is changed to J, f and f to J, f to f , etc., each fraction is changed to lowest terms. 3. Change to lowest terms : ABC D E a. i 4 o o o o b- A B" 6" A ta 4. Change to a whole or to a mixed number : ABODE . I * t 160 SECTION THREE Adding and Subtracting Fractions Preparatory Exercises To add or to subtract fractions having different denominators, the denominators must be made the same ; that is, the fractions must have a common de- nominator. 1. Add ^ and J. What is the common denominator? 2. Add i- and ^. What is the common denominator ? Zi D 3. (a) What is the common denominator of -|-, ^, andi? (6) Ofi,i,andJ? (c) Of^andi? (d) Of ? (e) Of J and*? (/) Of 1 J, andj? Sight Exercises Give answers : A 5 c D E F e a. J J i \ i 1 3 4 i + J- i +1 +1 i + | 6. 11 ' 1* 11 14 11 * A 4 if If i i +* + | +| +4 + 1 c. n if 4 1J If if If 1 1 +1 + f +f +J + 1 d 2J 2| 2| 2J 21 2f 2| i i i i 1 1 i e. I 1 1 1 11 1J If MIXED NUMBERS 161 Adding Mixed Numbers Sight Problems 1. How much cloth is needed for a suit of clothes if 4 yards are needed for the coat, 1^ yards for the vest, and 21 yards for the trousers ? 2. How many inches (a) in ^ foot ? (6) In J- foot ? (c) In l ft. + l ft.? (d) What part of a foot is 6 inches ? (e) What part of a foot is J ft. + J ft. ? t 3. (a) What is the smallest number that is divisible by 3, by 4, and by 6 ? (6) What is the least common denominator of ^, ^, and ? (c) How many 12ths in the sum of J, J, and J? (d) -|- 4- ^ + J- = how many fourths ? 4. After a man had sold ^- + ^- + ^ of his farm, () what part of his farm was sold ? (6) What part did he still own? (c) If he still owned 40 acres, how many acres did he own at first ? Written Exercises Add: ABODE a. 20| 16f 32 46| 72f 13 7f 8J 4 20f _3| 1_ 19J- Ji _6_ 6. 17J 30 14J 21f 8 13 18J- 5J 16 22J 162 SECTION THREE Subtracting Mixed Numbers Oral Problems 1. An airship remained in the air 58-| minutes. How many minutes is that less than an hour ? 2. After spending \ and J of his money, (a) what part of his money did a boy still have? (6) If he then had 10 cents, how much had he at first? 3. A man had to walk 10 miles. He walks 3^ miles the first hour and 3 \ the second hour. How many miles has he still to go ? 4. Out of 100 yards of cloth 921 yards have been used. How many yards remain ? 5. How much more than $ 50 will be the cost of a dress at $ 40 J and a trunk at $ 91 ? Written Exercises Subtract : A B c D E a. 30 43 74 65 72 -18* -19J -27| -38* -47| 6. 50 83 92 36 45 -24* -46f -25* -16| -26* c. 30| 42| 51* 62f 71| -181 -261 -391 -131 -221 MIXED NUMBERS 163 Addition Written Exercises i. Find the sum of 30f + 25f + 8f. PROCESS To the right of the add- ends draw a vertical line QQJ^ and from its top draw a 9 _ 5 horizontal line. Above the latter write 12 (the 12 9 10 8 least common denonina- Ans. 654- -?4- = 2 A = 24- fc X ^ J_ t t tor). Change the fractions to ^-, J-|, and -j^-, respectively, and write 9, 10, and 8 (the new numerators) under 12 (the new denominator). Add these numerators, and write 27 (their sum), over 12 (the common denominator), making the sum of the fractions f -J . Change this first to 2^-, and then to 1\. Write \ under the original fractions, and carry 2 to the whole numbers. 2. Find sums : ABC 30J 16| 5 251 8J 201 84- 394 84 336 60 9f 18f 6 3. Add: 2J 20} Ji 164 SECTION THREE Addition and Subtraction Written Exercises i. A field contained 35^- acres of land. 15} acres were sold. How much remained ? PROCESS Write 20 (the least common de- 20 nominator), as in addition, and 35* 3. 18 15 18 and 15 (the new numerators). Place 3 (their difference) over 20 u 4 (the common denominator), mak- Ans. ing the difference between the frac- tions ^. Write the latter under the original fractions, then subtract the whole numbers. 2. Subtract : ABODE a. 15| 30f- 27 54f 40} _8| -14| -9} -261 6. 64f 92f 87| 73f 50J -16f -24J- -59f 3. Find answers : a. 35| 35} 42| 42f 26| + 16f -16f 1. 26| 67| 67J _9 + c. 72J 72| 90f + 6| -6f +8f FRACTIONAL PARTS 165 Fractional Parts of Numbers Oral Exercises 1. How many ounces (a) in J- pound? (&) In -f pound ? (c) In f pound ? (d) In -J pound ? 2. Find (a) J- of 24. (6) f of 24. (c) f of 24. (d) % of 24. 3. How many minutes (a) in J- hour? (6) In f hour? (c) In f hour? (d) In f hour? (e) In hour? (/) In f hour? (^) In \ hour? (A) In 1- hour? (i) In | hour? (j) In J hour? () In | hour? 4. How many hours (a) in ^ day ? (6) In ^ day ? (c) In i day? (d) In f day? (e) In f day? (/) In l day? (g) In f day? (h) In f day? (i) In I day? (j) In 1 day? (A:) In f day? Sight Exercises Give answers : A z? r* /-> *-* D a. 4 of 168 J of 184 iof 250 \ of 248 1). i of 366 i of 396 Jof 369 i of 648 c. i of 440 i of 484 1 of 4 01 888 I of 848 d. J of 255 J- of 300 j-of 555 J- of 525 e. I of 606 i of 486 Jof 366 i of 420 f. i of 168 i of 248 Jof 328 fr of 480 9- 1 of 120 1 of 120 Jof 150 2 S" of 150 h. ] of 150 * of 150 |of 180 3 of 160 166 SECTION THREE Written Problems i. A man sold -J of his farm of 168 acres. How many acres did he sell ? PROCESS Find 1 eighth of 168 by dividing 168 by 8, J f Iff A. = 147 A. Am. canceling these numbers, and writing the quotient 21, over the former. Multiply 21 by 7 to find 7 eighths. The abbreviation for acres is A. ; for pounds, Ib. ; for yards, yd. ; for miles, mi. 2. How many pounds are there in f of a ton of 2,000 pounds ? 3. How many yards are there in J mile, a mile containing 1,760 yards ? 4. A planter sold -f of his crop of cotton consisting of 258 bales. How many bales did he sell ? 5. Of a grove of 360 trees, three fifths v are pine trees. How many are pine trees ? 6. How far has a train gone when it has gone f of the distance from New York to Washington, 224 miles ? 7. A man sold for |- of its cost a farm that cost him $ 2,800. What did he receive for it ? 8. An acre contains 160 square rods. How many square rods are there in a plot containing |- acre ? 9. A piece of cloth contains 56 yards. How many yards are left after j- of the piece is used ? MIXED NUMBERS 167 Multiplying by a Mixed Number Preparatory Exercises 1. If a horse travels at the rate of 6 miles an hour, (a) how far does he go in -|- hour ? (6) In an hour and a half ? 2. How many miles are 1J times 6 miles ? 3. At 12 cents a yard, what is the cost (a) of 2 yards of muslin ? (6) Of J yard? (c) Of 2J yards? 4. How many cents are 2J times 24 cents ? 5. At $ 40 per acre what is the cost (a) of i acre of land ? (6) Of 3i acres ? 6. How many dollars are 3i times $ 40 ? 7. How many bushels of oats can be raised (a) on } acre at the rate of 40 bushels to the acre ? (&) On If acres? 8. How many bushels are If times 40 bushels ? Sight Exercises Give products : A B C D E F a. 24 24 24 24 24 30 6. 60 60 40 60 80 80 c. 30 30 40 30 40 20 168 SECTION THREE Written Exercises i. A man makes 87 hats in a week. How many will he make in 2^ weeks ? PROCESS "7 nats Get 1 third of 87 by dividing ?S 87 by 3. Place the right-hand (i times 87) 29 figure of the product of 87 by 2 (2 times 87) 174 in the ones' column. ( 2 $- times 87) 203 hats 2. Find products. Consider the mixed number the multiplier. A B c D a. 86x21 84x31 96x21 34x41 6. 88x4J- 75x5j 72x31 25 x 3 c. 72x31 92x4J 84x51 48 x 2 J 3. Multiply 78 by 4f . 78 PROCESS 4JL First find | of 78, which (J of 78) 13 is 13. Multiply the latter (Multiply by 5) X 5 ^u- ^/^?' , (fof78) 65 Multiply 78 by 4 and \ 6 ' combine the two results. ( 4 times 78 ) 312 (4f times 78) 377 Ans. A B C D a. 84x4f 96x3f 96x4f 84 x 7| b. 48x5| 88x6^ 85x3f 84 x 8f c. 70x4f 96x5J 80 x 6f 84 x 9f MIXED NUMBERS 169 4. Multiply 84 by 24f . PROCESS After finding |- of 84, 84 multiply 84 by 4, plac- 24f ing the right-hand fig- (i of 84) 14 ure of the product in \ O ' x5 4-T-%A r\m ^Q i**/Vl n TTI n *f" n PTI 1 I M U 1 H ^> ( ( H 1 1 1 1 1 1 1 , I 1 1 1 1 multiply 84 by 2 tens, (1 of 84) 70 placing the right-hand (4 times 84) 336 figure of the product in (2 tens times 84) 168 the tens' column, etc. (24f times 84) 2086 Ans. ABC a. 36x211 42x33J 48 x 23J- I. 75 x 311 84 x 421 c. 108x15* 112x24f d. 180x24f 132xl4f 144x41f e. 176x34f 184 x 26* 140x35^ /. 140x51f 116x61| 170x52f g. 200x62| 150x71| 160x81| h. 130xl2| 240x23| 306 x 34* Sight Exercises Give products : A B 60 60 xlOf c 40 xlOf D 50 xlOf E 50 xlOf F 50 170 SECTION THREE Oral Problems 1. "What is the cost of a yard of dress goods when one half yard costs 32 cents ? 2. At 6 pairs of stockings for a dollar, how many pairs can be bought for $ 4.50 ? 3. When collars sell at 2 for 25 cents, what is re- ceived for a dozen collars ? 4. How many tons of hay are there in two stacks, one containing 10^ tons and the other containing 15|- tons? 5. A man has two farms each containing 64 acres. How many acres are there in both ? 6. A rod is 16^ feet. How many feet wide is a road that is 2 rods wide ? 7. At 60 cents a dozen, how much is paid for 36 roses? 8. How many are 5 J + 5 \ + 5J ? 9. When each house is 20 feet wide, how many houses can be built on a plot 200 feet wide ? 10. How many half-pint glasses of lemonade are there in 2 gallons ? 11. One third of a flock of sheep are in the barn. How many are there in the flock if there are 32 sheep in the barn ? 12. How many nails can be made from a yard of wire (a) when \ inch is needed for each nail ? (6) When 2 inches are needed ? REVIEWS 171 is. If a pail of water weighs 16 pounds and the pail alone weighs 2J pounds, how many pounds does the water weigh ? 14. If \ peck of wheat weighs 7 \ pounds, what is the weight (a) of a peck ? (6) Of 2 pecks ? (c) Of 4 pecks ? is. A freight car can carry 100 thousand pounds. How many tons of 2 thousand pounds each can it carry? 16. A steel freight car weighs 22 tons of 2 thousand pounds each. How many pounds does it weigh ? 17. Taking 1 pound as the weight of a pint of water, what is the weight of a 3-gallon pail of water, when the pail weighs 6 pounds ? is. At 25 cents a pound, how many pounds of coffee can be purchased (a) for $ 1 ? (5) For $2 ? (c) For $4? 19. If a gallon of paint will cover 300 square feet, how many gallons are needed to paint 1,800 square feet? 20. At the rate of 3 pounds of flour to 4 pounds of bread, how many pounds of bread will 30 pounds of flour make ? 21. If the dough weighs |- more than the flour, how many pounds of dough will 20 pounds of flour make? 22. The sum of 2 numbers is 60 ; one of the num- bers is 26. What is the other? 172 SECTION THREE Written Problems 1. What is the cost of a yard of cloth when one eighth of a yard costs 22 cents ? 2. At 3 pairs of stockings for 50 cents, how many pairs can be bought for $ 3.50 ? 3. When cuffs sell at 2 pairs for 45 cents, what is received for a dozen pairs ? 4. How many tons of hay are there in 3 stacks, containing, respectively, 10J tons, 12^ tons, and 14^- tons? s. A farm consists of 3 fields each containing 36|- acres. How many acres are there in the farm ? 6. How many feet wide is a road 4 rods wide, a rod being 3 yards of 5^ feet each ? 7. At $ 1 per dozen, how much is paid for 66 roses ? s. How many are IGI+IGJ+IGJ + IGJ+IGJ? 9. When each house is 25 feet wide, how many houses can be built on a plot 600 feet wide ? 10. How many half-pint glasses of lemonade are there in 5 gallons ? 11. One sixth of a flock of sheep are in the barn. How many sheep are there in the flock if there are 35 in the barn. 12. How many nails can be made from 5 yards of wire if ^ inch is used in making each nail ? WRITTEN REVIEW 173 13. If a barrel with the flour weighs 220^ pounds and the barrel weighs 24^- pounds, how many pounds of flour does it contain ? 14. When J peck of potatoes weighs 3J pounds, find the weight of the potatoes in a barrel contain- ing 2^- bushels. is. How many tons of 2,240 pounds each make 98,560 pounds? 16. How many pounds less than 44,000 does a car weigh that weighs 19 tons of 2,240 pounds each ? 17. Assuming that a pint of water weighs a pound, what is the weight of a 40-quart can filled with water when the can weighs 25 pounds ? is. My bill for coffee at 25 cents per pound was $ 11.25. How many pounds did I buy? 19. If a gallon of paint will cover 315 square feet, how many gallons will be needed to paint a fence containing 1,890 square feet? 20. At the rate of 3 pounds of flour to 4 pounds of bread, how many pounds of bread will 195 pounds of flour make ? 21. If the dough weighs 1|- times as much as the flour, how many pounds of dough will 195 pounds of flour make ? 22. The sum of two numbers is 117. One of the numbers is 79. Find the other number. 23. What number multiplied by 32 gives a product of 480 ? 174 SECTION THREE Denominate Numbers Long Measure 12 inches (in.) 1 foot (ft.) 3 feet 1 yard (yd.) 5^ yards 1 rod (rd.) 320 rods 1 mile (mi.) Oral Problems 1. How many houses, each 20 feet wide, can be built from First Street to Second Street if the build- ing space is 200 feet ? 2. "What is the distance from the middle of First Street to the middle of Second Street if each street is 50 feet wide, the sidewalk on each street is 7 feet wide, and the building space is 200 feet ? 3. How many houses, each 20 feet wide, can be built from Avenue A to Avenue B if the building space is 800 feet ? 4. What is the distance from the middle of Avenue A to the middle of Avenue B if each avenue is 60 feet wide, the sidewalk on each avenue is 10 feet wide, and the building space is 800 feet ? 5. What part of a mile is the length of a square field, if each side measures 80 rods ? 6. How many feet are there in a rod ? 7. If there are 20 streets to a mile, how far do you walk in going from 60th Street to 85th Street ? DENOMINATE NUMBERS 175 Written Problems 1. Fill the blanks in the following : 1 mi. = 320 rd. = yd. = ft. 1 rd. = 5lyd. = ft. 2. When a man's pace measures 3 feet, how many paces are there in J mile ? 3. If a boy's pace is 2^ feet, how many yards are equal to 120 paces ? 4. When railroad ties are 3 feet apart, how many ties are there to a mile ? 5. How many dollars will pay for 1,320 railroad ties at $ \ each ? 6. How many rails each 60 feet long will make a mile? 7. At 18 pounds to the yard, find the weight of one 60-foot rail. 8. Measure the length of a page of your book. Measure the length of the cover. Find the difference between the two lengths. 9. When streets are 264 feet apart how many streets are there to a mile ? 10. When avenues are 880 feet apart, how many avenues are there to a mile ? 11. How many miles does a person walk in going from Avenue A to Avenue I and from 18th Street to 33d Street ? 176 SECTION THREE Areas Oral Exercises 1. (a) How many square inches are there in a square 2 inches long? (6) In a rectangle 2 inches long, ^ inch wide ? (c) In a rectangle 2^ inches long, 2 inches wide ? 2. How many square feet in a blackboard 8^ feet long, 4 feet wide ? 3. How many square feet in a door 6 feet high, 2^ feet wide ? 4. How many square feet in two windows each 5 feet high, 2-J feet wide ? 5. How many square yards in a floor 6 yards long, 4^ yards wide ? 6. How many square yards in a piece of carpet 36 yards long, f yard wide ? 7. How many square inches in a square 12 inches long, 12 inches wide? In a square 1 foot long, 1 foot wide? 8. How many square feet in a city lot (a) 20 feet by 100 feet? (6) 25 feet by 100 feet? (c) 30 feet by 150 feet? 9. How many square feet are there (a) in 4 city lots, each measuring 20 feet by 100 feet ? (6) In 40 ? 10. There are 160 square rods in an acre, (a) How many square rods in 10 acres ? (&) How many acres in a square field 40 rods long ? AREAS 177 Written Exercises Square Measure 144 square inches (sq. in.) 1 square foot (sq. ft.) 9 square feet 1 square yard (sq. yd.) NOTE. To find the area in square inches, express each dimen- sion in inches; to find the area in square feet, express each dimension in feet ; to find the area in square yards express each dimension in yards. 1. Find the area of a rug 54 inches wide, 72 inches long. The number of square inches = 54 X 72. The number of square feet = 4^ x 6. The number of square yards = \\ x 2. 2. Find the number of square inches in each of the following rectangles : a. Length, 48 inches ; width, 42 inches. b. Length, 4 feet ; width, 42 inches. c. Length, 4 feet; width, 3^ feet. 3. Find the number of square feet in each of the following rectangles: a. Length, 4 feet ; width, 3^- feet. b. Length, 54 feet ; width, 60 inches. c. Length, 120 feet; width, 25 feet. 4. Find the number of square yards in each of the following rectangles : a. Length, 36 yards ; width, f yard. b. Length, 108 feet; width, 27 inches. c. Length, 48 yards ; width, 37J yards. 178 SECTION THREE Measurements Written Problems 1. A room is 18 feet long, 12 feet wide, 9 feet high. a. Find the number of square feet in the floor. b. How many square yards of carpet will cover the floor? c. Find the number of square feet in the ceiling. d. How many running feet of baseboard are there in four sides of the room, deducting 3 feet for the space taken by a door? e. How many square feet of baseboard are there in the room, when the baseboard is one foot high ? 2. A paper cracker box is 8 inches long, 3 inches wide, 3 inches high. a. How many square inches of paper are required for each of the square ends ? 6. How many square inches of paper are required for each of the four long faces ? c. How many square inches of paper are required for all six faces, making no allowance for overlapping ? d. How many crackers 3 inches square will it hold, if each cracker is J- inch thick ? 3. How many square feet of wall paper are there in a roll of paper 18 yards long, 18 inches wide? 4. At $ 1.50 per square yard, what will be the cost of the linoleum covering a kitchen 15 feet square ? s. A fence on the front of a lot is 25 feet long and 6 feet high. How many square feet does it contain ? VOLUMES 179 Cubical Contents Preparatory Exercises 1. (a) How many one-inch cubes can be placed on the bottom of a box 5 inches long, 4 inches wide? (6) If the box holds 3 layers of one-inch cubes, how many one-inch cubes will the box hold ? 2. A box 6 inches long, 5^ inches wide, and 7 inches high will hold exactly a gallon. How many cubic inches are there in a gallon ? 3. How many one-inch cubes can be put into a cracker box 3 inches by 3 inches by 8 inches ? 4. Carefully balance an empty pint measure by placing sand on the opposite pan of the scales, then fill the measure with water and ascertain the weight of the water. s. (a) Find the number of cubic inches in a cube 12 inches long, 12 inches wide, 12 inches high. (6) How many cubic feet does it contain? (c) How many cubic inches are there in a cubic foot? 6. (a) How many cubic feet are there in a cube 3 feet long, 3 feet wide, 3 feet high ? (6) How many cubic feet are there in a cubic yard ? 7. Give the capacity of a bin measuring, on the inside, 2 yards long, 2 yards wide, 2 yards high. 180 SECTION THREE Volumes To find the volume of a solid means to find the number of cubic inches, cubic feet, cubic yards, etc., it contains. A solid having six rectangular faces is called a rectangular prism. The opposite faces of a rectangu- lar prism are equal in area. When each of the six faces is a square, the solid is called a cube. A caramel is a rectangular prism, so is a brick, a box, a room, etc. A rectangular prism has three dimensions ; length, width, and thickness. To find the number of cubic inches, cubic feet, cubic yards, etc., in a rectangular prism, multiply the number of inches, feet, yards, etc., in its length, by the number of the same denomination in its width, and this product by the number of the same denomination in its thickness. Table Cubic Measure 1728 cubic inches (cu. in.) 1 cubic foot (cu. ft.). 27 cubic feet 1 cubic yard (cu. yd.). 1 cubic foot = 7J gallons (nearly). 1 cubic foot= J bushel (nearly). RECTANGULAR PRISMS 181 Written Exercises 1. How many cubic feet are there in a pile of wood 8 feet long, 4 feet wide, 4 feet high ? NOTE. A pile of wood 8 ft. by 4 ft. by 4 ft. contains a cord. 2. How many cubic feet of grain can be placed in a bin 5 feet long, 5 feet wide, 5 feet high ? 3. At ^ bushel to the cubic foot, how many bushels are there in 125 cubic feet ? 4. A cubic foot of water weighs 1,000 ounces. How many pounds does it weigh ? 5. At 62|- pounds to the cubic foot, what is the weight of 128 cubic feet of water ? 6. At 31 J pounds to the cubic foot, what is the weight of a cord of wood (128 cu. ft.) ? 7. If a cart body is 4 feet long and 3 feet wide, how high must it be to hold 27 cubic feet ? 8. When dry sand weighs 86 pounds to the cubic foot, what is the weight of a cart load containing 22 cubic feet ? 9. How many cubic feet are there in a wagon body 6 feet long, 4 \ feet wide, 1 \ feet high ? 10. What is the weight of a cubic yard of sand weighing 86 pounds to the cubic foot ? 182 SECTION THREE 11. How many cubic yards are there in a box 2 yards long, 1^- yards wide, \ yard high ? 12. How many cubic feet are there in 1^ cubic yards ? 13. At 7^ gallons to a cubic foot, how many gallons of water will fill a tank 8 feet long, 4 feet wide, 4 feet high ? 14. How many cubic feet of earth are removed in digging a cellar 50 feet long, 20 feet wide, and 9 feet deep ? is. How many cubic yards of air are there in a room 18 feet long, 12 feet wide, and 9 feet high ? 16. (a) How many cubic feet are there in a timber 16 feet long, 6 inches wide, 3 inches thick ? (6) what is its weight at 40 pounds to the cubic foot ? SECTION IV REVIEW AND SUPPLEMENTARY WORK Notation and Numeration For convenience in reading large numbers divide them by commas into periods of three figures each, beginning at the right. TABLE WHO WHO WHO 123, 456, 789 Millions' Thousands' Ones' Period Period Period In reading numbers omit the name of the ones' period. Read 789, seven hundred eighty-nine. Read 6,789, 6 thousand 789. Read 3,456,789, 3 million 456 thousand 789. The periods above the third are named billions 1 period, trillions' period, etc. Omit the name of a period consisting exclusively of ciphers. Thus, read 14,000,000 as 14 million ; 6,000,375 as 6 million 375. Do not use a comma in writing a number of four figures denoting a year. Numbers of four figures are frequently read as hun- dreds ; as, 19 hundred 10. The word hundred is also omitted ; as, nineteen ten. 183 184 SECTION FOUR Oral Exercises Read the following : a. 6. c. d. e. A 250,000 173,086 45,297 6,050 23,915 B 1,205,060 789,516 2,314,800 6,519,260 800,000 C 23,004,005 3,000,486 28,000,000 36,050,000 5,060,000 D 103,206,580 215,387,614 319,600,000 48,600,000 562,009,007 / 456,808 3,465,292 45,103,950 643,870,530 Written Exercises Express in figures : 1. Three hundred seventy-two thousand. 2. One million, forty-two thousand, six hundred. 3. Ten million, fifty thousand, seventy. 4. One hundred seventy-five million, two hundred three thousand, six hundred. 5. Eighty -four million, seven hundred twenty-two thousand, three hundred ninety-five. 6. Two million, five thousand, three. 7. One hundred six million, three hundred ten thousand, eight. 8. Nine hundred eighty-seven million, six hundred fifty-four thousand, three hundred twenty-one. 9. Five hundred million, two thousand, one. 10. Seven hundred eighty-five million, two hundred forty-four thousand, nine hundred fifty-six. REVIEWS 185 Addition and Subtraction Sight Drills 1. Add 128 to each of the following numbers : ABCDEFGH I J a. 93 26 92 66 84 74 96 15 87 78 b. 64 57 36 44 38 53 28 67 91 86 c. 46 34 83 25 77 88 56 85 76 37 d. 82 18 65 14 45 95 73 27 14 55 e. 25 73 47 58 98 17 35 68 94 48 2. Subtract each of the foregoing numbers from 210. 3. Add 350 to each of the following numbers: A B c D E F a. 190 125 56 1,050 2,700 155 b. 270 216 67 2,060 3,800 256 c. 350 137 78 3,070 4,900 357 d. 480 325 89 4,080 5,800 458 e. 560 445 95 5,090 6,700 559 4. Give answers rapidly : Think 221 + 100 + 50+6. Think 221 - 100 - 50 - 6. A B c a. 221 + 156 422-363 362 + 284 b. 221-156 540 + 451 362-284 c. 334 + 245 540-451 273 + 175 d. 334-245 431 + 362 273-175 e. 422 + 363 431-362 184 + 122 186 SECTION FOUR Addition and Subtraction Written Exercises 1. Find sums. Test : A B C a. 612,345 2,543,876 37,873,887 89,764 264,668 6,525,307 7,652 4,828,748 895,400 98,030 402,606 67,859 204,786 90,090 242,784 88,359 825,163 68,135 \ f P ^KvVS^ I. 842,735 7,369,644 78,642,371 94,681 377,180 1,945,852 27,658 92,143 647,481 248,569 869,515 25,635 60,370 4,625,790 847,162 85,396 608,405 3,836,505 c. 714,653 6,833,726 2,439,876 94,728 188,706 60,400 615,840 548,057 294,501 53,174 64,382 804,975 58,034 1,008,590 2,800,463 65,348 438,249 858,674 2. Find differences without rearranging : -83,604 -890,089 45,305,902 600,000 1,234,567 -7,408,904 REVIEWS 187 3. Add: A B c a. $123,456.78 $ 94,309.82 $767,088.00 90,312.20 172,074.55 126,435.79 6,734.87 99,873.00 205,719.65 539,314.60 67,021.45 80,482.36 619,512.14 216,835.50 198,060.00 6. $854,923.74 $270,050.66 $820,932.32 45,024.58 382,104.20 88,150.08 7,056.00 92,307.55 26,934.45 304,729.68 81,256.43 342,345.00 51,120.09 507,759.00 6,437.46 c. $ -82,559.24 $895,694.63 $450,880.77 76,385.64 96,438.00 57,028.24 371,626.00 42,740.08 96,462.50 27,842.96 138,128.72 792,664.82 381,730.53 69,569.06 48,125.82 78,473.75 70,591.76 38,484.57 4. Insert missing addend : A B C a. $ 157,943.87 ? $345,678.90 -1- ? + $983,654.63 f ? $ 200,000.00 $1,000,000.00 $ 1,234,567.89 6. $ 82,559.24 $895,694.63 ? 76,385.64 ? $ 350,880.77 ? 96,438.00 792,664.23 $ 200,000.00 $1,000,000.00 $ 1,234,567.89 188 SECTION FOUR Multiplication and Division Drill Exercises 1. Give products: Say twice two ten are four twenty, four times six ten are twenty-four forty. Omit the words hundred and thousand. A B C D a. 2 X 210 3 x210 4x 210 5 X 210 6. 2 X 310 3 X 310 4x 310 5 X 310 c. 2 X 410 3 X 410 4x 410 5 X 410 d. 2 X 510 3 X 510 4x 510 5 X 510 e. 2 X 610 3 X 610 4x 610 5 X 610 E F G H a. 6 X 210 7 X 210 8x 210 9 X 210 b. 6 X 310 7 X 310 8x 310 9 X 310 c. 6 X 410 7 X 410 8x 410 9 X 410 d. 6 X 510 7 X 510 8x 510 9 X 510 e. 6 X 610 7 X 610 8x 610 9 X 610 2. Give quotients and remainder, if any : Think two ten into fourteen seventy, etc. A B C D a. 210)1,470 210)1,890 210)1,260 210)1,680 6. 310)1,550 310)2,480 310)1,860 310)2,180 c. 410)3,700 410)1,640 410)2,500 410)2,050 d. 510)2,040 510)3,670 510)1,540 510)4,600 e. 610)5,500 610)1,830 610)3,070 610)4,880 / 710)5,000 710)1,430 710)6,400 710)5,680 REVIEWS 189 Oral Problems 1. A man went from New York to Baltimore, 184 miles, and then from Baltimore to Washington, 40 miles. How far did he travel from New York to Washington ? 2. At 32 miles per hour how long would it take a train to go 960 miles ? 3. What is the weight of 40 bushels of oats at 32 pounds to the bushel ? 4. Twelve hundred eighty pounds are how many pounds less than twenty hundred pounds ? 5. At 320 rods to the mile, how many rods are there in 4 miles ? 6. How many working days are there in a year of 365 days, excluding 52 Sundays and 4 holidays? 7. How many J-acre plots are there in a piece of ground containing 16 acres? 8. If a man walks 1\ miles to his work and 2J miles back, how many miles does he walk in a year of 309 working days in going to work and returning? 9. How many square feet are there in a tight fence 6 feet high around a square plot, each side measuring 200 feet ? 10. Find how much less than $ 100 will be the cost of 80 hammocks at $ 1.10 each. 11. How much change is given when a $ 20 bill is given for a purchase of $ 18.40 ? 190 SECTION FOUR Multiplication Written Exercises Multiply. Test : A B c a. 415 X 306 1,205 X 308 1,504 X 1,307 b. 421 X 509 1,307 X 345 1,498 X 1,406 c. 123 X 876 1,406 X 403 1,234 X 4,301 d. 234 X 543 1,508 X 512 1,407 X 2,048 e. 316 X 435 2,042 X 332 2,305 X 1,352 / 423 X 316 2,105 X 623 1,234 X 1,234 9- 538 X 243 3,016 X 712 1,357 X 2,468 h. 641 X 165 4,204 X 805 1,536 X 1,243 i. 812 X 143 1,234 X 916 2,135 X 1,354 j- 924 X 122 1,345 X 821 3,216 X 1,234 L 844 X 137 2,123 X 777 2,123 X 2,331 I 763 X 204 3,214 X 666 3,214 X 1,332 m. 627 X 212 4,125 X 555 4,125 X 1,665 n. 586 X 423 1,326 X 987 1,263 X 4,164 o. 437 X 514 2,314 X 876 2,143 X 2,468 P- 359 X 625 3,245 X 765 3,425 X 1,357 9,177-19 9,432-*- 18 9,129-^-17 9,120-5-16 b. 9,773-29 9,660-*- 28 9,828-27 9,386-26 c. 9,555-39 8,968-38 9,250-37 9,252-36 d. 9,163-5-49 8,400-48 9,071 -i- 47 9,062-46 e. 9,617-59 9,106-5-58 9,519-57 9,576-56 / 8,832-69 9,044-68 9,916-67 9,900-66 g. 5,925-79 5,148-78 9,625-77 9,652-76 h. 7,298-89 6,864-5-88 7,569-87 7,568-86 i. 6,237-99 8,134-98 9,894-5-97 9,984-96 \ 2. Divide. Test : A B C a. 8,159-199 61,854-793 608,006-5-634 b. 9,536-298 47,276-892 472,515-545 c. 9,131-5-397 33,694-991 336,528-456 d. 7,936-496 58,565-901 585,365-5-367 3. Divide. Test: A B a. 974,036-5-1,204 1,522,756-5-1,234 6. 827,431-5-2,033 4,704,070-2,345 c. 925,344-3,024 9,381,372-5-3,124 d. 953,428-4,006 8,417,040-5-4,126 e. 903,060-1-5,017 5,548,040-5-5,234 192 SECTION FOUR Oral Problems 1. A gross is 12 dozen. How many pencils (a) in \ gross ? (&) In \ gross ? (c) In J gross ? (d) In J gross ? 2. How many quarts of oil are there in a barrel containing 51 gallons ? 3. From a farm of 300 acres, 261 acres were sold. How many acres were not sold ? 4. How many quarter-dollars are there in $ 22 ? 5. Mr. Alphin sets out 88 hundred cabbage plants in 44 rows. How many plants are there in each row ? 6. If a wagon weighs 12 hundred pounds and the straw on it weighs 22 hundred pounds, what is the weight of both ? 7. From a barrel of oil containing 50 gallons 10 \ gallons were sold. How many gallons remain ? 8. How many days (a) in 50 weeks ? (6) In 2 weeks ? (c) In 52 weeks ? 9. How many dollars are there in 888 quarter dollars ? 10. How many days will 48 pounds of butter last if 16 pounds are used in a week ? 11. If a man earns $ 125 per month and his ex- penses are $ 94 per month, how many dollars can be saved (a) in a month ? (6) In a year ? 12. What is the weight of 1,000 common bricks at I Ib. each ? REVIEWS 193 13. At 160 bricks per day, how many days are re- quired to lay 960 bricks ? 14. What is the cost of a cement walk 3 feet wide, 100 feet long, at 8 cents per square foot? is. A man takes an automobile trip 40 miles and back at the rate of 8 miles per hour going, and 10 miles per hour returning. How long does he require for the trip ? is. I bought 20 yards of velvet for $ 70 and sold it at a profit of $ J per yard. How much per yard did I receive for it ? 17. Two boys start from the same point and travel in opposite directions at the rate of 2J- miles per hour. How far apart are they in 5 hours ? is. What is the cost of a horse and a wagon if the horse cost $250 and the wagon cost $125 ? 19. A field is 300 yards square. How many yards of wire will be required to inclose it with a fence 4 wires high ? 20. A man bought a lot for $650, giving 12 cows, for which he was allowed $50 each, and the balance in cash. How much cash did he give ? 21. After selling one half of his crop of cotton Judge Wilson has 324 bales left. Of how many bales did the crop consist ? 22. A boy used 224 of a bunch of 250 shingles in mending a roof. How many shingles were left ? 194 SECTION FOUR Written Problems 1. A great gross consists of 12 gross of 12 dozen each. How many pens are there in one half of a great gross ? 2. How many quarts of oil are there in 49 gallons ? 3. From a farm of 300 acres 137J acres were sold. How many acres remain ? 4. How many quarter dollars are there in $ 27J ? 5. If 9,180 cabbage plants are required for 45 rows, how many plants are there to a row ? 6. A wagon weighs 1,245 Ib. and the straw on it weighs 2,156 Ib. What is the weight of both ? 7. From a barrel of oil containing 50 gallons, three lots were sold of 10J gallons each. How many gallons remain ? 8. How many days are there in 26 weeks ? 9. How many dollars are there in 972 quarter dollars ? 10. How many days will 255 pounds of butter last if 17 pounds are used each week ? 11. If a man earns $ 112.50 per month and his ex- penses are $ 83.75 per month, how many dollars can he save in (a) 1 month ? (&) 1 year ? (c) 2 years ? 12. A girl makes a steamboat trip of 100 miles and back at the rate of 16 miles per hour going and 12 miles per hour returning. How many hours are re- quired for the round trip ? REVIEWS 195 is. What is the weight of 960 common bricks at 1^ pounds each ? 14. At 160 bricks per day, how many days are re- quired to lay 4,000 bricks ? is. At 10 cents per square foot, what is the cost of a cement walk 5 feet wide, 164 feet long ? 16. A dealer bought 32 yards of cloth for $ 80 and sold it at a profit of $ J per yard. How much did he receive for it per yard ? 17. Two boys start from the same point and travel in opposite directions, each going at the rate of 2j- miles per hour. How far apart are they in 8 hours ? is. What is the cost of a horse, a wagon, and a set of harness, if the horse cost $ 375, the wagon $ 150, and the harness $35 ? 19. A field is 256 yards square. How many, yards of wire will inclose it with a fence 4 wires high? 20. A man bought a lot for $800, giving 12 cows at the rate of $55 each and the remainder in cash. How much cash did he give ? 21. After selling three fourths of his crop of hay Mr. Whitlock has 123 tons left. How many tons were there in the crop ? 22. A boy uses 787 shingles out of 4 bunches of 250 shingles each. How many shingles are left? 23. How many tons of 2,240 pounds each do 1,600 paving bricks weigh if a brick weighs 7 pounds ? 196 SECTION FOUR Aliquot Parts of Dollar Oral Exercises 1. How many dollars are equal to 44 half dollars ? 2. What is the cost of 2 dozen baseballs at $^ each? 3. How much must be paid for 48 yards of dress goods at 50 cents per yard ? 4. A girl has saved 12 dollars and 12 half dollars. How much has she saved ? 5. Find the cost of 2 dozen straw hats at $1J each. 6. A man cut down 64 trees, each yielding 1^ cords of wood. How many cords of wood did he cut ? 7. What will be received for cutting 64 cords of wood at $1^ per cord? 8. -At 25 cents per dozen, what is received for 48 dozen eggs ? 9. What is the yearly cost of a newspaper at 25 cents per month ? 10. At 25 cents each, what is the cost of 88 readers ? 11. When meat is 25 cents per pound, how much does the butcher receive for 168 pounds? 12. If 4 pairs of cuffs cost a dollar, how many pairs can be bought for $ 42 ? is. At 4 for a dollar, how many baseballs can be bought for $ 12 ? 14. How many 50-cent baseballs will cost $ 24 ? REVIEWS 197 Sight Exercises i. Give the cost of each of the following items : a. 120 yards of silk @ $ 1.50 per yard. $120 + 1 of $120. 6. 248 gallons of sirup @ 50 cents per gallon. 248 half dollars. c. 484 pounds of coffee @ 25 cents per pound. 484 quarter dollars. d. 620 bushels of apples @ $1.50 per bushel. e. 840 handkerchiefs @ 25 cents each. f. 360 pairs of socks @ 25 cents per pair. g. 168 baseballs @ 50 cents each. A. 440 penknives @ $ 1.50 each. i. 120 gallons of oil @ 16 cents per gallon. 16 times 120. j. 210 pounds of sugar @ 6 cents per pound. k. 320 yards of velvet @ $ 3 per yard. L 24 carpet rugs @ $ 20 each. m. 180 dozen collars @ $ 1.50 per dozen. n. 44 bushels of wheat @ $ 1.02 per bushel. 44 times $1.02. o. 110 barrels of flour @ $ 6 per barrel. p. 15 horses @ $300 each. q. 2 automobiles @ $ 2,400 each. r. 200 acres of land @ $ 40 per acre. s. 488 pounds of butter @ 25 cents per pound. t. 620 geranium plants @ 4 cents each. 198 SECTION FOUR Cancellation Preparatory Exercises i. (a) How many pigs at $ 7 each will pay for 27 cows at $ 35 each ? (6) How many pigs at $ 7 each will pay for 28 cows at $ 34 each ? A LONG METHOD (a) $35 In (a) the value of the cows is 27 x ^ ' times $35, or $945. The quotient 245 of $945 by $7 gives 135 as the 70 number of pigs. $ 7)1T945 Ans. 135 (pigs) Find the answer to (6) by this method. Examples involving only multiplication and divi- sion can sometimes be shortened by cancellation. A SHORTER METHOD In (a) indicate the product of 35 by 27, by 35 x 27 placing the multiplication sign (x) between n them. Indicate that this product is to be di- vided by 7 by writing the latter under a line drawn below the former. Divide 35 by 7, canceling both, and writing the quotient 5 above 5 35 X 27 *ke f rmer - The prod- -= = 135 (pigs) Ans. uc t of 5 and 27 is the number of pigs, 135. Find the answer to (6) by this method. CANCELLATION 199 Written Problems i. How many yards at 18 cents per yard will pay for 21 pounds of butter at 24 cents per pound ? PROCESS Indicate the product of 24 (/) j * by 21 and the quotient of this re- rf x fffi _ 2 suit by 18. Divide 18 and 24 by ^ 6, canceling both, and writing the respective quotients 3 and 4. Cancel 3 and 21, writing the quotient 7. Write the product of 4 and 7 as the number of yards. 2. How many pounds of sugar at 6 cents a pound will pay for 15 pounds of butter at 24 cents a pound ? 3. How many pounds of butter at 25 cents a pound will pay for 75 pounds of sugar at 6 cents a pound ? 4. How many pounds of pork at 15 cents a pound can be exchanged for 27 bushels of potatoes at 50 cents per bushel ? 5. How many dozen of eggs at 33 cents per dozen will pay for 22 pounds of ham at 20 cents per pound ? 6. When a 6-acre field yields 95 bushels of wheat, what should be the yield of an 18-acre field ? 7. When 18 acres yield 276 bushels of wheat, what should be the yield of 6 acres ? 8. At the rate of 370 miles in 20 hours, how many miles would a freight train go in 15 hours ? 200 SECTION FOUR 9. If 16 men require 20 days to do a piece of work, how many days would it take 15 men to do it? PROCESS Indicate the time re- * I n x s(i\ quired by 1 man, 16 times rr __ 64 _ 20 days; and the quotient f-j> of this product by 15 as the time needed by 15 men. Divide 15 and 20 by 5, canceling both, and writing the respective quotients 3 and 4. Write the product of 16 by 4 above a line and 3 underneath it, making the improper fraction ^ 4 -, which is reduced to 21^, the number of days. 10 ; If 16 men require 15 days to dig a ditch, how many days would it take 12 men to do it ? 11. If 16 men can do a piece of work in 18 days, how many men would be needed to do it in 12 days? 12. If 13 men require 17 days to do a piece of work, how long would it take 39 men to do it ? 13. If 14 men require 19 days to do a piece of work, how long would it take 38 men to do it ? 14. When 135 bushels of corn are raised on 3 acres, how many bushels should be raised on 7 acres at the same rate ? is. At 60 pounds to the bushel, how many bushels of clover seed are there in 7 bags weighing 150 pounds each ? FRACTIONS 201 Relation of Numbers Preparatory Exercises 1. Find ^ of each of the following numbers : a. 24. W b. 48. c. 62. d. 86. e. 50. / 70. 2. Find ^ of each of the following numbers : a. 24. b. 36. c. 48. d. 84. e. 64. /. 96. 3. Find ^- of each of the following numbers : a. 18. 6. 24. c. 36. d. 48. 6. 69. / 96. 4. Find |- of each of the following numbers : a. 16. 1. 20. c. 32. d. 40. e. 48. /. 80. 5. Find | of each of the following numbers : a. 15. b. 21. c. 33. d. 63. e. 90. /. 96. 6. What fraction of 60 is each of the following : a. 30? b. 20? c. 15? d. 12? e. 24? /. 40? 7. What fraction of 48 is each of the following : a. 8? 6. 16? c. 24? d. 12? e. 36? /. 32? 8. What fraction of a foot is each of the following : a. 2 in.? b. 3 in.? c. 4 in.? d. 5 in.? e. 6 in.? /. 7 in.? g. 8 in.? A. 9 in.? i. 1 in.? / 12 in.? k. 10 in.? J. 11 in.? 9. What fraction of a pound is each of the following : a. 1 oz.? b. 2oz.? c. 3 oz.? d. 4 oz.? e. 5oz.? /. 6oz.? g. 7 oz.? 7i. 8 oz.? i. 9oz.? j. 10 oz.? &. 11 oz.? /. 12 oz.? m. 13 oz.? n. 14 oz.? o. 15 oz.? _p. 16 oz.? 202 SECTION FOUR 10. What fraction of 24 is : a. 1? 6. 5? c. 7? d. 11? e. 13? /. 17? gr. 19? h. 23? 11. What fraction of 24, in lowest terms, is : a. 20? 6. 9? c. 15? d. 21? e. 10? / 14? 0. 22? h. 18? 12. Find the number whose one-half is : a. 10. &. 15. c. 20. d. 25. e. 30. / 40. 13. Find the number whose one-fourth is : a. 10. 6. 12. c. 21. d. 25. e. 31. /. 42. 14. Find the number whose one-third is : a. 12. 6. 13. c. 15. 6?. 16. e. 21. /. 32. 15. Find 2 thirds of a number when 1 third is : a. 10. 6. 11. c. 14. d. 24. e. 33. /.. 44. is. Find 1 third of a number when 2 thirds are : a. 22. &. 30. c. 36. d. 44. e. 60. /. 80. 17. Find 3 thirds of a number when 2 thirds are : a. 22. 6. 30. c. 36. d. 44. e. 60. / 80. is. Find 4 fourths of a number when 3 fourths are : a. 12. b. 18. c. 24. d. 30. e. 60. /. 90. 19. Find the number when 4 thirds are : a. 12. 1. 16. c. 32. d. 40. e. 48. /. 84. 20. Find the number when 4 fifths are : a. 12. 6. 16. c. 32. d. 40. e. 48. /. 84. FRACTIONS 203 Oral Problems 1. If \ pound of butter costs 12 cents, what is the price of a quarter of a pound ? 2. If one half of a pie costs 10 cents, what is the price of a quarter of a pie ? 3. If a girl paid 15 cents for 3 quarters of a pie, what was the price of each quarter ? 4. How is the price of a quarter of a yard found when you know the price of a half yard to be 20 cents ? 5. How is the price of 1 fourth of a pie found, when you know the price of 3 fourths to be 15 cents ? 6. When 2 equal parts of a number are 60, what is each of the equal parts ? 7. When 3 of the equal fields of a farm contain 60 acres, how many acres are there in each ? 8. If 3 fourths of a number is 12, (a) what is 1 fourth of the number ? (6) What is the number ? 9. If 2 thirds of a yard of goods cost 22 cents, what will 3 thirds of a yard cost ? 10. If 9 in. of wire cost 18^, what will 12 in. cost? 11. If 3 fourths of a foot of wire cost 12 cents, what will 4 fourths of a foot cost ? 12. Find the cost of a gallon of oil when 3 quarts cost 12 cents. 204 SECTION FOUR Fractions A fraction is one or more of -the equal parts into which a thing is divided. A fraction consists of two numbers, called the terms of the fraction, one written above the other with a line between them. When an apple, a pie, a number, etc., is divided into two equal parts, each part is called a half; when it is divided into three equal parts, each part is called a third; etc. One half, one third, one fourth, etc., are written : l, i, 1, etc. If a pie, for instance, is divided into five equal parts, one of these parts is called one fifth ; a portion equal in size to two of these parts is called two fifths ; a portion equal in size to three of these parts is called three fifths ; etc. The number written below the line is called the denominator of the fraction ; it shows into how many parts a thing is supposed to be divided. The number written above the line is called the numerator; it shows the number of these parts that are taken. A fraction indicates division : the numerator being the dividend and the denominator the divisor. FRACTIONS 205 Proper and Improper Fractions A. proper fraction is a fraction in which the number expressing the numerator is smaller than the number expressing the denominator ; an improper fraction is one in which the number expressing the numerator is at least as large as the number expressing the denom- inator. i> f > 5~ ? ? e * c< > are P r P er fractions. -|, -|, J^-, J^, etc., are improper fractions. A mixed number is a whole number and a proper fraction written together. 1J, 3f , 19J, etc., are mixed numbers. An improper fraction may be reduced to a whole or to a mixed number by dividing the numerator of the fraction by the denominator. Sight Exercises 1. Change to whole or to mixed numbers : ABODE tt H V tt tt 2. Change to improper fractions : A B c D E 6. 206 SECTION FOUR Lowest Terms A change in the form of a fraction that does not alter its value is called a reduction. A fraction is reduced to lowest terms by dividing the numerator and the denominator by the largest number that will exactly divide both terms. Sight Exercises Reduce to lowest terms : ABODE & * * A A A c - 9" T6" "2"o~ 2T TS" ^ 12. 16. .2J) 1_8 15 Higher Terms A fraction is changed to higher terms by multiply- ing both terms by the same number. Sight Exercises 1. Change \ to 4ths. To 8ths. To 24ths. 2. Change \ to 6ths. To 9ths. To I5ths. 3. Change f to 6ths. To 9ths. To 18ths. 4. Change f to 8ths. To 16ths. To 24ths. 5. Change f to 12ths. To 24ths. To 30ths. 6. Change f to IGths. To 24ths. To 40ths. 7. Change f to IGths. To 24ths. To 40ths. a. Change f to lOths. To 20ths. To 30ths. FRACTIONS 207 Common Denominators To add or to subtract fractions, they must have the same denominator. When two or more fractions are changed to equivalent fractions each having the same denominator, they are said to be reduced to a com- mon denominator. Sight Exercises 1. Reduce each set to fractions having a common denominator : A B a- i> ib t i> t> TO- b- J> i> J "2? ^' To C ' i> ' F 2"' "6"' I'D" d- \-> i> i > ^? i 2. Find sums : Reduce improper fractions in results to mixed num- bers. ABC i+* i+* *+l 6- t+t i+i t+* e. i+* i+f i+* ;* *+* *+ t+f 3. Subtract: ABC i-t i-t i-t 208 SECTION FOUR Addition of Fractions Written Exercises i. A boy divided a strip of copper into four pieces measuring, respectively, f ft., j- ft., f ft., and |- ft. What was the length of the strip ? PROCESS Find the least common denominator of and -J, reject- ing the other denominators 3 and 4, each of which is a factor of 6 or of 8. Test successive multiples of 8, until one is found, 24, that is a multiple of 6 ; 24 is the required common denominator. Change each fraction to 24 3. 24ths, writing only the new numerators in a line under the common denominator. Write the sum of the nu- If merators,75, over the denom- - inator, 24. Change -f to Ans. 3" , and then to 3|. 16 18 20 21 .2-5. = Q_3_ _ 2. Add the following : ABC Hi+i i+S+I *+*+* i+l + l i + t+! i+i + i i+f+A i+l+A *+*+* +* *+l+A i+A+i 17- t + f + f * + * + * l + l + f *- i+f+f 4+f+A f+f+A FRACTIONS Addition of Mixed Numbers 209 i. A boy divided a rope into four pieces measuring 171 yd., 8| yd., f yd., 20f yd., and 9^ yd., respec- tively. What was the length of the rope ? PROCESS Reject the denom- 36 inators2,3,and4,each jyi 18 of which is a factor of ft 2 od. 9 or of 12. Test suc- cessive multiples of 9" ^" 12, until one is found, 20 J 27 36, that is a multiple 9_5_ J5 of 9 ; write 36 as the ^ ng ^Q^ jjy. common denominator. = 2|f = 2| Change the fractions to 36ths, find their sum , and reduce the result to 2f. Write f under the original fractions and carry 2 to the ones' column of the whole numbers. 2. Add the following : A B C D E a. 6 lOf 33J 53 17| 20J 5f 16 20f 12J 15 27i 2f 7J 13 3 A 19 !f 9 A r H r 3 g- b. 24J 30J 27 3f 16f 7^ 36 18f 17J 27f 16U 7i 124 28 8-1 20 7A 210 SECTION FOUR Subtraction of Mixed Numbers Preparatory Exercises 1. What fraction added to -J gives 1 as the sum ? 2. What fraction added to -| gives a sum ^ greater thanl? i+?=lf 3. What fraction added to J gives 1J- as the sum ? Sight Exercises Subtract : A B C D a. 1 zi li zi If zi b. 1 li 11 11 zi zi zi _ 5 5. c. 11 ij 11 If zi ri zi zi d. zi li .2. zi _1 e. 1* U li If # F H if zi zi if if zi zi if if zi zi zi zi if *i- _1 -.3. -1 -^ -5 4^ 2. 4. 2. 4. S / 2 i 3 i 4 i 5} 6| 7J zi zi zi zi zi zi g. 4J 5J 6J 7^ 8| 9i -31 -41 -51 -6* -74 -81 ,_.._-. O ^ ^ O ^ B O SUBTRACTION 211 Written Exercises i. From 60^- yards of iron cable a man cut 32| yards. How many yards were left ? PROCESS Since 36 is a multiple of 36 -321 17 53 32 21 _ 7 36 ~~ 12 9, 36 is the least common denominator. Write 36, and underneath it the numerator 17, and the new numerator 32. Since ff is greater than ^, increase the latter by 1, making it 1^, and change it to the improper fraction -||-, writing the numerator of the latter alongside for convenience in performing the sub- traction. Take 32 from 53, writing the difference, 21, over the common denominator. Reduce ^ to fa. Write -fa under the original fractions and carry 1 to the ones' figure of the subtrahend. Say 3 and 7 (writing 7) are 10 ; carrying 1 to 3, say 4 and 2 (writing 2) are 6. Test by adding the remainder and the subtrahend. 2. Subtract. Test : ABC a. 511 -27f 48J-27f 86J-37| 6. 80f-16| 35f-12f 63|-36f d. 75* -55| 67i-49fJ- 42f-36i O *x. V JL O v e. 30^-15^ 40f-14 T V 50-20f 23 T ^ 80J-70J 212 SECTION FOUR Sight Problems 1. From a tub containing 501 pounds of butter there were sold 20f pounds. How many pounds remain ? 2. What is the cost of 8 yards of muslin at 10 j cents a yard ? 3. How much is received for 10 J yards of velvet sold at $ 4 per yard ? 4. A coat takes 3^ yards of cloth and a vest -|- yard. How many yards are required for both ? 5. After walking 2J miles, how far must a girl still walk to complete 41 miles ? 6. If the distance around a running track is a quarter of a mile, how many times around it must a boy run to go 1\ miles ? 7. At 40J miles per hour, how far will a train go in 10 hours? 8. If it requires 3 bushels of seed wheat for 2 acres, how many acres can be sown with 30 bushels ? 9. (a) At $ ^ each, how many books can be bought for $ 8J ? (6) How many books at $ f each can be bought for $ 8J? 10. At If bushels to the acre, how many bushels of oats will be required to sow 40 acres ? 11. At the rate of 31 J gallons to a barrel, how many gallons are there in a hogshead of 2 barrels ? 12. Take three times 1 from 100. REVIEWS 213 Written Problems 1. From a tub containing 52J- pounds of butter 27^- pounds have been sold. How many pounds are left? 2. Find the cost of 8 yards of dress goods at 56|- cents per yard. 3. A man buys 23^ yards of velvet at $ 3.60 per yard. How much does he pay for it ? 4. How much cloth is required to make a suit if the coat takes 3-|- yards, the vest % yard, and the trousers 1|- yards ? 5. After riding 12|- miles, how many miles remain of a journey of 20^- miles ? 6. If the distance around a track is an eighth of a mile, how many laps must a bicycle rider make to complete 13|- miles ? 7. At 37J- miles per hour, how far will a train go in 10 hours? 8. If it requires 3 bushels of seed wheat to sow 2 acres, how many acres can be sown with 72 bushels ? 9. (a) At $ ^ each, how many books can be bought for $1SJ? (b) If the books are 3 quarter dollars each, how many can be bought for $ 18|-? 10. At If bushels to the acre, how many bushels of seed will be required for 56 acres ? 11. There are 5J yards in a rod. How many inches are there in a rod ? 214 SECTION FOUR Product of a Mixed Number and a Whole Number Preparatory Exercises 1. Add 1 and \. Add 12J and 12f Multiply 121 by 2. 2. How many quarters are 8 times 1 quarter? How many ones are 8 times J? Multiply $1 by 8. Multiply $ 1J by 4. 3. What will 8 baseballs cost at $ 1^ each ? 4. How many fourths are 8 times 3 fourths ? How many ones are 24 fourths ? |- of 8 = ? 8 times f = ? 5. Multiply 7f by 12. PROCESS 7fxl2 = 92 Ans. Think 12 times 2 thirds are 24 thirds, or 8. Carry 8 to 12 times 7 (84), making 92. Write 92. Test by dividing 92 by 12. Sight Exercises Give products : A B C D E F x2 x4 x6 x8 x8 x4 b. 3J 4J 5| 6 7| S\ x3 x6 x3 x9 x6 x9 c. 9J 8f 7f 6f 51 4j x5 x5 x5 x5 x6 x6 WRITTEN EXERCISES 215 Written Exercises i. A barrel of potatoes weighed 24 6f Ib. What would be the probable weight of 12 barrels? PROCESS Think 12 times f are - 3 ^-, or 9. Carry 9 to 12 times 6 (72), making 81, etc. Test by dividing 2961 by 12. A ^Lf 246f Ib. . 2961 Ib. 2. Find products A a. 25jx8 &. 46|x4 c. 75fx8 d. 18|xl2 B 17Jx8 28f x5 87f x6 26f x 10 c 331x6 66f x9 93|x8 42f x!2 3. Multiply 437J by 126. PROCESS 4371 Keduce ip, the product of by 126, 126 to 31 by dividing 126 by 4 (without rewriting these numbers), and change | ocoo to . Write 31f Multiply 437 by 6, placing the right- $74 hand figure of the product in the ones' 437 column. Place the right-hand figure of 55093i A.11S. the product by 2 tens and by 1 hundred in the tens' and hundreds' columns, respectively. Com- bine the partial products. 216 SECTION FOUR 4. Find products : A a. 36^x25 6. 42jxl8 c. 911x19 d. 721x23 s. Multiply 275 by 16. B C 291 x 17 63Jxl5 541x31 70x27 83^ x 16 65^x26 52^1- x 24 441 x 22 275| PBOCESS 16 Find the product of by 16 by mul- 6)80 tiplying 16 by 5, and dividing the 13()l result by 6, reducing the fraction to \. Under this write the product of 6 1650 times 275, etc. 275 44131 Ans. A B c a. 48| X 25 37J x 17 64f X 15 b. 73| X 16 82|- X 14 90f X 24 c. 56^ X 13 62* X 18 86 X 21 d. 70| X 19 58| X 23 49^ -X 22 e. 22f X 16 41| X 22 75f X 31 / 31| X 22 32f X 31 54f X 33 9- 63f X 31 23f X 18 43f X 21 h. 45f X 15 55f X 17 32| X 22 i. 54f X 20 34| X 13 21f X 14 MULTIPLICATION 6. Multiply 285 by 161 217 PROCESS Find the product of 285 by % by divid- ing 285 by 6. Find the other two partial products and combine the three. 4607^ Aw. 25x131 a. 6. 52x16^ c. 82xl9 B 37 X 141 63 x 171 98 7. Multiply 285 by 23f 98x211 C 49 x 151 74 x 181 87 x 45 T V 285 23J Find the product PROCESS of 285 by by 6)1425 OQ7/3\ 1 multiplying 285 by 5 and dividing the " ' \F/ 2" result by 6, reducing the fraction. 855 570 67921 Aw. A B c a. 98x23f 98 x 24f 78 x 25f .6. 67x26f 58 x 27f 49 x 32f c. 39x33^ 27 x 34^ 36x4l| d. 47x42| 54 x 43|- 63 x 44^ e. 72x51f 66 x 62$ 55 x 73^ 218 SECTION FOUR Multiplication of Fractions Preliminary Exercises 1. A boy earned $ \. He gave \ of it to his sister. What part of a dollar did he give her ? 2. Draw 3 circles. Divide one of them into 2 halves, one into 4 fourths, and one into 3 thirds. 3. What is (a) i of 1? (&) i of i? (c) 1 of 1? 4. Draw two circles. Divide each into halves. 5. What is (a) of ? WiofJ? (c)Jofi? (d) lofl? (e) lofj? (/) Jofi? 6. What is (a) of 2 thirds ? (6) of 3 fourths ? (c) 1 of 4 fifths ? (d) i of 4 fifths ? 7. At $ J per yard, find the cost of (a) 4 yards ; (b) 3 yards ; (c) 2 yards ; (d) ^ yard ; (e) 1^ yards. 8. (a) 4 times $J = ? (6)3x$J = ? (c)2x$J = ? MULTIPLICATION 219 Written Exercises i. Multiply 71 by (a) J; (6) J; (c) Ifc (d) 1J; PROCESS (a) 71 xl=i/x 1 = ^ = 31 Change 1\ to an improper fraction and rewrite the fractions. Place 15, the product of the numerators, over 4, the product of the denominators, making *-. Reduce this to a mixed number. Cancel 3 in a denominator and 15 in a numerator. Place the product of the numerators, as changed, over that of the denominators, etc. (c) 7ixli = -yLx-| = -^=lli Ans. Change both mixed numbers to improper fractions, etc. (d) 7ixl|-=^x|=10 Ans. Cancel 3 and 15 by 3, 2 and 4 by 2. Omit 1, the prod- uct of the changed denominators, and write only 10, the product of the changed numerators. (e) 7lx2i = ^xf = lfA=i 6 Ans. 2. Multiply : a. 220 SECTION FOUR Division of Fractions Preliminary Exercises 1. A man divided $ ^ equally between his 2 chil- dren. What part of $ 1 did each receive ? 2. If I have $ 6, how many times can I spend (a) $i? (6) $i? 3. If I have $ 6 in quarters, how many times can I spend 3 quarters ? 4. To save $ 6, how many times must I save (a) 5. (a) 6+-!=? (6) 6. Give quotients : (a) 2)12 (6) 1)12 (c) J)12 (d) })12 7. Give dividends : (a) 211_ (6) Jll_ (c) !)_?_ W J)J_ 12 12 12 12 8. Give divisors : (a) ?)48 (6) ?)18 (c) ?)24 (d) ?)10 24 24 48 40 9 . (a) 6-1 = 6 x? (6)6-*-i=6x? (c) 6 + f = 6x? (d) 6^-f=6x? (e) 6-*-f = 6x? 10. Give quotients : (a) f )12 (6) |)12 (c) f )10 (d) f)12 DIVISION 221 Written Exercises i. At $ 1 -J per yard, how many yards of cloth can .be bought for $ 9| ? PROCESS The number of yards is the quotient of 9f by 1. (a) (6) (c) 13 9}-*- 1* = - 5 -f=^xf =V = 6i (yd,) 2 In (a) indicate the division of the mixed numbers. In (&) rewrite the mixed numbers as improper fractions. In (c) indicate the multiplication of the dividend by the divisor inverted. Perform the indicated multiplication. TEST Multiply the quotient 6|- by the divisor 1. 2. Find quotients : ABC a. 9J-H6J. 19fH-29j- 83J+12J 6. 83J-J-6| 12J-H83J 71J-1Q1 c. 71|-5-3^ 10J-*-71f 87J H- 100 d. 87^ -6J 12J-J-87J e. 37^ + 3^ 12^-!-33J 222 SECTION FOUR Fractional Relations 1. The operation a. Of finding 2 times 8 is indicated thus : 2 x 8. &. Of finding 2J times 8 : 2J x 8. c. Of finding 2| times 8 : 2f x 8. d. Of finding \ times 8 : ^- x 8. 6. Of finding j- times 8 : f x 8. / Of finding f of 8 : f x 8. 2. Each of the foregoing is a multiplication exam- ple, even if one-half of 8 is obtained by dividing the latter by 2. 3. The operation of finding the number a. That produces 16 when multiplied by 2 is indi- cated thus: 16-*- 2. &. That produces 20 when multiplied by 2^: 20-21. c. That produces 22 when multiplied by 2f : 22+2f. d. That produces 4 when multiplied by \ : 4 -4- J. e. That produces 6 when multiplied by f : 6 + f . 4. The operation of finding the number a. Of which 4 is J is indicated thus : 4 -*- J. &. Of which 6 isf : 6-*-. 5. Each of the foregoing is a division example, even if 8-*- \ is obtained by multiplying the former by 2. REVIEWS 223 Oral Problems One Operation State the operation required to solve each of the following examples when the given numbers are employed. 1. When dress goods are 48 cents a yard, what is the cost of |- yard ? 2. When |- pound of coffee costs 18 cents, what is the price of a pound ? 3. When tea is 60 cents a pound, how much can be bought for 90 cents ? 4. What part of a book of 60 pages have I read when I have completed 40 pages ? s. What is ^ of 84 ? 6. What fraction of 84 is 49 ? 7. 40 is |- of what number ? 8. 24 is what fraction of 32 ? 9. 55 contains 15 how many times and what frac- tion of a time ? 10. 10 is what fraction of 15 ? 11. How many quarter pints of ice cream are there in 3 pints? If a boy eats |- pint, how many boys will eat 3 pints ? 12. How many times can I take a quarter out of a box containing 6 dollars in quarters? How many times can I take 3 quarters out of the same box? How many times can I take out 2 quarters ? 224 SECTION FOUR Written Problems NOTE. First indicate the solution by placing the proper sign between the given numbers. 1. When cloth is 96 cents a yard, what is the cost of |- yard ? 2. When j pound of tea costs 45 cents, what is the price of a pound ? 3. When dress goods are 64 cents a yard how many yards can be bought for 88 cents ? 4. What part of a book of 64 pages have I read when I have finished 24 pages ? 5. (a) What is JJ- of 96 ? (6) Jfofl44? 6. What fraction of 96 is (a) 44? (5) 72? (c) 36? 7. 72 is (a) if of what number? (6) ^ O f what? 8. 64 is what fraction (a) of 96 ? (&) Of 144 ? 9. How many times and what fraction of a time does 448 contain 160? 10. (a) 128 is what fraction of 160? (6) 156 is what fraction of 160 ? (c) 20 is what fraction of 64 ? 11. (a) How many quarter pints are there in 27 pints of ice cream ? (&) Among how many boys can 27 pints be divided if each receives 3 quarter pints ? 12. How many |- inches are there (a) in 12 inches ? How many are there (6) in 18 inches? (c) In 1 yard? DECIMALS 225 Decimals Preparatory Exercises 1. In 'writing $83.17, how many dollars does the 8 represent ? The 3 ? What part of a dollar is rep- resented by the 1 ? By the 7 ? 2. What part of a dollar is 17 cents ? How many cents are there in ^j- of a dollar ? 3. Express as dollars and cents : 4. Express as dimes and cents : $0.77. $0.49. $0.82. $0.03. 5. Express as tenths of a dollar. As hundredths of a dollar : $0.40. $0.10. $0.30. $0.50. 6. What part of a dollar is 30 cents, 70 cents, 10 cents, 90 cents ? 7. Express as hundredths of a dollar : $0.37. $0.13. $0.75. $0.84. 8. In the expression $ 333.33, how many dollars are represented by the left-hand 3 ? By the next 3 ? By the next 3 ? What part of a dollar is represented by the next 3 ? By the right-hand 3 ? 9. Read $ 333.33 as dollars and the fraction of a dollar. Read $ 33.30 as dollars and the fraction of a dollar, expressing the fraction in lowest terms. 226 SECTION FOUR Notation and Numeration Fractions such as ^, J, ^, etc., are called common fractions ; fractions whose denominators are, 10, 100, 1,000, and the like, are sometimes called decimal frac- tions, such as -ffr, -J^L, j^Q-, etc. These fractions are called decimals when they are written without denominators, the prefixed decimal point showing that they are fractions. M H O Q H M 111.11 The number 111.11 is read one hundred eleven, and eleven hundredth s,' the word "and" being used between the whole number and the decimal. The number 23.4 is read twenty-three, and four tenths. The number 256.03 is read two hundred fifty-six, and three hundredths. The number .7 is read seven tenths; the number .56 is read fifty-six hundredths; the number .09 is read nine hundredths. In the . expression $123.45, how many times is the value expressed (a) by 5 contained in the value expressed by 4 ? (6) The value expressed by 4 in that expressed by 3 ? REDUCING DECIMALS 227 Reading and Writing Decimals 1. Read the following : a. 103.27 / 983.41 1). .5 g. .22 c. 20.03 h, 6.15 d. .87 i. 32.08 e. 864.09 j. .7 2. Write as decimals or as whole numbers and decimals : a. One hundred nine, and nine hundredths. 1}. Twenty-seven, and seventy-three hundredths. c. Five tenths. d. Sixteen hundredths. e. Eight hundred sixty-five, and seven tenths. Reductions Sight Exercises Express the following decimals as common frac- tions in lowest terms : A B c D a. .02 .25 .75 .22 6. .5 .4 .04 .05 c. .08 .84 .45 .38 d. .95 .8 .12 .18 e. .6 .85 .2 .06 228 SECTION FOUR Written Exercises Write the following common fractions as decimals A B c D * i * A if ^ A- A * 41 * f ^" tt TD i. Give sums : Sight Exercises 2 + 3 .5 .5 2 + 3 .05 .05 2 + 3 .2 .25 Ans. Omit a final decimal cipher in giving the answers. 6 Ans. 5 .1 Ans. 5 .45 j 1.5 + 1.5 1 + 1 B .25 .75 c 2.5 + 1.25 D 3 + 2 .5 3 + 3 .15 .4 F 3.6 + 3.45 G 2.4 + 2.7 2. Give remainders : 5 3.5 4.75 Vacant deci- o .5 -1.5 1 25 ma ^ P^ aces i n 2 .5 Ans. 2. An Q _ . uie subualieiiu * are considered to contain ciphers. 9. 8.5 7.35 6.5 5.55 4.7 3. -4.5 - 1.25 -4.25 -2.5 -3.5 -1.35 -1.8 DIVIDING DECIMALS 229 Multiplying a Decimal by a Whole Number 3. Give products: 1.23 2.5 3.05 Place the decimal x3 x4 x2 point in the product 3.69 .4ns. 10. Ans. 6.1 Ans. when it is reached in multiplying. Omit unnecessary ciphers. 2.5 2.03 3.5 4.05 4 .05 4.5 .25 x3 x4 x2 x5 x 4 x 4 x 8 Dividing a Decimal by a Whole Number 4. Give quotients : 4)9. 2)8.1 3). 12 5)1. 2.25 Ans. 4.05 Ans. .04 Ans. .2 Ans. Place the decimal point in the quotient when it is reached in dividing. Be careful not to omit a necessary cipher in the tenths' place of a quotient. 4)8.4 4)8.2 5)9.1 6)9 2)7.1 5)6. Miscellaneous Exercises s. Give answers : ABC a. 2.6 + 3.4 1.25 + 1.5 2.05 + 1.7 6. 3.4-1.6 4.25-1.5 3.25-1.07 c. 3.4x5 1.02x5 1.24x3 d. 5.4-5-5 1.05-1-5 .1 -s-5 230 SECTION FOUR Written Exercises Perform operations with decimals in the same way as operations with dollar^ and cents. Addition and Subtraction of Decimals To add or to subtract decimals, write the decimal points in a vertical line, and the decimal point in the sum or the difference in this same line. In writing results, reject unnecessary ciphers. Thus, write 37 instead of 37.00, 28.5 instead of 28.50. i. Find sums : ABODE 24.03 48.94 32. 16.66 37.7 5.6 .67 5.85 .49 25.08 18. 13.25 .4 .8 3.4 2.95 1.4 27.75 58.95 11.29 2. Find remainders : a. 56.38 72.4 63. 86.42 90. -25.49 -5.96 -8.88 -19.72 - 25.08 b. 45.32 50.2 14.06 21.3 37.24 -22.66 -7.49 -.99 -1.85 -20.06 c. 95. 84.3 73.13 62.5 51.25 -5.06 -2.42 -8.9 -56.03 -6.66 DIVIDING DECIMALS 231 Multiplying a Decimal by a Whole Number 3. Find products : In multiplying or dividing a decimal by a. whole number write the decimal point in the re- sult when it is reached in performing the operations. ABODE a. 48.05 27.2 12.5 1.15 .24 x 4 x 5 x 6 x8 x_5 b. 37.09 56.4 23.5 3.25 .48 x7 x2 x4 x8 x_5 Dividing a Decimal by a Whole Number 4. Find quotients : E A a. 4)90. B 6)33. c 4)11. D 5)23.4 6. 25)3. 48)60. 32)88. 64)^2 64jl6. c. 4)4.2 6)5.7 8)10. 5)1.35 8)6. d. 25)1.5 36)81. 16)12. 88)4.4 48)12. 232 SECTION FOUR Miscellaneous Written Examples i. Find the sum of thirty-five and four hundredths, six and seven tenths, twenty-nine, three and ninety- six hundredths. ' 2. Find the difference between fifty-nine and eighty-three hundredths and seventeen and nine tenths. 3. Multiply by five the sum of six and ninety-four hundredths plus twelve and eighty-six hundredths. 4. Divide by forty-four the difference between eighty-four and ninety-five hundredths and eighty- seven and fifteen hundredths. 5. After walking 7.2 miles and riding 27.3 miles, how far has a man still to go to make a journey of 50 miles ? 6. At 38.2 miles per hour, how far does a train go in 5 hours ? 7. A French franc is equal to 19.3 cents. What is the value in our money of a 20-franc piece ? 8. How many cents are equal to a German mark if 25 marks are worth $ 5.95 ? 9. If 100 pounds sterling are worth $ 486.65, what is the value of 500 pounds sterling ? 10. A meter is 39.37 inches long. How many inches are there in 500 meters ? 11. A man owns 100 acres. How many acres has he after he buys 24.2 acres and sells 48.45 acres ? REVIEWS 233 Compound Numbers Oral Exercises 1. How many hours in .25 day ? 2. How many minutes in .2 hour ? 3. How many hours in .3 day ? 4. How many hours and minutes in .3 day ? 5. What fraction of an hour is 12 minutes ? 6. What decimal of an hour is 24 minutes ? 7. Add 2 Ib. 8 oz. and 2 Ib. 8 oz. 8. From 3 Ib. take 1 Ib. 8 oz. 9. Multiply 4 Ib. 8 oz. by 4. 10. Divide 4 Ib. 8 oz. by 4. 11. One fourth of 5 Ib. is how many pounds and ounces ? 12. How many ounces are there in 4 Ib. 8 oz ? is. Change .25 bushel (a) to pecks, (b) To quarts^ 14. What fraction of a bushel is (a) 1 quart ? (6) 2 quarts? (c) 3 quarts? (d) 4 quarts? (e) 6 quarts? (/) 8 quarts? (g) 12 quarts? (h) 16 quarts? (i) 18 quarts? (/) 24 quarts? (k) 28 quarts ? is. Change -| bushel to quarts. 16. How many half-pint glasses of lemonade are there (a) in a gallon? (6) In 10 gallons? 17. How many inches are there in 2 yards ? is. What fraction of a yard is 27 inches ? 234 SECTION FOUR Avenue 200 100 N 100 200 Avenue Measurements 1. The diagram represents a plot of ground 300 feet long, 200 feet wide. If the diagram is 1 J inches by 1 inch, how many feet tj are represented by each inch of w the diagram ? 2. (a) How many square feet are there in the whole plot ? How many square feet (b) in Ml (c) InP? (d) In.ZV? (e) In 0? 3. (d) M represents what fraction of the whole ? (b) What fraction does P represent? (c) Nt (d) 0? 4. (a) M is how many times JV? (b) JV is what part of M ? (c) ^- is what part of J- ? (d) J is how many times ^ ? 5. (d) Into how many building lots 25 feet front, 100 feet deep, facing the avenue, can M be divided ? (b) Into how many can P be divided ? (c) Into how many lots facing the street can jJVbe divided ? (d) 0? 6. (d) M and P together form what part of the whole plot ? (b) N and P ? (c) JV and ? 7. Divide a rectangle 4 inches by 2 inches into 8 one-inch squares, (a) What part of the whole rec- tangle is each square ? (6) How many square inches in one half of the rectangle? (c) In one fourth? (d) In one eighth? (e) How many times J is J? (/) How many times | is J? CUBICAL CONTENTS 235 Volumes 1. How many bricks are there in a pile 3 bricks long, 2 bricks wide, 2 bricks high ? 2. How long is the pile if a brick is 8 inches long, 4 inches wide, and 2^ inches high ? How wide is the pile? How high? ' 3. How many cubic inches in a pile of bricks 2 feet long, 8 inches wide, 5 inches high ? 4. How many cubic inches hi a peck if there are 2150.4 cubic inches in a bushel ? How many cubic inches in a quart ? 5. How many cubic inches will a box contain if it measures on the inside 4.2 inches by 4 inches by 4 inches ? 6. Find the contents of a box 4.48 inches by 5 inches by 3 inches. 7. (a) How many cubic inches are there in a block 12 inches long, 12 inches wide, 12 inches high? (b) How many cubic inches are there in a cubic foot ? 8. (a) How many cubic inches are there in 1J- cu. ft. ? (b) How many more cubic inches are there in 1 J cu. ft. than there are in a bushel ? 9. A gallon contains 231 cubic inches, (a) How many cubic inches are there in 7.5 gallons ? (b) How many more cubic inches are there in *l\ gallons than there are in a cubic foot ? 236 SECTION FOUR Business Forms Bills for Goods bought at One Time i. Copy the following bill, inserting the missing extensions and the footings : LOGANSPORT, IND., Dec. 20, 1914 Mrs. Wm. Priedigkeit Cypress Hills Bought of WALLACE GODETT Dealer in Toys 4 Dolls $1.50 6 pr. Dolls' Shoes .15 2 sets Dishes 1.05 1 Doll Carriage 2 boxes Tools .85 8 strings Beads .10 6 1 50 g> $> Received payment Dec. 20, 1914 Wallace Godett Per S. M. T. 2. Make out a bill for the following goods. Use your own name as the buyer, and the name of a local dealer as the seller. Receipt the bill, adding the ini- tials of a classmate as clerk. Do not use (3) in the bill. 31 Ib. Nuts If gal. Ice Cream 4| doz. Lemons 6 boxes Strawberries 1 Pineapple 3 Watermelons BUSINESS FORMS 237 Goods bought at Different Times 3. Copy the following bill, inserting omitted items : NEW YOKK, April 1, 1914 Mrs. L. C. Alphin 181 Livingston St. Bought of F. LOESER & Co. Dry Goods Fulton and Bond Sts. 1914 Mch. 2 2pr. Gloves $1.15 2 80 14 1 /2 yd. Muslin .12 7 3 1 / 3 doz. Buttons .15 15 4 Tablecloths 2.75 8 pc. Trimming .37 l /% 1 Hat 8 27 1 / 2 doz. Handkerchiefs 2.50 4 pr. Stockings .45 81 By Cash V 9 10 Balance Due The payment of $10 on Mar. 31 is called a payment on account. The bill is settled by the payment of the balance due, and is then receipted. 4. Supply dates, names, etc., for the following : | doz. Collars @ $ 1 .50 3 pr. Cuffs @ 25 1 1 Hat @ $ 2.75 2 Ties @ 37J0 2 pr. Gloves @ 75 t 238 SECTION FOUR Bill for Labor and Materials 5. Copy the following bill. Insert missing exten- sions and the footing. Receipt it of the date Oct. 3. CHAMPAIGN Co., ILL., Sept. 30, 1914 Mr. Daniel Carroll To FRED. CHAMBERS, DR. Steam Threshing Sept. 16 To 800 bu. Oats .02 630 " Wheat .04 s l u ton Coal 6. 2 Teams 3. 4 Men 1.75 16 $ The foregoing bill is rendered by Mr. Chambers, who threshes Mr. Carroll's grain. Besides the charge for the threshing, the bill includes the cost of the coal used in the engine and also the labor of extra teams and men supplied by Mr. Chambers. The abbreviation Dr., debtor, means that Mr. Carroll is in- debted to Mr. Chambers for the items specified. 6. Make out a carpenter's bill for making repairs. Use the following items: 1 man, 6 days, at $3.75; 1 helper, 2 days, at $ 2 ; 1 bunch shingles, at $ 1.25 ; 2 M boards at $ 30. 7. Make out a bill for 6 rolls of wall paper at 20 cents per roll ; 20 yards of border at 3 cents per yard ; hanging the paper at 25 cents per roll and 3 cents per yard for the border ; 4 gallons of paint, at $ 1.80 per gallon ; 2 days for painter, at $ 3 per day. BUSINESS FORMS 239 Miscellaneous 8. Make out bills for goods purchased as follows: Date each the first of the month succeeding the one in which the goods are bought. Use your own name as buyer, and the name of a local dealer or firm as seller. Receipt each bill, adding the initials of a classmate as clerk. Make all exten- sions without using another sheet. Abbreviate quantities, and omit such unnecessary words as " of," " at," etc. a. Jan. 5, 1915, bought 20 pounds of sugar at 5^ cents per pound ; 3 quarts of milk at 8 cents per quart; Jan. 11, 4 bars of soap at 15 cents per bar, 1 barrel of flour for $ 5.60 ; Jan. 18, 5 pounds of butter at 32 cents per pound, 6 gallons of oil at 16J cents per gallon. 1}. Nov. 2, 1915, bought 24 pounds of nails at 6 cents and 1 hatchet for 75 cents ; Nov. 8, 3 pails at 75 cents each; Nov. 11, 9 dozen screws at 18 cents per dozen ; Nov. 15, gave 4 dozen eggs at 45 cents a dozen in part payment. 9. Make out the bill of the owner of a dry-goods store for six items bought at one time. Give credit on the bill for butter and eggs supplied the same day by the purchaser of the dry goods. Find the balance which is paid in cash, and receipt the bill. NOTE. A bill showing credits is somtimes called a statement. 10. Make out a farmer's statement of hay and oats supplied a storekeeper at different dates. Credit the latter with goods delivered to the former. Receipt the statement. 240 SECTION FOUR 11. Copy the following letter : MADISON, Wis., Mch. 17, 1914 Martin Haupt & Sons 138-156 La Salle St. Chicago, 111. Gentlemen Kindly ship me the following by fast freight : 3 bbl. Portland Cement 10 M Shingles, Cedar 14 bbl. Wire Nails, 10's Your early attention to this order is respectfully requested. Truly yours COTTIER JONES R.F.D. No. 4 12. Copy the following reply : 138-156 LA SALLE ST. CHICAGO, ILL., Mch. 19, 1914 Cottier Jones, Esq. R.F.D. No. 4 Madison, Wis. Dear Sir We have shipped to-day by fast freight the goods ordered in your favor of the 17th inst. Inclosed please find bill. Yours sincerely MARTIN HAUPT & SONS is. Draw on the back of the first letter a rec- tangle 6^ inches by 3^ inches to represent an envel- ope. Write on it the address given on the next page. BUSINESS FORMS 241 MARTIN HAUPT & SONS 138-156 LA SALLE ST. CHICAGO ILLINOIS i*. On the back of the second letter draw an en- velope and write the proper address. is. Frank O'Malley has money in the National City Bank of Omaha. He owes Smith, Brown, & Co., of St. Louis, $ 154.36 for goods purchased from them. He draws his check as follows : A Bank Check No. 354- Omaha, Neb., fffiul g, t V-, 19? 5 Received of c$uw&/6 Q'TWaMeu check for (Swe /i-^vyldA&ci f^^^^^^^^-- ^L Dollars 100 on account $fOO 2