ga fasM n s wK wwww 1 OJCrf NEW ELEMENTARY ARITHMETIC THE NORMAL COURSE IN NUMBEI INDIANA- EDITION SILVER, BURDETT AND COMPAHY f "^^^ UNIVERSITY OF CALIFORNIA DEP ^ No, cT Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/cookarithmetic'OOcookrich _ ..•; •: : v , •... THE NEW *****'-*^*' ELEMENTARY ARITHMETIC BY JOHN W. COOK PRESIDENT OF ILLINOIS STATE NORMAL UNIVERSITY AND MISS N. CEOPSEY ASSISTANT SUPERINTENDENT OF CITY SCHOOLS, INDIANAPOLIS, INDIANA REVISED BY ROBERT J. ALEY PROFESSOR OF MATHEMATICS, INDIANA UNIVERSITY AND OSCAR L. KELSO PROFESSOR OF MATHEMATICS, INDIANA STATE NORMAL SCHOOL SILVER, BURDETT AND COMPANY NEW YORK BOSTON CHICAGO CopTRiGUT im:\, 1895, i^\r.), iixxi. By silver, BtJRDETT AND COMPANY EDUCATION DEPi. PEEFACE TO THE KEVISED EDITION The present edition of " The ]^ew Elementary Arith- metic" will be found to differ from the earlier edition ])rincipally in an increase in the amount of drill work given upon the fundamental operations. This change, it is believed, recognizes a tendency of present-day arith- metic work which has the approval of the best authorities. In a few other respects also the book has been changed, notably in the direction of simplification and the bringing of the book into harmony Avith the advance which has been made in the teaching of arithmetic since the first edition appeared. In making these changes, the arrangement of the mat- ter has been altered, a large number of new exercises have been added, and the problems in many cases have been rewritten or replaced. It is hoped that the present book will be found thor- oughly abreast of the best practice in the teaching of arithmetic, and that in its new form the work will con- tinue to enjoy the confidence and appreciation which have been so kindly extended to it up to the present time. 54IS33 PREFACE TO THE FIRST EDITION. IT has seemed to the authors of the Normal Course IN Number that there is room for another series of Arithmetics, notwithstanding the fact that there are many admirable books on the subject already in the field. The Elementary Arithmetic is the result of the ex- perience of a supervisor of primary schools in a leading American city. Finding it quite impossible to secure satisfactory results by the use of such elementary arith- metics as were available, she began the experiment of supplying supplementary material. An effort was made to prepare problems that should be in the highest degree practical, that should develop the subject systematically, and that should appeal constantly to the child's ability to think. The accumulations of several years have been care- fully re-examined, re-arranged, and supplemented, and are now presented to the public for its candid consideration. Not the least valuable feature of this book is the care- ful gradation of the examples, securing thereby a natural and logical development of number work. No space is occupied with the presentation of theory, — that side of the subject being left to the succeeding book. The first thoughts SLve ivhat and hoWj — these so presented that the processes shall be easily comprehended and mastered. Subsequently, the tchf/ may be intelligently considered and readily understood. INTRODUCTION. It has been said that the " new education " proceeds to give the child an experience, instead of presupposing one for him. Pupils become practical, not by learning forms of reasoning, but by exercising the reason upon their own plane of comprehension. In such a spirit this Elemen- tary Arithmetic has been prepared. It presents three- years' work, based upon carefully graded exercises which may be used as a means of training pupils to think, and of teaching at the same time the practical application of numbers to ordinary business transactions. It is very important that children should master the fundamental processes so thoroughly that they come to serve thought without loss of time or energy. The patient following of these graded exercises and drills should secure this result. In general, division and multi- plication, as converse processes, are followed by addition and subtraction on the same general plan. As the work becomes more complex, it is diiRcult to make this alterna- tion with perfect regularity without detriment to the efficiency of the work. Stress must be laid upon such a complex subject as long division, a very difficult subject for children, which requires an amount of practice that, at first view, might seem out of proportion to the practice given in other subjects. The primary facts of addition and subtraction are pre- sented in the first twenty pages. Neither accuracy nor rapidity in calculation can be secured until these combina- vi INTRODUCTION. tions can be given with readiness. Those facts are used again in the tables of ''endings," for ai)|)lication to num- bers above twenty. These tables have a practical value and should be as thoroughly applied as the multiplication and division tables. These tables in subtraction give an oppor- tunity for reviewing the primary facts, and for using this knowledge with numbers above tvoenty^hvX they have no such direct application as the tables in addition. Any reasonable system of teaching addition can be used with the graded examples of the Elementary Arithmetic. The first and hardest step in solving an arithmetical question is to determine the processes required ; the sec- ond, to state the different steps of the solution in proper arithmetical form. Children can give results long before they are conscious of the process by which the results are obtained. The statement of the process by means of arithmetical signs and figures is a new language to the pupil; it is not surprising that the mastery of tliis lan- guage takes time and skillful teaching. The statement of the result, in a concrete problem, is probably all that should be required in the second school year. Tt may be desirable to introduce a simple statement of the process early in the third year. Such a statement can be added to the sentence giving the result, as on- page 16, example 1 : 4 cents x 3 = 12 cents. No formula should be taught with the thought that it will do the thinking for the pupil. Let the problem be pictured, and this picturing followed by the expression in figures, before any formal expression in Avords is attempted. The object of pictur- ing problems is not to teach children to make pictures (though all this work should be done with reasonable care), but to give a method of representation by Avhich they can make their thoughts clear to themselves. It is INTRODUCTION. vii a means, not an end, and should be so regarded. When problems can be stated clearly and solved correctly there is no further necessity for picture representation, except as a means of testing the pupiPs comprehension of spoken or written forms. Let not objective work be undervalued, however. It is a very necessary means, which, rightly used, will secure accurate knowledge and a correct use of terms, thus saving much time and confusion later on. Pupils should learn early to show objectively the differ- ence between six and one-sixth of six, between one-sixth of six and one-sixth of 07ie, etc. Problems which may be worked out orally in the reci- tation will often be found too diificult for a written test. "Miscellaneous problems" should be used with discrimi- nation, the teacher selecting such as seem suited to the capacity of the class. All measures introduced should be learned by actual use. The standards in common use, such as the yard, foot, ounce, pound, quart, etc., can be obtained easily, and should form a part of the regular school supplies. Exer- cises in estimating volume and extension train the judg- ment while giving practical results in knowledge, and there is no time in the course when pupils can better afford to do this work than during the first years of the elementary school course. Eules may be made by the pupils after the process is learned from which the rule is derived. This book has grown from experience, and is offered to fellow-teachers as a systematic work-book. CONTENTS. PAGE Chapter I 1_24 Numbers Through Twelve 1 The One-Inch Square and the Inch 7 Pints and Quarts 8 Numbers Through Seventeen 9 Numbers from Ten to Twenty, as Tens and Ones . . .18 Writing Numbers to Ten 19 The Numbers Eighteen, Nineteen and Twenty .... 19 Comparison of Halves and Fourths 31 Chapter II . . . 25-41 Numbers from Twenty to One Hundred, as Tens and Ones . 25 Writing Numbers from Ten to Thirty 27 Addition and Subtraction .28 Quarts and Gallons 31 Multiplication and Division . 32 Chapter III 42-61 Reading and Writing Numbers : Hundreds . . . .42 Addition 45 Subtraction . . .46 Inch, Foot and Yard 49 Measuring Time . . 53 Multiplication and Division 55 Chapter IV 62-116 Reading and Writing Numbers : Thousands . . . .62 Roman Notation 64 Multiplication 65 Division 67 Multiplying and Dividing by 3 69 Multiplying and Dividing by 4, 5 and 6 70 United States Money 72 Addition and Subtraction by Endings : 1 + 8, 1 + 9, 2 + 5, 2 + 6, 3 -f 7 74 X CONTENTS. Dry Measures 79 Addition and Subtraction by Endings : 2 + 8, 2 + 9 . .83 Ounces and Pounds 85 Addition and Subtraction by Endings : 3 + 5, 0+6, 3 + 7, 3 + 8, 3 + 9 88 Comparison of Halves, Fourths and Eighths . . . .96 Multiplication and Division 100 Multiplying and Dividing by 7 and 8 103 Addition and Subtraction by Endings : 4 + 4. 4 + 5, 4 + . 109 Multiplication and Division Ill Chapter V 117-137 Reading and Writing Numbers 117 Multiplying and Dividing by 9, 10, 11 and 12 . . .119 Addition and Subtraction by Endings : 4 + 7, 4 + 8 . . . 124 Multiplication 128 United States Money 129 Addition and Subtraction by Endings : 4 + 9 . . . . 131 Halves, Thirds and Sixths 135 Chapter VI 138-202 Multiplication and Division 138 Addition and Subtraction by Endings : 5 + 5, 5 + 6, 5 + 7, 5 + 8 . 142 Multiplication and Division . 152 Addition and Subtraction by Endings : 5 + 9 . . , . 154 Multiplication and Division 160 Comparison of Halves, Thirds, Fourths and Sixths . . . 164 Division 167 Addition and Subtraction by Endings : 6 + 6. 64-7. 6 + 8, 6 + 9 . 168 Division 172 United States Money 174 Square Measure . . . , , 176 Addition and Snbtraction by Endings : 7+7, 7 + 8, 7+9, 8 + 8, 8 + 9 179 Cubic Measure 187 Multiplication and Division 190 Addition and Subtraction by Endings : 9 + 9 . . . 190 Division 195 Addition and Subtraction . ....... 195 Division 200 CONTENTS. xi I'AGE Chapter VII 203-241 Fractions , ^ 203 Like and Unlike Numbers 204 Reduction 207 Addition of Fractions 212 Subtraction of Fractions 215 Division of Fractions 219 Multiplication of Fractions 226 Chapter VIII 242-261 Decimal Fractions 242 Writing and Reading Decimals 246 Reduction 247 Addition of Decimals 249 Subtraction of Decimals 250 Division of Decimals 251 Multiplication of Decimals 256 Chapter IX 262-276 Compound Numbers . . . 262 Dry Measure 262 Liquid Measure 265 Avoirdupois Weight 265 Measures of Length 267 Square Measure 268 Cubic Measure .... 270 Time Measure .... 273 THE NEW ELEMENTARY ARITHMETIC. CHAPTER I. I2345bngq 1. Write the names of the numbers from one to ten. Write the figures which stand for these numbers. Count one hundred by ones; by tens. NUMBERS THROUGH TWELVE. [A review of work learned in the second school year.] 2. Count by ones through twelve; count by twos; by threes; by fours. Begin with twelve and name the numbers in their order to one. Name two numbers which together make four; two numbers which make six. NUMBERS THROUGH TWELVE. Addition and Subtraction u 3, Sums of any two numbers through eight 112 1 2 12 3 1 2 3 12 3 4 2 3 24 3 5 4 3 6 5 4 7 6 5 4 3 4 4 5 5 6 6 6 7 7 7 8 8 8 8 1 and 2 are — 5 less 3 is — 7 less 5 is 3 less 2 is — 1 and 5 are — 4 and 3 are — 1 and 3 are — 6 less 5 is — 7 less 3 is — 4 less 3 is — 4 and 2 are — 7 less 4 is — 2 and 2 are — 6 less 2 is — 6 and 2 are — 4 less 2 is — 3 and 3 are — 8 less 2 is — 1 and 4 are — 6 less 3 is — 8 less 6 is — 5 less 4 is — 5 and 2 are — 5 and 3 are — 2 and 3 are — 7 less 2 is — 8 less 3 is — 4. What two equ^l numbers make four? Separate four into two equal parts. Take away one of of the parts; what is left? Separate eight into two equal parts. Take away one of the parts; what is left? Separate eight into two unequal parts. Take away one of the parts; what number is left? 5, Sums of any two numbers through twelve. 1234 12345 12345 ^ 1 ^ ^ ^8_7_6 5 10_9876 9 9 9 9 10 10 10 10 10 11 11 11 11 11 ADDITION AND SUBTRACTION. 3 12 3 4 5 6 11 10 9 J 7 6 12 12 12 12 12 12 7 and 2 are — 7 and 3 are — 9 less 7 is — 10 less 3 is — 6 and 3 are — 10 less 7 is — 9 less 3 is — 6 and 4 are — 5 and 4 are — 10 less 4 is — 9 less 4 is — 10 less 6 is — 9 less 5 is — ^ 9 and 2 are — 8 and 2 are — 11 less 2 is — 10 less 2 is — 11 less 9 is — 8 and 3 are — 9 and 3 are — 11 less 3 is — 12 less 3 is — 11 less 8 is — 8 and 4 are — 6 and 5 are — 12 less 4 is — 11 less 5 is — 12 less 8 is — 11 less 6 is — 7 and 5 are — 7 and 4 are — 12 less 5 is — 11 less 4 is — 12 less 7 is — 11 less 7 is — 12 less 6 is — 6. Find the sunis^ giving results only: 24212324 124375513 4^563574865423496 Note to Teachers. — If the children are not very faniihar witli these fundamental facts, sufficient time must be given to secure a thorough mastery of them. If necessary, the work must be given with the objects. As soon as possible; however, children should becoriie independent of the use of objects. NUMBERS THROUGH TWELVE. 2 5 3 8 4 6 7 9 6 2 5 3 5 3 3 4 9 8 2 7 2 6 3 3 1 4 9 6 7 4 5 8 7 2 8 6 3 4 3 4 7 5 10 6 8 9 6 5 7 3 3 9 8 4 5 3 7 2 6 2 3 5 7 4 Supply the numbers omitted: 6 6234 6 2517 3 3654 10 11897129116121110989 4475987893 12 IT 10 12 11 12 IT lO 11 12 Subtract at sight, giving results only : 8675689 10 87 11 12 10 876 5432465 672 6 7 8522 11 9 10 11 12 9 10 7 10 12 5376523184 11 9 8 12 12 11 12 12 _484_7_8_8_9j; 7. 1. Henry bought a book for 8 cents and a pencil for 4 cents; he paid — cents for both. MULTIPLICATION AND DIVISION. 5 2. There were 4 boys and 6 girls in a class; together there were — children. 3. Helen had 11 roses and gave 4 of them to May; Helen then had — roses. 4. James earned 7 cents and George earned 5 cents; together they earned — cents. 5. Make problems for: 6 and 3 are 9. 10 less 6 is 4. 9 and 3 are 12. 11 less 5 is 6. Note. — The children should group objects and make the problems with the objects before them. This work is for the recitation, not the study period. Multiplication and Division. 8. (1) 2 twos are — 3 twos are — 4 twos are — 5 twos are — (2) 6 twos are — 2 threes are — 3 threes are — 4 threes are — (3) 2 fours are — 3 fours are — 2 fives are — 2 sixes are — (4) (6) (6) 6 is — twos. 12 is — threes. 8 is — fours. 8 is — twos. 12 is — fours. 6 is — threes. 9 is — threes. 12 is — sixes. 10 is — fives. (7) One-half of 4 is — One-half of 6 is — One-half of 8 is — (8) One-half of 10 is - One-half of 12 is - One-third of 6 is - 6 NUMBERS THROUGH TWELVE. (0) (lo) One-third of 9 is — J of 9 is — One-third of 12 is — i of 12 is — One-fourth of 8 is — i of 8 is — One-fourth of 12 is — J of 12 is — 9. 1. Frank bought 4 pencils at 3 cents each; for all he paid — cents. 2. Mary has 10 cents; oranges cost 5 cents each; she can buy — oranges. 3. William has 8 apples; he divides them equally be- tween his two brothers; each receives — apples. 4. Anna has 9 nuts; she divides them equally among three children; each child receives — nuts. 5. Anna has 12 roses, and gives 4 roses to each of her sisters; she has — sisters. 6. Make similar problems, using the following forms: 2 threes are 6. 9 is 3 threes. One-half of 12 is 6. 3 fours are 12. 10 is 5 twos. One-third of 9 is 3. 8 is 4 twos. 8 is 2 fours. One-fourth of 12 is 3. Note. — This work is to be given in tlie recitation and under the direction of the teacher. No attempt should be made to state the process either in words or figures. It is well to have children lay out the objects on the desk and make their statements with the objects before them. Buttons or small pasteboard tablets make convenient counters. In a problem like Number 5 above, let the children lay out buttons or grains of corn for the roses, and see that 3 fours can be taken out of 12. A stick or tooth -pick might be placed under each of the fours. These sticks represent the real answer to the problem, the number of sisters. THE ONE-INCH SQUARE AND THE INCH. 7 THE ONE-INCH SQUARE AND THE INCH. One Inch Square. One Inch. Three Inches. 10. 1 . Cut from paper a square which is one inch on each side. 2. Draw a line one inch long. Draw upon the board a line twelve inches long. Twelve inches are equal to one foot. 3. Cut from paper a measure one foot in length. Find one-half of a foot One-third of a foot. 4. Draw a square which is two inches on each side. How many inches is it around the square? 5. Cut this square from paper; fold it so as to show four small one-inch squares. 6. Place two one-inch squares side by side. Make a drawing which is two inches long, and one inch wide. 7. 6 inches are of a foot. 8. 3 inches are of a foot. 9. 4 times 3 inches are — inches. 10. 3 times 3 inches are — inches. 8 PINTS AND QUARTS. 11. i of 12 inches is — inches. 12. How much longer is a twelve-inch line than a six- inch line? 13. How much longer is a twelve-inch line than a nine- inch line? 14. How many are 3 times 3 square inches? 15. How many inches is it around a square which meas- ures two inches on each side? 16. Julia has a piece of ribbon 12 inches long, which she divides into 4-inch pieces; how many pieces are there? 17. Ella has a pencil eight inches long. When she has used 3 inches, how many inches will be left? Note. — In this review of the work of the Second Grade, all exercises must necessarily be much condensed. Children should be familiar with the one-inch square^ the inch and the foot. Begin with the cube and derive from it the one-inch square. Children should use these measures in their daily work, whenever opportunity offers. PINTS AND QUARTS. Pint. Quart. 11. 1. A quart of milk is how many pints? 2. A pint is what part of a quart? 3. How many quart measures can I fill with four pints of milk? 4. 6 pints equal — quarts. 6 quarts equal — pints. 5. 3 quarts equal — pints. 5 quarts equal — pints. ADDITION AND SUBTRACTION. 9 6. 10 pints equal — quarts. 8 pints equal — quarts. 7. 3 pints are how much more than a quart? 8. 5 pints are how much more than two quarts? NUMBERS THROUGH SEVENTEEN. 12. Count seventeen by ones. Write the names of the numbers through seventeen. Begin with seventeen and name the numbers in their order to one. Addition and Subtraction. 13. Sums of any two numbers through seventeen, 12 3 4 5 6 12 n 10 ^ _8 _7 13 13 13 13 13 13 12 3 4 5 6 7 13 12 11 10 9 8 7 14 14 14 14 14 14 14 12 3 4 5 14 13 12 11 10 6 9 7 8 15 15 15 15 15 15 15 12345678 1514131211 10 ^_8 16 16 16 16 16 16 16 16 12345678 16151413121110_9 17 17 17 17 17 17 17 17 10 NUMBERS THROUGH SEVENTEEN. Separate fourteen into two equal parts. Take away one of the parts what is left? Separate fourteen into two unequal parts. Take away one of the parts; what is left? What number must you add to 9 to make 14? What number must you add to 8 to make 13? 14. 1. 10 plus 3 equals 13. 2. 13 less 3 equals 10. 10+3=13. 13-3=10. 7 + 6=? 13-6=? 6+8=? 14-8=? 9+5=? 14-5=? 5+8=? 13-8=? 4+9=? 13-9=? 8 + 3=11. 4. 12-9= 3. 9 + 5=14. 13-2=11. 7 + 7=? 13-3=? 5+9=? 13-4=? 9+4=? 14-9=? 8+5=? 14-6=? 6+7=? 13-7=? Note. — Teach the signs + (plus), — (less), and = (equals). 5. Find the sums, giving results only : 763426496 776988554 566957564 838486758 ADDITION AND SUBTRACTION, H 6. Supply the numbers omitted : 6859467594 14121314111413121314 3786934798 11131413 12 1112131413 7. A man planted 8 apple trees and 9 pear trees; how many trees did he plant? 8. I had 14 dollars, and spent 9 dollars for a table; how many dollars had I left? 9. Ella has 6 roses and 7 violets; how many flowers has she? 10. John had 14 cents, and spent 6 cents for a top; how much money had he left? 11. Make problems for: 8 + 6=14 13-8 = 5 14-9 = 5 6 + 7=13 9 + 4=13 14-6=8 12-5 = 7 12 + 2=14 15. Add the horizontal lines from left to right, naming each sum (1) (2) (3) (4) (5) (6) 3,2,4 1,8,4 2,3,8 1,9,4 3,3,8 1,4,9 2,6,3 2,7,3 1,7,6 2,6,5 3,4,6 3,5,5 3,5,4 1,2,9 2,6,3 3,2,8 2,5,5 3,9,2 (T) (8) (S) (10) 10 + 5 = .? 9 + 6=? 16-8=? 16- - 9 = ? 7 + 8= = ? 7 + 9=? 15-7=? 15- - 8=? 8 + 8 = = ? 6+9=? 16-7=? 15- -10=? 9 + 8 = = ? 8+9=? 17-9=? 17- - 8=? 12 NUMBERS THROUGH SEVENTEEN. 1 1 . Find the sums, giving results only : 51325384 676253433 49682726334967585 4 9 8 6 2 4 2 1 2 3 2 4 12 4 3 7 7 2 3 3 4 3 5 6 3 5 7 4 8 6 5 4 2 5 3 4 3 4 7 5 10 6 8 9 6 5 7 4 9 7 3 9 8 4 5 3 7 2 6 2 3 5 7 4 9 5 7 5 396848 11 11 65878 11 6 8 10 47695 2 389798 59 12 9765584695899637 46879647957546784 9697886 10 89 7859799 678 12. Supply the numbers omitted: 9659867697695 12 13 14 16 15 13 12 16 16 16 14 13 12 56699779 16 13 15 17 15 15 12 15 5 8 9 9 8 8 7 8 14 12 14 17 15 17 16 17 MULTIPLICATION AND DIVISION. 13 13. Subtract the lower number: 8675689 10 87 11 12 10 876 5432455 672 6 7 8522 11 9 10 11 12 9 10 7 10 12 11 8 12 11 53765231844478 12 10 12 13 11 14 12 13 14 11 12 14 13 9 8 8 6 5 9 7 9 8 6 4 5 8 14 12 13 14 14 13 14 11 13 15 16 15 14 4 3 4 9 5 7 6 7 6 9 7 6 6 16 16 16 15 15 13 15 14 9 8 5 10 7 8 8 8 12 16 15 17 17 16 14 17 7 7 9 9 8 9 9 8 Multiplication and Division. 16. 7X2=14 14-h2 = 7 2X7 = 14 14^7 = 2 8X2=16 16^2 = 8 2X8=16 16-^-8 = 2 5X3=15 15h-3 = 5 3X5=15 15^5=3 14 NUMBERS THROUGH SEVENTEEN, i of 14=7 i of 16 = 4 i of 16=8 J of 15 = 5 Note. — The children should make these tables by means of objects, laying down two sevens, seven twos, etc. The sign X is always read ''multiplied by/' 6X2=12 is read ''6 multiplied by 2 equals 12/' 2X6=12 is read ''2 multiplied by 6 equals 12/' The sign -^ is read ''divided by." 12^2 = 6 is read "12 divided by 2 equals 6." 12 ^6 = 2 is read "12 divided by 6 equals 2." 17. 1. Albert works in his garden 2 hours each day; how many hours does he work in 6 days? 2. If a kite costs 8 cents, how many kites can be bought for 16 cents? 3. If a boy rides 8 miles an hour on his bicycle, how many hours will it take him to ride 16 miles? 4. A lady divided 16 pencils equally among her four children; how many pencils did each receive? Make problems orally for: (5) (6) (7) 3X5=15 16h-8=2 i of 14=7 4X4=16 16-^2 = 8 J of 15=5 2X8=16 15-^3 = 5 i of 16=4 Note. — Have the children make these problems in class, using objects. MULTIPLICATION AND DIVISION. 15 18. REVIEW. Note. — These tables may be used for oral recitation, the pupil giving answers rapidly; or for occupation during the study period ; or for oral tests, the work being dictated by the teacher. 7X2 = 3X3 = 2X8= 2X6= 10 -^ 2 = 9^3 = 8-^4 = 10 ^5= 3+5 = 7 + 3 = 3 + 4 = 3+7 = 16- 7 = 16-9= 16-8 = 15- 8 = 9+3 = 6 + 4 = 3 + 9= 6 + 4 = 15- 7 = 15- 8= 14- 5 = 14-9 = 4 + 8= 3 + 8= 2 + 6 = 8 + 3 = 13- 9 = 13- 4 = 13- 5= 13-8= 7 + 5 = 4+7 = 8 + 4 = 9 + 7= 13- 6 = 13- 7 = 3X2 = 2X3 = 3+7 = 5+6 = 5 + 7 = 7 + 8 = 6X2 = 2X4= 4X4 = 16^8= 8 + 8 = 8+7 = 9 + 5 = 5 + 9= 2X5 = 3X5 = 4X4= 2X7 = 7 + 9 = 7 + 7 = 6+9 = 9 + 6= i of 14 = i of 12 = i of 15 = iof 16 = 12 ^3 = 14 ^2 = 15 4-5 = 4X4 = 19. J of 15 = 5 may also be expressed: 15 -^ 3 = 5, or 3)15. 3115 .^ ^^^^ ,, ^g divided by 3 equals 5." 5 5 Read the following and solve : i of 12=? 12^2=? 2)12=? iofl6=? 2)16=? J of 8=? 8^2 = ? 2) 8=? iofl5=? 3)15=? iof 9=? 9^3=? 3) 9=? io"16=? 4)16=? 16 NUMBERS THROUGH SEVENTEEN. EXERCISE. 20. Give each answer in a statement. 1. At 4 cents a yard, what will 3 yards of ribbon cost? 3 yards of ribbon, at 4 cents a yard, will cost 12 cents. 2. There are 3 rows of trees in my yard; in each row there are 6 trees; how many trees are there in the yard? 3. Henry rode 9 miles in the morning, and 4 miles in the afternoon; how many miles did he ride in all? 4. The sum of two numbers is 8; one of the numbers is 2; what is the other number? 5. The sum of two numbers is 14; one of the numbers is 9; what is the other number? 6. I sold a pint of milk to each of four customers; how many quarts did I sell? 7. Mr. Jones sold six quarts of milk; how many pints did he sell? 8. A square which is 2 inches on each side contains how many square inches? NUMBERS THROUGH SEVENTEEN. 17 9. How many inches is it around a 2-inch square? 10. An oblong one inch wide and 3 inches long contains how many square inches? 11. What is the distance round an oblong 2 inches wide and 3 inches long? 12. Draw an oblong 2 inches wide and 5 inches long. 13. At 6 cents a yard, how many yards of lace can I buy for 18 cents? 14. John paid 15 cents for 3 pencils; what did one pencil cost? 15. Julia gave away 12 pinks to her sisters, giving 3 to each; how many sisters had she? 16. There are nine boys and 6 girls in Mary's class; how many children are there in the class? 17. George had 15 cents; he spent one-third of his money; how much did he spend? 18. 6 inches is half the length of John's ruler; what is the length of the ruler? 19. 5 cents is one-third of what I paid for a box of berries; what did I pay for the berries? 20. I have 9 cents; how many cents must I add to it to make 15 cents? 18 NUMBERS FROM TEN TO TWENTY. NUMBERS FROM TEN TO TWENTY, AS TENS AND ONES 21. Note. — Use tooth-picks, shoe-pegs, or any available ob- jects for this illustration work. nUJIII Ten ones are one ten. fli III 1111 One ten and one one are eleven. 10 and 1 are 11. One ten and two ones are twelve, 10 and 2 are 12. One ten and three ones are thirteen, 10 and 3 are 13. One ten and four ones are fourteen. 10 and 4 are 14. One ten and five ones are fifteen, 10 and 5 are 15. One ten and six ones are sixteen, 10 and 6 are 16. One ten and seven ones are seventeen. 10 and 7 are 17. One ten and eight ones are eighteen. 10 and 8 are 18. One ten and nine ones are nineteen. 10 and 9 are 19. WRITING NUMBERS TO TEN, 19 Two tens are twenty, 2 tens are 20. 23. 1. Eleven is how many tens and how many ones? Which figure in the number 11 stands for the one? Which figure stands for the one ten? 2. Write the numbers from one to ten in a column. 3. Write the numbers from ten to twenty in a column. 4. Which figure always stands for the ones, when ones and tens are written? (The right hand figure.) 5. Which figure in the number 18 stands for ones? Which stands for the tens? 6. Which figure stands for the tens in the number 20? Which figure stands for ones? is called nought or zero. WRITING NUMBERS TO TEN. 23. The letters I (one), V (five), and X (ten), are also used to represent numbers. Numbers to 10 are represented in three different ways, as follows : Wovds.' zero, one, two, three, four, five, six, seven, eight, nine, ten. Figures: 012 3 456 7 89 10 Letters : I II III IV V VI VII VIII IX X THE NUMBERS EIGHTEEN, NINETEEN, AND TWENTY. 34. Write the names of the numbers from one to twenty. Sums of any two numbers through tiventy. 123456789 17 16 15 14 13 12 n 10 ^ 18 18 18 18 18 18 18 18 18 20 NUMBERS EIGHTEEN, NINETEEN, AND TWENTY, 123456789 181716151413121110 19 19 19 19 19 19 19 19 19 12 3 4 5 6 7 8 19 18 171615141312 20 20 20 20 20 20 20 20 Add these columns, beginning at the top and naming each sum : 975886425442 660514871255 139165187526 211241376744 8594. 3 7940687 778769427689 675978893 698698628 823223697 7 10 12 112 2 3 2 8 1110 9 112 3 5 09867596796 18756478534 13 11112 2 3 12 43494436864 89867987698 54534543243 Add the columns above, beginning at the bottom. COMPARISON OF HALVES AND FOURTHS. 21 Note.— In addition and all other arithmetical processes, ac- curacy should be the first consideration. A reasonable degree of rapidity may be acquired by practice. COMPARISON OF HALVES AND FOURTHS. 25. 1. A whole melon can be divided into how many- halves? How many fourths? 2. Fold a paper square into two equal oblongs. One of the oblongs is what part of the square? 3. Fold the same square into two equal triangles. One of the triangles is what part of the whole square? 4. In one whole there are how many halves? 5. Fold a paper square so as to make four small squares of equal size. One of these small squares is one-fourth of the whole. In one whole there are how many fourths? 6. One half of the square is how many fourths? 7. If you should fold down one half of the large square, how many fourths would remain? 22 COMPARISON OF HALVES AND FOURTHS. 26. From the circles on page 21, find answers to the following questions: (1) (2) i + i=? i— i = how many fourths? J + i=? |— i = how many fourths? | + i=? 4 — i= how many fourths? | + i= ? |— J = how many fourths? (3) (3) { is contained in J, — times. \ is contained in f , — times. i of i = how many fourths? iX4= how many fourths? J X 3 = how many fourths? 4. Frank has half an orange, and Edwin one-fourth of an orange; they both together have — fourths of the orange. 5. George ate one-fourth of a pie; there were remaining of the pie. 6. There are three-fourths of a bushel of apples in one barrel, and one-fourth in another; in both barrels there is — . 7. Jennie had an apple and gave away one-fourth of it; she had left. EXERCISE. 27. Write the answers in statements. 1. Frank has 8 pigeons, 3 red birds, and 4 canaries; how many birds has he? Frank has 15 birds. EXERCISE. 23 2. Helen used 16 eggs in making 4 cakes; how many eggs did she put into each cake? 3. Mary drew four leaves on each of the four sides of her box; how many leaves did she draw? 4. Fanny had 19 shells; before she reached home she lost 5 of them; how many had she left? 5. Jennie sewed 6 buttons on her shoes, and had 9 buttons left; how many had she at first? 6. Each of my 3 brothers gave me 6 cents; how much money did all give me? 7. At 15 cents a yard, what must I pay for one-third of a yard of ribbon? 8. After spending 10 cents for paints, Frank had 5 cents left; how much money had he at first? 9. When milk is 6 cents a quart, how much must you pay for a pint? 10. Four children each spent 5 cents for car-fare; how much money did they all spend? 11. ISboys were coasting; there were three boys on each sled; how many sleds were there? a D D D D D Read the problem and the answer from tlie pic- ture. 12. At 3 cents each, how many pencils can I buy for 18 cents? 13. At 3 cents a pint, what will 4 pints of milk cost? 14. 4 pints are how many quarts? 15. 12 pints are how many quarts? 24 EXERCISE. 16. Draw an oblong 3 inches by 4 inches; how many square inches are there in the surface? 17. Divide an apple equally among 4 boys; what part does each boy receive? 18. George divided a melon into four equal parts, and gave away three of the parts; what part of the melon did he keep for himself? 19. Henry wishes to visit his cousin who lives 17 miles away; after riding 8 miles how much farther has he to go? 20. Anna, Carl, and Fred went nutting and gathered 12 quarts of nuts; if they divided them equally, what part did each receive? CHAPTER II. NUMBERS FROM TWENTY TO ONE HUNDRED, AS TENS AND ONES. 2 8. Write the numbers from ten to twenty. Which figures always stand for ones? Two tens are twenty — 20. Three tens are thirty — 30. Four tens are forty — 40. Five tens are fifty — 50. Six tens are sixty — 60. Seven tens are seventy — 70. Eight tens are eighty — 80. Nine tens are ninety — 90. Ten tens are one hundred — 100. 26 NUMBERS FROM TWENTY TO ONE HUNDRED. 29. Two tens and one one are twenty-one — 21. Two tens and two ones are twenty-two — 22. Two tens and three ones are twenty-three — 23. Two tens and four ones 3,re twenty-four — 24. Two tens and five ones are twenty-jive — 25. Two tens and six ones are twenty-six — 26. Two tens and seven ones are twenty-seven -27. Two tens and eight ones are twenty-eight -28. Two tens and nine ones are twenty-nine -29. Three tens are thirty — 30. 30. 1. Count by ones from thirty to forty. 2. Read these numbers: 31, 32, 33, 34, 35, 36, 37, 38, 39, 40. 3. Write the above numbers in a column and name the ones. WRITING NUMBERS FROM TEN TO THIRTY, 27 4. Read the following numbers : 40 50 60 70 80 90 41 51 61 71 81 91 42 52 62 72 82 92 43 53 63 73 83 93 44 54 64 74 84 94 45 55 65 75 85 95 46 56 66 76 86 96 47 57 67 77 87 97 48 58 68 78 88 98 49 59 69 79 89 99 100 5. How many ones are there in 69? How many tens and how many ones in 69? 6. Which is more, 83 or 74? 92 or 89? 7. Arrange these numbers in order: 70, 65, 69, 67, 66, 64, 68. 8. Arrange these in order: 81,79,83,78,80,82,77. WRITING NUMBERS FROM TEN TO THIRTY. 31. Numbers from 10 to 30 are represented by words, figures and Roman characters, as follows : Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen 10 11 12 13 14 15 16 X XI XII XIII XIV XV XVI Seventeen Eighteen Nineteen Twenty Twenty- one 17 18 19 so 21 XVII XVIII XIX XX XXI 28 ADDITION AND SUBTRACTION. Twenty-two Twenty-three Twenty-four Twenty-five Twenty-six 22 28 24 25 26 XXII XXIII XXIV XXV XXVI Twenty-seven Twenty-eight Twenty-nine Thirty 27 28 29 30 XXVII XXVIII XXIX XXX The Roman characters are not now employed in number work, but are chiefly used for numbering" chapters and lessons. They are used also to indicate the ditt'ercnt volumes of a series of books, and to mark the hours on the dials of clocks and watches. Write the numbers 31 to 39 in Roman characters. Read the following: VII XIX XV XVI XVIII IX XXIX XXV XXIV XXIII XI XXXIX XXXV XXXVII XXXIV ADDITION AND SUBTRACTION. 32. Give sums at sight, adding by tens : Thus, in adding 20 and 12 say: 20 and 10 are 30, and 2 are 32. (1) (2) (3) (4) (5) (6) (•7) 20 30 40 50 60 70 80 12 12 12 12 12 12 12 (8) (9) (10) (11) (12) (13) (14) 21 31 41 51 61 71 81 12 12 12 12 12 12 12 ADDITION AND SUBTRACTION. 29 (15) (16) (17) (18) (19) (20) (21) 21 31 41 51 61 71 81 14 14 14 14 14 14 14 (22) (23) (24) (25) (26) (27) (28) 24 34 44 54 64 74 84 15 15 15 15 15 15 15 Add the same numbers, giving the sum of the ones and then the sum of the tens. 33. Add 12, 49, and 33. 12 Add the ones first, naming results only ; thus: 3, 12, AQ 14 ones (1 ten and 4 ones). Write the 4 ones in ones' place below the line, and add the 1 ten with the tens. 4, 8, 9 tens. Write 9 tens in tens' place. 94 The sum is 94. Copy and add : (1) (2) (3) (4) (5) (6) (7) (8) (9) (lO) 12 25 36 44 16 29 34 28 25 19 14 10 11 14 25 16 23 16 26 52 16 12 14 10 12 11 39 51 33 22 (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) 28 39 28 18 29 67 58 45 19 35 16 14 26 54 14 17 13 16 52 13 2 3 13 17 23 2 ^ 14 26 48 (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) 19 17 25 38 48 36 14 12 25 24 26 26 10 25 11 14 28 27 31 12 31 33 29 31 26 35 31 28 47 45 30 ADDITION AND SUBTRACTION. (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) 23 28 14 22 42 41 32 21 84 21 10 10 15 18 15 25 14 39 13 37 28252237363747182619 34. Find differences, subtracting by tens: Thus, in subtracting 12 from 36, say: 36 less 10 = 26 ; 26 less 2 = 24. (1) (2) (3) (4) (5) (6) (7) 36 47 58 69 75 86 94 10 10 10 10 10 10 10 (8) (9) (10) (11) (12) (13) (14) 36 47 58 69 75 86 94 12 12 12 12 12 12 12 (15) (16) (17) (18) (19) (20) (21) 36 47 58 69 75 86 94 14 14 14 14 14 14 14 (22) (23) (24) (25) (26) (27) (28) 36 47 58 69 75 86 94 20 20 20 20 20 20 20 (29) (30) (31) (32) (33) (34) (35) 46 65 79 54 93 88 37 22 22 22 22 22 22 22 (36) (37) (38) (39) (40) (41) (42) 57 49 65 74 87 95 46 23 23 23 23 23 23 23 QUARTS AND GALLONS. 31 [43) (44) (45) (46) (47) (48) (49) 64 75 39 88 95 57 46 24 24 24 24 24 24 24 QUARTS AND GALLONS. Quart. Gallon. 35. 1. It takes 4 quarts to fill a gallon measure; one quart is what part of a gallon? 2. Two quarts are what part of a gallon? 3. Eight quarts are how many gallons? 4. Two gallons are how many quarts? 5. Five quarts are how much more than a gallon? 6. A gallon and two quarts are how many quarts? 7. I bought a gallon of milk on Tuesday, and half a gallon on Wednesday; how many quarts did I buy? 8. A man sells a quart of milk to each of ten customers; how many gallons does he sell? 9. Sixteen quarts are how many gallons? 10. Two and one-half gallons are how many quarts? 2 pints (pt.) = 1 quart (qt.). 4 quarts = 1 gallon (gal.). 32 MULTIPLICATION AND DIVISION. MULTIPLICATION AND DIVISION. 36. REVIEW. (1) (2) (3) (4) 1X2=? 6X 2=? 2h-2=? 12^ 2=? 2X2=? 7X 2=? 4h-2 = ? 14^ 2=? 3X2=? 8X 2=? 6^2=? 16^ 2=? 4X2=? 9X 2=? 8^2=? 18^ 2=? 5X2=? lOX 2=? 10^2=? 20^ 2=? 2X1=? 2X 6=? 6-^3=? 12 -^ 6=? 2X3=? 2X 7=? 10^5=? 14^ 7=? 2X4=? 2X 9=? 16h-8 = ? 18^ 9=? 2X5=? 2X10=? 8h-4 = ? 20^-10=? 37. Copy and learn: 11 X 2 = 22 22^ 2= 11 12X 2=24 24- 2 = 12 2X11 = 22 22 4-11 = : 2 2X12 = 24 24h-12= : 2 EXERCISE. 38. 1. At 2 cents each, what will 11 pen-holders cost? 2 cents X 11 = 22 cents 11 pen-holders will cost 22 cents. 2. If a man earns 2 dollars a day, how much will he earn in 12 days? 3. Mary wishes to buy some flowers for her mother's MULTIPLICATION AND DIVISION. 33 birthday; if she buys pinks at 2 cents apiece, how many can she buy for 22 cent J? 22 cents ^2 cents = 11 Mary can buy 11 pinks. Read the problem, and give the answer from the picture. f ^ f f ^ ^ ^ 4. At 2 cents each, how many pencils can you buy for 24 cents? 39. REVIEW (1) (2) (3) (4) 2X3=? 5X3=? 6^3=? 12 --3=? 3X3=? 3X5=? 9^3=? 15^3=? 4X3=? 6X3=? 18-^3=? 15^5 = ? 3X4 = ? 2X6=? 6h-2=? 12 -=-4=? 40. Copy and learn: 7X3 = 21 10X3=30 21^3=7 30- ^3=10 8X3 = 24 11X3=33 24-^3 = 8 33- ^3=11 9X3=27 12X3=36 27-^3=9 36- ^3=12 3X7 = 21 3X10=30 21^7=3 30- ^10=3 3X8=24 3X11 = 33 24^8=3 33- Hll = 3 3X9=27 3X12=36 27^-9=3 36- hl2=3 34 MULTIPLICATION AND DIVISION. EXERCISE. 41. In the written work^ give figures and statements. 1. John bought 7 peaches and paid 3 cents apiece; how much money did he spend? 3 centsX7 = 21 cents. John spent 21 cents. 2. At 3 cents each, how many oranges can I buy for 24 cents? (Make a picture.) 3. What will 9 chairs cost, at 3 dollars each? 4. How long does it take a man to earn 27 dollars, if he earns 3 dollars a day? 5. In going to school and returning, George walks 3 miles each day; how far does he walk in 12 days? 6. Helen learned 3 new words each day for 11 days; how many words did she learn? 7. Make problems for: 7X3 = 21 18^3= 6 6X3=18 9X3 = 27 12X3=36 12X2 = 24 24-3= 8 33-^3=11 18-^2= 9 Note. — This work should be done in the recitation period and under the direction of the teacher, the children first laying out the objects for each problem, or making the picture. 8. Write the multiplication table of 3's from 2X3 to 12X3. 9. Recite the table of 3's from memory. 10. Write the table from 3X2 to 3X12, and recite it. 11. Beginning with 3, count by 3's to 36. 12. Beginning with 36, subtract by 3's to 0. MULTIPLICATION AND DIVISION. 35 43. Copy and learn : 2X4== 8 5X4 = 20 8X4=32 11X4=44 3X4=12 6X4 = 24 9X4=36 12X4=48 4X4=16 7X4 = 28 10X4=40 Write these by placing 4 first. Thus, 4X2= 8, 4X3=12, etc. Recite the division table, from the multiplication table. Thus, 2X4=8; 8-^4 = 2, 3X4=12; 12-4=3, and 8^2 = 4. and 12 --3 = 4, etc. EXERCISE. 4:3. Twelve things make a dozen. 1. How many cakes can be made from a dozen eggs, if 4 eggs are used for each cake? 2. If there are 4 desks in a row, how many desks are there in 7 rows? 3. At 4 dollars each, what will 8 hats cost? 4. How many gallon measures can be filled from a can which holds 36 quarts? 5. What will 12 chairs cost, at 4 dollars each? 6. Frank is 32 miles away from home ; if he walks at the rate of 4 miles an hour, how long will he be in reaching home? 7. I put 28 quarts of oil into lamps holding 4 quarts each; how many lamps can I fill? (Make a picture.) 8. 9 gallons are how many quarts? 9. How many quart cans will be needed to hold 9 gallons of maple molasses? 36 MULTIPLICATION AND DIVISION, 10. Make problems for: 6X4 = 24 9X4 = 36 48^4 = 12 11X4 = 44 8X4 = 32 28^-4- 7 40h-4=10 36-^4= 9 7X4 = 28 Note. — This should be done in class. 11. Begin with 4 and count by 4^s to 48. 12. Begin with 48 and subtract by 4^s to 0. 13. Give answers at sight: Thus : 7 multiplied by 4 equals 28. 7X4 12X4 7X3 11x4 12X3 3X 9 4X12 3X 7 9X3 8X3 6X4 9X2 9X4 3X12 3X 8 3X11 6X3 8X4 12X2 9X3 9X3 2X 9 4X 7 4X11 14. Among how many boys can 12 oranges be divided, if each boy receives 4 oranges? This result may be stated in two ways : 12 oranges ^4 oranges = 3; or, 4 oranges)12 oranges ~~3 They can be divided among 3 boys. 15. Read at sight: 4 apples )24 apples 4 roses) 48 roses 3 books) 36 books 6 "^^^^^ ^^^ 4 nails ) 36 nails 3 pencils)21 pencils 4 gallons)44 gallons 4 pints) 32 pints 3 lemons)24 lemons 3 dollars)27 dollars FINDING ONE OF THE EQUAL PARTS OF A NUMBER. 37 Finding One of the Equal Parts of a Number. 44. I wish to divide 21 nuts equally among three chil- dren; how many nuts will each receive? We count off one to each child in turn, until we have given away all the nuts. First child, ©©©©©©© Second child, ©©©©©©© Third child, ©©•©©©©© J of 21 nuts is 7 nuts. Each child will receive 7 nuts. EXERCISE. 45. 1. Divide 21 apples equally among 3 children. How many apples will each receive? (Make a picture, and give figures and statement.) 2. Divide 27 shells equally among 3 children; how many will each receive? 3. Divide 28 roses equally among 4 sisters; how many will each receive? 4. A gardener takes 24 plants to market and sells one- third of them; how many does he sell? 5. Frank had 30 cents and spent ^ of his money for a top; how many cents did he spend? 38 FINDING ONE OF THE EQUAL PARTS OF A NUMBER. 6. Divide 23 apples equally between two boys. How many will each receive? First boy, OOOOOOOOOOOO Second boy, OOOOOOOOOOOO i of 23 apples equals 11 J apples. ' Each boy receives 11^ apples. 7. Henry paid 25 cents for two pounds of butter; what is the cost of one pound? 8. Divide 21 pears equally among 4 children. What part of all the pears will each receive? How many pears will each receive? (Make a picture.) 9. I had 40 cents and spent J of it for a yard of muslin; how much did I spend? 10. John has a tape measure 36 inches in length; one- third of the measure is how many inches? 11. If I divide a melon among three boys, so that their shares are equal, what part of the melon does each boy receive? 12. Divide 16 cakes equally among 3 playmates; how many cakes will each receive? First, O O O O O ^ Second, O O O O O ^ Third, O O O O O ^ 13. Find J of 10, 15, 16, 18, 19, 21, 24, 25, 28 and 36. 14. Find J of all numbers from 12 to 25. 15. Find i of 16, 17, 20, 21, 24, 28, 32 and 36. FINDING ONE OF THE EQUAL PARTS OF A NUMBER. 39 16. Make problems for : J of 15 = 5 i of 24 = 6 i of 21 = 7 J of 18 = 6 i of 30 = 10 i^ofl9 = 6J Note, — To be done in class. Here are two ways by which we may express the finding of one of the three equal parts of 18 dollars. J of 18 dollars = 6 dollars. 3)18 dollars 6 dollars 17. Give answers at sight: 3)18 cents 4)36 inches 3)28 pencils cents inches pencils 2)24 apples 3)30 yards 4)32 roses apples yards roses 3)21 nuts 2)25 apples 4)48 dollars nuts apples dollars 46. REVIEW. 1X2= 2 1X3= 3 1X4= 4 2X2= 4 2X3= 6 2X4- 8 3X2= 6 3X3= 9 3X4=12 4X2= 8 4X3 = 12 4X4 = 16 5X2 = 10 5X3 = 15 5X4 = 20 6X2 = 12 6X3 = 18 6X4 = 24 7X2 = 14 7X3 = 21 7X4 = 28 8X2 = 16 8X3 = 24 8X4 = 32 9X2 = 18 9X3 = 27 9X4 = 36 10X2 = 20 10X3 = 30 10X4 = 40 r 11X2 = 22 11X3 = 33 11X4 = 44 12X2 = 24 12X3 = 36 12X4 = 48 Write these tables with the 2's, 3's and 4's first. 40 EXERCISE. EXERCISE. 47. 1. On Monday Albert had 30 cents in his savings bank; on Tuesday he earned 10 cents by selUng papers. How much money had he then? 2. George had 24 cents and spent 6 cents for an orange; how many cents had he left? 24 cents — 6 cents = 18 cents. George had 18 cents left. 3. Frank paid 24 cents for a kite and 6 cents for a top; how much more did he pay for the kite than for the top? 24 cents — 6 cents = 18 cents. Frank paid 18 cents more for the kite than for the top. 4. Robert had 3 dozen nails and gave one dozen to Henry ; how many dozen had he left? How many nails had he left? 5. Margaret gathered 28 pond-lilies and Mabel gathered 10; how many did both gather? 6. A boy having 40 cents, paid 8 cents for a ball; how many cents had he then? 7. Grace wrote 11 lines in her copy-book on Monday, 6 on Tuesday and 3 on Wednesday; how many lines did she write in all? 8. Mary wishes to buy a picture which costs 25 cents ; she has only 10 cents. How many more cents must she have to buy the picture? 9. One week a chair-maker made 2 dozen parlor chairs and 8 office-chairs. How many chairs did he make in the week? EXERCISE. 41 10. Alfred had 40 cents; he paid 9 cents for pencils; how many cents had he then? 11. A grocer bought 4 dozen boxes of strawberries; he sold one dozen boxes. How many boxes had he left? 12. Eight gallons are how many quarts? 13. If 8 quarts of cider are sold from an 8-gallon cask, how many quarts are left? CHAPTER III. READING AND WRITING NUMBERS: HUNDREDS One Hundred to Five Hundred. 48. Take counters and find 1 hundred, 2 tens, 4 ones. Write the number. Note. — Use toothpicks or shoe-pegs, as heretofore suggested. Find with the counters and write : 1 hundred, 3 tens, 7 ones. 1 hundred, 9 tens, ones. ^ a 5 a g u ^ u Q u ^ u 1 ^' 8 ^^ 3 ^^ 1 " 1 ^< lone. How many ones are there in the first number? How many ones in each of the other numbers? Begin with 1 hundred, and write in figures all the num- bers through 1 hundred, 9 tens, and 9 ones. Add 1 to 199; how many tens have you? How many ones? Write the number. 49. What does the figure 4 mean in the num- 134 ber 134? Because the figure 4 means 4 ones, it 216 is written in ones^ place. For what do the 6, 9, |^^ and 8 stand? 5 and 7 stand for what? They are written in tens^ place. The figure 1 in the first number means what? The figure 2 in the second number means what? READING AND WRITING NUMBERS: HUNDREDS. 43 Because the figure 1 means one hun- P eriod of ones. dred, it is written in hundreds^ place. Hundreds' Tens' ones' ' ^ ^ place. place, place. In what place is the figure 2 written? 13 4 Hundreds are always written in hun- 2 16 dreds^ place, tens in tens' place, and ones 15 9 in ones' place. Ones^ place, tens' place, and hundreds' place make the period of ones. 50. 1. Copy and read these numbers: 124 101 116 173 119 112 186 111 105 113 191 121 210 198 115 131 109 200 181 106 137 103 129 201 2. Find with the counters and write the numbers from 200 to 300. 3. Write in figures: 2 hundreds, 5 tens, 3 ones. 3 hundreds, tens, ones. 4 '' 6 " 7 " 3 '' 2 '' 2 '' 3 " 7 ^^ 9 ^^ 5 " " '' 4. The first number equals how many ones? 5. Which is greater, 3 hundred or 3 tens? 6. 5 hundred is how many more than 3 hundred? 7. Read the following : 311 222 202 210 333 212 413 331 221 125 313 211 8. '414 is how many more than 313? 330 is how many more than 230? 450 is how many less than 500? 314 384 404 341 448 401 413 444 309 431 414 319 44 READING AND WRITING NUMBERS: HUNDREDS. Five Hundred to One Thousand. 51. 1. Find with the counters 5 hundreds, 9 tens, and 9 ones. Add one one. How many hundreds have you? 2. Write in figures 6 hundreds. Write 6 hundreds, 4 tens; 6 hundreds, 5 tens, and 2 ones. 3. Find 7 hundreds. Write the number in figures. 4. Read: 643 611 721 505 707 606 751 610 712 689 770 660 589 601 702 799 777 666 5. Which is the greater number, 659 or 569? 697 or 769? 571 or 751? 6. Find 8 hundreds, 8 tens, and 8 ones. 7. Find 9 hundreds, and represent the number by figures. 8. Find 9 hundreds, 9 tens, and 9 ones. 9.- Read: 845 984 901 991 999 880 862 936 909 919 808 881 933 847 990 900 888 818 52. The greatest number that can be expressed by three figures is 999. Add 1 to 999; how many hundreds have you? Ten hundreds make 1 thousand. The number one thousand is expressed by writing the figure 1 in thousands' place, to the left of the hundreds; thus, 1,000. Ten hundreds equal one thousand. ADDITION. 45 ADDITION. . EXERCISE. 53. (a) Add at sight, naming each sum: (b) Copy and find the answers: (1) (2) (3) (4) (5) (6) (7) (8) (9) 31 27 20 11 12 31 51 12 32 21 18 31 41 50 19 21 41 12 29 40 28 16 17 25 19 16 27 383127251634171815 (10) (11) (12) (13) (14) (15) (16) (17) (18) 23 12 13 51 31 61 41 32 62 23 25 23 14 29 14 24 43 16 19 16 39 18 46 17 59 5S 17 234424254324351453 (19) (20) (21) (22) (23) (24) (25) (26) (27) 13 51 82 13 41 79 54 8 14 58 36 14 48 36 10 26 25 36 66 49 28 66 49 28 7 92 27 42 3443433443531482 (28) (29) (30) (31) (32) (33) (34) (35) (36) 79 99 89 38 38 97 38 56 2 13 03 13 04 16 49 17 57 12 66 25 25 72 72 21 82 23 49 21 51 41 62 52 11 41 41 63 11 11 12 13 12 11 12 11 93 46 SUBTRACTION (37) (38) (39) (40) (41) (42) (43) (44) (45) 76 36 55 51 50 98 71 72 34 64 34 24 34 59 57 15 45 75 23 73 75 72 34 12 64 41 22 12 33 21 14 31 12 25 16 12 121111191110191557 54. Finding the sum of two or more numbers is called Addition. The sign of addition ( + ) is called plus. The numbers between which it s placed are to be added. 8 + 6=14 is read, ^' 8 plus 6 equals 14. '^ SUBTRACTION. 55. 1. The sum of two numbers is 14; one of the num- bers is 8. What is the other number? 2. The sum of two numbers is 30; one of the numbers is 5. What is the other number? 3. Separate 40 into two equal parts, and take out one of the parts; what remains? 4. From 40 take 27. 5. What are the two parts of 40 in problem 4? 6. Subtract at sight : 40 40 50 60 85 75 60 95 88 20 10 20 12 15 5 30 12 14 7. Read answers, subtracting ones first: 126 138 146 137 129 158 169 187 54346574 SUBTRACTION. 47 8. 42 is part of 161. If we subtract 42 from 161, what remains? Write ones under ones, tens under tens. (Place the bundles of sticks, 1 hundred, 6 tens, 1 one, over the figures 161.) Subtract ones first. 2 ones cannot be taken out of 1 one. Take 1 ten from the tens, leaving 5 tens. (Show with the sticks.) The 1 ten which we have taken is equal to 10 ones, which we add to the 1 one to make 11 ones. 2 ones from 11 ones leave 9 ones. 9 is written in ones' place below the line. 4 tens from 5 tens leave 1 ten, which is written in tens' place below the line. No hundreds from 1 hundred leave 1 hundred, which is written in hundreds' place below the line. 119 is the part of 161 which we wished to find. The two parts, 42 and 119, make what number ? 5 10 1^1 4 2 1 1 9 5Q. Subtract: EXERCISE. (1) (2) (3) (4) (5) (6) (7) (8) 171 281 261 252 263 361 260 392 32 42 34 26 35 28 44 58 (9) (10) (11) (12) (13) (14) (15) (16) 391 282 271 390 471 392 282 373 54 68 69 72 54 75 67 58 (17) (18) (19) (20) (21) (22) (23) (24) 184 153 174 183 168 173 163 177 45 35 146 156 59 49 138 148 (25) (26) (27) (28) (29) (30) (31) (32) 340 265 292 290 156 182 280 180 123 48 147 137 49 55 168 146 48 SUBTRACTION. (33) (34) (35) (36) (37) (38) (39) (40) 291 294 162 161 165 261 183 190 44 58 136 138 59 48 157 129 (41) (42) (43) (44) (45) (46) (47) (48) 290 192 191 187 281 191 271 288 74 89 168 158 67 69 136 169 (49) (50) (51) (52) (53) (54) (55) (56) 294 390 295 365 293 372 281 390 69 85 147 148 175 148 167 178 (57) (58) (59) (60) (61) (62) (63) (64) 204 312 315 305 206 304 305 306 141 191 184 144 156 142 143 164 (65) (66) (67) (68) (69) (70) (71) (72) 300 216 321 222 311 214 304 301 85 97 84 74 94 65 68 192 Note. — Just as soon as possible have tlie pupils subtract with- out rewriting the minuend. 5*7. Taking a part of a number out of it, to find the remainder, is called Subtraction. The number to be diminished by taking one of the parts is called the Minuend. The part taken out of the minuend is called the Subtra- hend ; the part left is called the Remainder. The sign of subtraction ( — ) is called minus or less. 14—6 = 8 is read, ''14 minus 6 equals 8'^; it means that 14 diminished by 6 equals 8. INCH, FOOT AND YARD. 49 INCH, FOOT AND YARD. One Inch, 58. What is the length of the first Hne? Of the second? The length of the first line is what part of the length of the second? The second line is what part of the third? Cut a piece of paper 12 inches long and 1 inch wide. Draw a line 12 inches long. 12 inches make a measure that is called one . How many six-inch sticks of candy can you cut from a stick 12 inches long? How many 3-inch sticks? 3 inches are what part of a foot? How many 4-inch lead-pencils can be made from a piece of lead 12 inches long? 4 inches are what part of a foot? 8 inches? One inch? How many feet are there in fifteen inches? In eighteen? 59. Draw a line 3 feet long. Three feet make one yard. Mention some things that are sold by the yard. How many inches are there in one yard? 1 foot is what part of a yard? J a yard is how many inches? How many feet? i of a yard is how many inches? | of a yard? 12 inches are what part of a yard? 4J inches? 50 INCH, FOOT AND YARD. Ella has a yard of silk with which to dress 4 dolls for the fair. What part of the silk will she use for each dress, if she divides it equally? How many inches for each? If a yard of ribbon is divided for badges equally among 6 boys, what will be the length of each piece? In 2 yards there are how many feet? In 3 yards? How many half yards in 2 yards? How many in 3 yards? 60. How many feet tall are you? What is the height of the teacher's table from the floor? (Estimate first, and then measure.) What is the length of the table? How far is it from the top of your desk to the floor? What is the height of the transom from the floor? Width of window-sash? Height of the clock from the floor? Length, in feet, of front blackboard? Length of room? Width of room? Tie a knot for every foot in a piece of twine 6 feet long. Tie a double knot for every yard. Estimate length, width, and height of things outside of the schoolroom, and then measure: height of a barrel; of a common wooden bucket; length of an ear of corn. 12 inches (in.) = 1 foot (ft.). 3 feet =1 yard (yd.). MISCELLANEOUS PROBLEMS. 61. 1. At 10 cents each, how many pineapples can you buy for 30 cents? 2. Mary wishes to plant some pinks in a triangular garden bed. How many plants will she use, if she plants 10 on each of the 3 sides? (Picture.) MISCELLANEOUS PROBLEMS. 51 3. How many 5-cent pieces make 30 cents? 4. A church is Hghted by 8 lamps of 4 burners each. How many burners are there in all? 5. George planted some hyacinth bulbs in 6 boxes, planting 4 bulbs in each box. How many did he plant in all? 6. A boy walks 10 squares in 30 minutes. At that rate, how long is he in walking one square? 7. I bought 4 verbenas for 32 cents. What did each plant cost, if they are of equal value? 8. Ella found 4 eggs each day for a week; how many did she find? 9. How many petals have 8 violets, if each flower has 5 petals? 10. John worked four weeks. How many days did he work? 11. How much money did John earn if he earned a dollar each working day? 12. Eleanor had 3 dimes; she spent 6 cents for a pencil. How much money had she left? 13. 27 feet is three times the length of a ladder; what is its length? 14. Mary made 25 sponge-cakes; she divided them equally among 5 brothers and sisters; how many did each receive? 15. Divide 30 crackers equally among 5 boys. 16. At 12^ cents a can, what will 2 cans of corn cost? 17. Some children were gathering goldenrod. They found that they had gathered in all 24 branches. They divided them equally and each had 8 branches. How many children were there? 52 MISCELLANEOUS PROBLEMS. 18. I wish to put 36 quarts of milk into cans holding 4 quarts each; how many cans will be needed? 19. Divide 32 quarts of milk equally among 4 customers; how many quarts will each receive? 20. 48 pounds of honey were packed in 4 jars of equal size ; how many pounds were in each jar? 21. I bought some muslin for 25 cents and had 25 cents left; how much money had I at first? 22. John bought a pair of skates for 75 cents and sold them for 50 cents. Did he gain or lose? How much? 23. Edgar sold a knife for 30 cents; this is 5 cents less than he paid for it. How much did he pay for the knife? 24. A peck of apples costs 30 cents; I must borrow 5 cents in order to pay for them; how much money have I? 25. At a picnic 44 cups of lemonade were passed to 4 rows of children; how many cups were passed to each row? 26. How long will 48 pounds of butter last, if used at the rate of 4 pounds a week? 27. If you have 40 pansies tied in bunches of 10 each, how many bunches have you? 28. A family used 32 bushels of apples in 8 months; at that rate, how many bushels were used in one month? 29. If 36 tuberoses are planted in 4 equal rows, how many are there in each row? 30. 48 quarts of ice-cream are how many gallons? 31. If Frank and James each can mow the lawn in 2 hours, in how many hours can Frank and James together do the same work? (Will it take a longer or a shorter time?) 32. Grace hemmed six aprons in 4 hours; in how many hours can Grace and Mabel together do the work, if Mabel works as fast as Grace? MEASURING TIME. 53 33. On a journey of two days, I traveled 28 miles the first day and 67 miles the second day; how many miles did I travel? 34. A man earns 125 dollars in one month and spends 18 dollars for rent and 30 dollars for groceries; how much money has he left? 35. Edna gathered 30 pinks, 42 roses and 24 violets; how many flowers did she gather? 36. Mabel spent 25 cents for some pencils, 15 cents for paper, and 40 cents for a book; how many cents did she spend? 37. Emma spent 5 cents for a pencil and had 42 cents left; how much money had she at first? 38. I bought ^ a yard of velvet and found that I needed ^ of a yard more; how much should I have bought at first? 39. After using | a yard of ribbon, I had IJ yards left; how many yards had I at first? IIEASURING TIME. 62. Are there any other ways of measuring time than by the clock and the hour-glass? Have you ever seen a sun-dial? 54 MEASURING TIME. How many minutes is the long hand in passing from one figure to another? The space between the figures is divided into five equal parts. The long hand is a minute in passing over one of these smallest spaces. See how many times you can walk across the floor in a minute. Sit still and watch the clock a minute; notice how much space the long hand has passed over. Is any smaller portion of time than a minute measured by the clock? Some clocks tick 60 times in a minute. Sixty seconds make a minute. 30 seconds are what part of a minute? 63. How long does it take the minute hand to move en- tirely round the face of the clock? Count the small spaces on the face of the clock. Sixty minutes make an hour. How many minutes are there in 2 hours? 5 minutes are what part of an hour? What time is it by the clock on page 53? If the long hand were moved forward to the figure 1, what time would the clock show? Where will the short hand of the clock point when the minute hand points to 6? Draw a picture of the clock. What time does it show? How many hours are there from 6 in the morning until noon? How many hours from noon until midnight ? Twenty-four hours make a day, 64. How many days make a week? How many weeks make a month? Name the months in order. Name those MULTIPLICATION AND DIVISION. 55 which have 30 days. How many days are there in Feb- ruary? 60 seconds (sec.) = l minute (min.). 60 minutes =1 hour (h.). 24 hours =1 day (d.). 7 days =lweek(w.). 4 weeks = 1 month (m.). 12 months 1 365 days I =lyear(yr.). MULTIPLICATION AND DIVISION. 65. Copy and learn: 5X5 = 25 7X5 = 35 9X5 = 45 11X5-55 6X5 = 30 8X5 = 40 10X5 = 50 12X5 = 60 Write the table with the 5's first. Recite the division table from the multiplication table. Thus,25-5 = 5 30^6 = 5, etc. 30^5 = 6 EXERCISE. 66, 1. At 5 dollars a pair, what will 5 pairs of shoes cost? 2. If a family uses 5 pounds of butter in one week, in how many weeks will 30 pounds be used? 3. At 5 cents a pound, how many pounds of sugar can be bought for 45 cents? 4. 40 loaves of bread will last a camping party how many days, if they use 5 loaves a day? 5. At 5 cents a spool, how many spools of thread can be bought for 60 cents? 56 MULTIPLICATION AND DIVISION. 6. If a yard of cloth costs 5 dollars, what will 7 yards cost? 7. If a man earns 5 dollars a day, how many dollars does he earn in a week, counting 6 working days? 8. Mabel gathered 35 roses which she divided among her friends, giving 5 roses to each; among how many friends did she divide them? 9. At 5 cents a paper, what will a dozen papers of needles cost? 10. If it takes 5 yards of cloth to make a suit of clothes, how many suits can be made from 55 yards? 11. Make problems for : 9X5=45 5)45 5)35 8X5=40 7X5=35 9 7 6X5 = 30 12X5=60 5)55 5)50 5X6=30 5X8=40 11 5 11X5=55 Note. — Do this in class. 67. Copy and learn: 6X6 = 36 8X6 = 48 10x6 = 60 12X6=72 7X6 = 42 9X6 = 54 11X6 = 66 Write the table with the 6^s first. Recite the division table from the multiplication table. EXERCISE. 68. 1. How many yards of fringe will be needed for 7 rugs, if 6 yards are used for one rug? 2. When melons are selling for 12 cents each, what will 6 cost? FINDING ONE OF THE EQUAL PARTS OF A NUMBER. 57 3. At 6 dollars each, how many flags can be bought for 48 dollars? 4. If I use 6 small flags for decorating one window, how many shall I need for 9 windows? 5. In how many months can I pay for a sewing machine which costs 60 dollars, if I pay 6 dollars a month? 6. How many inches are there in a yard? How many badges 6 inches long can be cut from one yard of ribbon? 7. A merchant sold a dozen silk umbrellas at 6 dollars each; how much money did he receive in payment? 8. At 6 dollars a dozen, what will 5 dozen spoons cost? 9. I received 66 dollars for 11 barrels of apples; how much is that a barrel? 10. If 6 gallons of oil are used in a month, how long will 48 gallons last? 11. Make problems for: 9X6 = 6)48 6)36 11X6= 8 6 .2X6 = 6)54 6)^ 7X6= 9 12 Note. — Do this in class. Finding One of the Equal Parts of a Number. EXERCISE. 69. 1. If 6 yards of cloth cost 30 dollars, what is the cost of one yard? 2. Frank walks 35 miles in 5 days; at that rate how many miles does he walk in one day? 58 FINDING ONE OF THE EQUAL PARTS OF A NUMBER. 3. 42 pounds of butter are packed in 6 jars of equal size; how many pounds are put in each jar? 4. I paid 60 cents for a dozen oranges; at that rate, what is the cost of one orange? 5. George planted 45 tulip bulbs in 5 equal rows; how many did he plant n one row? 6. A man earned 72 dollars in 6 weeks; at that rate how much does he earn in one week? 7. Six children were gathering shells; they found that they had gathered 54 in all. If they divided them equally, how many did each child receive? 8. A boy walks 9 squares in 27 minutes; how long is he in walking one square? 9. If 9 pounds of rice cost 72 cents, what is the cost of one pound? 10. A gardener takes 4 dozen plants to market and sells only one-sixth of them; how many plants does he sell? 11. Make problems for: i of 60=10 5 )30 apples 60h-6 = 10 6 apples i of 40 = 8 6 )72 cents 42-6 = 7 12 cents i of 54 = 9 5)60yards 72-12 = 6 12 yards Note. — Do this in class. 70. Copy and complete : 5^5 = 1 8-h5 = 11^5 = 6 -^5= 1, and 1 remaining 9-5 = 12^5 = 7 -r5 = 1, and 2 remaining 10-^5 = 13-^5 = REVIEW OF MULTIPLICA TION, 59 14^5 = 21^5= 28^5 = 15^5 = 22 --5 = 29^5 = 16-4-5= 23^5 = 30h-5 = 17-^5= 24h-5 = 31^5= 18^5 = 25^5 = 32^5= 19h-5 = 26^5 = 33^5 = 20^5 = 27-^5 = 34^5= 35h-5= Divide all numbers from 6 to 36, by 6. 71. REVIEW. 1X2= 2 1X3= 3 1X4= 4 2X2= 4 2X3= 6 2X4= 8 3X2= 6 3X3= 9 3X4 = 12 4X2= 8 4X3 = 12 4X4 = 16 5X2 = 10 5X3 = 15 5X4=20 6X2 = 12 6X3 = 18 6X4 = 24 7X2 = 14 7X3 = 21 7X4 = 28 8X2 = 16 8X3 = 24 8X4 = 32 9X2 = 18 9X3 = 27 9X4 = 36 10X2 = 20 10X3 = 30 10x4 = 40 11X2 = 22 11X3 = 33 11X4 = 44 12X2 = 24 12X3 = 36 12x4 = 48 1X5= 5 7X5 = 35 2X5 = 10 8X5 = 40 3X5 = 15 9X5 = 45 4X5 = 20 10X5 = 50 5X5 = 25 11X5 = 55 6X5 = 30 12X5 = 60 60 ADDITION TABLE. 1X6= 6 7X6 = 42 2X6 = 12 8X6 = 48 3X6 = 18 9X6 = 54 4X6 = 24 10X6 = 60 5X6 = 30 11X6 = 66 6X6 = 36 12X6 = 72 Without rewriting, read these with 2, 3 4, 5 and 6 first. 72. ADDITION TABLE. 1 2 32 43 543 1 1 12 12 12 3 654 7654 8765 123 1234 1234 98765 9876 12345 2345 9876 987 987 3456 456 567 9 8 9 8 9 9 6 7 7 8 8 9 Note. — The 45 sums given above must be learned as the basis for accuracy and rapidity in addition. 73. SUBTRACTION TABLE. 2 33 444 5555 1 12 123 1234 SUBTRACTION TABLE. 61 66666 777777 12345 123456 9 9 1 2 9 9 9 9 3 4 5 6 9 9 7 8 10 10 10 1 2 3 10 10 10 4 5 6 10 10 7 8 10 9 11 11 11 11 11 11 11 11 23456789 12 12 12 12 12 12 12 3 4 5 6 7 8 9 13 13 13 13 13 13 14 14 14 14 14 4 5 6 7 8 < ) 5 6 7 8 9 15 15 15 15 6 7 8 9 16 7 16 8 16 9 17 17 18 8 9 9 Note. — These 74 primaiy facts of subtraction should be thoroughly learned. These tables should be frequently reviewed. They should be placed on the board or on a chart so that they may be readily used. CHAPTER IV. READING AND WRITING NUMBERS : THOUSANDS. 74. You have learned that the number one thousand is expressed by writing the figure 1 to the left of hundreds' place. Read the following numbers : 1,500 1,230 1,400 1,670 1,873 1,999 1,220 1,864 1,748 1,976 1,449 1,650 The period of ones is separated from the thousands by a comma. Write in figures: two thousand, three thousand, five thousand, eight thousand, nine thousand. Read the following numbers: 3,000 7,000 6,350 3,200 5,102 8,008 4,340 2,501 8,108 8,650 7,206 0,888 9,241 7,777 5,230 The greatest number that can be expressed by four figures is 9,999. Write in figures: Three thousand seven hundred fifty. Eight thousand two hundred two. One thousand eleven; one thousand one. Five thousand five; five thousand fifty. Four thousand thirty-five; four thousand five. 4,500 9,400 1,111 0,444 1,001 2,020 1,100 4,009 1,004 9,999 READING AND WRITING NUMBERS: THOUSANDS. 63 Express in figures numbers composed of : thousands 6 hundreds 7 tens and 4 ones. 3 l( 3 U 3 ii li 3 li 9 iC 8 iC 5 (C (C 6 11 8 a u {( a 7 cc 5 (( 9 IC 9 (C C( IC 15. Write one thousand in figures. In what place does the figure 1 stand? If we wish to express a number ten times as great as 1,000, how shall we represent it? One ten- thousand is ten times as great as one thousand. We ex- press the 1 ten-thousand by writing the figure 1 to the left of thousands, in ten-thousands' place; thus, 10,000. Note. — A box of small toothpicks may be used in bundles of tens, hundreds, and thousands, to show the ten-thousand. 1. Write 2 ten-thousands. 2 ten-thousands are how many ones? 2. Write 3 ten-thousands and read the number in two ways. (How many thousands? How many ones?) 3. Read the following numbers: 30,000 25,400 15,021 10,010 50,000 36,303 21,048 11,001 90,000 47,350 16,743 15,005 41,000 54,707 28,096 15,015 65,000 90,900 11,110 99,999 4. How many ones are there in each of the last five num- bers? 64 ROMAN NOTATION. 5. Write the following in figures: 27 thousand 600 ones. 30 thousand 500 ones. 70 thousand 350 ones. 60 thousand 70 ones. 6 thousand 70 ones. 95 thousand 200 ones. 80 thousand 8 ones. 8 thousand 8 ones. _ 6. Write in figures: Seventeen thousand seven. Twenty thousand two. Eighty thousand eighty-one. Twelve thousand twenty- one. Eleven thousand one. Eleven thousand one hun- dred ten. Seventeen thousand seven- teen. Ninety thousand nine. Twelve thousand twelve. Fifty-six thousand one hun- dred fifty-six. Ten thousand ten. Eleven thousand eleven. •76, 30 40 XXX XL ROMAN NOTATION. 50 60 70 80 90 100 L LX LXX LXXX XC C When a letter is repeated, its value is repeated. When a letter is placed after one of greater value, its value is added; when placed before, its value is subtracted from the greater. Express the following numbers by figures: XXXIX LIX XC XCIX XLIX LXV XCI XCVIII XLVIII LXX LXXXIX LXXIX XIX XLIV XCVIII XLIV XXIX LXXX LXXXVIII LXVI M ULTI PLICA TION. 65 Express the following numbers by letters : 45 94 42 49 51 87 58 59 68 61 75 99 73 49 95 83 MULTIPLICATION. YTo Two times 24 cents are how many cents? 2 times $80=? 2 times 396=? Two times 6 ones are 12 ones. 12 ones equal 1 ten 396 and 2 ones. Write the 2 ones in ones' place. 2 2 times 9 tens are 18 tens ; adding 1 ten we have 19 tens, equal to 1 hundred and 9 tens. Write 9 tens 792 in tens' place. 2 times 3 hundreds are six hundreds; adding" one hundred we have 7 hundreds, which we write in hundreds' place. 2 times 396 equal 792. Find the same result by addition, and notice the number of ones added ; the number of tens, etc. 396 is called the Multiplicand; it is the number to be multiplied. 2 is called the Multiplier; it is the number which shows how many times the multiplicand is taken. 792 is called the Product; it is the result obtained by multiplying. The multiplicand and the multiplier are called Factors (makers) of the product. The sign of multiplication is X; read " multiplied by.^' 396X2 = 792 is read, '' 396 multiplied by 2 equals 792/' 66 MULTIPLICATION. EXERCISE. 78. 1. If a man travels 96 miles in a day, how far, at that rate, will he travel in 2 days? 96 miles, distance traveled in 1 day. 2 192 miles, distance traveled in 2 days. 2. At $2 a box, what will 87 boxes of lemons cost? $2, cost of 1 box. 87 $174, cost of 87 boxes of lemons. 87 times $2 = $174. Multiply 87 by 2, and call the result dollars. 3. What will 126 pairs of shoes cost, at $2 a pair? 4. If a train runs 328 miles in a day, how far will it run in 2 days? 5. How many feet have 837 men? 6. How many eyes have 918 men? 7. At $2 a yard, what will be the cost of 528 yards of cloth? 8. What will 929 barrels of apples cost, at $2 a barrel? Multiply: (9) (10) 378 856 (11) 504 (12) 978 (13) (14) 709 768 (15) 980 2 2 2 2 2 2 2 (16) (17) 309 2023 (18) 986 (19) (20) 4507 4659 (21) 4709 2 2 2 2 2 2 DIVISION. 67 22. 468 multiplied by 2= ? 25. 349 multiplied by 2 = ? 23. 763 '' 2=? 26. 786 " 2=? 24. 849 " 2=? 27. 605 " 2=? DIVISION. 79. How many times can 2 cents be taken out of 50 cents? How many times out of 80 cents? Out of 90 cents? How many 2's can be taken out of 9 tens 8 ones? Show with the counters that 45 twos can be taken out of 9 tens, or 90 ones, and that 4 twos can be taken out of 8 ones. 49 twos can be taken out of 9 tens 8 ones. Show that 9 tens (or 90) hold 2 ones 4 tens (or 40) times, with 1 ten remaining. The 1 ten is equal to 10 ones. 10 ones and 8 ones are 18 ones. 18 ones hold 2 ones 9 times. How many 2's can be taken out of 972? 2 is contained in 9 hundred 4 hundred times, with 2)972 1 hundred remaining undivided, which is equal to 10 AQa tens. 10 tens and 7 tens are 17 tens. 2 is contained in 17 tens 8 tens times, with 1 ten remaining, which is equal to 10 ones. 10 ones and 2 ones are 12 ones. 2 is contained in 12 ones 6 times. 2 can be taken out of 972 486 times, or 486 twos can be taken out of 972. How many $2 are there in $972? At $2 a barrel, how many barrels of potatoes can be bought for $972? $2 ) $972 486, number of 2-doIlars is $972. 486 barrels of potatoes, at $2 a barrel, can be bought for $972. 68 DIVISION, 972 is called the Dividend. 2 is called the Divisor. A divisor is called an Exact Divisor when it is contained in the dividend without a remainder. Divide 73 by 2. 2)_73 36—1 1 is the remainder and the division is not exact. 486 is called the Quotient ; it is the result of the division. The divisor and quotient are Factors of the dividend. The product of the divisor and the quotient, plus the remainder, is equal to the dividend. In the problem above, 2 is divisor, 36 quotient, and 1 re- mainder. 36 X 2 = 72 ; 72 + 1 = 73. 73 is the dividend. Division is expressed in three ways. Each of the ex- pressions, 24 -2 = 12, V = 12, and 2)24, is read, ''24 divided by 2 equals \2r 12 80. Divide by 2: 1. 7398 4. 8604 7. 7176 10. 2015 2. 8249 5. 7170 8. 1257 11. 4819 3. 9781 6. 9410 9. 6729 12. 9197 EXERCISE. 81. 1. One half of 90 cents is how many cents? 2. Divide 9 dimes equally between 2 boys; how many dimes will each receive? 3. Find ^ of 9 tens 8 ones. i of 9 tens is 4 tens, with 1 ten remaining, which is equal to ten ones. Ten ones and 8 ones are 18 ones. \ of 18 ones is 9 ones. One-half of 9 tens 8 ones is 49 ones. (Show by counters.) MULTIPLYING AND DIVIDING BY 3. 69 4. A man divided $972 equally between his two children; how much money did each receive? 2 ) $972, money to be divided. I486, money each child received. Find J of : 5.9875 7.9347 9.3098 11. $8101 13. 7003 bushels. 6. 6001 8. 7190 10. 5729 12. $7900 14. 5045 pecks. EXERCISE. 82. 1. If a man travels 286 miles in 2 days, at that rate how far will he travel in one day? 2. A clock strikes 312 times in 2 days; how many times does it strike in 1 day? 3. How many times must we take the number 2 to make 652? 4. If a man earns $2 a day, how long will it take him to earn $550? 5. A bookseller paid $114 for photograph albums at $2 each; how many did he buy? 6. A gardener had 750 strawberry plants, and sold J of them; how many did he sell? 7. What number multiplied by 2 will produce 1680? MULTIPLYING AND DIVIDING BY 3. 83. Find the products of: 1. 3086X3 4. 3006X3 7. 3009X3 10. 4123X3 2. 3097X3 5. 3246X3 8. 2549X3 11. 2867x3 3. 2786X3 6. 3269x3 9. 3369X3 12. 3009X3 13. 3 multiplied by 2738=? 14, 3 multiplied by 3108=? 70 MULTIPLYING AND DIVIDING BY 4, 5, AND 6. Find the quotients of : 15. 3687^3 17. 7891 -^3 19. 2501^3 21. 9108^3 16. 3456-7-3 18. 5476-^3 20. 7057^3 22. 8310-^3 EXERCISE. 84. 1. How many 3-cents are there in 3564 cents? 2. How many yards are there in a coil of wire which contains 2500 feet? 3. What will 687 yards of cloth cost, at $3 a yard? 4. A man saved $3 a week; in how many weeks, at that rate, will he save $450? 5. If a steamer can run 278 miles a day, how far can it run in 3 days? 6. $241 is i of my money; how much money have I? 7. Three times $395 is the price of a lot; what is the value of the lot? 8. Find dividends: 3) 3) 3) 3) 241 335 35J 680 9. How is the dividend found, when divisor and quo- tient are given? What are the factors of the dividend? MULTIPLYING AND DIVIDING BY 4, 5, AND 6. 85. Find products of: 1. 856X4 3. 968X4 5. 2079x4 7. 1976X4 2. 2798X4 4. 989X4 6. 2098X4 8. 1678x4 MULTIPLYING AND DIVIDING BY 4, 5, AND 6. 71 Find quotients of: 9. 6789 -^4 10. 2135 -f-4 11. 14009-^4 12. 15203 H-4 86. Find products of: 1. 1856X5 3. 2765X5 2. 2708X5 4. 3769X5 Divide by 5: 9. 19290 12. 18605 10. 94806 13. 43441 11. 31433 14. 38024 87. Multiply by 6: 1. 9874 4. 5907 2. 3009 5. 8679 3. 0068 6. 1948 Divide by 6: 13. 32430 16. 44445 14. 34850 17. 46847 15. 37838 18. 49250 13. 31033 -f-4 14. 67890 -^4 15. 34035 -4 16. 39393 ^4 5. 3579X5 6. 1978X5 15. 46979 16. 37300 17. 54306 7. 7308 8. 5897 9. 6087 19. 56457 20. 58259 21. 57456 7. 1948X5 8. 1067X5 18. 47464 19. 89180 20. 34744 10. 9396 11. 2958 12. 1769 22. 17171 23. 19191 24. 31433 88. CLASS EXERCISE. Note. — Each, pupil in turn should multiply or divide one num- ber by 6 and add the number carried over, or give the remainder. 1. 431024516819278 2. 843765074957 6 6 3. 6)468471029872109 4. 6)23453840196 72 UNITED STATES MONEY. UNITED STATES MONEY. 89. Draw a one-cent piece. Draw a dime. How many cents equal a dime? How many cents make a dollar? How many tens make one hundred? How many dimes make a dollar? One dollar is written, $1. 10 cents = 1 dime. 10 dimes = $1. Half a dollar is how many cents? 50 is what part of 100? 60 cents is what part of $1? 25 cents is what part of 100 cents? What part of $1? What part of 50 cents? f of $1 are how many cents? ^ of $1 is how many cents? i of $1 is how many dimes? i of $1 is how many dimes? Half a dime is how many cents? If I spend I of $1, how many fourths of $1 shall I have left? How many cents? Find with counters (buttons or circular discs) 100 cents; find 5 more cents How many cents have you? How many dollars and cents? To show that we have one dollar and five cents, we write it in this way: $1.05, placing a period between dollars and cents. Write one dollar and three cents; one dollar and six cents. The cents are written at the right of the dollars, with a period between the dollars and cents. Two places are required to express cents when the dollar sign is used. UNITED STATES MONEY, 73 90. 1. Begin with $1, and write all the dollars and cents up to $1.25. 2. Write the following in figures : One dollar fifty cents; one dollar and sixty-nine cents; one dollar one cent; one dollar and ninety-nine cents. 3. Read the following: $0.06 $1.02 $1.10 $1.70 $0.03 $1.00 $1.09 $1.01 $1.71 $1.44 $1.07 $1.90 $1.75 $1.17 $1.50 $1.88 $1.17 $0.98 $1.27 $1.80 $1.11 $1.60 $1.36 $1.05 $1.08 4. Write the above numbers and add. Add as in simple numbers and separate dollars from cents by a period. 5. How many cents are there in $2? In $3? In $4? 6. Write the following in figures : Two dollars seven cents ; two dollars twelve cents ; three dollars forty cents; four dollars ninety cents; five dollars nine cents; seven dollars seven cents. 7. Read the following : $6.08 $0.01 $5.05 $6.15 $7.71 $20.05 $9.10 $8.01 $4.01 $6.51 $7.07 $30.50 $7.05 $9.09 $3.10 $0.05 $10.50 $29.16 $4.04 $5.50 $6.11 $7.17 $10.05 $40.12 8. How many cents are there in two dollars ninety-five cents? 9. How many dollars in six hundred fifty cents? 10. How many hundreds in seven hundred ninety? 74 ADDITION AND SUBTRACTION BY ENDINGS. 11. How many 50's in 200? How many 50-cent pieces in $2? 12. Put down a dollar for eael^ hundred cents in ten dollars. How many hundred cents make ten dollars? 13. How many cents make seven dollars seven cents? 14. Find the sum of $9.06 and $12.20. 15. Find the sum of $15.25 and $4.30. 16. Subtract: $12.00- $5.00. $20.00— $6.00. $15.50- $12.50. $9.30— $8.30. Subtract as in simple numbers, and separate dollars from cents by a period. ADDITION AND SUBTRACTION BY ENDINGS. 91. Write ten numbers ending in 4; add 2 to each of these numbers. 14 24 34 44 54 64 74 84 94 104 222222222 2 With what figure does each sum end? Make a subtraction table by using the results of addi- tion obtained above and subtracting 2 from each. Thus: 16 26 36 46 56 66 76 86 96 106 JJ 2 2 2 2 2 2 2 2 With what figure does each remainder end? Write ten numbers ending in 3, and add 4 to each num- ber. What is the ending figure? Write the results of additions obtained above and sub- tract 4 from each number. What is the ending figure? ADDITION AND SUBTRACTION BY ENDINGS. 75 93. 1 + 8. Add, giving first the ending figure of the sum, and then the whole sum : 8 18 28 38 48 58 68 78 88 98 IJLJJL IJ. 1 1_1_1 1 11 21 31 41 51 61 71 81 91 Write the results of addition obtained above and sub- tract 1 from each number. Subtract 8 from each. Add, beginning at the left: 1, 8, 1, 1, 8, 1, 1, 8, 1, 1, 8, 1, 8, 1, 1, 8. 8, 1, 1, 8, 1, 1, 1, 8, 1, 1, 8, 1, 8, 1, 1, 1. 9, 6, 3, 1, 1, 1, 8, 1, 8, 1, 1, 8, 1, 1, 1, 8. 6, 8, 4, 1, 1, 8, 1, 1, 1, 8, 1, 8, 1, 1, 1, 8. 93. 1+9. Add 9 to numbers ending in 1 : 1 11 21 31 41 51 61 71 81 91 Make a subtraction table, taking 9 from each of the re- sults of addition obtained above. Add, beginning at the left : 1, 9, 9, 1, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9. 9, 1, 1, 9, 9, 1, 1, 9, 9, 1, 1, 9, 1, 9, 9, 1. 8, 8, 4, 1, 9, 1, 9, 8, 1, 1, 9, 1, 9, 1, 9. 9, 8, 3, 9, 1, 1, 9, 1, 9, 9, 1, 1, 9, 5, 1. 8, 7, 5, 9, 1, 8, 1, 1, 9, 1, 1, 9, 1, 9, 9, 1. 76 ADDITION AND SUBTRACTION BY ENDINGS. Add, beginning at the bottom of the hne : (1) (2) (3) (4) (5) (6) (7) (8) (9) (lO) (11) (12) 9 1 1 9 2 9 8 9 8 9 9 1 1 9 1 1 8 1 1 1 1 1 1 9 9 1 8 9 1 9 8 3 9 1 9 1 1 9 1 9 8 1 1 6 1 9 1 1 1 1 8 1 1 1 9 9 1 9 1 5 8 1 1 9 1 1 1 1 9 1 8 4 1 9 9 1 9 8 1 1 9 1 1 1 9 9 1 1 1 1 8 9 1 9 9 9 1 1 1 9 1 9 1 1 1 1 1 1 1 4 1 5 8 7 7 3 1 5 9 7 9 8 9 9 1 4 5 7 3 7 6 6 9 8 9 6 9 9 8 9 5 7 5 6 Copy and 1 add: (1) (2) (3) (4) (6) (6) (7) (8) (9) (10) 99 99 99 88 98 19 99 89 99 91 91 11 11 11 91 91 99 19 11 19 11 11 11 98 19 11 11 91 99 91 19 88 19 11 91 18 11 11 11 19 99 91 81 99 11 89 19 99 11 11 11 19 90 11 19 10 89 11 91 91 11 11 18 11 89 91 11 11 98 19 81 19 11 31 11 19 15 81 14 16 93 46 99 39 99 94 47 98 68 46 75 95 61 99 71 67 98 79 98 98 ADDITION AND SUBTRACTION BY ENDINGS. 77 94. 2 + 5 and 2 + 6. Add 5 and 6 to numbers ending in 2. Read the ending figure first, then the whole sum : 2 12 22 32 42 52 62 72 82 92 102 2 12 22 32 42 52 62 72 82 92 102 6 666666666 6 Make subtraction tables by using the results of addition and subtracting 5 from each. Subtract 6 from each. Copy and add: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 15 16 16 81 16 22 15 51 69 66 62 11 52 11 12 11 61 21 11 22 21 69 21 16 51 91 21 16 19 11 19 21 11 62 21 16 16 12 11 91 11 11 18 21 16 12 12 61 51 16 81 16 61 11 92 61 81 21 26 12 11 62 21 14 11 21 11 11 12 61 15 25 16 27 11 11 12 14 19 29 65 97 89 57 69 99 68 67 65 97 97788380997898968683 95. 2 + 7. Add: 2 12 22 32 42 52 62 72 82 92 102 7777777777 7 Make a table, subtracting 7 from each of the results of addition above. 78 ADDITION AND SUBTRACTION BY ENDINGS. Add, beginning at the left: 2, 7, 1, 9, 1, 2, 7, 1, 1, 9, 2, 7, 1, 2, 7. 9, 8, 2, 1, 9, 1, 2, 7, 1, 2, 7, 1, 9, 1, 2. 6, 6, 5, 2, 1, 2, 6, 1, 1, 2, 5, 2, 1, 2, 7. Copy and add: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 97 87 72 11 67 79 11 71 11 19 12 12 21 77 21 21 71 27 79 71 71 11 17 22 17 11 26 12 21 27 27 61 92 11 72 26 12 71 97 92 12 26 11 79 21 52 71 27 12 19 91 12 79 21 19 21 27 12 19 11 12 71 21 11 91 12 12 91 91 91 3 22 11 92 17 74 29 18 14 11 77 98 79 98 77 28 45 77 78 79 97 89 89 78 85 75 96 84 78 89 EXERCISE. 96. Subtract: (1) (2) (3) (4) (5) (6) 4221 5121 7011 3121 6321 4112 132 223 213 232 332 123 7. 9131-322=? 13. 2121-312=? 8. 4011-303=? 14. 4118-223=? 9. 5210-223=? 15. 5102-213=? 10. 3101—222=? 16. 6110-123=? 11. 5119-223=? 17. 8112-213=? 12. 3122-123=? 18. 7231-322=? DRY MEASURES. Subtract : (19) (20) (21) (22) (23) (24) 4310 5301 8234 6202 8402 5021 234 404 344 334 314 334 79 25. 5012- 223 = ? 31. 9139-244=? 37. 9301-3544=? 26. 4610 - 132 = ? 32. 81 23 - 334 = ? 38. 4302 - 1443 = ? 27. 7051- 233=? 33. 5210-334=? 39. 6201- 344=? 28.6311- 233=? 34.9401-424=? 40.9032- 334=? 29.5012- 213 = ? 35.5204-325=? 41.6020-2344=? 30. 6300-1311 = ? 36. 9103-234=? 42. 8023-3254=? CLASS EXERCISE. 07. Note. — Let each pupil subtract one number. 1. 820431290425613 2. 76498364271503 624153423542134 53726547326174 3. 87365419854721 28158274936294 4. 3746937100 2587394837 DRY MEASURES. 8 Pint. Quart. Peck. 98. 1, A quart of berries is how many pints? 2. A peck of beans is how many quarts? Eight quarts make a peck. 80 DRY MEASURES. 3. Where have you seen these measures used? Name some things which you have seen measured by them. 4. Half a peck of nuts is how many quarts? 5. i of a peck is how many quarts? 6. 4 quarts of berries are what part of a peck? 7. 6 quarts are what part of a peck? 8. f of a peck of oats are how many quarts? 9. John sowed |^ of a peck of blue-grass seed; how many quarts were left out of a peck? 99. Four pecks make a bushel. 1. ^ a bushel of potatoes is how many pecks? 2. I of a bushel are how many pecks? 3. Half a bushel of cranberries is how many quarts? 4. Two bushels are how many pecks? 5. Estimate the capacity of a box or basket brought into the schoolroom. 6. IJ bushels of walnuts are how many pecks? 7. Henry gathered a bushel of beans from his garden, and sold half of them for 25 cents a peck; how much money did he receive? 100. Grains, fruits, vegetables, and some other things that are not liquids, are sold by these measures. They are called Dry Measures. 2 pints (pt.) = 1 quart (qt.)- 8 quarts = 1 peck (pk.). 4 pecks = 1 bushel (bu.). MISCELLANEOUS PROBLEMS. 81 MISCELLANEOUS PROBLEMS. 101. 1. How many feet are there in a yard? How many inches in a yard? 2. ^ of a yard is how many inches? How many feet? 3. We have a measure which holds just 8 quarts; what is the measure called? How many quarts are there in a peck of corn? 4. A party of boys went nutting and gathered 2\ pecks of nuts; how many quarts did they have? 5. 32 quarts of strawberries are how many gallons? 6. At 6 cents a yard, how many yards of muslin can you buy for 72 cents? 7. How many spools of thread can I buy for 35 cents, at 5 cents a spool? How many at 4 cents a spool, and how many cents remaining? 8. ^ of 24 acres of land is planted in sugar-corn, \ in potatoes, \ in oats, and the remainder in meadow; how many acres are planted in meadow? 9. A man is rowing down the river 8 miles an hour; at that rate, how long will he be in going 48 miles? 10. At 4 cents a pound, how many pounds of oatmeal can you get for 60 cents? 11. 12 bushels are how many pecks? 12. How many quarts are there in 5 pecks? 13. A boy earned $1.65, and his father gave him 35 cents; he paid 50 cents for a scrapbook; how much money had he left? 14. Bought 10 yards of cloth at 4 dollars a yard, and sold it for $8 less than I gave for it; how much did I get for it? 82 MISCELLANEOUS PROBLEMS. 15. A boy earned 75 cents a day, and paid 50 cents a day for his board; how much did he save each day? 16. How much did this boy save in 6 days? 17. Max has a quarter of a dollar, a dime, a 5-cent piece; how much money has he? 18. James has J as much money as Max; how nmch has he? 19. We paid for a Christmas tree, $2; for tapers, 40 cents; for candy, 75 cents; for netting for candy bags, 10 cents; for toys, $1.20; for books, $3.60; what did all cost? 20. How many yards of fringe will be needed to go round a rug 5 ft. long and 3 ft. wide? (Make a drawing.) 21. A class of children made 69 holly wreaths to trim a schoolroom, and used all but 10; how many did they use? 22. If I of a yard of ribbon costs 2 cents , what is the cost of a yard? How many yards can I buy for 48 cents? 23. A man bought 60 boxes of peaches, but found { of them unsound; how many boxes were sound? 24. Mary's aunt gave her a doll for which she paid $4; for the doll's house she bought a set of chairs for which she paid $1.50, a sofa for $1, a bedstead for $1.20, and a little bureau for 90 cents; what did all cost? 25. How many feet must a boy walk in going around this lot? 51. ft SSft, ADDITION AND SUBTRACTION BY ENDINGS. 83 ADDITION AND SUBTRACTION BY ENDINGS. 103. 2 + 8. Make a table, adding 2 to numbers ending in 8. Add 8 to numbers ending in 2. Make a table, subtracting 2 from the results of this addition. Subtract 8 from each of these results. Copy and add: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 91 98 97 89 86 82 89 78 97 95 17 12 12 28 12 29 18 22 12 12 22 81 78 82 11 81 72 89 89 79 68 27 22 29 87 28 29 21 21 21 22 92 89 91 22 12 91 98 98 80 18 18 21 18 18 19 18 10 12 28 72 82 91 82 72 81 82 82 82 92 24 22 12 26 28 28 26 29 25 16 88 69 88 95 84 45 87 87 89 86 78998969889777748488 Add: 11. 27, 62, 29, 81, 28, 12, 75, 27, 88, and 80. 12. 68, 22, 89, 21, 98, 12, 80, 28, 63, and 99. 103. 2 + 9. Add: 2 12 22 32 42 52 62 72 82 92 9999999999 84 ADDITION AND SUBTRACTION BY ENDINGS. 9 19 29 39 49 59 69 79 89 99 2222222222 Make a table, subtracting 9 from each of the results obtained above. Subtract 2 from each. Add: 9, 2, 8, 2, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 8, 2. 6, 6, 9, 1, 9, 1, 9, 1, 6, 2, 2, 6, 1, 1, 7, 2. 9, 9, 2, 2, 9, 1, 9, 1, 9, 9, 2, 9, 9, 2, 2, 2. 8, 4, 9, 9, 2, 9, 1, 7, 2, 9, 2, 6, 2, 9, 2, 7. Copy and add: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 89 89 99 89 99 29 89 82 72 89 22 28 81 29 81 69 92 97 21 20 98 12 28 92 28 22 99 21 97 99 92 79 92 99 92 89 19 99 92 92 29 29 99 29 99 29 92 92 29 29 99 92 29 92 29 92 28 29 89 92 92 99 12 91 82 99 92 91 22 91 29 29 22 22 24 29 19 25 94 24 54 68 49 98 67 65 65 97 18 97 97847959897786584857 Add: 11. 29, 79, 22, 98, 92, 29, 99, 92, 28, 99, and 43. 12. 98, 82, 29, 99, 92, 29, 92, 96, 29, 64, and 80. OUNCES AND POUNDS. 85 EXERCISE. 04. Find the differences: 1. 8013- - 334=? 15. 8054- -3505=? 2. 7321- - 233=? 16. 7033- -2204=? 3. 8122- - 124=? 17. 7021- -3402=? 4. 6124- - 334=? 18. 9041- - 405=? 5. 9324- - 405=? 19. 2043- - 535=? 6. 8432- - 445=? 20. 7040- - 534=? 7. 6242- -2345 = ? 21. 7031- - 435 = ? 8. 6301- -3444=? 22. 6304- -1325 = ? 9. 6413- -1434=? 23. 5402- -3444=? 10. 8234- -1135=? 24. 5204- -2245=? 11. 8012- - 245 = ? 25. 5302- - 325=? 12. 4210- - 245=? 26. 9303- -5454=? 13. 6243- -2345=? 27. 9000- -2001 = ? 14. 8544- -2035 = ? 28. 9011- -7017 = ? OUNCES AND POUNDS. 105. 1. If I put the pound weight on one side of the scales, how many ounces must I put on the other side to balance it? A pound is 16 ounces. 86 MISCELLANEOUS PROBLEMS. 2. J of a pound is how many ounces? 3. If I wish to buy a quarter of a pound of tea, how many ounces must be put upon the scales to balance it? 4. 4 ounces of ginger arc what part of a pound? 5. At 5 cents an ounce, what will i of a pound of celery seed cost? 6. At 2 ounces for 5 cents, how many ounces of pepper can be bought for 20 cents? 7. \\ pounds of figs are how many ounces? 8. f of a pound of maple sugar are how many ounces? 16 ounces (oz.)= 1 pound (lb.). 106. MISCELLANEOUS PROBLEMS. 1. A man paid $72 for a wagon and $8 for repairs, then sold it so as to gain $9; how much did he receive for it? 2. Three men bought a horse, the first man paying $36, the second man $15, and the third man as much as the first two; how much did the horse cost? 3. If I buy 11 yards of velvet at $3 a yard, and sell it at $4 a yard, how much shall I gain? 4. If a man earns $12 a week and spends $7, how much will he save in 6 weeks? 5. Willie gathered a bushel of chestnuts; he gave his brother 10 quarts, kept 6 quarts, and sold the remainder; how many quarts did he sell? 6. When Alfred reads 8 pages more he will have finished his story book, which contains 90 pages; how many pages has he read? 7. A man gave a watch and $10 in money for a horse worth $75; what is the value of the watch? MISCELLANEOUS PROBLEMS. 87 8. Two persons start from the same point and travel in opposite directions; one travels 26 miles and the other 38 miles; how far apart are they? 9. A man saved 24 dollars one month, half as much the next month, and 6 dollars the third month; how much money had he saved at the end of the three months? 10. If 82 feet of wire are already used in making a fence, and 9 feet more are needed, how much wire will be used? 11. James shoots an arrow which does not reach the mark by 9 feet. If the mark is 51 feet away, how far is the arrow from James? (Make a drawing.) 12. Two persons start from the same place and travel in the same direction; one travels 40 miles an hour, and the other 35 miles an hour; how far apart will they be in 1 hour? (Show by drawing ) How far in 6 hours? 13. Charles gets $6 a month for selling a daily paper; Henry gets \ as much for selling a weekly paper; how much will both have earned in 5 months? 14. From a chest of tea containing 60 pounds, 9 pounds v/ere sold at $1 a pound; what was the value of the re- mainder, at the same rate? 15. I bought a bushel of tomatoes for 70 cents, a half- bushel of turnips for 20 cents, and a peck of beans for 10 cents; what I paid for all was half the cost of a barrel of potatoes. What did the potatoes cost? 16. A box contains 134 oranges, and a barrel contains 64 more than the box; how many oranges does the barrel contain? 17. I bought a horse and sleigh for $150; the sleigh cost $45; what did the horse cost? 18. After spending $80 for a pony, George found that he 88 ADDITION AND SUBTRACTION BY ENDINGS. had $65 left in his savings bank; how much money had he at first? 19. In an orchard there are 150 apple trees; this is 50 more than the number of peach trees; how many peach trees are there? 20. A man having 190 young orange trees, bought 89 more, and then sold 50; how many had he left? 21. Add three hundred nine to seven hundred eleven, and subtract twenty-nine from the sum. 22. A farmer bought 40 sheep for 144 dollars at one time, and 50 sheep for 155 dollars at another time; how much did the sheep cost him? 23. A boy shot an arrow 145 feet up the road, and another 149 feet down the road; how far were the arrows apart? (Make a drawing.) 24. What will 150 tons of coal cost, at $6 a ton? ADDITION AND SUBTRACTION BY ENDINGS. 107. Make a table, adding 3 to numbers ending in 3. What is the ending figure of each sum? Add 3 to numbers ending in 4. Add 3 to numbers ending in 5. What is the ending figure? Make subtraction tables, taking 3 from the results ob- tained in each of the three addition tables above. 108. 3 + 6. Add, beginning at the left: O, o, ^, O, o, iU, O, o, ^, o, O, ^, O, o, ^. 9, 6, 3, 2, 3, 5, 1, 2, 2, 5, 2, 5, 3, 2, 5. 8, 7, 3, 2, 5, 3, 1, 2, 4, 3, 2, 3, 5, 1, 2. ADDITION AND SUBTRACTION BY ENDINGS. 89 Add, beginning at the bottom of the column : (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 95 39 53 23 85 99 29 82 55 59 23 92 39 19 23 92 89 95 22 39 59 15 99 51 99 29 22 23 92 92 31 23 92 32 91 91 12 29 92 92 92 52 29 95 22 12 55 59 23 25 95 35 91 13 95 25 33 32 94 23 23 93 12 21 13 53 12 92 10 59 22 24 25 32 24 38 23 24 24 39 59 68 57 48 58 64 57 56 57 54 99988679989878987998 109. 3 + 6. Add: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 66 36 89 38 52 99 26 69 29 96 33 83 22 99 31 21 63 92 62 29 91 29 66 92 16 66 39 26 36 62 26 92 33 26 63 33 92 63 93 36 63 26 99 13 31 99 23 39 29 93 39 63 29 69 96 22 36 92 32 29 92 32 62 32 23 67 56 99 68 69 26 93 36 96 32 36 27 23 27 36 67 97 55 95 59 67 67 68 67 58 96 88 99 78 79 99 90 89 67 98 110. 3 + 7. Make a table, adding 7 to numbers ending in 3 and 3 to numbers ending in 7. 90 ADDITION AND SUBTRA CTION BY ENDINGS. Make a table, subtracting 7 from the results of the ad- dition tables. Subtract 3 from the same numbers. Add: 9, 9, 2, 3, 7, 2, 9, 9, 3, 7, 3, 6, 2, 9, 3, 7. 6, 6, 7, 2, 9, 3, 7, 2, 9, 9, 3, 7, 2, 9, 9, 5. 9, 7, 4, 3, 7, 2, 9, 9, 3, 6, 2, 9, 3, 7, 2, 9. 8, 7, 5, 2, 9, 9, 3, 7, 3, 6, 2, 9, 3, 5, 2, 3. Add: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 69 99 99 92 99 62 29 96 73 78 32 79 22 26 99 36 90 73 37 32 76 32 56 73 29 73 22 39 93 97 33 96 33 39 82 30 68 19 99 93 77 24 99 99 27 99 32 92 20 29 33 97 29 22 73 9 77 97 9 72 99 73 62 63 30 92 33 23 72 36 9 32 37 38 65 23 92 57 38 95 96 89 74 79 6 98 99 7 65 98 66697980996979969770 REVIEW. 111. Add: (1) (2) (3) r4) (5) (6) (7) (8) (9) (10) 57 17 28 28 97 16 52 55 86 58 73 42 42 53 32 51 37 42 73 22 32 53 33 36 52 32 31 28 39 29 26 37 35 81 26 97 59 52 11 61 62 51 33 22 92 11 21 31 78 23 23 23 43 35 17 29 27 35 22 37 89 89 98 57 98 76 55 68 21 42 88 97 86 96 85 85 98 86 24 35 ADDITION AND SUBTRACTION BY ENDINGS. 91 EXERCISE. 113. Subtract: (1) (2) (3) (4) (5) (6) 5514 5065 7617 708 8119 8004 1445 5050 3455 3055 5505 4345 7. 8301-2034 13. 7322- -6543 8. 9312-3543 14. 6411- -5524 9. 6431-2332 15. 3501- -3032 10. 8433-3544 16. 8032- -6503 11. 9441-5034 17. 9020- -5003 12. 9400—3623 18. 8042- -5604 Find the differences : 19. 9000-1445 28. 8122- -4435 20. 8000-4405 29. 5333- -1045 21. 3111-2445 30. 7303- -4045 22. 5111-2405 31. 4313- -4144 2-3. 7011-4435 32. 5113- -4245 24. 9112-4345 33. 9313- -4344 25. 7414-4425 34. 8041- -1445 26. 4444-3345 35. 5414- -1415 27. 9041-4445 36. 8434- -1435 CLASS EXERCISE. 113. Note. — Each pupil should subtract one number. 9602410131120561 17954362801756 5254443513453545 7932425360939 92 MISCELLANEOUS PROBLEMS. MISCELLANEOUS PROBLEMS. 114. 1. James has 9 cents, and John has three times as many less 6; how many has John? 2. One day a man traveled 25 miles by railroad, 34 miles by steamboat, and returned 10 miles; how far was he then from his starting place? 3. I bought 4 yards of silk at $2 a yard, and 2 shawls at $10 each; how much change ought I to receive from 2 twenty-dollar bills? 4. A grocer paid $165 for 30 barrels of flour, and $50 for 20 barrels of potatoes; how much did he pay for both? 5. A boy who had 53 marbles loaned 20, and afterwards borrowed 9 more; how many marbles had he then? 6. A boy paid $20 for a team of goats, $8 for his carriage, and $4 for harness; he sold them so as to lose $4; for how much did he sell them? 7. I bought 3 yards of cloth at $7 a yard, for a coat; the buttons and cord cost $2, and the making of it $10; what was the cost of the coat? 8. A pole is 40 feet long. If J of it is in the ground, and the rest in the air, how many feet are in the air? 9. A man gave a carriage and $110 in money for a lot worth $400; what was the value of the carriage? 10. Some children returned from the lake with a basket of shells; after giving half of them away, they had 95 left. How many had they at first? 11. A party of school children went to Fairview Park. $1.75 is 50 cents less than they paid for carfare; how much did they pay? 12. John bought a pack of shingles, and used 175 in ADDITION AND SUBTRACTION BY ENDINGS. 93 mending the roof. He had 75 left; how many did the pack contain at first? 13. A farmer having $156 paid one-half of his money for a horse, and one-half of the remainder for a cow; how much did each cost? 14. How much money has the farmer left after paying for both horse and cow? 115. 3 + 8. Make a table, adding 8 to numbers ending in 3, and 3 to numbers ending in 8. Write the results of the addition above and subtract 8 from each. Subtract 3 from each number. Add: 6, 7, 8, 2, 8, 9, 3, 8, 9, 3, 8, 9, 3, 8, 9. 9, 9, 3, 9, 3, 8, 9, 2, 9, 9, 3, S, 9, 0, 9. 7, 4, 9, 3, 8, 9, 3, 8, 9, 2, 9, 9, 0, 3, 7. 9, 8, 3, 3, 7, 3, 8, 9, 2, 9, 9, 3, 7, 3, 8. Add: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 82 30 93 89 88 59 93 7 89 90 39 99 99 32 33 32 99 87 32 23 93 8 89 76 97 99 29 33 9 69 87 83 32 33 83 98 72 79 78 8 39 37 99 7 39 23 37 28 33 33 72 93 98 93 78 97 93 82 97 79 29 99 23 89 33 83 89 32 83 33 8 23 94 38 99 36 38 93 33 99 86 59 89 65 94 57 56 8S 58 89 77 99 77 88 67 97 97 88 99 80 94 ADDITION AND SUBTRACTION BY ENDINGS. Add: 11. 99, 78, 33, 97, 93, 29, 98, 83, 38, 44, and 98. 12. 88, 33, 99, 89, 32, 99, 92, 28, 98, 84, and 69. 13. Find the sum of: 93, 35, 33, 53, 94, 39, 43, 43, 77, and 98. 14. 62 + 39 + 89 + 23 + 93 + 35 + 43 + 43 + 87 + 77=? 116. 3 + 9. Add: 3 13 23 33 43 53 63 73 83 93 9 9 9 9 9 9 9 9 9 9 9 3 19 3 29 3 39 3 49 59 69 3 3 3 79 89 3 3 99 3 Make a subtraction table, taking 9 from each of the results of addition above. Subtract 3 from each number. Add; beginning at the left: 9, 4, 9, 8, 2, 9, 9, 3, 9, 8, 3, 8, 9, 9, 1. y, y, o, y, o, y, y, y, o, y, o, o, y, y, y. 8, 5, 9, 9, 9, 3, 7, 3, 8, 9, 3, 9, 8, 0, 9. 8, 9, 3, 3, 7, 3, 8, 9, 3, 9, 7, 2, 9, 9, 2. 51+9=? 62 + 7=? 72-3 = ? 70-2 = ? 72 + 6=? 73 + 7 = ? 91-2=? 90-1 = ? 81 + 8=? 92 + 8=? 81-3=? 80-3=? 83 + 9=? 93 + 5=? 32-3=? 91-3=? ADDITION AND SUBTRACTION BY ENDINGS. 95 Add: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 99 99 89 99 89 99 90 8 88 97 93 38 39 39 93 99 39 39 93 99 30 93 93 79 39 39 99 83 99 33 78 77 99 33 98 93 93 99 39 99 99 30 98 99 93 89 29 39 99 89 33 89 33 98 97 38 99 92 83 32 9 93 9 93 33 93 82 89 37 86 99 8 83 37 93 24 37 35 6 90 96 33 99 54 98 67 55 86 57 6 6659 69 89696998609078 Add: 11. 98, 89, 33, 79, 39, 80, 9, 92, 38, 98, and 54. 12. 99, 20, 83, 97, 33, 99, 8, 83, 34, 78, and 78. 13. Find the sum of: 3, 35, 39, 53, 94, 34, 49, 43, 89, and 79. 14. 95 + 39 + 33 + 45 + 83 + 39 + 63 + 34 + 97 + 67 = ? EXERCISE. 117. Find the differences • 1. 9453- 545=? 8. 3323-2554=? 2. 8453- 544=? 9. 9203-3405=? 3. 9341—5445 = ? 10. 8801- 134=? 4. 8341-5345 = ? 11. 7640-2534=? 5. 9623-3545=? 12. 3141-2445=? 6. 7344-3545=? 13. 3001-2154=? 7. 6412-4534=? 14. 6011-5403=? 96 COMPARISON OF HALVES, FOURTHS AND EIGHTHS. 15. 3043-1415=? 23. 7004- -1345 = ? 16. 8001—5045=? 24 9043- -1534=? 17. 7002—5435 = ? 25 3842- -1435=? 18. 4003-3004 = ? 26. 5101- -4434=? 19. 6301-4345=? 27. 8600- -5534=? 20. 7304-3025=? 28. 8122- -6035 = ? 21. 9801- 534=? 29. 5043- -1415=? 22. 8074-2135=? 30. 4312- - 243 = ? 118. Find the differences: 1. 26345-2556=? 10. 59203- -2505 = ? 2. 15043-1546=? 11. 68304- -1056=? 3. 19411-1506 = ? 12. 70005- -6056=? 4. 27522-3615=? 13. 20003- -2455=? 5. 35432-4546=? 14. 49503- -4546=? 6. 43543-1456 = ? 15. 30533- - 36=? 7. 23543-1554=? 16. 30334- -3455=? 8. 50354-5646 = ? 17. 23451- - 56=? 9. 47352-4353 = ? 18. 10052- -1554 = ? COMPARISON OF HALVES, FOURTHS, AND EIGHTHS. x^ ~%\ \"s\ /%\ V's/ \'/s) \\ jy 119. 1. A whole melon can be divided into how many- halves? How many fourths? COMPARISON OF HALVES, FOURTHS, AND EIGHTHS. 97 2. Fold a paper square into two equal triangles. One of the triangles is what part of the whole square? 3. Fold the same square so as to make four equal triangles. What part of the whole square is one of the small triangles? 4. i is equal to how many fourths? 5. i and { make how any fourths? 6. i and f equal how many fourths? 7. Fold your square so as to make eight equal triangles. One of the triangles is what part of the whole square? Two of the triangles are what part of the whole? 8. If three of the triangles were cut out of the square, what part of the whole would be left? 9. i of the square is equal to how many eighths? ^ is equal to how many? | are equal to how many eighths? 10. I of a whole cheese are equal to how many eighths of the cheese? 11. I + I are how many eighths? 4 + 1 = ? 12. l^ + f are how many eighths? Note. — Use objects freely in the work with fractions. 130. 1. From your folded squares find the answers to the following questions : Kf=? l-i=? i-i=? i-i=? l+i=? |-|=? *-i=? i-f=? i+t=? l-i=? i-f=? i-t=? 2. Draw a square and divide it into eight equal oblongs; find from the drawing the answers to the following ques- tions : 98 COMPARISON OF HALVES, FOURTHS, AND EIGHTHS. 2 times i = ? |X3=? iX4 = ? |X5 = how many wholes? - 2 times f = ? tX2 = ? iX3=? |X3 = how many wholes? 2 times t=? |X2 = ? fX2 = ? 1X2 = how many wholes? 4X4=? iX3=? iofi = ? |X4=? iofi = ? fX2=? iX2 = ? iofi = ? iX4=? 3. Mary made a cake for tea; half of it was eaten, and the remainder was divided equally among four visitors. What part of the whole cake did each visitor receive? 4. George had a ball of twine for his kite ; he used half of it, and divided the remainder equally between two other boys. What part of the whole ball of twine did each boy get? 131. Take two equal squares of paper. Fold each into four equal smaller squares; call them square crackers, and give them to four children, so that they shall have equal shares. Note. — Distribute all the parts of one square first. 1. What part of the two large squares does each child re- ceive? 2. What part of one square? 3. To how many children did you give the two squares? 4. One of the four equal parts of anything is called what? Place the small squares together again so as to form the two large squares. COMPARISON OF HALVES, FOURTHS, AND EIGHTHS. 99 5. J of 2 squares is what part of one of the squares? 6. i of 2 pies= ? i of 2 apples- ? i of 2 melons = ? 7. Divide two sticks of candy equally among four boys. What part of the whole will each boy receive? What part of one stick is that? 133. Take three equal squares of paper. Divide these equally among four children. (Fold each into four smaller squares.) 1. What part of the three large squares does each child receive? 2. What part of one square? 3. Into how many equal parts did you divide the three squares? 4. One of these equal parts is called what? Place the small squares together again so as to form the three large squares you had at first. 5. I of 3 squares is what part of one square? 6. What part of one whole toothpick is ^ of 3 tooth- picks? 7. Divide 3 oranges equally among 4 boys. What part of the 3 will each receive? | of 3 is what part of 1? 8. Suppose you plant 3 potatoes in 4 hills. If you divide them equally, what part of 1 potato will be in each hill? (Make a picture to show this.) 9. Give 3 bananas to 4 girls, dividing them equally; what will each girl receive? 10. If you divide 23 melons equally among 4 boys, what is each boy's share? (Picture.) 11. I wish to put 27 quarts of blackberries into 4 jars, putting the same number of quarts into each; how many quarts will each jar contain? 100 MULTIPLICATION AND DIVISION. MULTIPLICATION AND DIVISION. 133. Copy and learn: 7X7 = 49 lOX 7 = 70 8X 8 = 64 llx 8 = 88 8X7 = 56 llx 7 = 77 9X 8 = 72 12x 8 = 96 9X7 = 63 12X 7 = 84 lOX 8 = 80 8X11 = 88 7X8 = 56 7X10 = 70 8X 9 = 72 7X12 = 84 7X9 = 63 7X11 = 77 8X10 = 80 8X12 = 96 Recite the division table from the multiplication table. MENTAL EXERCISE. 134. 1. How many desks are there in a schoolroom which has 7 rows of desks, and 7 desks in each row? 2. If a pair of curtains cost 7 dollars, what will 9 pairs cost? 3. Eight boys are building a snow fort; if each makes 7 balls of snow, how many balls will they have in all? 4. What is the cost of 8 yards of muslin at 8 cents a yard? 5. One peck is how many quarts? John gathered 9 pecks of chestnuts; how many quarts had he? 6. If 8 loaves of bread are used in one week, how long will 96 loaves last? 7. If melons are selling at 8 cents each, what will 9 cost? 8. May had 75 cents to spend for lace at 8 cents a yard; how many yards did she buy? How many cents had she left? 9. At 8 dollars a ton, how many tons of coal can be bought for 96 dollars? FINDING ONE OF THE EQUAL PARTi^jQJ^\XWM¥^^^i^rX^l' 10. At 7 cents a roll, how many rolls of wall paper can be bought for 56 cents? 11. Make problems for: 10X8 = 80 12X7 = 84 88-^8 = ll 56^7 = 8 11X7 = 77 12X8 = 96 70-^7 = 10 63-^-7=9 Note. — Do this in class. Finding One of the Equal Parts of a Number. EXERCISE. 135. 1. If 9 melons cost 72 cents, what is the cost of one ? 2. One man working alone can do a piece of work in 56 hours; in how many hours can 7 men do the work? 3. If 8 tons of coal cost 72 dollars, what is the cost of a ton? 4. I paid 96 cents for 8 dozen eggs; how much were they a dozen? 5. If one man works alone, it will take 63 hours to dec- orate a hall; if 9 men work together, in how many hours can the work be done? 6. A merchant paid 84 dollars for 12 rugs; at that rate, what was paid for one rug? 7. Mary paid 96 cents for 12 dozen buttons; how much were they a dozen? 8. 64 trees were planted in 8 equal rows ; how many were planted in one row? 9. How many pecks are there in 72 quarts? 10. A man earned 84 dollars in 7 weeks; at that rate, how many dollars did he earn in one week? «>©• "S', J IX^Z m^'Bim (iNp, QF THE EQUAL PARTS OF A NUMBER. 11. How many cents are there in half a dollar? In J of a dollar? In i^j^? J? 12. 25 cents is what part of $1? 20 cents is what part of $1? 13. f of $1 are how many cents? | of $1? f of $1? |? 14. Frank spent tV of a dollar for pencils and J of a dollar for drawing paper; how much money did he spend? 15. Helen spent f of $1 for muslin, and i of $1 for thread; how many cents did she spend? 16. Make problems for: ^ of 49 = 7 i of 96 = 12 8 )64 bushels 12 )96 dollars 8 bushels 8 dollars Jof72 = 9 +of 70=10 7 )56 cents 10 )80 nails 8 cents 8 nails Note. — Do this in class. 17. Begin with 49 and find one-seventh of all numbers to 63. Thus: i of 49 = 7. | of 50 = 7 and 1 remaining. ^ of 51 = 7 and 2 remaining, etc. 18. Find one-eighth of all numbers from 64 to 88. 126. TABLES FOR REVIEW. 1X6= 6 1X7= 7 1X8= 8 2X6 = 12 2X7 = 14 2X8=16 3X6 = 18 3X7 = 21 3X8 = 24 4X6 = 24 4X7 = 28 4x8 = 32 5X6 = 30 5X7 = 35 5x8 = 40 6X6 = 36 6X7 = 42 6X8 = 48 MULTIPLYING AND DIVIDING BY 7 AND 8. 108 7X6=42 7X7 = 49 7X8 = 56 8X6 = 48 8X7=56 8X8 = 64 9X6 = 54 9X7 = 63 9X8=72 10X6 = 60 10X7=70 10X8=80 11X6=66 11X7 = 77 11X8 = 88 12X6 = 72 12X7 = 84 12X8 = 96 Without rewriting, read these with 6, 7, and 8 first. MULTIPLYING AND DIVIDING BY 7 AND 8. 127. Multiply by 7: 1. 6948 3. 9485 5. 6098 7. 10769 2. 5769 4. 7906 6. 6937 8. 11894 Divide by 7: 9. 19539 12. 39648 15. 54957 18. 17799 10. 18049 13. 18563 16. 13607 19. 68009 11. 17825 14. 28359 17. 27620 20. 67265 128. Multiply by 8: 1. 3849 3. 6957 5. 6384 7. 6094 9. 8649 2. 8539 4. 9384 6. 3947 8. 7483 10. 5973 Divide by 8: 11. 41443 13. 56457 15. 21391 17. 29019 19. 62808 12. 51652 14. 58259 16. 39036 18. 78863 20. 39013 104 MISCELLANEOUS PROBLEMS. CLASS EXERCISE. 139. Note. — Let each pupil multiply one number by 7 and add the number carried over. 1. 42952091897634017926 7 2. 53104293697012369876 3. 7 )19539021650103121798 4. 8)51652019098723651904 MISCELLANEOUS PROBLEMS. 1 30. 1. If in half a day a man picks 4 bushels of apples, and a boy 2 bushels, how many bushels will both pick in a day? In 5 days? 2. A newsboy having 42 papers, sold all but i of them; how many did he sell? How many had he left? 3. Henry's age, which is 7 years, is 1 seventh of his father's age; how old is his father? 4. What measure holds 4 pecks? 48 pecks of cran- berries are how many bushels? 5. A boy having 45 cents spent i of his money for drawing paper and -5^ for pencils; how many cents did he spend? 6. How many wheels have six freight cars, if each car has 8 wheels? 7. A farmer's boy fed to his colt ^ a peck of oats each day for eight days; how many bushels is that? MISCELLANEOUS PROBLEMS. 105 8. John had 5 dimes; he spent 15 cents for stamps, and with the remainder took 7 car rides; what was each fare? 9. I bought a steak weighing a pound and a half; how many ounces did it w^eigh? 10. 3 pounds of coffee make how many ounces? 11. A bushel of wheat weighs 60 pounds; how many pounds does a peck weigh? 12. Wheat bran weighs 20 pounds to the bushel; what is the weight of a peck? 13. Does a pound of wheat weigh more than a pound of bran? Which is the larger bulk? (See problems 11 and 12.) 14. At 6 cents a pound, how many pounds of rice can you buy for 50 cents? 15. If John gives 3 pecks of corn to twelve horses divid- ing it equally, how much corn does each horse receive? 16. How many hours are there in 2^ days? 17. If a boy is 3 minutes late at school, how many seconds has he lost? 18. For our school gardens we spent $1.50 for foliage plants, $2.10 for geraniums, $1 for tulip bulbs, and $2 for roses and pansies. How much money had we left out of $10, after paying for all? 19. A farmer raises 850 bu. of corn, 920 bu. of oats, 560 bu. of wheat, 390 bu. of barley, 78 bu. of buckwheat; how much grain has he in all? 20. I had in bank $1125, and drew out $415; how much have I left in bank? 21. Johnson & Co., after selling 2000 cans of sugar corn, had 1500 cans left; how many cans were on sale at first? 106 REVIEW OF ADDITION. 22. If I have $230, how much must I add to it to be able to buy a horse and buggy worth $550? 23. A man receives $700 for his fruit crop this year, which is $150 more than he received last year; how much did he receive last year? 24. Add 300 to 500, and from this sum subtract the difference of the numbers. 25. If I borrow at one time $327, and at another time $783, how much do I owe aft^r paying $221? 26. The greater of two numbers is 419, and the less 244; what is their difference? 27. Henry's father was born in 1859; how old is he now? 28. The sum of two numbers is 650; one of the numbers is 200; what is the other? 131. REVIEW OF ADDITION. (1) (2) (3) (4) (5) (6) (7) (8) 93 62 22 99 96 69 93 33 35 39 49 29 33 29 33 33 33 89 39 63 39 32 34 45 53 23 28 35 52 56 53 99 94 93 33 93 96 93 92 23 39 35 39 39 33 39 35 64 43 43 32 42 49 43 49 34 43 43 93 43 43 43 43 93 77 87 87 88 89 87 99 98 98 77 79 78 79 78 69 79 REVIEW OF ADDITION. 107 (9) (10) (11) (12) (13) (14) (15) (16) 89 29 93 26 97 32 23 23 79 63 94 33 29 96 64 56 38 34 34 68 63 23 94 39 93 84 68 89 35 39 29 23 37 28 22 21 93 63 63 65 43 92 99 99 29 25 35 23 43 89 83 83 62 43 93 83 93 33 33 33 33 43 38 23 98 79 87 89 88 99 76 87 76 79 78 76 78 68 66 87 (17) (18) (19) (20) (21) (22) 1 (23) (24) 73 62 36 98 29 82 98 96 26 26 68 32 39 99 99 99 29 53 92 39 62 39 39 33 63 37 39 58 72 42 42 65 95 93 43 93 34 49 49 23 23 39 45 39 94 99 78 89 62 42 83 42 89 83 33 22 34 41 28 43 33 33 93 93 95 89 98 77 97 79 88 97 79 79 64 79 79 76 79 79 ;25) (: 26) (27) (28) (29) (30) (31) (32) (33) (34) 29 99 95 72 93 39 98 78 65 99 69 12 29 36 28 63 93 29 99 39 22 37 29 72 53 34 33 79 98 83 99 52 52 29 45 85 25 92 33 85 21 22 33 52 33 39 43 38 72 34 33 93 42 46 66 29 32 49 23 35 79 99 97 97 86 98 89 99 97 99 108 MISCELLANEOUS PROBLEMS. (35) (36) (37) (38) (39) (40) (41) 87 83 9 936 924 989 484 938 983 692 983 756 933 724 383 315 593 979 364 346 496 517 399 836 399 725 953 442 993 622 363 633 492 342 273 332 934 743 494 434 373 944 999 789 998 699 997 798 799 658 66 878 877 278 367 936 M'SCELLANEOUS PROBLEMS. 132. 1. I bought for Christmas presents a calendar, for which I paid $1, a bronze inkstand for $1.50, a paper weight for 90 cents, and an album for $2.50; w^hat did I pay for all? 2. I received $148 for fruit trees, and $260 for shade trees; the expense of raising the fruit trees was $40, and the shade trees $50; what were the profits on each? 3. Bought a house, lot, horse, and buggy for $1400. If I paid $600 for the lot, and $200 for the horse and buggy, how much was paid for the house? 4. An agent during the year traveled 921 miles by rail- road and 234 miles by boat ; how much farther did he travel by railroad than by boat? 5. A man had $5424. To his son he gave $965, and the remainder to his wife; what was his wife's share? 6. A father and his two sons earned $1843 in a year, the elder son earning $628, and the younger $456; how much did the father earn? 7. What year will it be, in 10 years from this time? In 20 years? In 150 years? ADDITION AND SUBTRACTION BY ENDINGS. 109 8. A merchant drew out of bank $650 one day, $327 the second, $474 the third, and then had $564 in bank; how much money had he in bank at first? 9. A man bought 23 barrels of flour for $138, 27 barrels for $135, and 36 barrels for $144; how many barrels did he buy, and how many dollars did he pay? 10. A merchant living 18 miles out of Chicago, goes to the city every morning and returns in the evening; how many miles does he travel in 6 days? 11. A shoe merchant sold four dozen pairs of shoes for $192; this is $24 more than they cost him; what did they cost? 12. Holt & Co. sold 620 pairs of gloves this month, which is 20 pairs less than they sold last month, how many pairs were sold last month? 13. A real estate agent sold 6 lots of land for $9,600; if the lots were of equal value, how much did he receive for each ? ADDITION AND SUBTRACTION BY ENDINGS. 1,33. 4 + 4. Add: 4 14 24 34 44 54 64 74 84 94 4444444444 Make a subtraction table by using the results of the above addition and subtracting 4 from each. 110 ADDITION AND SUBTRACTION BY ENDINGS. (1) (2) (3) (4) (5) (6) ■ 212 444 929 424 394 793 444 943 393 492 834 339 422 222 994 333 919 449 244 779 933 629 343 333 444 322 243 333 424 934 422 988 928 837 233 382 244 232 933 292 392 893 444 424 397 486 398 329 243 787 683 766 629 829 937 347 677 226 552 458 134. 4 + 5. • Add: 14 24 34 44 54 64 74 84 94 104 5 5 5 5 5 5 5 5 5 5 Make a subtraction table by using the results of the above addition and subtracting 5 from each. (1) (2) (3) (4) (6) (6) 425 9 2 54 219 494 554 54 594 294 442 535 141 745 224 591 234 240 514 293 451 435 335 445 445 492 234 954 498 493 151 335 334 839 513 132 415 632 243 332 354 989 552 933 328 535 735 219 326 379 289 232 221 853 396 618 902 893 449 508 MULTIPLICATION AND DIVISION, 111 135. 4 + 6. Add: 4 14 24 34 44 54 64 76 84 94 104 Make a subtraction table, taking 6 from each of the results of the above addition. (1) (2) (3) (4) (5) (6) 389 644 246 454 695 946 983 655 564 645 244 354 439 262 441 242 363 241 690 344 454 464 448 664 834 465 635 345 452 395 376 544 384 369 224 26 946 996 726 892 994 994 562 326 549 839 539 618 55 897 73 686 73 43 7 557 28 154 58 9 MULTIPLICATION AND DIVISION. 136. Copy and learn: 9X9= 81 10X10 = 100 11X11 = 121 10X9= 90 11X10 = 110 12X11 = 132 11X9= 99 12X10 = 120 12X12 = 144 12X9 = 108 Write these with the 9, 10, and 11 first. Recite the division table from the multiplication table. 112 MULTIPLICATION AND DIVISION. 137. MENTAL EXERCISE. 1. At 9 cents a pound, what will 12 pounds of raisins cost? 2. A newsboy makes 10 cents a day by selling papers; how much will he earn in 11 days? 3. If 12 grape-fruits cost 144 cents, how much are they apiece? 4. When eggs are selling at 9 cents a dozen, what will 9 dozen cost? 5. A jeweler received in one week $132 for clocks which were valued at $11 each; how many clocks did he sell? 6. A street-car makes a trip of 11 miles in one hour; at that rate, how many miles will it run in 12 hours? 7. 12 men working together can do a piece of work in 9 days ; in how many days can one man working alone do the same work? 8. How long will it take an excursion party to complete a journey of 121 miles, walking 11 miles a day? 9. 12 dozen foliage plants were used in the border of a garden walk; how many plants were used? 10. At 10 cents a yard, how many yards of muslin can be bought for $1.20? 11. At 12 cents a pound, what will 9 pounds of maple sugar cost? 12. Make problems for: lOx 10 = 100 $11)$121 12 bushels)132 bushels 11 10 11 X 9= 99 11 miles )110 miles 12 trees ) 108 trees 10 9 Note. — Do this in class. MULTIPLICATION AND DIVISION. 113 13. Divide all numbers from 9 to 36 by 9, naming each undivided remainder. 14. Divide all numbers from 11 to 44 by 11. 15. Divide all numbers from 12 to 60 by 12. 16. 641009187695398205965897083 17. 541798160708943009685479187 9 EXERCISE. 138. 1. A gardener sold 10 geraniums for $1 ; what was the value of each plant? 2. Robert spent | of $1 for a book; how many cents did he spend? 3. I paid $1 for 3 pounds of coffee; how much was it a pound? 4. If 4 pounds of sugar cost 25 cents, what is the cost of one pound? 5. George had 90 cents and spent to of his money for nails; how many cents did he spend? 6. 12 miles is to of my journey; what is the whole dis- tance? 7. 132 pounds of prunes were packed in 11 boxes of equal size; how many pounds were put into each box? 8. I paid 12 dollars^ which was i of my money, for some peach trees; how much money had I? 9. Carl says, '' If my marbles were divided equally 114 MULTIPLICATION AND DIVISION, among 12 boys, each boy would receive 11 marbles/' How many marbles has he? 10. One man working alone can do a piece of work in 132 days; in how many days can 12 men do the same work? 11. A watch costs $90, and a chain i as much; what is the value of the chain? Of both watch and chain? 12. A farmer sells 144 bushels of potatoes to 12 cus- tomers ; if they are divided equally, how many bushels does each receive? 13. Make problems for: 9 )90 pounds 9)108 miles tV of $120 = $12. 10 pounds 12 miles 1 1 )99 cents 1 1)132 dollars iV of 1 10 miles = 10 miles. 9 cents 12 dollars 1 2)84 cents 12 )144 inches tV of $108 = $9. 7 cents 12 inches Note. — Do this in class. CLASS EXERCISE. 139. 1. 6 )29855217448115692558121854109792352 2. 6 )41909373891581097635338589870093874 3. 7 )5523470092665607315858723102163 4. 7)433547467495487876969201320134 MULTIPLICATION TABLE. 115 140. MULTIPLICATION TABLE. 1X2= 2 1X3= 3 1X4= 4 2X2= 4 2X3= 6 2x4= 8 3X2= 6 3X3= 9 3X4 = 12 4X2= 8 4X3 = 12 4X4 = 16 5X2 = 10 5X3 = 15 5X4 = 20 6X2 = 12 6X3 = 18 6X4 = 24 7X2 = 14 7X3 = 21 7X4 = 28 8X2 = 16 8X3 = 24 8X4 = 32 9X2 = 18 9X3 = 27 9X4 = 36 10X2 = 20 10X3 = 30 10X4 = 40 11X2 = 22 11X3 = 33 11X4 = 44 12X2 = 24 12X3 = 36 12X4 = 48 1X5= 5 1X6= 6 1X7= 7 2X5 = 10 2X6 = 12 2X7 = 14 3X5 = 15 3X6 = 18 3X7 = 21 4X5 = 20 4X6 = 24 4X7 = 28 5X5 = 25 5X6 = 30 5X7 = 35 6X5 = 30 6X6 = 36 6X7 = 42 7X5 = 35 7X6 = 42 7X7 = 49 8X5 = 40 8X6 = 48 8X7 = 56 9X5 = 45 9X6 = 54 9X7 = 63 10X5 = 50 10X6 = 60 10X7 = 70 11X5 = 55 11X6 = 66 11X7 = 77 12X5 = 60 12X6 = 72 12x7 = 84 116 MULTIPLICATION TABLE. 1X8= 8 1X9= 9 1X10= 10 2X8 = 16 2X9= 18 2x10= 20 3X8 = 24 3X9= 27 3X10= 30 4X8 = 32 4X9= 36 4x10= 40 5X8 = 40 5X9= 45 5X10= 50 6X8 = 48 6X9= 54 6x10= 60 7X8 = 56 7X9= 63 7X10= 70 8X8 = 64 8X9= 72 8X10= 80 9X8 = 72 9X9= 81 9X10= 90 10X8 = 80 10X9= 90 10X10 = 100 11X8 = 88 11X9= 99 11X10 = 110 12X8 = 96 12X9 = 108 12X10 = 120 1X11= 11 1X12= 12 2X11= 22 2X12= 24 3X11= 33 3X12= 36 4X11= 44 4X12= 48 5X11= 55 5X12= 60 6X11= 66 6X12= 72 7X11= 77 7X12= 84 8X11= 88 8X12= 96 9X11= 99 9X12 = 108 10X11 = 110 10X12 = 120 11X11 = 121 11X12 = 132 12X11 = 132 12X12 = 144 Be sure to read these tables both ways. CHAPTER V. READING AND WRITING NUMBERS. Two Periods: Units and Thousands. 2d period. Ist period. Thousands. Units (ones). 141. Ten ten-thousands are equal to one hundred- thousand. One hundred-thousand is how many times ten thousand? Hundred-thousands are written in the first place to the left of ten-thousands. 405,623 is read, " four hundred five thousand six hundred twenty-three/' The figure 4 expresses the number of hundred-thousands. In the number 405,623, in what place, or order, does the figure 6 stand? The figure 4? 2? 0? Ten units of any order make one of the next higher order. Expressing numbers by means of figures is called Nota- tion. Expressing numbers in words is called Numeration. 142. Write 6 ciphers and separate them into periods. Place 3 in hundred-thousands' place, 2 in thousands' place, and 4 in hundreds' place. Read the number you have written. 118 READING AND WRITING NUMBERS Read the following numbers: 401,392 500,020 800,005 110,111 503,001 909,008 850,050 101,001 648,406 763,204 616,016 111,101 Express the following in figures : Two hundred thousand sixty-three. Seven hundred seven thousand eighty-one. Five hundred fifty-one thousand one. Eight hundred eighteen thousand six. One hundred eleven thousand eleven. Two hundred thousand twelve. Nine hundred ninteen thousand nineteen. Three Periods: Units, Thousands, and Millions. 3d period. 2d period. lat period. Millions. ThouBands. Units. fl«.5 c«.-§ fl«5 ^ S ^ ^ S a S ^ s 000 000 000 143. The third period of figures expresses ones of mil- lions, tens of millions, and hundreds of millions. Write 9 ciphers and separate them into periods. Place 3 in ten-thousands' place, 6 in ten-millions' place, 8 in tens' place, 4 in thousands' place, and 7 in milUons' place. Read the number. Read the following numbers: 100,000,000 150,004,150 19,300,019 1,000,000 50,040,040 9,999,000 4,700,630 804,307,321 11,110,011 20,343,101 10,010,001 10,111,101 MULTIPLYING AND DIVIDING BY 9, 10, 11, AND 12. 119 Write in figures: Fifty-six million one hundred seventeen thousand six hundred nine. Three hundred eight thousand three hundred eight; six million sixteen. Ten million one hundred eleven thousand one. MULTIPLYING AND DIVIDING BY 9, 10, 11, AND 12. 144. Multiply by 9: 1. 8439 6. 5968 11. 6874 16. 9005 21. 6298 2. 7095 7. 6374 12. 3758 17. 8161 22. 2759 3. 6394 8. 4738 13. 8647 18. 7463 23. 8463 4. 8007 9. 6834 14. 9376 19. 6389 24. 3874 5. 6398 10. 4958 15. 4837 20. 8476 25. 6438 145. 9-^9 = 10-^9 = 1, Irem. 11-9 = 1, 2 rem. 12-9 = 1, 3 rem. 13-9 = 1, 2 rem. 14-9 = 1, 5 rem. 15-9 = 1, 6 rem. 16-^9 = 1, 7 rem. CLASS EXERCISE. 17^9 = 1, 8 rem. 18^9 = 2 19-^9 = 2, Irem. 20^9 = 2, 2 rem. 21-9 = 2, 3 rem. 22-9 = 2, 4 rem. 23-^9 = 2, 5 rem. 24-9 = 2, 6 rem. 25-7-9 = 2, 7 rem. 26^-9 = 2,8 rem. 27^9 = 3 28^9 = 3, Irem. 29-^9 = 3, 2 rem. 30-^9 = 3, 3rem. 31^9 = 3, 4 rem. 32-9 = 3, 5 rem. Beginning with 33, complete the table to 108 -r- 9, each pupil giving one number divided by 9. Note,— Do this in class. 120 MULTIPLYING AND DIVIDING BY 9, 10, 11, AND 12, Divide : 8)39806956597477590077495805479723136 8 )7767000543372858384567121413212 9 )44782826172173538837182781164688528 9 )17343806471078661983608026751009801 146. Divide by 9: 1. 15443 7. 13540 13. 39128 19. 53080 25. 29123 2. 17867 8. 44547 14. 29109 20. 44064 26. 20367 3. 27364 9. 88432 15. 24389 21. 41229 27. 23389 4. 72351 10. 76302 16. 66093 22. 45562 28. 35198 5. 11128 11. 68134 17. 75623 23. 89054 29. 55555 6. 55408 12. 47562 18. 64224 24. 76323 30. 44444 147. Multiply: 1. 7865X10 8079X10 80563X10 96532x10 Short Method. When the multiplier is ten, the product is obtained by annexing zero to the multiplicand. 2. 89736X10 78895x10 45838X10 40009X10 Find quotients : 3. 28930^10 26845-^10 870470-10 693879^10 Short Method. Cut oflP one figure from the right of the divi- dend. The part cut off is the remainder and the rest of the dividend is the quotient. 4. 7630456^-10 3987652-10 3101487 -f- 10 MULTIPLYING AND DIVIDING BY 9, 10, 11, AND 12. 121 148. Multiply by 11: 1. 89723 2. 65049 3. 830976 4. 394857 5. 385047 6. 629875 7. 748693 8. 480019 149. 11 12-1 13^1 14-1 15 = 1 16 = 1,1 rem. 17^1 = 1,2 rem. 18-^1 = 1,3 rem. 19^1 = 1,4 rem. 20^1 MENTAL EXERCISE. 1 = 1,5 rem. 21-11 = 1, 10 rem. = 1,6 rem. 22-11=2 = 1,7 rem. 23-^11=2, 1 rem. = 1,8 rem. 24 -11 =2, 2 rem. = 1, 9 rem. 25 -11=2, 3 rem. Beginning with 26, complete the table to 132—11, each pupil giving one number divided by 11. 150. Divide by 11: 1. 25826 2. 20441 3. 37838 4. 899604 5. 283563 6. 190009 7. 567802 8. 900456 9. 404040 10. 9800457 11. 2394836 12. 1938479 151. Find the products: 1. 2. 3. 4. 7809X12 9489X12 7618X12 9284X12 5. 29848X12 6. 72952X12 7. 47836X12 8. 39647X12 9. 63749X12 10. 34952X12 11. 6784X12 12. 56900X12 13. 61809X12 14. 19072X12 15. 72839X12 16. 85535X12 122 MISCELLANEOUS PROBLEMS. 152. 12-12 = 1 13 14-12 = 1, 2 rem. 16 17 12 = 1, Irem. MENTAL EXERCISE. 18^12 = 1, 6 rem. 24-12 = 2 19 20 15h-12 = 1, 3rem. 21^12=1, 9 rem. 27 12 = 1, 4 rem. 22 12 = 1, 5 rem. 23 12 = 1, 7 rem. 25 12 = 1, 8 rem. 26 12 = 1, 10 rem. 28 h- 12 = 2, 4 rem. 12 = 1, 11 rem. 29^12 = 2, 5 rem. 12 = 2, Irem. 12 = 2, 2 rem. 12 = 2, 3 rem. Complete the table to 144-12. 153. Divide by 12: 1. 6384 2. 2952 3. 29548 4. 98345 5. 54389 6. 87432 7. 49673 8. 83440 9. 970836 10. 483974 11. 298375 12. 483762 13. 7864532 14. 1111111 15. 9999999 16. 3568903 17. 2494967 18. 8607859 19. 9860004 20. 7581924 21. 9676893 22. 6892853 23. 4786900 24. 6927465 MISCELLANEOUS PROBLEMS. 154. 1. John says, '' Four times my money is $1.00"; how many cents has he? 2. What will 12 pounds of soap cost at 12 J cents a pound? 3. 75 cents is ^ of my money; how much money have 1? 4. What was received in payment for 867 desks sold at $9 each? 5. Rugs which cost $7.50 each, were sold for $8.00 apiece ; what was the profit on each rug? On 12 rugs? 6. What is the cost of 36 bolts of ribbon at $9 a bolt, and 25 yds. of velvet at $5 a yard? MISCELLANEOUS PROBLEMS. 123 7. A merchant bought 9 pieces of merino, each piece containing 45 yards. After selHng 135 yards, how many dress patterns of 9 yards each had he left? 8. A clerk saves $9 a month; how many months will it take him to save $684? 9. Bought 882 acres of woodland. After clearing one- ninth of it, I sold the remainder at $9 an acre; how much did I receive? 10. If you have $238 when you are 18 years old, and save $49 each year until you are 27, how much money will you then have? 11. A lady having $125, bought a cloak for $75, and 7 yards of silk at $2 a yard; how much money had she left? 12. A coal-dealer bought 11 tons of coal for $126.50 and sold it at $12 a ton; did he gain or lose? How much? 13. A hotel-keeper bought 98 pounds of crackers at 8 cents a pound, and 138 loaves of bread at 4 cents a loaf; how much did he pay for both? 14. At 5 cents a quart, what is the value of a barrel of cider containing 31^ gallons? 15. A manufacturer received $2688 for gloves, at the rate of $12 per dozen pairs; how many dozen pairs did he sell ? 16. How many days are there in eleven years? 17. A shoe merchant received $288 for 12 dozen pairs of shoes; what was the value of one dozen pairs? Of one pair? 18. $37,863 were received in three months for coal sold at $9 a ton; how many tons were sold? 19. An office building valued at $600,000 is owned by a company of 12 men; the rental received each year is $30,000. What is each man's share of the rent? 124 ADDITION AND SUBTRACTION BY ENDINGS. 20. How many barrels of flour, at $9 per barrel, will pay for 60 cords of wood, at $12 per cord? 21. 12 men can do a piece of work in 20 weeks; in how many weeks can one man do the same work? 22. How many revolutions will be made by a wheel 12 feet in circumference in running 52,800 feet? ADDITION AND SUBTRACTION BY ENDINGS. 155. 4 + 7. Add: 4 14 24 34 44 54 64 74 84 94 Make a subtraction table, taking 7 from each of the results of the above addition. (1) (2) (3) (4) (5) (6) 634 646 689 956 497 90 476 443 344 734 973 994 634 574 765 275 748 427 446 437 337 347 480 99 667 649 784 564 39 749 743 767 369 449 3 969 367 424 244 992 989 793 473 396 936 738 927 739 452 787 879 788 697 77 849 567 652 253 47 64 7. 477 + 743 + 267 + 344 + 598 + 442 + 675 + 484 + 834 + 646=? ADDITION AND SUBTRACTION BY ENDINGS. 125 8. Add 594, 764, 432, 474, 544, 347, 854, 334, 788, and 568. 9. Find the amount of 9, 93, 838, 297, 944, 469, 93, 739, 479, and 60. 10. Find the sum of 9, 34, 897, 378, 949, 983, 639, 84, 1, 78, and 78. 11. 44 + 987 + 909 + 738 + 493 + 989 + 37 + 704 + 989 + 44 + 7=? 12. 839 + 799 + 3 + 488 + 937 + 784 + 478 + 842 + 649 + 83 +9+9=? 13. Add 39, 899, 980, 97, 734, 97, 473, 648, 783, 68, 4, 7. REVIEW. 156. Add rapidly, giving the ending figures first, then the whole sum: 69785979 22 32 42 52 62 72 82 92 9 8 7 6 9 7 8 9 23 33 43 53 63 73 83 93 5 4 6 7 5 7 6 7 24 .34 44 54 64 74 84 94 4 4 4 4' 4 4 4 4 34 46 55 67 74 85 95 37 Add the following lines, beginning at the left: 7, 7, 4, 2, 4, 4, 2, 5, 4, 1, 4, 5, 1, 5, 3, 3, 3. 8, 8, 4, 6, 4, 4, 4, 2, 5, 4, 1, 5, 4, 2, 8, 3, 2, 9. 126 ADDITION AND SUBTRACTION BY ENDINGS. 5, 4, 4, 9, 8, 4, 5, 1, 4, 4, 2, 9, 3, 7, 2, 3, 6, 4. 7, 4, 5, 4, 7, 4, 7, 3, 3, 6, 5, 4, 1, 9, 3, 2, 6, 7. 8, 3, 7, 3, 4, 4, 2, 8, 3, 3, 4, 3, 8, 7, 4, 8, 3, 2. 9, 3, 9, 9, 8, 3, 7, 3, 6, 4, 4, 4, 1, 7, 4, 5, 4, 9. 157. 4 + 8. 4 14 24 34 44 54 64 74 84 94 104 8888888888 8 Make a subtraction table, taking 8 from each of the results of the above addition. (1) (2) (3) (4) (5) (6) 448 283 184 444 979 89 662 224 626 666 887 48 448 484 484 444 222 994 662 644 226 636 348 479 848 882 484 433 563 749 262 244 826 447 387 984 448 866 284 384 744 936 880 288 841 327 399 994 526 447 28 773 393 649 38 267 939 668 968 99 7. Add 384, 384, 348, 314, 958, 328, 733, 493, 889, and 269. 8. Find the sum of 799, 423, 853, 124, 489, 534, 928, 298, 988, and 464. 9. 84+47 + 978+288 + 784 + 349 + 899+892 + 699 + 34 + 48 + 84= ? 10. Find the amount of 83, 387, 938, 974, 949, 889, 398, 794, 448, 983, 348, 87, 4. ADDITION AND SUBTRACTION BY ENDINGS, 127 158. Subtract: 1. 2442-2546 2. 13123-1666 3. 43454-1456 4. 23455-1666 5. 35656-4666 6. 60003-2666 7. 53320-1777 8. 44320-2676 9. 39131-7777 10. 68442-6767 11. 49563-6777 12. 18004-7777 13. 37404-6476 14. 18805-3076 15. 99405-6066 16. 97503-6707 17. 19505-6007 18. 13006-1007 MISCELLANEOUS PROBLEMS. 159. 1. Two numbers, taken together, make 1000. One of the numbers is 320; what is the other number? 2. Find the difference between 534 and 3034. 3. Subtract 12 from 1000. 4. An excursion train left Chicago for Niagara Falls with 543 passengers. On the way 254 passengers left the cars and 162 came aboard; how many were on the train when it reached Niagara Falls? 5. Morris paid 50 cents for a hammer, $1.25 for a saw, 75 cents for a file, 25 cents for a gimlet, 25 cents for a screw- driver, $1 for an auger, 50 cents for a chisel, and $2 for a plane; how much did his tools cost him? 6. If he should sell his tools for $6, would he gain or lose? How much? 7. The sum of 3 numbers is 1345. Two of the numbers are 300 and 400; what is the third? 8. From a cask containing 900 gallons of kerosene I sold at different times 200 gallons, 165 gallons, and 150 gallons; how many gallons remained in the cask? 128 MULTIPLICATION. 9. A gentleman bought a lOOO-mile ticket on a railroad for the use of his wife, his daughter, his son, and himself. His wife rode 233 miles, his daughter 289 miles, his son 221, and he himself rode the remainder; how many miles did he ride? 10. A farmer raised 225 bushels of blue-grass seed. He sowed 74 bushels, and sold 95 bushels; how many bushels had he left? 11. A has $629, B has $865, C has $786, and D has as much as A, B, and C; how many dollars has D? 12. A has $2400, B has $500 less than A, C has $150 less than B; how much money has C? MULTIPLICATION. When the Multiplier Consists of More Than One Order. 160. Multiply 234 by 25. 234 (multiplicand) (1) 234 units multiplied by 25 (multiplier) 5-1170 uuits. (2) 234 units multiplied by 5 times 234 =1170 2 tens (or 20) = 468 tens, or 20 times 234 = 468 4680 units. The 8 tens in this or x- oo^ ror/^ , , .. product are written in tens' 25 times 234 = 5850 (product) ^ ..u ^ u j j • u place, the 6 hundreds in hun- dreds' place, and the four thousands in thousands' place. (It is not necessary to write the zero as the right hand figure of this product, as 8 tens means 80 units.) (3) 1170 + 4680 =5850. Note. — The process of multiplication may be taught as given above without further explanation. If it is thought best to make a more extended study of the process, the following method may be useful : UNITED STATES MONEY. 129 5 times 234 = 1170 (Ist partial product) 20 times 4 = 80^ 20 times 3 tens = 600 [ 20 times 2 hundreds; =4000 ' 4680 (2<1 partial product) 1170 + 4680 = = 5850 (product). Find products: 1. 4697X26 13. 1086X97 2. 8309X28 14. 8594X85 3. 3597X34 15. 9457X69 4. 4318X28 16. 3749X58 5. 7906X47 17. 4008X37 6. 6708X39 18. 8096X59 7. 8009X95 19. 9085X68 8. 7926X87 20. 6927X74 9. 0193X68 21. 9619X47 10. 9658X76 22. 6538X87 11. 9037X98 23. 7108X98 12. 8395X79 24. 8693X80 In example 24, multiply by 8 and annex one cipher. UNITED STATES MONEY. 161. Read the following: $426.37 $4003.90 $50035.05 $200.02 $9040.09 $16200.15 $187.07 $1919.19 $70017.17 Express in figures: Nine hundred sixty-seven dollars eight cents. Fifty-two thousand eleven dollars seven cents. 130 UNITED STATES MONEY Forty-one thousand eleven dollars seven cents. Eleven thousand one hundred dollars one cent. What will 4 barrels of flour cost, at $6.80 a barrel? $6.80, cost of one barrel. 4 $27.20, cost of 4 barrels. Multiply as in simple numbers, and if there are cents in the multiplicand, point off two places for cents in the product. Find products : 1. $16.15X3 4. $286.04X3 7. $0.89x4 2. $26.10X2 5. $480.70X4 8. $0.75X3 3. $45.01X5 6. $0.85X5 9. $0.90X5 Multiply: 10. $9.50 by 48 13. $8.72 by 75 16. $16.87 by 68 11. $10.54 by 36 14. $7.96 by 87 17. $20.35 by 95 12. $12.54 by 65 15. $11.84 by 96 18. $35.62 by 76 EXERCISE. 163. 1. There are 24 rows of trees in an orchard and 196 trees in each row; how many trees does it contain? 2. There are 24 hours in a day; how many hours are there in a year? 3. If a ton of coal costs $9.75, what must I pay for 95 tons ? 4. If there are 78 school buildings in a city and an average of 690 pupils in each building, how many pupils are there in all the schools of the city? 5. There are 144 pens in a box; how many pens are there in 75 boxes? ADDITION AND SUBTRACTION BY ENDINGS. 131 6. If flour is selling at $5.75 a barrel, what is the value of 85 barrels? 7. A man earns $175 a month; how much does he earn in 3 years? 8. If there are 196 pounds of flour in a barrel, how many pounds do 65 barrels contain? 9. Albert earns $10.50 a week; how much does he earn in a year? 10. There are 5280 feet in a mile; how many feet are there in 98 miles? ADDITION AND SUBTRACTION BY ENDINGS. 163. 4 + 9. Add: 4 14 24 34 44 54 64 74 84 94 104 9999999999 9 Make a subtraction table, taking 9 from each of the results of the above addition. (1) (2) (3) (4) (5) (6) 422 222 499 998 979 694 649 494 911 499 943 948 961 616 144 794 468 498 199 949 966 948 797 989 311 161 149 497 343 894 449 494 941 979 879 347 261 616 124 994 698 984 994 949 943 649 44 498 444 337 228 47 9 43 565 717 828 4 50 38 132 ADDITION AND SUBTRACTION BY ENDINGS. 7. Add 98, 949, 899, 981, 444, 96, 974, 443, 78, and 68. 8. Find the amount of 949, 377, 994, 899, 448, 79, 794, 948, 98, and 744. 9. Find the sum of 998, 874, 949, 467, 94, 848, 78, 894, 749, 87, 44, and 9. 10. 78+90+949+478+44 + 909 + 887+94+989+708+ 79+3 = ? 11. 84+939+47+874+478+848+869 + 44 + 989 + 787 + 44 + 20=? 12. 44+969+448+99+794+447 + 74+878+849+94+ 49+489=? 164. Subtract: 1. 24321- 2878 18. 39705- 16706 2. 16653- 388 19. 10000- 6117 3. 37304- 5888 20. 69676- 5767 4. 48765- 4878 21. 49075- 7076 5. 29530- 4878 22. 29640- 6777 6. 14878- 7807 23. 90653- 3767 7. 88676- 17878 24. 20412- 7777 8. 19644- 888 25. 10586- 5767 9. 20595- 5468 26. 70001- 4006 10. 60677- 6878 27. 80076- 2076 11. 69004- 10768 28. 70604- 5075 12. 78002- 7834 29. 60243- 4767 13. 44320- 1667 30. 59456- 7537 14. 36543- 4757 31. 17653- 2767 15. 17654- 5667 32. 69114- 7116 16. 28065- 6776 33. 80000- 67 17. 38345- 14677 34. 91011- 8927 ADDITION AND SUBTRACTION. 133 EXERCISE. 165. 1. A man bought a coat for $24, a hat for $5, a pair of shoes for $6, and a cravat for $1.50; how much did they all cost? 2. I give a fifty-dollar bill in paying an account of $36.37; how much change should I get? 3. The difference between two numbers is 1160. The smaller number is 8340; what is the larger number? 4. Washington was born in 1732; in what year was he 57 years old? 5. 260 bushels of potatoes is 55 bushels more than a grocer sold during the month of September; how many bushels did he sell? 6. Bought 30 yards of cloth for $96.90, 20 yards of carpet for $40, and two pairs of curtains for $16.50; what did all cost? 7. Bought a farm for $13716, and sold it for $13379; did I gain or lose? How much? 8. A saleswoman earns $0.89 a day, and her expenses are $3.75 a week; how much does she save in a week? 9. I bought a house for $6500, spent $1876 in improve- ments, and then sold it for $9155; how much did I gain? 10. Of a railroad 2465 miles long, 1266 miles are double track; how many miles are single track? MENTAL EXERCISE. 166. 1. How many days are there in 9 weeks? 2. How many six-inch pencils can you cut from 54 inches of lead? 134 MENTAL EXERCISE. 3. The minute hand goes round the dial in an hour; how many minutes is it in passing over iV of this space? 4. A confectioner put up 52 pounds of candy in 8 boxes of equal size; how many pounds did each box contain? 5. The water in a tank is 72 inches deep; how many feet deep is it? 6. 7^ pecks of beans are how many quarts? 7. Bessie had 54 cents; she spent 9 cents for envelopes. The remainder of her money will pay for how many street- car rides, if she pays 5 cents each time? 8. A farmer having 56 bushels of potatoes, planted \ of them ; how many bushels did he plant ? How many pecks ? 9. Frank had 70 cents; he spent 4" of it for a ball of twine, 4 for some nails, and with the remainder he bought a Reader; what did his Reader cost? 10. If a cook uses 6 eggs each day, how many days will 8 dozen last? 11. How many sides have two triangles? How many plants will be needed for two triangular garden plats, if 9 are planted on each side? (Make a drawing.) 12. If a man works at his trade nine hours a day, how many hours does he work in a week? 13. 16 bushels of oats are how many pecks? 14. I bought eight yards of muslin at 7 cents a yard, and gave in payment a fifty-cent piece and a ten-cent piece; what change ought I to receive? 15. At the rate of 72 pages in 7 days, how many pages do I read in a day? 16. Ten cents, which Horace paid for his drawing book, was one-eighth of his money; how much had he? HALVES, THIRDS, AND SIXTHS. 135 17. A florist having 7 dozen roses, sold one-fourth of them; how many did he sell? 18. If a peck of berries costs 96 cents, how much are they a quart? 19. If you have collected 8 dozen stamps, of which I are 6-cent stamps, how many 6-cent stamps have you? 20. If 5 cents is paid for a cup of coffee, 4 cents for fish, and 2 cents for bread, what will 6 such breakfasts cost? 21. If I earn 54 dollars a month, and save } of it, in how many months will I save 72 dollars? 22. How many inches are there in 8 feet? 23. 6 dozen rosebuds will be enough for how many bouquets, if 9 are used for each one? 24. I bought 2 yards of flannel for 75 cents; what was the cost of 1 yard? 25. If 96 tiles are used for a fireplace, how many dozen are used? HALVES, THIRDS, AND SIXTHS. Note. — Use objects freely in work with fractions. 16 7. 1. One third of an orange is equal to how many sixths? 136 HALVES, THIRDS, AND SIXTHS. 2. One half is how many sixths? 3. Fold a square of paper into two equal oblongs; one of the oblongs is what part of the whole? 4. Measure and draw (parallel to the line made by fold- ing) lines which shall divide the paper square into three equal oblongs One of these oblongs, made by drawing, is what part of the whole square? 5. How does one of these oblongs compare in size with J the square? Which is larger J or J? 6. A third and half of a third will make what part of the whole square? 7. Fold your square into six equal oblongs. One of the oblongs is what part of the whole square? 8. I are what part of the whole? J is equal to how many sixths? 9. I are what part of the whole? 10. Take away J of the square; how many sixths are left? 168. 1. Draw an oblong on your slate; divide it into six equal oblongs. One of these is what part of the whole? 2. Three of the small oblongs are what part of the large one? From your drawing find the answers to these questions: 3. ^ and } are how many sixths? 4. I + J are how many sixths? 5. i + J are how many sixths? 6. *+4=? 10. i + i=? 14. |-^=? 18. t-J=? 7. l+i=? 11. 4 + f=? 15. f-4 = ? 19. *-*=? 8. * + § = ? 12. f + i=? 16. ?-J = ? 20. !-§=? 9. 4Xi = ? 13. 3Xi=? 17. 6Xi = ? 21. 3X| = ? HALVES, THIRDS, AND SIXTHS. 137 22. i are how many thirds? 23. I are how many halves? 24. f are how many thirds? 169. 1. A grocer bought a cheese, of which he sold i on Monday, and J on Tuesday; what part of the whole cheese remained unsold? 2. Henry is 24 miles from home. In returning, he rides J of the distance on his bicycle, | on horseback, and walks the remainder; how many miles does he walk? 3. 12 cents is half my money; how many cents have I? Two times the half of anything equals what? 4. 6 cents is J of EUa^s money; how many cents has she? 3 times J equals what? 5. Find J of 2. Take two squares. Fold each into three equal oblongs. Divide these two squares equally among three children. One child receives what part of the whole? 6. Place the oblongs so as to form the two squares again^ and find the answers to these questions : J of 2 squares is what part of one square? J of 2 is what part of 1? 7. i of 2 cakes is what part of one cake? J of 2 pine- apples is what part of 1 pineapple? 8. Divide 2 pies equally among three visitors; how much will each receive? (Picture.) 170. Learn the following table: i of 10 = 2 1 of 25 = 121 iofl00 = 33i i of 10 = 21 iof25= 6i fofl00 = 66f iofl0 = 3i iof50 = 12i iofl00 = 12i f of 10 = 61 |of50 = 37i TVoflOO= 8J i of 100 = 161 CHAPTER VI. MULTIPLICATION AND DIVISION. ni. Long Division. Divide 2688 by 12. The process of Long Division is the Short Method same as that of Short Division, exeept- i"& ^^^^ ^^® w^ork is written in full. 12 ) 2ooo 224 (quotient) ^^^^ contains 12, two hundred times, with 2 hundreds remaining undivided. 28 tens contiiiiis 12, two tens times Long Method. ^^o times), with 4 tens remaining un- 12 ) 2688 ( 224 (quotient) divided. 24 48 units contains 12, four times. The — — result is 200 + 20 + 4 = 224. 24 12)2688(200; ^^^ 2400 20} = 224 4^ 288 48 240 48 48 00 J 20 [: 4^ EXERCISE. 172. Divide the following numbers by 21: Use the left hand figure of the divisor as the trial divisor. 1. 672 5. 961 9. 9266 2. 655 6. 8862 10. 7654 3. 483 7. 6552 11. 9794 4. 252 8. 7205 12. 8319 MULTIPLICATION AND DIVISION. 139 13. 2043 14. 3791 15. 8359 16. 2043 17. 7482 18. 6944 19. 8752 20. 9342 21. 3484 22. 5184 23. 1249 24. 1988 25. 6732 26. 8493 27. 3048 28. 8065 29. 9271 30. 1839 31. 1692 32. 9437 33. 8874 34. 1986 35. 2016 36. 2019 37. 2026 38. 9032 39. 4876 40. 7654 41. 5437 42. 3999 43. 8763 44. 4472 45. 9652 46. 3698 47. 4271 48. 2039 EXERCISE. 173. Divide 32019 by 31. 31 ) 32019 ( 1032 31 101 93 89 62 27 rem. Tlie divisor is not contained in the second partial dividend. Write zero in the quotient and annex the next fig-ure of tlie dividend, which gives the jmrtial dividend 101. Divide the following numbers by 31 : 1. 33065 2. 34139 3. 32098 4. 34129 5. 34149 6. 34108 7. 36128 8. 30192 9. 30298 10. 30179 11. 30421 12. 30568 13. 30671 14. 30897 15. 30086 16. 31008 17. 18297 18. 28269 19. 19134 20. 17698 21. 15982 22. 25769 23. 22109 24. 19987 25. 20081 26. 30109 27. 41006 28. 50963 29. 81900 30. 90191 31. 20018 32. 11009 140 MULTIPLICATION AND DIVISION. EXERCISE. 174. Find products: 1. 3845X64 17. 9106X78 2. 2762X85 18. 8009X41 3. 9381X36 19. 5970X93 4. 7469X94 20. 6715X28 5. 8309X87 21. 8510X99 6. 7670X98 22. 9296X70 7. 9875X36 23. 5438X51 8. 4319X71 24. 2914X98 9. 3007X69 25. 9009X76 10. 6219X92 26. 6597X98 11. 6101X76 27. 7850X49 12. 1990X81 28. 9687X19 13. 9799X19 29. 4896X87 14. 8423X84 30. 8910X93 15. 1906X65 31. 9768X95 16. 7254X37 32. 8007X97 EXERCISE. 175. Divide by 21: 1. 15960 4. 13242 7. 64692 10. 13455 2. 17640 5. 10933 8. 147856 11. 9879 3. 9898 6. 12790 9. 190274 12. 14291 Divide by 31 : 13. 23584 16. 14287 14. 21405 17. 29159 15. 15207 18. 21407 19. 281170 22. 279930 20. 187240 23. 218567 21. 126520 24. 188815 MULTIPLICATION AND DIVISION. 141 Divide by 41 : 25. 31529 26. 38800 Divide by 51 : 35. 34578 36. 43163 27. 28059 28. 32570 31. 370667 32. 388001 37. 47655 38. 38175 41. 19357 42. 46217 29. 34826 30. 249690 33. 94826 34. 149690 39. 35565 40. 25319 43. 34704 44. 360060 EXERCISE. 176. Divide by 24: 1. 8747 4. 15571 7. 19384 10. 18975 13. 19678 2. 30313 5. 23225 8. 8897 11. 97697 14. 89394 3. 17712 6. 18456 9. 22580 12. 8758 15. 75639 Divide by 34: 16. 26735 17. 25475 18. 29165 19. 20266 22. 26746 23. 26887 20. 33082 21. 28807 24. 28943 25. 86742 EXERCISE. 177. 1. At $21 an acre, how many acres of land can be purchased for $5187? 2. What will 1296 acres of land cost at $45 an acre? 142 ADDITION AND SUBTRACTION BY ENDINGS. 3. If a man earns $24 a week, in how many weeks can he earn $1248? 4. What is the cost of 9560 pounds of butter, bought by a commission house, at 18 cents per pound? 5. Mr. A spent $1464 for trees at $24 a dozen; how many dozen did he buy? 6. If a man saves $31 a month, in how many months can he save $1488? 7. What is the value of 950 bushels of tomatoes at 70 cents I3er bushel? 8. If I travel at the rate of 35 miles an hour, in how many hours can I complete a journey of 1225 miles? 9. A commission house sold 2980 bushels of potatoes at 65 cents per bushel; what was the amount of money received? 10. 45 feet is the width of a lot, valued at 85 dollars per foot; what is the value of the lot? 11. The multipUcand is 964; the multiplier is 98; what is the product? 12. The product is 9867480; the multiplier is 95; what is the multiplicand? 13. The divisor is 95; the quotient is 9684; what is the dividend? 14. The dividend is 1667120; the quotient is 65; what is the divisor? ADDITION AND SUBTRACTION BY ENDINGS. ms. 5 + 6 and 5 + 6. Add 5 15 25 35 45 55 65 75 85 95 5 5 5 5 5 5 5 5 5 5 ADDITION AND SUBTRACTION BY ENDINGS. 143 5 15 25 35 45 55 65 75 85 95 6 6 6_^_^_^_^_^_^_^ Make a table, subtracting 5 from each of the results above. Subtract 6 from the same numbers. ' (1) (2) (3) (4) (5) (6) 536 266 694 594 598 964 554 544 557 692 657 798 656 556 654 887 454 457 455 654 265 446 559 754 561 465 725 95 656 489 543 545 645 769 982 835 655 556 565 625 694 496 445 634 42 843 848 758 644 426 487 89 87 458 142 665 67 64 74 54 Add: 7. 444, 788, 656, 565, 989, 936, 482, 744, 568, 7, 54. 8. 99, 459, 855, 595, 644, 976, 65, 626, 848, 89, 53. 9. 75, 896, 559, 60, 969, 982, 444, 688, 655, 57, 859. 10. 458, 764, 997, 456, 762, 534, 678, 745, 756, 57, 3. 179. 5 + 7. Add: 5 15 25 35 45 55 65 75 85 95 2 1 1 J. 1. 1 111 1 Make a subtraction table, taking 7 from each of the results of the above addition. 144 ADDITION AND SUBTRACTION BY ENDINGS. Add: (1) (2) (3) (4) (5) (6) 567 312 553 469 495 354 537 735 525 795 898 557 753 375 735 986 975 585 355 535 355 748 756 526 757 557 755 275 389 959 373 753 335 475 75 147 535 377 555 855 726 743 555 533 651 581 847 494 583 876 958 68 87 299 254 516 778 8 57 359 Add: 7. 789, 572, 757, 484, 979, 834, 548, 674, 668, 898. 8. 457, 756, 973, 724, 596, 745, 485, 839, 679, 74. 9. 479, 620, 799, 239, 497, 775, 48, 797, 872, 99, 4. 10. 79, 20, 745, 284, 497, 872, 954, 787, 844, 79, 59. 11. 975, 726, 548, 875, 775, 239, 443, 878, 797, 775, 90. 180. B + 8. Add: 5 15 25 35 45 55 65 75 85 95 Make a subtraction table, taking 8 from each of the results of the above addition. ADDITION AND SUBTRACTION BY ENDINGS. 145 Add: (1) (2) (3) (4) (5) (6) 325 225 455 799 297 589 855 555 555 578 855 898 255 255 385 888 485 942 558 355 225 55 874 984 452 835 555 988 447 978 855 225 555 499 869 429 225 855 865 955 88 857 558 555 201 447 652 542 852 353 555 75 785 89 543 345 437 6 68 4 Add: 7. 979, 944, 577, 647, 962, 875, 225, 848, 2, 88, 955. 8. 945, 973, 878, 223, 755, 274, 855, 955, 84, 89. 9. 475, 647, 779, 247, 362, 875, 57, 878, 585, 89, 9. 10. 75 + 426 + 858 + 962 + 289 + 528 + 872 + 824 + 648 + 87 + 54=? Find the sum of: 11. 859, 354, 46, 975, 98, 887, 25, 997, 79, 8, 4. 12. 789, 290, 459, 878, 782, 437, 894, 53, 607, 6, 5. MENTAL EXERCISE. 181. 1. If 4 pounds of chocolate cost 50 cents, how many cents is it a pound? 2. If a man travels 15 miles in 3 hours, how far will he travel in one hour? In 9 hours? 3. How many ounces are there in a pound? What will I of a pound of candy cost, at 3 cents an ounce? 146 MENTAL EXERCISE. 4. Bought i bushel of apples and ^ bushel of peaches; what part of a bushel have I? % 5. Bought 4f pounds of grapes at 6 cents a pound; how ^uch did they cost? 6. If 3 pounds of almonds cost 25 cents, what will one pound cost? What will 5 pounds cost, at the same rate? 7. A boy gave to his sister ^ of an orange, and to his brother ^ as much as he gave to his sister; how much did he give to his sister? 8. If two pounds of cheese cost 36 cents, what will 1 pound cost? What will half a pound cost? 9. A bushel of corn weighs 56 pounds ; what is the weight of a peck? a half peck? 10. A boy living J of a mile from school, who goes home to dinner, will walk how many miles each week in going to and from school? 11. Buy 4 dozen pencils at 30 cents a dozen, and sell them at 4 cents apiece; what is gained? 12. With what you have gained, buy 2 dozen erasers and sell them at 6 cents apiece; how much do you gain this time, and how much money have you altogether? Solve: 13. |+i = ? 17. f + i=? 21. |-i=? 25. 4xi=? 14. i + f=? 18. f-| = ? 22. i-i=? 26. 3XJ = ? 15. Ki=? 19. l-i = ? 23. |-| = ? 27. 6Xi=? 16. i + | = ? 20. |-i=? 24. |-i=? 28. 3Xf=? ADDITION AND SUBTRACTION. 147 REVIEW. 182. Add: 1. 88, 492, 744, 799, 277, 558, 772, 534, 887, 87, 65. 2. 599, 540, 489, 775, 957, 898, 388, 745, 764, 88, 88, 3. 544 + 868 + 454 + 334 + 558 + 663 + 854 + 156 + 594 + 288=? 4. 878 + 925 + 848 + 89 + 295 + 975 + 424 + 989 + 529 + 98 + 973=? Find the sum of : 5. 989, 587, 659, 884, 497, 958, 52, 598, 844, 68, 65. 6. 799, 947, 864, 577, 959, 795, 495, 844, 577, 58, 4. .7. 559, 675, 576, 543, 76, 345, 975, 34, 486, 98, 965. MENTAL EXERCISE. 183. Subtract: 21 32 43 54 61 72 83 94 100 >55555555 5 21 32 43 54 65 90 100 6 6 6 6 6 6 6 21 32 43 54 65 76 100 7 7 7 7 7 7 7 Note..— Do the following" work in class: Subtract : 91027300467193450014 17263398720 4 56901029 230091402130680970013520 128940080956198296746936 148 ADDITION AND SUBTRACTION. Subtract : 33310201010063452960013 12100617289113856972869 EXERCISE. 1 84. Find the difference : 1. 4287003 2. 3010604 3650169 1720456 4. 8001641 7100956 7. 3596001 2100998 5. 9340106 5340508 8. 2684302 1008697 3. 4284006 3639109 6. 5017603 979184 9. 3001801 2018697 REVIEW. 185. Read sums rapidly : 32 42 52 62 83 93 63 53 43 33 103 869796875 98 44 54 64 74 85 95 65 35 25 105 _9_6_8J7_9^_8^_9_7 Read the differences in the above rapidly. Add: 5, 9, 9, 2, 9, 9, 8, 4, 9, 8, 2, 7, 3, 9, 7, 9. 9, 9, 4, 9, 9, 5, 9, 7, 4, 8, 2, 9, 7, 7, 2, 2. 7, 5, 8, 9, 5, 8, 9, 4, 9, 7, 4, 6, 4, 7, 0, 8. 9, 5, 8, 9, 7, 5, 4, 5, 3, 9, 8, 4, 5, 3, 0, 7. 3, 9, 4, 4, 6, 1, 4, 6, 4, 3, 5, 3, 9, 4, 4, 3, 2, 5, 3. 4, 7, 6, 2, 3, 8, 9, 1, 5, 4, 1, 7, 4, 8, 2, 6, 4, 3, 5. 9, 9, 2, 9, 2, 9, 7, 4, 5, 4, 6, 4, 5, 4, 1, 9, 3, 8. ADDITION AND SUBTRACTION. 149 (1) (2) (3) (4) (5) (6) (V) (8) 854 697 576 797 799 436 478 975 376 482 648 346 423 644 584 476 785 748 542 553 853 751 558 729 456 284 345 474 124 244 867 589 755 459 856 426 489 865 984 593 556 789 464 745 534 496 949 549 752 429 984 585 928 724 579 685 596 789 459 259 296 998 844 548 756 687 695 389 988 713 859 895 456 757 659 534 464 249 679 245 9. 798 + 557 + 789 + 985 + 557 + 78 + 895 + 559 + 849 + 96 + 85=? Add: 10. 589, 457, 855, 587, 658, 545, 758, 89, 599, 99, 54. 11. 58, 79, 594, 957, 85, 474, 545, 874, 689, 77, 8. 12. 599, 759, 575, 557, 254, 788, 357, 785, 587, 78, 4. 13. 557, 640, 555, 579, 459, 808, 879, 955, 909, 87, 4. 14. 45, 575, 678, 554, 508, 370, 757, 545, 959, 86, 54. 15. 987, 895, 956, 967, 485, 54, 875, 580, 97, 9, 3. EXERCISE. 186. 1. In what year was your schoolhouse built? How many years have passed since that time? 2. How many years have passed since the discovery of America b}^ Columbus in 1492? 3. A farmer went to town with a load of wood, which he sold for $8. He bought 25 pounds of sugar for $2, eight pounds of raisins for $1, two pounds of tea for $1.50, and six pounds of coffee for $2.10; how much did he spend? Did he receive for his wood enough to pay for his groceries ? 150 ADDITION AND SUBTRACTION. 4. A grocer bought 50 barrels of apples and 100 boxes of peaches, for which he paid $225. If he paid $75 for the peaches, how much did he pay for the apples? 5. Bought: 4 lb. butter, @ 22 c. 21b. cheese, '' 18 c. 3 doz. eggs, '' 15 c. 9 qt. milk, " 6 c. 2 bu. potatoes, '^ 65 c. 2 bu. carrots, ^' 60 c. What was the amount of my bill? 6. Bought: 9 lb. rice, 2 " tapioca, 3 '^ sago, 5 " sugar, 7 '^ prunes, 3 " figs. (3 7 c. '' 15 c. '' 13 c. '' 9 c. '' 9 c. '' 15 c. What was the amount of my bill? 7. Bought: 41b. tea, @ $1.25 . 2 '' coffee, '' .42 . 2 '^ raisins, '^ .11 . 7 '^ currants, '^ .09 . 5 '^ crackers, ^^ .12 . 7 " sugar, ^^ .08 . What was the whole amount? ADDITION AND SUBTRACTION. 151 8. Fill in the total: ]3cMA>^ c4 13. 5. dttoTV ^ Gcv., I .54- .c]0 M-.80 .80 \.25 2 u>UXcMA> tpoub^kd^, @ Li-5^ I n/Ji/rY\/YYi<> . . I tcX^OTV TVUMAKAy 1 \/jJ(tV I a-jaxuie/ 13. S. OXte/n. ^ (V. 9. Complete the bill: Nashville, Tenii., Feb. 10, 1904. Mr. John Mitchell, Bought of J. D. Hunt & Co., 2 lb. coffee, @ 32 c $ 6 ^^ crackers, ^Ml c 3 '' honey, ^M8 c 1 ^^ Japan tea .98 1 doz. oranges .40 1 sack flour 3.89 Received payment, J. D. HUNT & CO. 152 MULTIPLICATION AND DIVISION, MULTIPLICATION AND DIVISION. EXERCISE. 187. 1. Multiply 478 by 624. 478 X 4 units = 1912 units 478 478 X 2 tens = 956 tens 524 478 X 6 hund. = 2868 hundreds Jg J2 478 X 624 = 298272 units QKA The second partial product = 9560 units. The oogg third partial product = 286800 units. It is not necessary to write the zeros, since the place in 298272 which each figure is written gives its value ; 6 in tens' place is the same as 60 units. 2. Multiply the following numbers by 624: 396, 489, 279, 486, 295, 197, 176, 294, 395, 284, 692, 986. 3. Multiply by 798: 276, 347, 468,987, 692, 985, 569, 696, 459, 879. 4. Multiply by 718: 698, 437, 329, 492, 682, 694, 987, 962, 764, 829, 677, 276. 5. Multiply by 691 : 218, 912, 986, 497, 319, 489, 637, 956, 739, 895, 989, 759. 6. Multiply by 976: 1241, 3124, 2312, 1342, 2137, 7125, 1259, 2138, 3216, 4132, 5123, 3495, 4287, 5196. EXERCISE. 188. Divide by 43: 1. 34228 4. 29001 7. 302720 10. 261895 2. 29439 5. 34675 8. 389580 11. 174604 3. 21389 6. 347010 9. 34718 12. 302290 MULTIPLICATION AND DIVISION. 153 Divide by 53: 13. 51756 16. 44877 19. 50880 14. 34363 17. 32305 20. 41376 15. 25349 18. 33920 21. 480180 Divide by 54: ' 25. 21457 28. 41565 31. 43445 26. 37729 29. 37719 32. 32934 27. 41541 30. 48978 33. 41564 22. 42888 23. 39723 24. 36994 34. 21456 35. 37728 36. 32933 EXERCISE. 189. Divide by 35: 1. 16231 4. 26581 7. 31737 10. 32848 2. 22664 5. 34510 8. 31176 11. 24455 3. 32847 6. 24456 9. 16800 12. 26582 Divide by 45: 13. 35839 16. 41024 19. 43650 22. 28379 14. 31032 17. 31708 20. 38700 23. 36288 15. 43847 18. 36289 21. 22077 24. 43651 Divide these numbers by 55 and 65. EXERCISE. 190. 1. What is the value of 8950 bushels of wheat, at 95 cents per bushel? 2. A commission house sold 75 bushels of cranberries, at $2.75 cents per bushel; how much money was received? 3. $4320 was paid to 54 men for one month's work in building a bridge; how much money did each man receive? 154 ADDITION AND SUBTRACTION BY ENDINGS. 4. B paid $25668 for land, at $46 an acre; how many acres did he buy? 5. What is the value of 870 barrels of apples, at $3.75 per barrel? 6. A wholesale house sold 1260 pairs of blankets in one month, at $4.50 a pair; how much money was received from these sales? 7. There are 1272 pupils in a school building and 53 pupils in each room; how many rooms are there in the building? 8. The distance from A to B is 936 miles. If I travel at the rate of 36 miles an hour, in how many hours can I com- plete the journey? 9. When oranges are selling at $3.50 a box, what must be paid for 598 boxes? 10. $1479 was spent for rugs, at an average cost of $17 each; how many rugs were purchased? ADDITION AND SUBTRACTION BY ENDINGS 191. 6 + 9, Add: 5 15 25 35 45 55 65 75 85 95 9999999999 Make a subtraction table, taking 9 from each of the results of the above addition. ADDITION AND SUBTRACTION BY ENDINGS. 155 Add: (1) (2) (3) (4) (5) 797 958 87 768 355 950 885 855 55 478 545 878 87 557 587 457 559 995 495 55 575 949 4 954 967 84 750 775 569 98 895 565 957 45 755 55 947 79 479 889 497 787 598 89 898 67 655 85 82 62 Add: 6. 895, 258, 978, 45, 554, 645, 546, 795, 606, 8, 4. 7. 958, 545, 758, 789, 478, 959, 570, 295, 906, 59, 4. 8. 989, 959, 575, 487, 55, 597, 897, 905, 897, 687, 75. 9. 895, 587, 798, 855, 566, 855, 975, 989, 589, 95. 10. 989, 455, 464, 955, 587, 768, 555, 789, 587, 898, 75. 11. 989, 597, 855, 867, 558, 485, 986, 505, 798, 597, 74. (12) (13) (14) (15) (16) (17) 325 515 585 995 959 899 598 959 515 587 575 575 512 151 954 798 689 856 955 555 151 455 552 789 195 599 959 787 784 585 915 911 191 378 499 897 255 155 515 595 585 594 457 535 995 556 348 854 575 287 277 688 787 89 977 242 347 565 465 65 156 ADDITION AND SUBTRACTION BY ENDINGS. REVIEW. 192. Read sums rapidly: 32 42 52 62 83 93 63 53 43 33 103 44 54 64 74 85 95 65 35 25 105 Read differences rapidly. Add: 5, 9, 9, 2, 9, 9, 8, 4, 9, 8, 2, 7, 3, 9, 7, 9. 9, 9, 4, 9, 9, 5, 9, 7, 4, 8, 2, 9, 7, 7, 2, 2. 7, 5, 8, 9, 5, 8, 9, 4, 9, 7, 4, 6, 4, 7, 0, 8. 9, 5, 8, 9, 7, 5, 4, 5, 3, 9, 8, 4, 5, 3, 0, 7. 3, 9, 4, 4, 6, 1, 4, 6, 4, 3, 5, 3, 9, 4, 4, 3, 2, 5, 3. 4, 7, 6, 2, 3, 8, 9, 1, 5, 4, 1, 7, 4, 8, 2, 6, 4, 3, 5. 9, 9, 2, 9, 2, 9, 7, 4, 5, 4, 6, 4, 5, 4, 1, 9, 3, 8. (1) (2) (3) (4) (5) (6) (7) (8) 854 697 576 797 799 436 478 957 376 482 648 346 423 644 584 476 785 748 542 553 853 751 558 729 456 284 345 474 124 244 867 589 755 459 856 426 489 865 984 593 556 789 464 745 534 496 949 549 752 429 984 585 928 724 579 685 596 789 459 259 296 998 844 548 756 687 695 389 988 713 859 895 456 757 459 534 464 249 679 245 ADDITION AND SUBTRACTION BY ENDINGS. 157 9. 798 + 557 -f 789 + 985 + 557 + 78 + 895 + 559 + 849 + 96 + 85=? Add: 10. 589, 457, 855, 587, 658, 545, 758, 89, 599, 99, 54. 11. 58, 79, 594, 957, 85, 474, 545, 874, 689, 77, 8. 12. 599, 759, 575, 557, 254, 788, 357, 785, 587, 78, 4. 13. 557, 640, 555, 579, 459, 808, 879, 955, 909, 87, 4. 14. 45, 575, 678, 554, 508, 370, 757, 545, 959, 86, 54. 15. 987, 895, 956, 967, 485, 54, 875, 580, 97, 9, 3. REVIEW. 193. Add, beginning at the left: 9, 9, 4, 9, 9, 4, 7, 8, 4, 8, 3, 7, 3, 8, 6, 4. 7, 7, 9, 9, 8, 4, 8, 8, 3, 7, 3, 8, 7, 4, 6, 1. 4, 6, 7, 3, 9, 4, 9, 2, 8, 2, 7, 9, 4, 8, 8, 7. 4, 9, 9, 3, 4, 4, 6, 4, 8, 3, 7, 3, 8, 9, 9, 2. 6, 6, 6, 4, 8, 9, 4, 6, 4, 8, 7, 4, 5, 4, 7, 3. 4, 8, 9, 9, 8, 4, 6, 3, 6, 4, 3, 7, 3, 8, 2, 6. 6, 7, 6, 2, 4, 8, 3, 9, 4, 4, 9, 4, 8, 9, 3, 5. Add: (1) (2) (3) (4) (5) (6) (V) (8) (9) 483 495 497 794 298 646 594 384 799 628 418 782 288 418 443 764 384 423 982 499 839 839 499 574 435 348 853 148 744 242 274 744 437 474 314 124 482 394 879 997 394 649 544 958 489 644 424 449 434 424 767 347 328 534 966 494 894 939 494 424 854 733 928 188 428 324 977 428 396 334 493 298 447 787 998 799 787 787 788 889 988 357 653 338 764 653 567 568 269 462 158 ADDITION AND SUBTRACTION. (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) 436 989 798 873 889 999 934 498 496 787 644 322 979 845 944 844 844 844 844 444 751 877 292 149 233 455 955 467 984 485 244 713 933 952 724 488 488 327 985 575 865 442 244 347 243 319 544 943 438 824 496 983 759 732 863 942 223 168 578 493 724 524 282 444 143 243 935 484 314 899 998 554 444 244 933 354 194 343 993 422 713 627 489 536 259 386 938 348 119 644 249 978 879 878 778 979 968 798 688 549 REVIEW. 194. Read sums rapidly: 21 32 43 54 65 76 87 100 888 8 888 8 21 32 43 54 65 76 87 98 100 99999999 9 Read differences rapidly. Subtract : 132964021347625010026 92467395689706110958 86213507010013426598312 63152710236647258698595 Note. — Do this in Class. ADDITION AND SUBTRACTION. 159 195. Subtract: 1. 5010026 2110958 4. 2937111 1930086 7. 6010028 4110869 10. 3674021 1690084 13. 8100118 1909099 EXERCISE. 2. 3004210 1910096 5. 6101034 3908956 3. 9721011 8120699 8. 7010039 6110699 11. 2901003 1889019 14. 7010165 3927896 EXERCISE. 6. 4010024 3110859 9. 8621011 7620388 12. 1807007 918069 15. 5900011 2574698 196. 1. Three men buy some land for $75000; the first pays $25000, and the second pays $17500; how much does the third man pay? 2. How much must I add to $12690 to enable me to buy a farm valued at $15000? 3. A has $125, B has $79 more than A, and C has as much as A and B; how much money have A, B, and C together? 4. A train travels 758 miles a day for the first three days of the week, and 695 miles a day for the remaining four days; how far has the train traveled in a week? 5. I paid $6800 for a lot and built a house which cost $9500; the street improvements cost me $500. For how much must I sell the house and lot in order to gain $2000? 160 MULTIPLICATION AND DIVISION. 6. Add 49170, 33040, 45215, and 8315; and take the sum from 265780. 7. The sum of two numbers is 185674; one of the num- bers is 92750; what is the other number? 8. The difference between two numbers is 18698; one of the numbers is 9740; what is the other number? 9. The remainder is 6793; the subtrahend is 6755; what is the minuend? 10. The minuend is 7848; the remainder is 6702; what is the subtrahend? MULTIPLICATION AND DIVISION. 19*7. Copy and complete the following tables: 3X13= 3X14 = 3X15 = 3X16 4X13 = 4X14 = 4X15 = 4X16 5X13= 5X14 = 5X15 = 5X16 6X13= 6X14 = 6X15= 6X16 7X13 = 7X14= 7X15= 7X16 8X13= 8X14 = 8X15= 8X16 9X13= 9X14 = 9X15= 9X16 3X17 3X18 3X19 = 4X17 4X18 4X19 = 5X17 5X18 5X19 = 6X17 6X18 6X19 = 7X17 7X18 7X19 = 8X17 8X18 8X19 = 9X17 9X18 9X19 = These tables may be used as an aid in finding any term of the quotient. MULTIPLICATION AND DIVISION. 161 EXERCISE. 198. 1. Divide by 13: 3844, 1456, 2899, 3241, 4869, 12980. 2. Divide by 14: 4899, 1386, 3287, 1642, 1196, 10896. 3. Divide by 15: 3453, 1296, 1484, 4192, 1483, 14080. 4. Divide by 16: 4686, 2825, 4242, 4339, 1509, 12243. 5. Divide by 17: 4198, 2649, 9497, 1562, 1678, 14909. 6. Divide by 18: 3584, 3291, 6183, 7139, 1796, 89010. 7. Divide by 19: 3764, 3698, 7501, 1368, 1509, 94055. EXERCISE. 199. Divide by 26: 1. 12447 4. 8046 7. 19492 2. 24336 5. 25239 8. 24338 3. 19491 6. 12443 9. 25230 Divide by 36: 10. 24461 13. 28251 16. 38440 19. 28252 11. 32331 14. 26666 17. 33497 20. 38441 12. 35507 15. 14689 18. 32330 21. 33495 Divide by 46: 22. 22421 25. 34353 28. 32422 31. 44620 23. 34481 26. 41216 29. 41704 32. 39100 24. 18325 27. 28033 30. 36369 33. 28034 162 MULTIPLICATION AND DIVISION. EXERCISE. 200. 8094X208=? 8094 208 208 =200 + 8 8094 X 8 = 64752 8094 X 200 = 1618800 64752 16188 1683552 (product). Find products: 1. 2965X204 2. 3472X409 3. 5409X508 4. 5696X607 5. 2897X906 6. 3587X609 201. 8094 X 208 = 1683552 product. It is not necessary to write the zeros in the second ])artial product. We write the 8 in hundreds' place. 8 in hundreds' place has the same value as 800 units. 7. 3098X709 8. 4037X694 9. 6089X358 10. 2064X708 11. 2022X109 12. 4967X907 EXERCISE. 13. 3048X308 14. 6497X309 15. 3859X276 16. 9294X709 17. 6789X608 18. 3008X907 To multiply by 10, 100, 1000, etc. , annex as many zeros to the multiplicand as there are zeros in the multiplier. Find products : 1. 345X10 4. 783X300 7. 249X1000 10. 369X4000 2. 386X100 5. 846X500 8. 728X2000 11. 484X7000 3. 985X100 6. 782X900 9. 689X5000 12. 299X6000 202. EXERCISE. To divide by 10, 100, 1000, etc., cut off from the right of the dividend as many figures as there are ciphers in the divisor. The part cut olf is the remainder ; the rest of the dividend is the quotient. MULTIPLICATION AND DIVISION. 163 Divide each of the following numbers by 10, 100, and 1000: 1. 24865 2. 35642 3. 21870 4. 64823 5. 46704 6. 39405 7. 38400 8. 96099 9. 10398 10. 26405 11. 30240 12. 43442 13. 65143 14. 84291 15. 75028 16. 63000 EXERCISE. 203. Divide by 27: 1. 9845 4. 23751 1 . 21349 10. 207700 2. 12844 5. 20770 8. 22680 11. 237511 3. 17478 6. 281430 9. 218430 12. 984500 Divide by 37: 13. 17205 16. 32426 19. 29940 22. 18130 14. 23514 17. 31442 20. 32930 23. 335604 15. 24959 18. 35817 21. 299349 24. 335609 Divide by 47 : 25. 17129 28. 41332 31. 32308 34. 413320 26. 36438 29. 28530 32. 40847 35. 285300 27. 37520 30. 32845 33. 41830 36. 375250 EXERCISE. 204. Divide by 57: 1. 48306 4. 28229 7. 483066 10. 282290 2. 39540 5. 51098 8. 395400 11. 510980 3. 48279 6. 55794 9. 482799 12. 557944 164 HALVES, THIRDS, FOURTHS, AND SIXTHS. Divide by 67: 13. 23765 18. 32126 14. 43102 15. 31040 16. 31901 17. 57945 19. 56768 20. 42814 21. 56731 22. 65554 23. 58266 24. 60157 25. 63543 26. 46793 27. 32868 EXERCISE. 205. Divide by 68: 1. 29631 6. 31805 2. 23517 3. 43343 4. 36331 5. 50084 7. 44015 8. 51978 9. 56955 10. 50900 11. 59489 12. 46754 13. 66504 14. 61059 15. 64367 28. 408059 29. 540690 30. 467688 31. 635219 32. 648560 16. 65317 17. 46240 18. 59179 19. 65307 20. 59177 COMPARISON OF HALVES, THIRDS, FOURTHS, AND SIXTHS. 206. Into how many equal parts is this circle divided? One of the twelve equal parts of any thing is called what? One half of the circle is how many of these parts? One third of the circle is how many twelfths of the whole circle. i is how many twelfths? I is how many twelfths? Which is more, J of a cake or J? ^ or i? 207. Look at the circle and find the answers to these questions : Note. — This may also be studied with a circle cut from paper and folded into halves, then into sixths, then into twelfths. HALVES, THIRDS, FOURTHS, AND SIXTHS. 165 1. ^ and i are how many twelfths? 2. ^ and j are how many twelfths? 3. i+i=? 6. f + i = ? 9. t + A=? 12. i|-A=? 4.1+1=? 7. i + J=? 10. f + A=? 13. 1-A = ? 5. i+iV=? 8. | + i = ? 11. f-A=? 14. 1- |=? 15. Alice cut out ^ of a cake to take to a picnic; her mother used J of the cake for tea. What part of the whole cake was left? 16. Edgar used ^ of a ball of twine, and his brother Carl used J of the ball; what part of the whole ball was left? 208. Look at the circle, or draw a circle, and find answers : 1. A = how many sixths? 4. A = how many fourths? 2. A = how many fourths? 5. ^t = how many sixths? 3. tV = how many halves? 6. Find J of ^ of the circle. 7. I of i is what part of the whole? 8. i of ^ is what part of the whole? 9. iofi=? 10. iofA=? 11. 4ofi=? 12. j% is found in j% how many times? 13. tV is found in VV how many times? 14. ii contains -i\ how many times? 15. 1%^ contains t% how many times? 16. 2 times | are how many wholes? 17. 4 times f are how many wholes? 18. 3XtV are how many twelfths? How many wholes? 19. 4Xf are how many wholes? 20. 4X|=? 21. Which of these forms is most used: If or §? iV or I? 166 MISCELLANEOUS PROBLEMS. MISCELLANEOUS PROBLEMS. 209. 1. Four boys worked together, and received $3 for a day^s work. If they divide the money equally, what part will each receive? How many cents will each receive? 2. $9 is i of my money ; how much money have I? 3. 5 qt. is J of all the berries James has to sell; how many quarts has he? 4. George and his two cousins received a present of 2 watermelons; they divided them equally; what was the share of each? (Make a drawing.) 5. May, John, and Ella gathered 2 pecks of nuts and divided them equally; what was each one's share? 6. George says, ^^61 marbles is 14 more than all I have.'' How many marbles has he? 7. A boy standing 30 feet from the edge of the water, shot an arrow to an island 40 feet from the shore. How far must he go in walking and rowing to get the arrow? 8. How far must he go to get the arrow and return to the place of starting? 9. Bought 10 yd. of silk for $9.50, and lOJ yd. of cloth for $5.25; how much more did the silk cost than the cloth? 10. A man owing $1000 made 2 payments, one of $180 and one of $260; how much remained unpaid? 11. A planing-mill sells 680 ft. of pine lumber, 845 ft. of poplar, 398 ft. of cherry, 480 ft. of ash, 560 ft. of walnut, 746 ft. of maple. How many feet are sold? 12. A farmer sold 26 dozen eggs at 22^ cents a dozen, and 16 pounds of butter at 28^ cents a pound. How much did he receive for them? DIVISION 167 DIVISION. EXERCISE. 210. Divide by 49: 1. 17353 2. 22725 3. 22922 4. 31093 5. 43280 6. 31297 7. 42972 8. 47479 9. 37247 10. 43610 11. 398410 12. 333200 Divide by 59: 13. 38072 14. 27582 15. 46393 16. 40030 17. 41162 18. 46393 19. 40041 20. 41182 21. 29311 22. 509170 23. 377600 24. .476189 EXERCISE. 311. Divide by 69: 1. 24442 2. 30053 3. 36894 4. 32085 6. 32104 7. 46506 8. 25991 9. 47442 11. 59871 12. 58443 13. 66723 14. 51718 5. 25142 10. 53015 15. 59999 16. 61871 17. 65469 18. 558210 19. 54547 20. 654690 EXERCISE. 212. Divide by 74: 1. 25597 2. 32227 3. 54503 4. 34330 5. 34596 6. 63955 7. 58201 8. 71669 9. 70125 10. 255970 11. 345966 12. 701250 168 ADDITION AND SUBTRACTION BY KNDINOS. Divide by 78: 13. 36140 14. 28439 15. 57079 16. 35595 17. 52641 18. 36480 19. 58841 20. 53625 21. 58413 22. 75458 23. 60040 24. 53769 25. 68598 26. 54500 27. 73991 28. 67820 29. 30622 30. 685980 31. 754580 32. 355951 ADDITION AND SUBTRACTION BY ENDINGS. 313. 6 + 6. Add: 6 16 26 36 46 56 66 76 86 96 6-666666666 Make a subtraction table, taking 6 from each of the results obtained above. Add: (1) (2) (3) (4) (5) (6) 96 965 955 7659 6989 4665 594 224 436 4444 5659 7654 662 655 596 7363 8346 4469 259 386 695 6735 3653 7653 569 596 39 . 6646 673 4986 756 765 566 8686 968 7866 684 939 64 7895 6366 6459 566 646 995 5563 9681 2794 437 356 668 8349 7988 9778 79 788 999 6879 7687 6899 ADDITION AND SUBTRACTION BY ENDINGS. 169 314. 6 + 7. Add: 6 16 26 36 46 56 66 76 86 96 Make a subtraction table, taking 7 from each of the r(^sults obtained above. Add: (1) (2) (3) (4) (5) (6) 996 967 768 997 896 6886 767 953 437 746 9565 4553 577 667 762 654 433 5665 877. 336 255 979 7766 636 676 57 646 469 6568 8387 554 656 656 463 9895 656 983 364 765 539 4647 9439 767 65 276 697 676 7475 127 889 797 678 7568 8518 899 79 959 988 989 798 Add: 7. 979, 969, 787, 696, 969, 878, 997, 788, 979, 89. 8. 76, 967, 899, 798, 697, 876, 968, 79, 577, 87, 9. 9. 78, 969, 697, 786, 978, 869, 979, 779, 6, 89. 10. 707, 966, 979, 799, 689, 76, 867, 978, 706, 66. 215. 6 + 8. Read endings; then sums: 26 46 36 96 56 76 66 86 106 170 ADDITION AND SUBTRACTION BY ENDINGS. Make a subtraction table, taking 8 from each of the results of the preceding addition. Add: (1) (2) (3) (4) (5) (6) 969 688 98 666 67 58 667 363 887 664 932 9986 536 666 686 686 6666 4965 594 563 439 357 6256 7667 468 678 967 949 4949 6364 666 774 936 967 6677 5635 546 668 866 842 6773 6764 788 286 546 556 6758 7825 98 789 797 959 6998 899 8 869 789 688 7957 9 Add: 7. 89, 966, 878, 696, 788, 966, 787, 989, 89, 95. 8. 899, 889, 869, 688, 986, 788, 769, 969, 88, 86. 9. 689, 869, 788; 966, 687, 869, 978, 798, 789, 989. 10. 899, 998, 866, 689, 969, 789, 669, 898, 678, 668. 216. 6+9. Add: 6 16 26 36 46 56 66 76 86 96 _9_9^_9999999 Make a subtraction table, taking 9 from each of the results obtained above. MISCELLANEOUS PROBLEMS. 171 Id: (1) (2) (3) (4) (5) (6) 889 67 799 896 6 6 896 936 981 9695 65 963 986 194 616 5338 636 9616 696 619 996 664 7464 7366 868 766 268 6937 39 6987 769 759 399 9996 6699 7586 996 996 663 6668 8965 9936 897 399 567 977 8898 7999 97 717 868 7196 8766 9965 6 89 978 878 988 899 MISCELLANEOUS PROBLEMS. 217. 1. -\ of 21 + J of 40 are how many? 2. Bought 6 bars of soap for a quarter of a dollar; what will 12 bars cost at the same rate? 12 bars are 2 times 6 bars ; then, 12 bars cost 2 times 25 cents, or 50 cents. 3. James had 72 cents. He spent ^ of it for a new book, and ^ for pencils; what part did he spend? How many cents has he left? 4. A colt was bought for $60, and sold for IJ times its cost; what was the gain? 5. What will f of 35 pears cost, at 3 cents each? 6. What will f of a gallon of vinegar cost, at 9 cents a quart? 7. A market woman bought 4 quarts of berries for 40 cents, and sold them at 6 cents a pint; how much did she gain? 172 DIVISION. 8. How many minutes are there in tV (or j) of an hour? 9. 3i dozen are how many times 7? At the rate of 7 marbles for 9 cents, what will 3J dozen cost? 10. I have 66 cents. If I spend A of it for a pound of butter, how much will I have left? 11. If 12 cents is \ of the cost of a book, what will 6 books cost? 12. At 7i cents an ounce, what will 4 ounces of nutmegs cost? 13. A grocer buys 8 barrels of apples, 7 times as many barrels of potatoes, and i as many barrels of turnips as potatoes. How many barrels of turnips does he buy? 14. Frank had $2.80. He spent i of it for a cap, \ of it for a ball, and with the remainder bought a book; how much did the book cost? DIVISION. EXERCISE. 218. Divide by 79: 1. 27985 6. 28814 11. 59160 16. 74734 2. 34394 7. 42392 12. 66881 17. 54469 3. 36081 8. 53274 13. 56441 18. 75651 4. 50993 9. 51130 14. 66976 19. 60818 5. 44619 10. 54292 15. 66191 20. 77922 EXERCISE. 219. Divide by 84: 1. 38971 4. 47676 7. 71069 10. 79548 2. 29192 5. 62693 8. 64597 11. 476760 3. 54881 6. 57557 9. 81315 12. 813150 DIVISION. 173 Divide by 87: 13. 39691 16. 56338 19. 56423 22. 42369 14. 36193 17. 58716 20. 82465 23. 587160 15. 66603 18. 59582 21. 69145 24. 845800 230o Divide by 89: 1. 30718 2. 38784 3. 57300 4. 47574 5. 32415 EXERCISE. 6. 56507 7. 50244 8. 57481 9. 58280 10. 56574 11. 59965 12. 66431 13. 60034 14. 68072 15. 42443 16. 67303 17. 60380 18. 78040 19. 60418 20. 77212 21. 57749 22. 65729 23. 86137 24. 86910 25. 61379 Divide by 97: 26. 44987 29. 46240 32. 66723 35. 67784 27. 61565 30. 73397 33. 84186 36. 87096 28. 62838 31. 76290 34. 77296 37. 841860 221. EXERCISE. 1. Divide by 200: 36472,22365,96284,87986,76384. 2. Divide by 300: 39672,44281,67243,88752,67971. 3. Divide by 120: 36448,29676,32439,28795,78134. 4. Divide by 130: 72941,63214,72811,93214,81719. 5. Divide by 125 : 76255, 83245, 96312, 84354, 26989. 6. Divide the numbers in problem 5 by 135; by 150. 174 UNITED STATES MONEY. UNITED STATES MONEY. 222, What is the cost of a bushel of apples, if 5 bushels are sold for $6? This means finding one of the five equal parts of $6, or 600 cents. 5 ) $6.00, cost of five bushels. $1.20, cost of one bushel. If the dividend contains no cents, annex two ciphers, separated from dollars by a period. Divide as in simple numbers, and separate dollars from cents in the quotient. 1. * of $416.35=? 4. i of $219.18=? 7. | of $625.17=? 2. 1 of $312.24=? 5. i of $916.25=? 8. i of $909.20=? 3. J of $700.17=? 6. i of $813.24=? 9. i of $805.10=? 223. 1. When sugar is selling at 5 cents a pound, how many pounds can be bought for $6? This means finding the number of times 5 cents ($.05) are found in 600 cents, or $6. $.05 ) $6.00 120 120 pounds of sugar at 5 cents a pound can be bought for $6. 2. At $1.30 a pair, how many pairs of gloves can be bought for $6? $1.30) $6.00 (4 5.20 80 cents remaining. 4 pairs of gloves can be bought, with 80 cents remaining. 3. At $1.30 cents a yard, how many yards of cloth can be bought for $6.00? UNITED STATES MONEY, 175 $1.30)16.00(4 5.20 80 cents remaining. Four yards of cloth can bo bought for $6.00, with 80 cents remaining. If we make a complete division and spend all the money, we have : $1.30)6.00(4A 6.00 80 4 yards and A of one yard can be bought for $6. EXERCISE. 224. 1. If 6 boxes of oranges are sold for $21, what is the value of one box? 2. I bought 20 yards of carpet, for which I paid $9.80; what was the price of one yard? 3. At 49 cents a yard, how many yards of flannel can be bought for $10.50? 4. When wheat is selling at 87 cents a bushel, how many bushels can be bought for $1200? 5. At $1.50 a barrel, how many barrels of potatoes can be bought for $60? 6. At $1.87 a yard, what will 325 yards of carpet cost? 7. $391.50 was paid for 87 bushels of clover seed; what was the cost per bushel? 8. If the clover seed was sold at $5 per bushel, what was the gain on 87 bushels? 9. Watermelons are selling at wholesale for $8 per hun- dred; what is the value of one melon? 10. If these melons are sold at retail at 15 cents each, what is the gain on 180 melons? 176 SQUARE MEASURE. SQUARE MEASURE. One square yard. 'I'i'i'i'i'i ii'i'i'i'i'i 225, Mark on the schoolroom floor, or on the black- board, a square which shall measure a yard on each side. This is called a square yard. Divide the square yard into 9 equal squares, as shown in the above figure. Each one of these squares measures how much on each side? Each one of these squares is called a square foot. A square yard is how many square feet? 9 sq. ft. = 1 sq. yd. SQUARE MEASURE. 177 226. 1. 1 square foot is what part of 1 square yard? 2. If you should set out 9 geraniums in a garden bed a yard square, how much ground could you allow for each plant, allowing the same amount for each? (Make a draw- ing.) 3. How many square feet are there in 2 square yards? 4. How many square feet in 3 square yards? 5. Draw on the board, or on the floor of the schoolroom, 3 square feet; inclose a space 3 feet square. Which is the larger space? How many times as large? 6. Draw and compare 2 square feet with a space 2 feet square. 227. Cut out of paper a square which is 1 foot on each side. How many inches is it on each side? Cut a square which is one inch on each side. This is 1 inch. called a square inch. Fold your square foot of paper into square inches. First, into how many 1-inch strips shall you fold it? One square foot is how many square inches? 1 square inch. This figure is a square incli iu size. 1 sq. ft. = 144 sq. in. 228. 1. Inclose a space on the board, which shall be 1 foot square. Divide this square foot into square inches. How many square inches are there? 2. Find out how many small squares of patchwork, each four inches square, can be cut from a square foot of calico. 178 MISCELLANEOUS PROBLEMS. 3. How many of these squares can be cut from a square yard of calico? 4. My slate is 9 inches long and 7 inches wide ; how many square inches of surface has it? There are 7 rows of 9 square inches. 7 times 9 square inches = 63 square inches. 5. Find the area (surface) of a flower bed which is 6 feet long and 2 feet wide. 6. The length of a flower bed is 5 feet; the area 15 square feet. What is the width? (Make a drawing.) MISCELLANEOUS PROBLEMS. 229. 1. How many pounds of sugar at 6 cents a pound can be bought for 9 yards of calico at 12 cents a yard? 2. How many pairs of shoes at $3.00 a pair must be given in exchange for 30 bushels of potatoes at 50 cents a bushel ? 3. A farmer sold to a grocer 19 bushels of apples at 75 cents a bushel, and took his pay in coffee at 30 cents a pound. How many pounds did he receive? 4. A milkman sells daily 50 quarts of milk at 4 cents a quart. How many yards of carpet at $1.00 a yard can be bought for the milk sold in a month of 30 days? 5. For 12-days' work a workman received $24. At that rate, how much would he receive for 18-days^ work? 6. Mr. Jones sold 121 pounds of beef at 14 cents a pound, and took his pay in potatoes at 77 cents a bushel; how many bushels did he receive? 7. A grocer sold 150 pounds of sugar, at 8 cents a pound. How many pounds of tea must he sell, at 60 cents a pound, to equal the amount he received for the sugar? ADDITION AND SUBTRACTION BY ENDINGS. 179 8. How much can I save in a year, if I earn $140 each month for ten months, and spend $68.63 each month for 12 months? 9. What will 2 bushels of berries cost, at 12^ cents a quart? 10. A man bought 28 boxes of lemons at $5.25 per box, and sold them at $4.68 per box; how much did he lose? 11. 12 X 12 X 12 = ? 12. If I save 5 cents a day, how much shall I save in 19 years? 13. A commission house spends $30 a day for telegrams; how much is spent in 65 days? 14. If 12 men earn $72 in one week, how much will 18 men earn in the same time? 15. How many square feet of surface has the floor of the room represented by this drawing? 16. How many square yards of oil- cloth will be required to cover the floor? Addition and subtraction by endings. 230. 7 + 7. Read endings; then sums: 7 27 37 47 57 67 77 87 97 107 Make a subtraction table by using the results of the above addition and subtracting 7 from each. 180 ADDITION AND SUBTRACTION BY ENDINGS. 1: (1) (2) (3) (4) (5) (6) 777 795 779 996 998 766 377 548 637 576 8979 8735 747 877 765 455 6613 7269 496 743 659 979 6267 7637 673 676 575 192 5772 572 357 834 924 626 6386 9826 787 797 777 473 7676 7466 774 272 565 856 8476 887 637 956 776 69 997 7988 789 879 986 9 859 978 231. 7 + 8. Read endings; then sums: . 7 27 37 47 57 67 77 87 97 107 888888888 8 Make a subtraction table by using the results of the above addition and subtracting 8 from each. (1) (2) (3) (4) (5) (6) 888 888 778 669 789 6789 777 382 223 179 6846 6677 443 627 879 768 7973 9826 978 365 458 795 4769 3667 726 967 764 987 6268 9547 789 194 228 876 8666 6978 621 797 977 754 9568 8779 775 777 842 478 7599 978 647 949 498 729 975 74 689 589 798 978 89 9 MISCELLANEOUS PROBLEMS, 181 232. 7 + 9. Read endings; then sums: 7 17 27 37 47 57 67 77 87 97 107 9 9 9 9 9 Make a subtraction table by using the results of the above addition and subtracting 9 from each. Add: (1) (2) (3) (4) (5) (6) (•7) 999 977 988 789 794 9779 97 776 296 979 189 9685 9979 686 245 872 777 756 8546 6877 399 777 187 448 257 8767 6777 769 833 956 776 764 6794 864 267 999 797 129 657 9467 6468 988 632 427 794 775 1895 9736 233 769 984 669 867 8736 798 777 647 227 778 978 6759 9797 97 989 898 976 598 988 986 89 MISCELLANEOUS PROBLEMS. 233. 1. Bought: 25 lb. of sugar, @ 7c. 11 '' tea, '' 48c. 12 '' coffee, '' 23c. 22 '^ raisins. '' He. 19 '' currants. " 9c. 18 ^' crackers. '' 12c, What is the amount of my bill? 182 MISCELLANEOUS PROBLEMS. 2. Bought: 12 lb. of dried apples, 14 doz. " eggs, 32 qt. " milk, 9 bu. " potatoes, 12 lb. '' butter, 11 lb. ^^ cheese, What is the amount of my bill? 3. Bought: 9 bbl. of apples, 12 bu. ^' plums, 9 ^' " peaches, 20 '' '' cherries, 12 '' '' pears, 11 '^ ^^ quinces. What was the whole amount? @ @ 9c. " 15c. " 6c. " 65c. " 22c. '' 18c. $2.15 1.20 1.75 1.05 1.35 1.50 4. Complete the bill: Cincinnati, O., Aug. 27, 1903. Mr. John Norris, Bought of Charles E. Scott & Co., 3 student lamps, @ $3.75 ... . . $ 1 doz. knives and forks, " 4.25 1 doz. plated teaspoons, '^ 2.65 .... 1 refrigerator, 12.75 1 lawn mower, 6.10 2 rakes, $0.68 and $0.93, 1 step-ladder, 1.75 Received payment, CHARLES E. SCOTT & CO. per John M. Austin. MISCELLANEOUS PROBLEMS. 183 5. Complete the bill: Cincinnati, O., Oct. 31, 1903. Mr. James C. Martin, Bought of Lloyd, Watson & Co., 9 yards of cassimere, @ $2.85 12 yards of pressed flannel, ^^ .58 . . 11 yards of black silk, ^' 1.65 2 pairs of hose, '' .75, $1.25 1 cloak, 1 pair of blankets, 6 handkerchiefs, ^' .50 . . 9 linen towels, '^ .35 18.00 6.75 Received payment, LLOYD, WATSON & CO. w. 6. A lady bought 2 yards of ribbon at 37 cents a yard, 6 yards of muslin at 19 cents a yard, 3 yards of flannel at 35 cents a yard, 5 yards of lace at 98 cents a yard, some needles for 31 cents, and a belt for 75 cents; what did her purchases amount to? Make out the bill. 7. Bought a pair of boots for $8.50, an umbrella for $3.62, a pair of gloves for $1.25, some collars for $0.75, and a hat for $4; what did all cost? Make out the bill. 8. Bought 8 yards of velvet at $1.25, 4 yards of satin at $1.85, 6 yards of Spanish lace at $0.87, and 7 yards of sateen at $0.38. Make out the bill. 9. Mr. John R. Holt bought of Hains & Co., 6 dozen oranges at 28 cents a dozen, 4 pounds of tea at 75 cents a pound, 8 lamp chimneys at 10 cents each, 10 pounds of crackers at 9 cents a pound, 5 pounds of coffee at 35 cents 184 ADDITION AND SUBTRACTION BY ENDINGS. a pound, and 8 pounds of starch at 20 cents a pound. Make out the bill. 10. The value of my farm is J the value of my house and lot. If the farm is worth $3600, what is the value of the house and lot? 11. If the remainder is 17, the quotient 75, and the divi- dend 45767, what is the divisor? 12. A man, having $18432, deposited in bank $558, and with the remainder bought land at $54 an acre ; how many acres did he buy? 234. 8 + 8. Read endings; then sums: 8 18 28 38 48 58 68 78 88 98 Make a subtraction table by using the results of the above addition and subtracting 8 from each. Add: (1) (2) (3) (4) (6) (6) (7) 787 698 988 878 878 97 77 439 797 675 636 889 7719 9998 978 685 799 987 386 8778 8794 828 878 479 957 879 8653 7347 584 154 988 969 787 4877 5865 748 488 843 768 687 7777 7179 886 728 788 856 995 4859 2759 686 871 788 949 297 8796 9578 738 348 919 786 798 89 628 988 899 899 877 768 7 HS ADDITION AND SUBTRACTION BY ENDINGS. 185 235. 8+9. Read endings; then sums: 8 18 28 38 48 58 68 78 88 98 _?_^_?_^_^J 1 ^ ^ ^ Make a subtraction table by using the results of the above addition and subtracting 9 from each. 1. Add 899, 283, 998, 158, 895, 887, 728, 993, 947, 989. 2. Find the sum of 78, 8887, 9988, 9763, 8989, 8989, 8799, 95, 9887, 48, 988. 3. Add 767, 6512, 9899, 8269, 768, 6938, 9799, 8967, 937, 8788. 4. Find the amount of 89, 6478, 9878, 7468, 9826, 9676, 9832, 7989, 899, 7. 5. 679 4- 695 + 977 + 889 + 649 + 877 + 778 -f 898 + 879 + 879=? Add: (6) (7) (8) (9) (10) (11) (12) 899 8 9 78 767 89 679 283 839 979 8887 6512 6478 695 998 SS8 83 9988 9899 9878 977 158 889 848 9763 8269 7468 889 895 393 889 8989 768 9826 649 887 868 738 8799 6938 9676 877 728 474 996 95 9799 9832 778 993 987 897 9887 8967 7989 898 947 859 58 48 937 899 879 989 79 69 988 8788 7 879 386 SUBTRACTION. MENTAL EXERCISE. 236. Subtract: 21 32 43 54 65 76 100 81 95 64 676678 8987 91 84 42 67 96 87 45 83 100 87698967 9 243510098290003881 184376591462388887 304009623076592800069 126993611159677863599 Note. — Do this in class. Find remainders : 1. 62101011 52781096 2. 66330490 29001695 3. 32505607 23809108 4. 35210101 5. 75004132 6. 8849060 2671908 6217779 157386 7. 97800110 1901906 8. 87096247 3504768 9. 93808706 76709809 10. 62001091 41901698 11. 32100901 12901967 12. 30103055 22768996 CUBIC MEASURE. 187 CUBIC MEASURE. 237. How many faces has a cube? What is the form of each face? How many edges has a cube? How many corners has a cube? Find a cube whose edges are each one inch long. A cube whose edges are each one inch long is called a cubic inch. 238. 1. Build a post of one-inch cubes; how high a post will 3 such cubes make? 2. One cubic inch is what part of the post? 2 cubic inches ai^ what part of the post? 3. Make a post of one-inch cubes; how high a post will 4 such cubes make? 4. One cubic inch is what part of the post? 2 cubic inches are what part of the post? 5. How many cubic inches are there in 2 posts, if each contains 4 cubic inches? 6. How many one-inch cubes are there in a block 3 inches long, 3 inches wide, and 3 inches high? (Build with inch cubes.) How many 1-inch cubes in J of the block? 7. How many cubic inches in a block of wood 4 inches long, 1 inch wide, and 1 inch thick? 8. How many cubic inches in a block 4 inches long, 2 inches wide. ,y y y hm^ 188 MISCELLANEOUS PROBLEMS. and 1 inch thick? How many rows ^^^^ ^^ y of 4 cubic inches each? 9. How many cubic inches in a block 4 inches long, 2 inches wide, and 2 inches thick? 2 times 4 cubic inches = 8 cubic inches. 2 times 8 cubic inches = 16 cubic inches. 10. Build a solid 4 inches long, 4 inches wide, and 4 inches high; how many cubic inches will it contain? Measure the distance round it. 11. How many 1-inch cubic blocks can you pack in a box which is 4 inches long, 4 inches wide, and 4 inches high? 12. How many 1-inch cubes of candy can you place in a box 6 inches long, 4 inches wide, and 4 inches high, measur- ing on the inside of the box? 13. Build a solid of one-inch cubes which shall be 12 inches long, 12 inches wide, and 1 inch high; how many cubes are used? MISCELLANEOUS PROBLEMS. 239. 1. If 40 men can do a piece of work in 10 days, in what time could 8 men do the same work? 2. My farm contains 120 acres; yV of it is in meadow, W in wheat, and the rest in woodland. What part is wood- land? How many acres are woodland? 3. A stationer bought 12 dozen pens at 5 cents a dozen, and sold them at two for a cent; what did he gain? 4. I had $120. I spent \ of it for a watch, \ of it for an overcoat, and i% of it for board; how much had I left? MISCELLANEOUS PROBLEMS, 189 5. A man had a dozen boxes of candy, each box con- taining 10 pounds. If he makes of it packages containing one-half pound each, how many packages will he have? 6. A man carried 4| pecks of cherries to market, and sold them at ten cents a quart; how much did he receive for them? 7. At 2 cents a square foot, what will 8 square yards of wire cloth cost? 8. Find the cost of 10 yards of calico at 14 cents a yard, and 8 yards of ribbon at 20 cents a yard. 9. A lady paid J of a dollar for a thimble, | of a dollar for braid, and -^^ of a dollar for thread ; how much money did she spend? 10. James had $100, and spent \ of it for a watch and ipo for a coat; how much money did he have left? 11. If 11 cents is ^ of the cost of a basket, what will 5 baskets cost? 12. If 6 apples cost 5 cents, how many apples can I get for 50 cents? 50 cents is 10 times 5 cents ; 10 times 6 apples, or 60 apples, can be bought for 50 cents. 13. What will be the cost of natural gas for 8 months on one cook-stove at $1 a month, two grates at $1.25 each per month, and one base-burner at 90 cents per month? 14. If a boy earns $12 a month, how much will he earn in a year? If he spends i of his earnings for clothes and board, how much will he have left? 15. Bought 10 bushels of peaches at $1 a bushel, and sold them at 30 cents a peck; how much was gained? 16. How many quarts of berries, at 12 cents a quart, will it take to pay for 8 yards of cloth, at 16 J cents a yard? 190 MULTIPLICATION AND DIVISION. MULTIPLICATION AND DIVISION. EXERCISE. 340. Find quotients: 1. 133215 -h107 11. 444280-^232 21. 766080^315 2. 347655^-215 12. 519013^319 22. 660303^423 3. 809437^-621 13. 923257^862 23. 735289+599 4. 217892 H-493 14. 707861-^639 24. 603972 -^224 5. 1130493^533 15. 753533^671 25. 487228-^827 6. 653219-^394 16. 219763-^995 26. 701101 +901 7. 676175^215 17. 3518599^ 59 27. 684938^ 98 8. 1603008-^198 18. 4519760-^196 28. 6503188+798 9. 1529012^-189 19. 5291234 H- 189 29. 1319229+189 10. 4805019^789 20. 8008191 -f- 129 EXERCISE. 30. 6536479+129 241. Find products: 1. 12486X907 6. 70009X907 11. . 67201X1000 2. 63579X786 7. 280963X746 12, , 26487X3002 3. 39889X642 8. 247560X985 13. , 39865X1008 4. 98270X876 9. 476394X457 14. , 90834X9020 5. 62498X805 10. 480976X805 15. 19598X8009 ADDITION AND SUBTRACTION BY ENDINGS. 242. 9+9. Read endings; then sums: 9 19 29 39 49 59 69 79 89 99 Make a subtraction table by using the results of the above addition and subtracting 9 from each. ADDITION AND SUBTRACTION BY ENDINGS. 191 Add: 1. 9998, 6799, 8798, 9789, 8989, 9987, 8899, 7899, 7027, 698. 2. 7978, 5887, 7646, 9687, 9596, 6988, 8799, 7996, 7968, 967. 24tS, Subtract and read endings: 11 43 28 32 54 65 79 87 96 109 999999999 9 Subtract : 1. 88764- ■ 2969 6. 40031- 9594 11. 10002- 2999 2. 49875- ■ 2789 7. 58431- 3989 12. 68003- 9095 3. 37953- - 1896 8. 19052- 9298 13. 90087- 5069 4. 90585- 13989 9. 90745-11989 14. 19864- 10989 5. 60103- - 389 10. 70001- 9867 15. 90003- 7648 344. Add: (1) (2) (3) (4) (5) (6) (7) (8) 28 88 38 56 29 28292 13977 44189 92 32 89 69 99 94919 88945 99899 19 99 93 95 92 98189 98288 65288 81 88 38 56 29 28922 33947 76879 28 33 89 69 99 94889 89885 88968 92 89 93 95 92 98328 99689 47399 19 98 38 56 29 28994 33641 89863 81 33 89 69 99 94418 98888 66258 28 89 93 95 92 98999 88697 98898 92 91 38 56 29 28884 33635 75364 19 38 89 69 99 94937 89889 84959 81 99 93 95 92 98488 99398 66895 28 83 38 56 29 28992 33533 78386 92 39 88 69 99 94838 88489 95939 19 83 95 94 95 89985 98982 58897 192 MISCELLANEOUS PROBLEMS, Subtract: 9. 1000101- 345879 12. 90148003- 9876435 10. 80118181- 698197 13. 67100011- 400968 11. 864121133-36849762 14. 810890890-20987689 MISCELLANEOUS PROBLEMS. 245. 1. A hardware store sold wire amounting to $161.46. How many pounds were sold, if wire was worth $.18 a pound? 2. A hotel-keeper paid $52.44 for 38 table-covers. How much was paid for each one? 3. Mr. Irvin spent $17.92 for burlap to cover the walls of his library. How many yards were used, if burlap cost $.29 a yard? 4. Mr. Adam collected $104.49 from a dry-goods house for several bolts of sheeting sold to it. How many yards were bought, if the sheeting was sold at $.27 a yard? 5. In a certain city there are 23,283 school children. How many teachers must be employed to teach them, if each room averages 39 children? 6. The Agricultural Department in Washington bought enough flower seed to fill 476,160 packages. How many pecks of seed were needed, if 96 packages were filled from each peck? 7. Last September a factory received $309.66 from the sale of penholders in Chicago. They were worth $.78 a gross. How many gross were sent there? (A gross is 12 dozen.) 8. A retail store paid $192.78 for 3 bolts of cloth, each containing 54 yards. How much was paid for one yard? MISCELLANEOUS PROBLEMS 193 9. The city assessed my property $194.04 for 98 square yards of asphalt on our street. How much was paid for every square yard of asphalt? 10. A man paid $18,144 for a farm of 96 acres. What was the price of each acre? 11. A contractor paid $67.15 for one-inch nails. The nails were woHh $.85 per hundred pounds. How many hundred pounds were bought? 12. There are 172 feet of fine wire to a pound. A rail- road company used 66,392 feet during the year. How many pounds were used? 13. The Atlas Engine Works spent $1,724.80 on 5i-inch bolts, at $17.60 per hundred. How many hundred bolts were used? 14. The distance from Indianapolis to Chicago is 196 miles. Last year an engineer covered 30,576 miles of ground on his trips to and from Chicago. How often did he cover the distance? 15. At $2.98 a pair, how many pairs of shoes must a dealer sell to receive $551.30? 16. If a young man earns $36 a month, in how many months would he earn $5608? 17. If 43 bu. of corn cost $15.91, what does 1 bu. cost? 18. How many wagon loads of corn containing 41 bu. can be filled from a bin containing 1066 bu.? 19. It is 1392 mi. from here to a certain place. How long will it take to get there, if we travel at the rate of 48 mi. an hour? 20. If 41 men do a piece of work for $69.29 a day, what does one of these men earn? 194 MISCELLANEOUS PROBLEMS. 21. If a bin of wheat is worth $5086.90, how many bushels does it contain, when wheat is selUng at $.65 per bu.? 22. How many bushels of wheat worth $.64 are in a bin valued at $469.92? 23. If 69 bu. of corn cost $50.96, what does 1 bu. cost? 24. If it is 2990 mi. from here to San Francisco, how long will it take to get there, traveling at the rate of 46 mi. an hour? 25. A man had $65060; he spent $1905, and purchased land at $65 per A. with the remainder. How many acres did he buy? 26. A farmer having 91 acres of land sold ^ of it for $30940. What did he receive per acre? 27. I sell I of my farm of 168 A. for $5796. What is the price per A.? 28. A man sold 86 baskets of grapes for $30.10. For how much did he sell one basket? 29. A surface contains 1431 sq. yd. and is 27 yd. wide; how long is it? 30. How wide is a hall that contains 414 sq. ft. and is 23 ft. long? 31. A floor 13 ft. wide contains 871 sq. ft. How long is it? 32. A surface contains 1677 sq. ft. and is 43 ft. long. How wide is it? 33. I sold 84 barrels of potatoes for $134.40 and lost $12.60. What was the cost per bbl.? 34. The cost of raising potatoes was $159.12. If they are sold in 36 bbl. at a gain of $1.75 per bbl., what is the selling price per bbl.? ADDITION AND SUBTRACTION. 195 35. The cost of 46 bu. of apples is $62.10. They are sold at a gain of $.50 per bu. What is the selling price? 36. I worked 18 weeks at $15.75 a week, and saved $69.30 during the time. How much did I save per week? What were my weekly expenses? DIVISION . Find quotients: 1. 774648 - -186 9. 614307- hl99 2. 295470- -190 10. 4722354 - ^178 3. 937387 - -184 11. 2966607 - ^189 4. 7210473 - -187 12. 713513- ^179 5. 8043840- -194 13. 2154003 - ^399 6. 842877 - -179 14. 1604083- =-987 7. 145260- -108 15. 685176- ^197 8. 1874774- -172 16. 1260524- hl59 17. 17820- -294 18. 632008 - :-196 19. 657320 - -178 20. 845679 - :-168 21. 2474420 - =-307 22. 15604064- Hl96 23 . 583700- =-395 ADDITION AND SUBTRACTION. REVIEW. 247. 1. Add: 898, 983, 698, 867, 886, 259, 618, 886, 989, 762 and 479. 2. Add: 464, 399, 987, 999, 878, 466, 598, 694, 726, 899, 668 and 987. 196 ADDITION AND SUBTRACTION. 3. 695 + 944 + 899 + 978 + 627 + 489 + 398 + 772 + 786 + 948 + 499 + 437 + 748 + 454=? 4. 869 + 254 + 497 + 967 + 669 + 494 + 362 + 368 + 349 + 688 + 547 + 174=? (5) (6) (7) (8) (e) 278 989 868 795 768 985 747 497 867 848 374 298 682 456 494 689 884 376 967 939 567 419 488 678 826 496 568 217 893 677 439 677 168 745 988 797 486 954 878 144 765 819 849 459 899 988 948 756 868 786 642 157 867 787 278 897 764 498 214 527 (lO) (11) (12) (13) (14) 526 987 278 278 2789 788 676 148 879 6187 144 762 126 726 9481 889 829 475 845 8276 496 414 987 794 5769 968 948 276 868 8787 684 687 279 489 9981 845 771 476 596 6688 496 869 278 648 1578 989 494 219 489 4444 878 926 798 964 9276 727 677 694 857 8296 MISCELLANEOUS PROBLEMS. 197 MISCELLANEOUS PROBLEMS. 248. 1. If a man earns $8 a week, in how many weeks will he earn $96? 2. I lost $50 in selHng a piano for $280; what was the value of the piano? 3. 8 men together paid $100 for some wheat; if they received equal shares of the wheat, what should each man pay? 4. A lady bought a bushel of sweet potatoes for $2.25, and gave in payment a five-dollar bill; how much change should she receive? 5. Bought 20 yards of carpet for $40, 30 yards of cloth for $75, and 2 pairs of curtains at $16 a pair; what did I pay for all? 6. A real-estate agent bought some land for $2000; how much will he gain, if he divides the land into 4 lots, and sells them for $600 each? 7. If it takes one man 100 days to do a piece of work, in how many days could 2 men do the same work, working at . the same rate? 8. If I pay 6 cents for the use of one dollar, what should I pay for the use of 5 dollars, at the same rate? What must I pay for the use of 12 dollars? 9. I borrowed $100 for a year, and paid 6 cents on the dollar for its use; how much did I pay? 10. A car line is 5 miles long; if a car makes 12 round trips daily, how many miles will it run in ten days? 11. In a school of 45 pupils, | are present; how many are absent? 198 MISCELLANEOUS PROBLEMS. 12. 5 gallons of cream were sold at 10 cents a pint; how much did it bring? 13. 2 lemons can be bought for 5 cents; at that rate, what is the cost of 2\ dozen? 14. At 30 cents a peck, what will 2J bushels of apples cost? 15. My lot is 50 feet wide, and four times as long; how many yards of fence will enclose it? 16. What will 2 pounds 4 ounces of tea cost at 80 cents a pound? 17. My slate has a surface of 72 square inches. It is 12 inches long; how wide is it? 18. I have a box 6 inches long, 4 inches wide, and 2 inches deep; how many cubic-inch blocks will it hold? 19. A sheet of paper which is 8 inches wide, has a surface of 96 square inches; find the length. 20. What will a roast of 6 pounds of beef cost, at 12 J cents a pound? 21. The transom above the door is 3 feet long and 2 feet wide; how many panes of glass will it require, if each pane is 1 foot square? (Drawing.) 22. How many cubic inches are there in a block of wood which is 7 inches long, 4 inches wide, and 2 inches thick? (Drawing.) If the block were 3 inches thick, how many cubic inches would it contain? If four inches thick, how many cubic inches? REVIEW. 199 249. REVIEW. (1) (2) (3) (4) (5) 271 274 4278 2789 2718 487 169 8691 2219 476 916 425 9547 6928 8679 629 676 2648 9476 2764 784 918 7927 4629 406 848 215 9687 8742 579 297 776 4948 7887 8719 666 574 8976 6996 2778 779 876 7476 9469 2789 884 217 6216 1754 5476 498 289 7364 4178 4769 867 417 9785 2889 8769 (6) ("7) (8) (9) (lO) 7248 722 125 8627 27694 8961 871 672 4578 9678 4476 981 897 9694 5761 9941 494 778 5278 94876 6698 348 444 2419 2765 8269 287 998 6265 47698 4884 666 626 8555 12789 5446 999 269 4278 47687 4998 455 855 3987 2767 9824 844 114 1876 71001 8767 788 748 2947 6843 7287 667 856 3782 9874 200 DIVISION. Subtract: • 11. 80005 14. 103070 17. 90005 20. 307561 67421 8524G 79008 298728 12. 100051 87643 13. 70005 38729 15. 60004 38965 16. 107302 56927 18. 207305 189649 19. 81012 69299 21. 78419 69.593 22. 415007 387321 DIVISION. 250. Find quotients: 1. 246573 H- 1212 2. 745201 ^2373 3. 1793257 ^6253 4. 4175959 ^7329 5. 9180257^6351 6. 7221483 h-992 7. 1250921 H-9253 8. 27263579-^1371 863973 915761 3621487 8724165 2153897 14. 11853221 15. 5995871 16. 42507633 9. 10. 11. 12. 13. ^2652 ^2483 ^7193 ^3998 ^8253 ^8123 ^6751 ^8952 17. 631253 18. 2187923 19. 4267942 4250963 793621 22. 2170821 23. 84371285 24. 97239643 20. 21. ^3251 ^2473 ^8198 ^9876 ^9957 ^6125 ^ 695 ^9853 MISCELLANEOUS PROBLEMS. 201 MISCELLANEOUS PROBLEMS. 251. 1. How many years is it from the time of the Centennial exhibit at Philadelphia in 1876 to that of the Columbian exhibit at Chicago in 1893? 2. In a field of turnips there are 296 rows, and each row yields 18 bushels; how many loads of 30 bushels each does the field yield? 3. $557283 added to a certain number of dollars will produce $1157003; what is the number? 4. If 68 pounds of coal are consumed in carrying a train one mile, how many pounds will be consumed, at that rate, in going 1894 miles? 5. What time elapsed from the battle of Lexington, 1775, to the firing on Fort Sumter, 1861? 6. If I buy real estate for $854657, agreeing to pay for it in yearly payments of $37159 each, how many payments shall I make? 7. The improved land of the United States is estimated at 207198720 acres; how many townships of 23040 acres each could be made from this land? 8. In a pile of 4701265 bricks, how many loads are there, if each load contains 1000 bricks? 9. 145310 -1085= ? 10. A miller purchased 2149 bushels of wheat, weighing 128940 pounds; what was the weight of 1 bushel? 11. A road was constructed at a cost of $4328 per mile, and the total cost was $8331400; how many miles long is the road? 12. Find the sum of sixteen million one thousand twenty, twelve million one hundred twenty-eight, nine million 202 MISCELLANEOUS PROBLEMS. thirteen thousand two, seven milUon sixteen thousand seven, and three hundred million nine. 13. The President of the United States receives $50,000 a year; how much is that a day? 14. Fairview Park consists of 480 acres, for which $180,- 000 was paid; how much was that per acre? 15. If 46 acres of land produce 2,484 bushels of corn, how many bushels will 120 acres produce? 16. There are 30000 voters in a city; counting this as one fourth of the population, what is the population of the city? 17. Lafayette was born in 1757, and entered the Ameri- can army in 1777; how old was he at that time? 18. How many cubic inches are there in a block of ice 2 feet long, 2 feet wide, and 1 foot thick? 19. The population of Chicago in 1890 was 1,099,850; in 1900 it was 1,698,575. Find the increase for ten years. CHAPTER VII. FRACTIONS. 252, What name is given to numbers which represent parts of things, as J of a dollar or | of a field? By means of these figures, review the relations of halves, fourths and eighths. How many halves are there in a whole? How many fourths in a whole ? In a half? How many eighths in a whole ? In a half ? In a fourth? Make other drawings showing halves, thirds, sixths and twelfths, and compare these fractions. 253. % : — What is the name of the parts in this fraction? (Eighths.) How many eighths have we? (Six.) Write the number in such a way as to show that it is a number with a name. (6 eighths.) $6; 6 books; 6 quarts: — Give the name, or the denom- ination, of each of these quantities. What is the denomination of 6 eighths? , (Eighths.) 204 FRACTIONS. The number which shows the name or the denomination of a quantity is called the denominator. What is the denominator in f? (Eight.) Since 6 tells the number of things that we have, we may call it the ''numberer '' or the numerator. What is the numerator in f? Give the numerator and the denominator in the following: 8 pencils; 5 dresses; 16 pints; $7; |; f ; r\; 4 hours; LIKE AND UNLIKE NUMBERS. 254. Numbers which have the same denominator or name are like. Numbers which have different denomina- tors or names are unlike. Processes Which may be Performed with Like Numbers. 2^^. $6 and $2; 6 books and 2 books; 6 quarts and 2 quarts; 6 eighths and 2 eighths: — These quantities may be added, subtracted, divided, or compared by subtraction and division. 1. John's coat cost $6 and his hat $2. How much did both cost? 2. Mrs. Jones received $6 for her weekly expenses. She had $2 left. How much did she expend? 3. At $2 apiece, how many books can be bought with $6? 4. Harry received $6 for a Christmas present. Willie re- ceived $2. How much more did Harry get than Wllhe? How much less did Willie get than Harry? 5. I paid $6 for a hat and $2 for a pair of gloves. The LIKE AND UNLIKE NUMBERS, 205 hat cost how many times as much as the gloves? The cost of the gloves is what part of the cost of the hat? Note. — Show that the same processes may be performed with fractions, if they have the same name or denominator. Have the pupils make problems illustrating all of the fundamental processes, first with integers and then with fractions. 356. Unlike numbers cannot be added, subtracted, di- vided or compared. Can these processes be performed with 6 pencils and 2 books? 257. Can you add 6 bushels and 2 pecks? (Yes.) What must be done before these quantities can be added? (They must be changed to the same measure.) 6 bushels = 24 pecks. 24 pecks + 2 pecks = 26 pecks. 24 pecks— 2 pecks = 22 pecks. 24 pecks -^2 pecks=12. 24 pecks are 22 pecks greater than 2 pecks. 2 pecks are 22 pecks less than 24 pecks. 24 pecks are 12 times 2 pecks. 2 pecks are V2 of 24 pecks. Can these processes be performed with ^ and J? Change the fractions to the same measure, say fourths. Note. — Teachers should give original problems showing that fractions not having the same name may be changed to the same name, and then added, subtracted, divided, or compared. 258. Add, subtract, divide, and compare the follow- ing fractions by means of the figures on p. 206, pointing to all fractional parts named : 206 FRACTIONS, '/3 % 1 1 H 1 1 I j 1 t and \ f and \ 1 and iV \ and i^ \ and \ and J and \\ f and iV f and tV I and iV 1 and ^ 1 and \ \ and \ \ and f 1 and \ \ and i \ and f i and f \ and I i and \ f and H \ and iV i and ^^ \ and 1^ i and \\ I and 1^ I and tV I and tV f and H i and iV 1 1 1 1/, Vj 1/2 h ^/c •ls\ 5i 1 1 1 1 • 1 1 j ^/J \ and f \ and f i and \ \ and i \ and f f and \ 1 and J 1 and \ \ and i \ and i i^ and -^ \ and tV i and \\ \ and iV I and iV I and tV I and li 4 and \ \ and f f and J REDUCTION, 207 REDUCTION. To Change Integers and Mixed Numbers to Fractions. 259. 1. How many fourths are there in an orange? How many fourths in 6 oranges? 2. How many fifths in a melon? In 3 melons? In one melon there are five fifths ; in 3 melons there are three times ^ve fifths, or 15 fifths. 3. How many sevenths are there in 4? In 6? In 8? 4. How many eighths of an apple are there in 2f apples? 2| apples = 2 apples and | of an apple. In one apple there are eight eighths. In two apples there are two times eight eighths, which are sixteen eighths. Sixteen eighths and three eighths are nineteen eighths. There are nineteen eighths in 2| apples. Change : 5. 3J oranges to fifths of an orange. 6. 2f apples to fourths of an apple. 7. If cakes to eighths of a cake. 8. 5g feet to fifths of a foot. 9. 3f yards to sevenths of a yard. 10. 3 1 gallons to ninths of a gallon. 11. 1^ gallons to fourths of a gallon. 12. f quarts to eighths of a quart. 208 FRACTIONS. 260. A whole number, as distinguished from a fraction, is called an Integer. A Mixed Number is an integer and a fraction united; as, An Improper Fraction is a fraction whose numerator is as large as, or larger than, its denominator. It is equal to one, or more than one. f, |, V, and ^ are improper fractions. 261. Change to improper fractions: 1. 2i 3i 51, 4f, 4i 2. 6f, 5f, 7i, 31, 5i. 3. 3f, 6t, 4f, 31, 5i 4. Change to eighths: 3, 2i, 3i, 4 J, 2f. 5 Change to twelfths: 3i, 2, 21, 4J, 5J, 3A, 3f, 5, 25, 5i 6. Change to halves: 7, 4|, 2^ 4|, 3 A- To Change Improper Fractions to Integers or Mixed Numbers. 262. 1. How many pears are there in 12 half-pears? In 13 half -pears? Tliere are two half-pears in one pear. 2 halves are in 13 halves 6i times. Therefore, there are 6i pears in 13 half-pears. 2. How many gallons are there in 6 half-gallons? 3. How many bushels in 17 half-bushels? 4. How many melons in 18 thirds of a melon? 5. How many yards in 17 thirds of a yard? 6. How many ones in 20 fifths? In 28 fourths? REDUCTION. 209 Change to integers or mixed numbers: 7. i V, ¥, ¥, V. 8. V-, ¥, ¥, ¥, V. 9. ¥, ¥, V, «, ¥. 10. iJ. f i ¥, ¥, ¥. To Change Fractions to Higher Terms. 263. To reduce to higher terms is to change a fraction having large-sized parts to a fraction of equal value having small-sized parts. 364. 1. How many fourths are there in J of a pie? There are 4 fourths in one pie. In i of a pie tliere are i of 4 fourths, which is 2 fourths, i of a pie = f of a pie. Reduce : 2. J to eighths, J to sixths, | to twelfths. 3. ^ to sixths, f to sixths, ^ to ninths. 4. I to twelfths, t to fifteenths, f to eighteenths. 5. ^ and J to sixths. 6. I and f to twelfths. 265. In changing to higher terms, does the size of the parts increase or decrease? Is the number of parts increased or decreased? The form of a fraction may he changed without changing its value, hy multiplying both terms hy the same number. Thus : 1_1><4_4 2^2><3_6 2 2X4~8 3~3X3~9 210 FRACTIONS. 266. 1. Reduce J, ^y and ^ to equivalent fractions hav- ing the same sized parts. (What is the meaning of the word ''equivalent'^?) By what fractional part can these fractions all be measured? Determine the common denominator from the drawing. i-i. 1 1 1 1 1 1 i i i = f. 1 1 1 1 1 1 1 t^ = tV Change to equivalent fractions having a common measure or denominator : 2. h h I 3. i, I h 4. I h f. 5. h h h 6. I I I 7. f, tV, iV- 8. A, h A. 9. i i\, i. 10. i, \h ^. 11. i I |. To Change Fractions to Lower Terms. 267. To reduce to lower terms is to change a fraction having small-sized parts to a fraction of equal value hav- ing large-sized parts. 268. 1. Two quarters of a dollar are the same in value as what single coin? 2. Two quarters of an apple, when placed together, are the same as what part of an apple? REDUCTION, 3. Answer the following from the drawings: 211 'k v^ \ ^/^ Vs! 'A\ ! i ^ ? i J 9 JL _ ? f = ? t = ? A = ? V. 1 = ? A = ? A = ? 1 = ? A = ? if = ? !o In the answers to the above questions, has the size of the parts been increased or decreased? Has the number of parts been increased or decreased? 269. Th£ form of a fraction may he changed without changing its value j by dividing both terms by the same number. Thus: 2232^1 ^_^±^_^ 4"4-2 2 T5~15T3~5 270. 1. Reduce 1% and tV to fourths. 2. Reduce 2^1, 21, and J| to sevenths. 3. Reduce if, H, if, and H to eighths. 4. Reduce ^\, A., if, and A to ninths. 271. When a fraction is reduced to an equivalent frac- tion with smaller terms, it is reduced to lower terms. 212 FRACTIONS. A fraction is in its lowest terms when no number, except 1, will exactly divide both its numerator and its denominator. Reduce ^ to lowest terms. 12h-2~6~6h-2~3 ^^ 12^4 3 f is the fraction in its lowest terms. 272. Reduce to lowest terms: 1. tV, tV, i, A, A 2. it, A, A, A, H 3. H, ii H, H, A 4. ii fi M, M, H 5. If, J^, f§, if, U Note. — Pupils should now receive much practice in addition, subtraction, and division of fractions, basing the work on the principle previously developed. Fractions must be changed to the same measure or denomina- tion before they can be added, subtracted, or divided. Occa- sionally call for a drawing to illustrate a given problem. ADDITION OF FRACTIONS. 373. 1. Add I and |. 2X4^ 8_ ^2 3X4 3X3 4X3' All answers must be changed to their simplest form. ADDITION OF FRACTIONS, 213 Find the sum of: 2.1, i. 5. f, h 3. h |. 6. i, i. 4. h h h 7. h f , 274. To add mixed numbers, add the integers and the fractions separately and combine the results. Add 2f , ^, 3i. 2| = 2A 4i=4rV 3i = 3A 9^| = 10i 275. Find the sum of: 1. ^, 3f, H. 4. 4i 8f, lOjJj. 2. 9f, 3i 5|. 5. 15A, 31, m. 3. 8, 5f, 7i. 6. 7f, 51, 12A. MENTAL EXERCISE. 276. 1. Mary bought f of a yard of ribbon on Tuesday and i of a yard on Wednesday. How much ribbon did §he buy all together? 2. Mary gave J of her money to her sister, and ^ of it to her brother. What part of her money did she give away? 3. One lot contains | of an acre and another lot | of an acre. How many acres are there in both lots? 4. The difference between two fractions is f . One frac- tion is I; what is the other? 5. f+^ + | = ? 214 FRACTIONS. 6. I needed 2^ feet of wire to hang one Japanese lantern, 3J feet for another, and 2iV feet for another. How many feet of wire were needed? 7. Si + 5i+^=? 8. 3i + 2| + 3J=? 9. 2KH + t = ? EXERCISE. 277. 1. By means of drawings, show the sum of J and f . 2. A farm is divided into 3 fields. The first contains 8i acres, the second 12 J acres, and the third 6§ acres. How many acres does the farm contain? 3. f + f + r\=? 4. Find the sum of 2J, 4?, 5^ and 2j\. 5. If a tailor uses SJ yards for a coat, 2^ yards for trous- ers, and I of a yard for a vest, how many yards are used in all? 6. Mrs. Thomas made 4 dresses. For one she used 2f yards of embroidery; for another, 1^ yards; for another, 4| yards; for another, 2§ yards. How many yards in all did she use? 7. Wilham walked 3f miles on Monday, 4^ miles on Tuesday, and 2f miles on Wednesday. How far did he walk in the three days? 8. 29K58f + 77A=? 9. Last Saturday Mr. Ray, a coal-dealer, sold 5f tons, 8 A tons and 6| tons of coal. How many tons did he sell? 10. 7i + 5i + 8r% + 6t=? SUBTRACTION OF FRACTIONS. 215 SUBTRACTION OF FRACTIONS. EXERCISE. 278, 1. Helen had | of a yard of ribbon; she gave J of a yard to Eleanor; what part of a yard had she left? 5x1—5 6X1 — & 1 X3 a She had J of a yard left. Subtract: 2. §-i = ? 6. f-! = ? 10. H-f=? 3. t-i = ? 7. TV-i = ? 11. f-l=? 4. f-f = ? 8. |-4=? 12. «-!=? 5. W-i = ? 9. if-4=? 13. I-|=? EXERCISE. 279. 1. Mr. Smith earns $15 a week and spends $7| week. How much does he save in one week? $15=$14t m He has left $7i Subtract : 5. 89-49tV=? i|=? 6. 57-18tV=? 4. 8-3t = ? 7. 52-271 =? 216 FRACTIONS. EXERCISE. 280. 1. Mary had 4J apples and gave If apples to her brother. How many did she have left? 2i 3 fourths of an apple from 2 fourtlis of an apple cannot be taken. Take 1 apple from the 4 apples, which leaves 3 apples. This one apple equals 4 fourths of an apple. 4 fourths of an apple and 2 fourths of an apple are equal to 6 fourths of an apple. 3 fourths from 6 fourths leave 3 fourths. One apple from 3 apples leaves 2 apples. Mary has left 2i apples. Subtract : 2. 4§ ~2i=? 6. 4J -2i=? 10. 5i - 2§ =? 3. 81 -3J = ? 7. 6i -2§ = ? 11. 15i -10tV=? 4. 4x^-1* = ? 8. 5f -1J=? 12. lOiV- 3J =? 5. 10| -2t=? 9. 6i^i7-2|=? 13. 24i -18i =? 14. lOi - 8J=? 15. 144 - 9| = ? 16. 25i^-16f=? MENTAL EXERCISE. 281. 1. I bought groceries amounting to | of a dollar. How much change did I receive if I gave the grocer J of a dollar? 2. I pay tV of a dollar for a book and sell it for i of a dol- lar. How much do I lose? SUBTRACTION OF FRACTIONS, 217 3. f-| = ? 4. f-*=? 5. |-f=-? 6. Mary had $3^ and spent $1|. How much had she left? 7. The sum of two numbers is 5 J. One number is 2|; what is the other? 8.1-1 = ?' 9. Mrs. Bush bought 12 pounds of sugar. She used 6J pounds in making jelly. How many pounds has she left? 10. My mother gave me a ten-dollar bill for my birthday present. I spent $6f for a hat. How much had I left? EXERCISE. 282. 1 . Mrs. Jones owned 3^ lots near our school-house. She sold li lots for $2,200. How many lots has she left? 2. A piece of cloth contains 18^^ yards. How many yards will be left after 13f yards are used? 3. Find the difference between 12^ and 23^. 4. I spent $21| for a table and a chair. The chair cost me $12J. Find the cost of the table. 5. A man bought a horse for $50, and sold it for $45f . Find the amount of loss. 6. I owe $6f . If I pay | of a dollar, how much shall I then owe? 7. If from $8^ there be taken $6|, how much will remain ? 8. A table which cost Mr. Howe $6.75 was sold for $7^. What was the gain? 9. What fraction added to | will make H? §18 FRACTIONS, EXERCISE. 383. 1. A tailor bought 8} yards of cloth. He sold Si yards for a coat, | of a yard for a vest and 2f yards for trousers. How many yards had he left? 2. A lady bought a pair of gloves for $lf , a hat for $7J, and some lace for $1^. She gave the clerk a twenty-dollar gold piece. How much change should she receive? 3. A coal-dealer bought 25} tons of coal. He sold 4J tons, 5J tons, 6f tons and 3J tons. How many tons had he left? 4. A man bought wood for $6 J, hay for $9| and feed for $7i; how much did all cost? 5. James had a distance of 85 miles to ride. He rode 31f miles on the first day and 24J miles on the second day. How many miles has he still to travel? 6. From $23f take the difference between $8i and $10f . 7. A grocer cleared $10 last Friday; $lf was cleared on vegetables; $2 J on fruits; $3f on flour; the remainder was made on small articles. How much did he clear on small articles? 8. From the sum of 10| and 8} take their difference. 9. Take the difference between 31 and 8 J from 10 J. 10. From the sum of 6^ and 3| take their difference. 11. A man divided his property among his five chiklren, giving J of it to the first, J to the second, i to the third, and iV to the fourth; what part did the fifth child receive? 12. I of my library is history, tV poetry, } science, and the remainder fiction; what part is fiction? 13. A man did J of his work the first day, J of it the sec- ond day, and ^ of it the third day. What part was left to do on the fourth day? DIVISION OF FRACTIONS, 219 DIVISION OF FRACTIONS. Division by a Fraction. EXERCISE. 284. 1. To how many children can I give J of an apple, if I have ^ of an apple? I can give to as many children as i of an apple is contained times in i of an apple. i of an apple = f of an apple. I of an apple -f- i of an apple = 2. I can give i of an apple to each of 2 children. In the following problems, reduce the fractions to the same denominator and then divide the numerators. 2. i 3. i 4. i 5. f 6. I f = ? * = ? 7. l-f 9. i 10. I 11. * f = ? -4 = ? i=? 12. f 13. |-f = ? 14. I 15. i 16. f 4 " EXERCISE. 285. 1. A man has 2 acres which he wishes to divide into lots of i of an acre each. How many lots will he have? He will have as many lots as i of an acre is contained times in 2 acres. 2 acres = t acres. f acres -j- i of an acre = 4. He will have 4 lots. 10 FRACTIONS. 2. 3. 4. $3H-$i=? 5. 2oz.-^|oz. = ? 2 bu. ^i bu. = ? 6. 10 lb. ^|lb. = ? 3 qt. ^f qt. = ? 7.5 apples -^ | of an apple = ? 8. 8pt.-^Jpt. = ? 9. 6 acres ^§ acres=? 10. 9iii.^iin. = ? EXERCISE. 286. 1. A woman bought 2§ pounds of candy to give to her nephews. She placed it in bags each holding \ of a pound. Into how many bags did she put the candy? ^ 21 pounds=f pounds=\^ pounds. ■y^ pounds -i-i pounds =16. She put the candy into 16 bags. 2. $3iH-$i=? 6. $3fH-$A = ? 10. $2i^$J = ? 3. 2f ^ i=? 7. 2^4- 1 =? 11. 2f^ | = ? 4. 3f ^ f=? 8. 5i^ \ =? 12. 4i^ f = ? 5. 2i-^ i = ? 9. 5i-^ % =? 13. 2f- § = ? Division by a Whole Number. EXERCISE. 287. 1. At $2 a yard, how much velvet can be bought with Hi? You can get I of a yard. i = $t. DIVISION OF FRACTIONS. 221 2. 4i bu. --2 bu. = ? 6. 2i in. -3 in. = ? 3. 5f pt.-3pt. = ? 7. lift.H-5ft. = ? 4. $2i -$3- ? 8. 7-1 bbl. -^3 lb. = ? 5. $2f H-$2 = ? 9. 51 lb. -r-7 lb. = ? EXERCISE. 388. 1. At $3 a barrel, how many barrels of apples can be bought for J of a dollar? tpo — 4. $14. ^4 — 1^ — 4. At $3 a barrel, i of a barrel can be bought for $|. 2. At $2 a pound, what part of a pound of writing paper can be bought with $^? $2 =: If. Ii-lf=i. You can buy ^ of a pound. 3. $i--S2 = ? 6. 1 4-5=? 9. 4-2 = ? 4. i- 4=? 7. i-3=? 10. 1^-5=? 5. |-H- 4=? 8. t^2=? 11. t'o-3 = ? Division by a Mixed Number. 289. 1. At $3| a bushel, what part of a bushel of cran- berries can be bought for $2? «p,C' — T 5 • ^05 — qp 6 • $J,Q - $1,^ = ig = I. You can buy f of a bushel. 222 FRACTIONS, 2. I can save $1^ from my weekly earnings. How long will it take me to save S3|? $31 = $V. $1^ = $f . $1/ - $f = 3. It will take me 3 weeks. 3. At $1| a gallon, what part of a gallon of cream can be bought with $§? $!=$i^. Ill = $i = m. m-^%u = ii = ^^ You can get A of a gallon. 290. EXERCISE. 1. 3 yd. -2iyd.= ? 6. $2§^i J3i=? 2. 5ft.H r6f ft. = ? 7. 3J lb. -i-2ilb.=? 3. $3^$5i = ? 8. ?pt.H -5Jpt. = ? 4. $li- $4i = ? 9. «-2i. = ? 5. 9ibu .^2ibu.= = ? 10. f-lf = ? 11. 6 oz. H-2^ oz.= = ? 12. 4f- ^li=? 13. 1- 3i=? 14. 3i- v-6i=? 15. 3i ^2i = ? MENTAL EXERCISE. 391. 1. f-f=? 2. |^f=? 3. How many pounds of butter can be bought with | of a dollar, if one pound costs | of a dollar? 4. How many badges, each -^q of a yard long, can be cut from I of a yard of ribbon? DIVISION OF FRACTIONS. ' 223 5. How many dozen oranges, at f of a dollar a dozen, can I buy with $2? 6. f^| = ? 7. John earns J of a dollar a day. How many days will it take him to earn $8? 8. How many pounds of coffee, at | of a dollar pound, can be obtained with $6? 9. One basket of peaches holds f of a bushel. How many baskets will hold 4 bushels? 10. How many tables can be covered with 5 yards of cloth, allowing | of a yard for each table? 11. How many times may | be subtracted from 3? 12. Among how many families can you divide 4t tons of coal, if each family receives ir of a ton? 13. At J of a dollar a pound, how many pounds of butter can be bought with $2^? 14. 5i^t==? 15. How many books can be covered with 2f yards of canvas, if you allow 4 of a yard for each book? 16. H^| = ? 17. A man divided $2f among his children, giving each child tV of a dollar. How many children had he? 18. I have $1^ with which to buy ribbon, at | of a dol- lar a yard. How many yards can I buy? 19. l|-^i^? 20. What part of a barrel holding 4 bushels can be filled with 1| bushels? 21. 2^-2 = ? 22. It takes 2 bushels of wheat to sow a field. What part of the field can be sown with f of a bushel? 23. f-6=? 224 • FRACTIONS. 24. li^4i = ? 25. 2f-3i=? 26. 2f-H = ? EXERCISE. 292. 1. To how many families does a grocer sell 6 bush- els of apples, if each family buys } of a bushel? 2. When sugar is 5J cents a pound, how many pounds can be bought with $66? 3. 13f-^5i = ? 4. John earns $1^ a day. How long will it take him to earn $1^? 5. How many vests, each containing 1 J yards, can be cut from lOJ yards of cloth? 6. H^5i=? 7. Anthracite is worth $7^ a ton; $31^ will buy how many tons? 8. 46-^11^=? 9. A merchant spent $13^ for apples at f of a dollar a bushel. How many bushels did he get? 10. When flour is $7f a barrel, how many barrels can be bought with $62? 11. A farmer received $175 for rye at |- of a dollar a bushel. How many bushels did he sell? 12. How many steps must be taken in walking a mile, or 5280 feet, if each step is 2^ feet long? 13. A piece of cloth is 45^ yards long. How many pieces, each containing If yards, can be cut from it? 14. At $.08^ a quart, how many gallons of strawberries can be bought with $2^? DIVISION OF FRACTIONS. 225 EXERCISE. 293. 1. A boy earns $5f one month and $4| the next month. How many chairs, at $1| apiece, could he buy for his mother with the money? 2. Add together 3| and 2^^^ and divide the sum by |. 3. Mary had 7j yards of gingham and bought 8^ yards more. How many aprons can she cut from both pieces, if each apron requires 2 J yards? 4. A man bought 12 J acres of land at one time, and 13| acres adjoining, at another time. Into how many lots of 1| acres each could he divide his land? 5. Jane spent the difference between $9 and $5^/ for a hat. How many such hats could be bought with $14? 6. Mr. Harold raised 50 bushels of wheat. He sold 7J bushels and placed the remainder into bags, each holding 2 J bushels. How many bags were necessary? 7. A grocer has 36^ pounds of flour in 1 barrel and 38 A pounds in another. He decides to pack it in sacks each holding 12^ pounds. How many sacks will be necessary? 8. A farmer had $86. He paid a bill amounting to $24^, and with the remainder bought sugar at iV of a dollar a pound. How many pounds did he receive? 9. A skilled carpenter earned $49t% at one time and $25^ at another. How many days did he work, if he received $3 per day? 10. A furniture dealer had $170| and spent $40tV for bed- steads. How many chairs, at $1^ a piece, can he buy with the remainder? 11. I wished to buy cloth for a dress, at $3^ a yard. My mother gave me $7f and my brother, $8J. How many yards could I buy with my money? 226 FRACTIONS. MULTIPLICATION OF FRACTIONS. 294. 1. How many inch squares are there in this figure? 2. In i of it? 4. In f of iti 3. In J of it? 5. In i of it? 7. 8. In t of it? 9. i of i is what part of the whole? 10. i of -J- is what part of the whole? 6. In I of it? In 1 of it? MULTIPLICATION OF FRACTIONS. 227 11. iofi=? 14. iofi=? 17. |oft=? 12. iofi=? 15. f ofi---? 18. |ofi=? 13. iofi=? 16. ioft=? 19. iof i=? EXERCISE. 295. 1. Mary divided | of a yard of ribbon equally among 3 girls; what part of a yard did each girl receive? Each girl received ^ of I of a yard, i of f of yard=i of a yard. 2. If a man mows f of an acre in a day, how much does he mow in ^ of a day? 3. What is the cost of | of a yard of flannel, at ^ of a dol- lar a yard? 4. If silk is worth | of a dollar a yard, what is | of a yard worth? 5. If a knife is worth to of a dollar, and a slate | as much, what is the slate worth? 6. At f of a dollar a pound, what is i^ of a pound of tea worth? 7. fofA=? 8. What is the cost of f of a gallon of milk, at S.20 a gal- lon? 9. A man^s field contained 90 acres of land. He planted I of it in corn. How many acres of corn had he? 10. The cost of my dress is f of $7^. Find the cost of the dress. 11. A tailor used 5t yards of cloth for a suit; f of this was used for the coat. How many yards of cloth were used for the coat? 12. Four spellers are worth $.60. What part of this are 3 spellers worth? How much money? 228 FRACTIONS. 13. Find the cost of 12 pencils, if 10 pencils are worth $0.25. 14. 2J barrels of apples are worth $10. Find the cost of 1 barrel. 2i barrels = | barrels. We wish to find the cost of J of a barrel. 5 ) $10 = cost of \ barrels. $2 — cost of i of a bari^el. 2 |4 = cost of 1 barrel. Or better: f of a barrel = | of f barrels. 4 of $¥-= $4. One barrel will cost $4. Dividing 5 into 10, as shown above, is called cancellation. It should be used whenever possible. 15. What is the cost of 1 yard of cloth, if 4^ yards are worth $9? 16. Mr. Smith paid $16 for 2f tons of anthracite. How much was it per ton? 17. What must a dealer pay for 1 bushel of cranberries, if 3i bushels are worth $6.10? 18. I live 10 blocks from the church. This is 2J times my distance to school. How far am I from school? 19. Z\ baskets of peaches were sold for $4^. What was the selling price per basket? 20. I paid f of a dollar for 2^ pounds of candy. How much was it per pound? 21. A man bequeathed $7500 to his son. This was 3f times as much as he gave to his daughter. Find the daugh- ter's share. 22. 25|^4i=? 23. 57i^l3J = ? MULTIPLICATION OF FRACTIONS. 229 296. From the drawing answer the following, express- ing the answers in the simplest form : 1. 2 2. 2 3. 3 4. 4 5. 2 6. 2 7. 2 8. 2 9. 2 10. 3 11. 4 12. 6 13. 8 times times times times times times times times times times times times times i=? f=? ~H? I = ? *=? tV=? tV=? 14. T^X10=? 15. AX2=? 16. AX2 = ? i%X2=? ? 17. 18. 19. AX3 = ? AX 4 20. AX3 = ? 21. /jX3 = ? 22. AX2 = ? Multiplication of a Fraction by an Integer or a Mixed Number. EXERCISE. SO?. 1. If a yard of cloth costs § of a dollar, what will 3 yards cost? Three yards cost $3. 230 MULTIPLICATION OF FRACTIONS, 2. At 1^ of a dollar a yard, how much must be paid for 1^ yards of ribbon? 1\ yards = t yards. ■f of i of a dollar = i of a dollar. 1\ yards cost i of a dollar. 3. fX5=? 7. IX 6=? 11. 5iX f=? 4. JX6=? 8. HX f=? 12. 5JX | = ? 5. iX4=? 9. 5iX i = ? 13. 4|X f = ? 6. |X9=? 10. HX16=? 14. §X15=? Make, in class, concrete problems from the above examples. Multiplication of Integers and Mixed Numbers. EXERCISE. 298. 1. What will 3 bushels of oranges cost, at $5 J a bushel? $5i=$^^ $^^x3 = $16. 3 bushels of oranges will cost $16. Multiply: 2. b\ X 8. 6. 16 XltV 10. 9 X3i 3. 2tV X 5. 7. 1tVX7. 11. 3^X4. 4. 4i X 6. 8. 10 X5i 12. 5 X7i 5. 12 X Z\. 9. 2^ X16. 13. 4gX8. MULTIPLICATION OF FRACTIONS. 231 Multiplication of a Mixed Number by a Mixed Number. EXERCISE. 299. 1. What must be paid for IJ pecks of English wal- nuts, at $11- a peck? li pecks = f pecks. $li = If. 2 li pecks of English walnuts cost li-i. Multiply: 2. llbyli 6. 5i by3f. 10. 2|by5i 3. 3fbyli 7. 3f by2i 11. 7iby3f. 4. Hbyli 8. 2\ by5i 12. Ubyli 5. If by H. 9. mby4i 13. 6iby4f. MENTAL EXERCISE. 300. 1. A boy walked 2 1 miles in 1 hour. How far can he walk, at that rate, in 10 hours? 2. Find the value of 12 bushels of corn, at f of a dollar a bushel. 3. At $.06i a yard, what will 7^ yards of braid cost? 4. A boy is able to read 5j pages of his book in an hour. How much can he read in 14 hours? 5. At f of a dollar a yard, what will 8 yards of cloth cost? 6. I bought 5J yards of cloth at | of a dollar a yard* What did I pay for it? 7. Find the cost of 2f pounds of sugar, at 5^ cents a pound. 232 FRACTIONS. 8. What must I pay for 12 barrels of flour, at $4^ a bar- rel? 9. 3|X5i = ? 10. My distance from the nearest grocery is 3f squares. John must walk li times as far. How far does John walk to the grocery? EXERCISE. 301. 1. How many acres are there in 56 lots, each con- taining f of an acre? 2. 2|X5i = ? 3. 4JX3f=? 4. 22iX3f=? 5. Mr. Smith had 96 bushels of potatoes, which he sold for I of a dollar a bushel. How much did he receive for them? 6. 5 men can do a piece of work in 10| days. How long will it take one man? 7. The distance from Indianapolis to Chicago is 192 miles. The distance to St. Louis is lH times as great. How far is it from Indianapolis to St. Louis? 8. A man travels 12f miles a day. How far will he travel in 5 J days? MISCELLANEOUS PROBLEMS. EXERCISE. 302. 1. Mr. Jones took 4^ bushels of peaches to market on Tuesday, 3| bushels on Thursday, and 4| bushels on Saturday. How much did he receive, if he sold them at $li a bushel? MISCELLANEOUS PROBLEMS. 233 2. John gathered 1 J dozen, f dozen and 2^ dozen of eggs at different times. He sold them at $.18 a dozen. How much did he get for them? 3. A grocer bought a barrel of sugar holding 190 pounds. He kept 40 pounds for his own use. He sold the rest at $.05^ a pound. How much did he receive for it? 4. Multiply the difference between 36| and 27| by 7^. 5. Multiply 2| by 1^, and divide the product by l\. 6. Divide the product of 2f and 8| by 3|. 7. Multiply lOf by 9J and divide the product by 5J. 8. What must be paid for 10 pairs of gloves, if one pair is worth f of S3i? 9. I bought 3f yards of velvet at %2\ a yard, and 3| yards of silk at $/o a yard. What was my bill? 10. I have $3| in my purse. I spend \ of it for my din- ner. How much must I pay for 11 such dinners? 11. I used 4^ yards of broadcloth for my skirt and 2f yards for my jacket. How much more did the cloth of my skirt cost than that of the jacket, if the cloth is worth %2\ a yard? MENTAL EXERCISE. 303. 1. What is meant by the following: f of an apple; f of a cake; | of a book? 2. Change 12f oranges to fifths of an orange; 2| apples to eighths of an apple. 3. How many weeks are there in V weeks? 4. How many pecks in %^ pecks? 5. How many cents in | of a dollar? 6. How many inches in I of a yard? 234 FRACTIONS. 7. Compare | of a yard with ^ of a yard, by means of inches. 8. Compare | of a bushel with | of a bushel, by means of quarts. 9. Compare J of a dollar with f of a dollar; f of a dollar with f of a dollar. 10. Show that § and 3^2 are of the same value. 11. I of 63 is how much greater than f of 42? 12. Three hours is what part of a day? Five hours? 13. Forty minutes is what part of an hour? 14. A train runs 30 miles an hour. How far will it run in 40 minutes? 15. In which of the fractions, j and |, are the parts smaller? 16. Change |f to its lowest terms. Has the size of the parts increased or decreased? Has the number of parts in- creased or decreased? 17. i of 2 apples is what part of one apple? 18. Which is greater, J of 4 or J of 5? How much greater? 19. My flower garden is f of a rod long and | of a rod wide; how many rods around it? 20. I drew a triangle which was 5f inches on one side, 41i inches on another, and 5f inches on another. How many inches around it? 21. What is the difference between 5Vo and 2|? 22. The sum of two numbers is 16f . One number is 8f ; what is the other? 23. f-i=? 24. The difference between two fractions is jj. One of the fractions is ^; what is the other? MISCELLANEOUS PROBLEMS. 235 25. To how many people can you give 5| barrels of flour, if you give J of a barrel to each person? 26. How many badges, each tV of a yard in length, can I cut from 1| yards of ribbon? 27. I wish to put 4f pounds of candy into eighth-pound packages. How many packages can I make? 28. If 10 oranges cost | of a dollar, what will 1 orange cost? 29. At $8i a yard, what will | of a yard of velvet cost? 30. There are 16^ feet in one rod. How many feet are there in 5 rods? 31. I of a quire of paper made one note-book; how many quires will be used in making 40 such books? 32. What will 10^ pounds of sugar cost, at 6J cents a pound? 33. 2iX5J=? 3iX6i=-? 34. Bananas sell at the rate of f of a dozen for ^ of a dollar. At that rate, what will 60 bananas cost? 35. 3 oranges are sold for a dime ; what must I pay for 2\ dozen? 36. Horace earns $1^ a day. In how many days, at that rate, can he earn $50? 37. Divide 25 by | of 3|. 38. $8i-$|=? 39. A man owning f of a mill sells f of his share; what part of the mill does he still own? 40. If a jar holds | of a gallon of fruit, how many jars will be required to hold 6 gallons? 41. If 3 pounds of coffee are sold for $1, what part of 3 pounds should be sold for 25 cents? What part of one pound? 236 FRACTIONS. 42. A cake of ice f of a foot thick floats J of a foot above the water; what part of a foot is below the surface? How many inches are below the surface? 43. A boy bought | of a bushel of chestnuts for $2, and sold them for 10 cents a quart; how much did he gain? 44. Take ^ of a dollar from | of a dollar, and with the .remainder buy oranges at | of a dollar a dozen. How many dozen can you buy? 45. A man is 42 years old, and ^ of his age is J of the age of his son. How is old his son? 46. 3 times J of f is how many times |? 47. Three-fifths of a ton of coal cost $6. What is the cost per ton? 48. 42 is 4 of what number? 49. I of a cord of wood was sold for $4.50; how much is it a cord? 50. 15 is 4 of a number; what is ?- of the same number? 51. If f of a yard of ribbon cost 25 cents, what will i of a yard cost? 52. One-eighth of a ton of coal costs | of a dollar. What will f of a ton cost? 53. Three-fourths of a yard of ribbon cost 12 cents. Find the cost of 3 yards. . 54. Five-sixths of a yard of cloth cost $2J. Find the cost of 2 yards. 55. Five-eighths of the cost of to-day's meals are ^ of $2^. What is the cost of to-day's meals? 56. $24 are | of my money. Find J of it, without find- ing all of it. 57. $1| is the cost of | of a pound of tea. Find the cost of i\ of a pound, without finding the cost of 1 pound. MISCELLANEOUS PROBLEMS. 237 58. John had 48 marbles in his bag, which were | of all his marbles. He gave away \ of his marbles. How many did he give away? 59. I pay $.09 for | of a yard of ribbon. How much must I pay for | of a yard? 60. Mr. Hoover receives $.15 for f of a pound of coffee. How much would he receive for 3^ pounds? 61. I of a yard of cloth cost | of a dollar. What will x^^ of a yard cost? 62. I pay $5^ for | of a barrel of flour. Find the value of 4 of a barrel. 63. A man is 63 years old. \ of his age is \ of the age of his son. How old is the son? 64. \ of my money is -^ of yours. I have $24. How much have you? 65. $20 is i of a man's salary. Find J of his salary. 66. A man bought a cow for $35, and 4 of that sum was \ of what he paid for a horse. What did he pay for the horse? EXERCISE. 304. 1. In three pieces of carpeting that contain 44| yards, 39f yards, and 53J yards, there are how many yards? 2. I sold a horse for $185i and thereby lost $9^. How much did the horse cost? 3. A miner digs 16|, 21|, and 18^ ounces of gold. He loses 3f ounces in washing. How much gold has he left? 4. Add 219f, 407J and 328f, and from the sum take 4581. 5. A farmer sold two loads of hay, one for $15| and the 238 FRACTIONS. other for $18i, and received $29^ down; how much is still due? 6. From 10^ take the difference between 3J and 8^. 7. Among how many families can 93i pounds of flour be divided, if each family gets 6^ pounds? 8. At $5 1 a cord, how many cords of wood can be bought with $72i? 9. My father gave me $5f , and my mother $6|. How many books, at %1\ apiece, could I buy with the money ? 10. I had 25J acres of land. After giving a number of acres to my son, I sold the remainder for $225, at $22^ per acre. How many acres did I give to my son? 11. A man paid $63 for 5^ tons of coal. What was the price per ton? 12. A woman paid $51J for 13f yards of satin. How much was it a yard? 13. What will one basket of peaches cost, if 13^ baskets cost $16K 14. I paid i of $4 J for a plate. How many such plates could be bought with $20tV? 15. A man having $|| spent i^r of it. How much had he left? 16. Find the value of ^ of f of a sailboat which is worth $720. 17. I paid $626 for 8 lots. How much, at that rate, are 7 lots worth? 18. Seven plates are worth $8f . What are 9 such plates worth? 19. Samuel walked | of llf miles. Thomas only trav- eled TT as far as Samuel. How far did Thomas walk? MISCELLANEOUS PROBLEMS. 239 20. A lady had $35 and spent f of it for a watch. How much had she left? 21. A grocer bought 63 gallons of oil and sold ^ of it. How many gallons had he left? 22. Mary had $15 and spent | of it for lace. How much money had she left? 23. Mr. Smith bought 12 gallons of vinegar, and used | of a gallon. How many gallons were left? 24. Divide A of 3f by | of 3i 25. From 240 acres of land, 43f acres are sold to one man, and J of the remainder to another. How many acres re- main unsold? 26. If 9i tons of hay cost $95, how many tons can be bought for $120? 27. 28fXl3H-? 28. 13| + 19K11§ + 18H? 29. At the rate of 9J miles an hour, how far can a boy travel on a bicycle, riding 3tV hours in the forenoon and 2f hours in the afternoon? 30. Bought 47 yards of cloth; kept 8^ yards, and sold the remainder at $3 a yard. What did I get for it? 31. 13tX6i^l6f=? 32. H-^2fX8H=? 33. Multiply li -21 by f -4i. 34. Find the value of | of 2|-- (1|-|). 35. What is the value of f of U -i of 5i? 36. How many weeks will it take to spend $182, if my weekly expenses are $22f ? If my income is $37^ a week, how much do I save in that time? 37. 13fX4J^18f = ? 240 FRACTIONS. 38. If a man travels 630 miles in SJ days, how far would he travel in 5^2 days, at the same rate? 39. The value of | of a farm is $4,746. Find value of the whole farm. The value of 1 of the farm = $4,746. The value of all of the farm = $4,746 x f = $5424. 40. I own I of a store. I sell | of my share for $120. Find the value of the store. 41. How much cloth can be bought for $27, if J of a yard cost $2|? 42. A man drives 45 miles in 4J hours. At that rate, how long will it take him to drive 75 miles? 43. A man receives $100 for 5| w^eeks^ labor. How much should he receive for working 8f weeks? 44. 4§Xl4i-3i=? 45. Mr. Harrow sold f of his land and had 104 acres left. How many acres had he at first? 46. A boy used | of the nails in a paper bag, and found on counting them that the bag still contained 553 nails. How many were in the bag at first? 47. I spent I and | of my money and had $220 left. How much had I at first? 48. After spending ^, f , and | of my money, I had $783 left; how much had I at first? 49. I withdrew | of my money from bank, leaving $725. How much did I withdraw? 50. I had a certain distance to walk. I walked J of it in the morning, ^ of it in the afternoon and the rest, which was 5 miles, in the evening. How far did I walk all together? 51. Mr. Atwood sold J of his farm at one time, J of it at MISCELLANEOUS PROBLEMS. 241 another, \ of it at another, and had 53 acres left. How many acres did he sell in all? 52. Mary had 224 beads. This was \ more than Jane had. How many had Jane? 53. Mr. Holman sold a piece of furniture for $50.80, gaining ^ of the cost. What did it cost him? 54. Mr. Smith has property valued at $1600. | of this is J of the value of Mr. Joneses property. How much is Mr. Jones's property worth? 55. My father's farm produced 625 bushels of wheat; J of this is I of what our neighbor raised on his farm. How many bushels did our neighbor raise? 56. I paid $8.75 for | of a ton of anthracite. The next month I bought | of a ton. How much did I have to pay the second time, at the same rate? CHAPTER VIII. DECIMAL FRACTIONS. 305. How many rows are there in this square? One row is what part of the whole ? Write the fraction which expresses this. What is the denomina- tor? AVhat is the numerator? ^ of anything may also be written .1. $.1 is one way of writing xV of a dollar, or $rV, which is one dime. $.10 may be read as ^^one dime and no cents," and has the same value as $.1, for both are A of a dollar. In this way of writing fractions, removing the cipher from the right does not change the value. Express -^ in another way. (.2.) Express in the same way ^, i^, i%, y^. DECIMAL FRACTIONS. 243 306. Two rows of this square are what part of the whole square? Show another way of expressing this same fraction. Two rows are also what part of 10 rows? (i.) Give three ways in which you may express the relation of 2 rows to the whole square. The first place to the right of the decimal point expresses rows of the square, or tenths of the square. Note. — Have the class work with 4 rows, 5 rows, 6 rows, 8 rows, and 10 rows. 307. How many squares are there in each row? How many squares are there in the large square? One small square is what part of the whole square? What is the denominator? What is the numerator? How would you express to o of a dollar? (1 cent or $.01.) ito of a dollar? y^u of a dollar? I'oi) of a dollar? Compare .1 of a dollar and .10 of a dollar. What is another way of expressing too of a dollar? r|o of anything? .01 of the whole square is how many small squares? .03 of the whole square? .09 of the whole square? 308. Fractions whose denominators are 1 with zeros annexed are called Decimal Fractions. rV, ih^ and to oo are decimal fractions. The denominator of a decimal frac- tion is usually not written, but the idea is expressed by the numerator and the decimal point. The usual form of writing the decimal fractions given above is : .1, .03 and .0011. Decimal is from a Latin word meaning ten. 244 DECIMAL FRACTIONS. 309. Fractions whose denominators are not 1 with ciphers annexed are called Common Fractions, f , I, H; and ^^ are common fractions. 310. Ten squares are what part of the whole square? (iWr.) Express this fraction as a decimal. Ten small squares are the same as what? (1 row.) We know that 1 row is what part of the whole square? Show the four ways in which you may express the relation of 10 small squares to the whole square. 311. Twenty-five small squares are what part of the whole square? Express it decimally. Twenty-five small squares are how many rows? 2J rows are what part of the whole? (.2^ or {.) Give four ways of expressing the value of 25 squares. WTr=.25 = .2i = i) Note. — Have the class work with 50 squares, 75 squares, 100 squares. 312. One row of squares is what part of the whole? (.1.) One small square is what part of the whole? (.01.) One row and one small square is what part of the whole? (.11.) Compare the value of the figure in the second place to the right of the decimal point with that of the figure in the first place to the right of the decimal point, describing them as parts of the square. (1 row = 10 small squares; 1 small square is tV of 10 small squares.) DECIMAL FRACTIONS. 245 Compare the value of the number in the second decimal place with that of the number in the first decimal place. (.1 = .10; .01 = iV of .1.) Fundamental Processes. Note. — The class should review work showing the processes which may be performed with like numbers ; also the work showing that the processes of addition, subtraction, division and comparison may sometimes be performed with unlike numbers ; as 6 bushels and 2 pecks. (§ 257.) 313. Add, subtract, divide, and compare .6 and .2, in the language of the square. 6 rows and 2 rows = 8 rows. 6 rows less 2 rows = 4 rows. 6 rows -T- 2 rows = 3 (times). 2 rows = i of 6 rows. 6 rows are 4 rows more than 2 rows. 2 rows are 4 rows less than 6 rows. Perform these processes in the language of United States money. Add, subtract, divide, and compare .6 and .2, as decimal fractions. 314. Add, subtract, divide and compare .6 and .15 in the same manner as shown in § 313. 6 rows = 60 small squares. 60 small squares and 15 small squares = 75 squares. 6 dimes = 60 cents. 60 cents + 15 cents = 75 cents. 246 DECIMAL FRACTIONS. Name the least number of coins you can have when you have 75 cents. What other coins taken together will make 75 cents ? 6 tenths = 60 hundredths. 60 + 15 = 75. Note. — The teacher should make original problems. WRITING AND READING DECIMALS. 315. Can you imagine an oblong that is 100 inches long and 10 inches wide? Note. — This could be drawn on the blackboard — 10 inches high, 100 inches long. How many one-inch squares would there be in this ob- long? (1000.) One one-inch square is what part of the oblong? Express this decimally. What is the denominator? What is the numerator? If a dollar sign were placed in front of this, in what two ways might it be read? (One thousandth of a dollar or 1 tenth of a cent.) 316. Ten one-inch squares are what part of the oblong? u 000-; Ten one-inch squares are the same as what? (1 row.) One row is what part of the oblong? ( rko or .01 .) Give the four ways of expressing the value of 10 one- inch squares. 317. Five hundred one-inch squares are what part of the whole? dVA or .500.) REDUCTION. 247 Five hundred one-inch squares are what part of the 1000 one-inch squares? (J.) Five hundred one-inch squares are the same as how- many rows? Fifty rows are what part of the oblong? {-f^^ or .50.) Give five or more ways of expressing the relation of 500 one-inch squares to the 1000 one-inch squares. 318. Express the following in decimal form: tV tV, i¥o, tVo, t¥A, iVA, 6A, 25t*o, 36rAo. 6 hundredths. 19 hundredths. 60 thousandths. 25 hundredths. 40 thousandths. 9 and 7 tenths. 301 thousandths. 97 hundredths. 6 and 7 hundredths. Read the following decimals : .5 .06 .145 3.45 .700 .05 .60 .265 4.89 4.900 .15 .56 .103 5.07 4.009 .30 .84 .047 7.008 6.800 ,45 .96 .006 9.037 6.080 REDUCTION. To change Decimals to Common Fractions. 319. Change .6 to a common fraction in lowest terms, having the same value. To change a decimal to a common fraction, write the de- nominator under the decimal ^ omit the decimal point, and change the fraction to its lowest terms. 248 DECIMAL FRACTIONS. EXERCISE. 330. Change to equivalent common fractions or mixed numbers: 1. .8. 8. .015. 2. .75. 9. .275. 3. .9. 10. .048. 4. .60. 11. .009. 5. .625. 12. 5.36. 6. .35. 13. 3.25. 7. .15. 14. 15.75. To Change Common Fractions to Decimals. 1 331. 1. Drawa line and divide it into halves. Express the divided line fractionally, (f.) 1 ' 1 1 — -I 1 1 1 ' 2. Draw a line and divide it into 10 equal parts. Express it fractionally. (U or 1.0.) 3. Imagine the line cut into 100 equal parts; express it fractionally. (iU or 1.00.) We see that 1 = 1.0 or 1.00 or 1.000, etc. 4. Express 5 as tenths, hundredths, thousandths. (5 = 5.0 or 5.00 or 5.000.) 5. Express i decimally. (Jof Horjof 1.0 = .5. i=.5.) 6. Express i decimally, (i of 1.000 = .125.) 7. Express f decimally. (| of 1.00 = .75; or i of 3.00= .75.) ADDITION OF DECIMALS. 249 To change a common fraction to a decimal, express the numerator decimally and divide by the denominator. EXERCISE. 322, Change to equivalent decimals: 1. i. 2. i- 3. 1- 4. f. 5. i- 6. I. 7. h 8. I 9. I. 10. I 11. i 12. f . 13. i 14. A. 15. if. 16. A. 17. 3i. 18. 7|. 19. 12i^. 20. ^2^, 21. 26f. ADDITION OF DECIMALS. Note. — Eeview addition as given in § 313 and § 314. 3«3. 6 tenths + 15 hundredths =? 6 tenths = 60 hundredths- .60+ .15 = .75 or .60 .75 Find the sum of 25.4, 120.7, 216.009, and .496. 25.4 1*^^^'7 Write the numbers so that units of the same order 216.009 stand in the same column. Begin at the right and .496 ^^^ ^^ ^^ addition of integers. 362.605 250 DECIMAL FRACTIONS. EXERCISE. 324. Find the sum of: 1. .680, .729, .006, .3, .40, and .400. 2. 65.789, 36.908, 45.8, and 3001.601. 3. 8.675, 34.604, .007, .897, and 189.3. 4. 1009.09, 3040.60, 10001.345, .009, and 987. 5. 62.5 yards + 95.7 yards f 67.25 yards + 9.48 yards. 6. 9 and 101 thousandths, 7 and 3 tenths, 15 and 75 hundredths, 38 and 25 thousandths. 7. One hundred eleven thousandths, two hundred twenty-five thousandths, sixteen tenths, one hundred five and one hundred five thousandths, three hundred fifty and three hundred thousandths. 8. Add as decimals: 56i 49 A, 42^, 39i 15 J. SUBTRACTION OF DECIMALS. Note. — Keview subtraction as given in §313 and §314. 325. From .04 take .005. .4 = .400 .400 - .005 = .395. or .400 .005 .395 From 45.75 take 26.9. 40. /o Write the subtrahend under the minuend, so that 26.9 units of the same order shall stand in the same column, 18.85 and subtract as in the subtraction of integers. DIVISION OF DECIMALS. 251 From 64.7 take 19.013. 64.700 If there are more decimal places in the subtrahend 19 01 S than in the minuend, fill the vacant decimal orders of the minuend with ciphers. EXERCISE. 326. Find the difference between: 1. 303.48 and 199.09. 2. 87.076 and 65.005. 3. 1005.15 and 105.015. 4. .8 and .08. 5. 9 tenths and 9 thousandths. 6. 101.009 and 81.998. 7. 1616.161 and 987.90. 8. 7 hundredths and 7 thousandths. 9. 90 hundredths and 90 thousandths. 10. From 80 thousand and 80 thousandths take 8 thous- and and 8 thousandths. 11. Find the difference between seven and seven tenths, and seven and seven thousandths. 12. A man walked 42.5 miles the first day and 17.875 miles the second. How much farther did he walk the first day than the second? DIVISION OF DECIMALS. Note. — Review division as given in § 313 and § 314. EXERCISE. 327. 1. At $15 apiece, how many books may be bought with 6 dimes? 252 DECIMAL FRACTIONS. 6 dimes = 60 cents. 60 cents -t- 15 cents = 4. You can get 4 books. 2. $.8 ^$.02 = ? 8. $5.5 -=-$.25 = ? 3. $.02 ^$.005 = ? 9. $2.1 ^$.35 = ? 4. 15 acres -J- .5 acres =? 10. 15.3 gal. -^5.1 gal. = ? 5. $2 ^$.05 = ? 11. $4.6 ^$1.15=? 6. 5bu.-^2.5bu. = ? 12. $.5^$.00J = ? 7. 8peeksH-1.6pecks=? 13. $3.8 ^$.76=? EXERCISE. 328. 1. What part of a gallon of milk can be bought with $.04, if milk sells for .2 of a dollar per gallon? .2 of a dollar = 20 cents. 4 cents -t- 20 cents = ^ = i = .2. or $.04 ^ $.2 = $.04 H- $.20 = .2. .2 of a gallon of milk can be bought for $.04. 2. $.08-T-$.4=? $.08 H- $.4 = $.08 ^ $.40 = -/o = ^ = .2. 3. .04 -.2=? 5. $.4 ^$.2=? 4. .5^2.5=? 6. .63^4.5=? .63 -^ 4.5 = .63 -J- 4.50 = -,%% = /o = S7 = -l^. 7. 25 bushels -5 bushels= ? 11. $1.2 -$24= ? 8. 2 pints -25 pints=? 12. 2-2.5=? 9. $1.5 -$2.5= ? 13. 1.2-2.4=? 10. 2.04-25.5=? 14. 3.6^7.2=? DIVISION OF DECIMALS. 253 EXERCISE. 329. 1. A man earns .2 of a dollar in one hour. In how many hours can he earn .5 of a dollar? . 5 of a dollar -^ .2 of a dollar = 2h 2i = 2.5. It will take him 2. 5 hours. 2. .9-^.4=? .9-i- A = I =2i = 2.25. 3. .7- - .5=? 7. 9 -^3.6 =? 11. 16.8^3.5=? 4. 2- - .8=? 8. 1.25 ^ .5 =? 12. 15.5-^5 =? 5. 5- - 4=? 9. 3.125^ .25 = ? 13. 36.4^8 =? 6.16- -2.5=? 10. 6.25 ^2.5 =? 14. 29.7-^9 =? 330. 1. Divide 16.048 by 3.4. 2. Divide 9.5 by .25. 3.4)16.048(4.72 .25(9.50(38 13 6 7 5 2 44 2 00 2 38 2 00 68 • 9.5 = 950 hundredths -f- 25 hun- 68 dredths = 38, an integral number. Divide as in the division of integers, and point off as many decimal places in the quotient as the number of decimal places in the dividend exceeds the number in the divisor. 1. When the dividend has fewer decimal places than the divi- sor, annex ciphers to the dividend. 2. When the quotient has not enough decimal figures, prefix ciphers. 3. When there is a remainder, the division may be continued by annexing ciphers to the dividend. 254 DECIMAL FRACTIONS. EXERCISE. 331. 1. I have $.72 with which to buy starch at $.045 a pound. How many pounds can I buy? 2. A candy dealer has 35 pounds of candy to place in bags, each holding .875 of a pound. How many bags will be needed? 3. Mr. Brown bought 60 acres of land and divided it into lots, each containing .75 of an acre. How many lots does he have? 4. A furniture dealer has $330 with which to buy chairs at $7.5 apiece. How many chairs can he buy? 5. Flannel sells for .625 of a dollar per yard. At that rate, how many yards could I buy with $11.25? 6. A man who owns a paper stand makes $16.35 per week. How many papers does he sell in a week if he makes .0025 of a dollar on each paper? 7. .0007-^.45=? 8. At $9,875 an acre, how many acres of land would I re- ceive for $63.99? Divide : 9. 34.5 by .15. 10. 34.5 by .015. 11. 5.5 by 1.25. 12. 5.5 by .0125. 13. 450.5 by 1.75. EXERCISE. 332. 1. One-half of $6=? One-half of 6 cents = ? Of .06=? Of .006=? DIVISION OF DECIMALS. 255 2. A man divided .16 of a square mile of land between 2 men. What part of a square mile did each receive? 2 ). 16 of a square mile. .08 of a square mile. Each man received .08 of a square mile. 3. Mr. Smith bought .625 of an acre of land. He divided it so as to have 5 equal garden plots. What part of an acre did each plot contain? 4. I paid $.75 for 4 yards of ribbon. How much was it per yard? 5. Eight bushels of peaches sell for $6. How much are they per bushel? 6. I paid $.09 for .3 of a yard of embroidery. How much was it a yard? 3 ) $.09 = cost of .3 of a yard. $.03 = cost of .1 of a yard. 10 (10 times .1=1). $.30 = cost of 1 yard. 7. $.80 is the cost of .25 of a yard of silk. Find the cost of a yard. 8. Eighteen cents are .9 of what I paid for some tablets. What did I pay for them? 9. .8 of a gallon of cream cost $.30. How much does a gallon cost? 10. I paid $.50 for 2.5 yards of ribbon. Find the cost of 1 yard. 2. 5 ) $.500 , cost of 2.5 yards. $.20, cost of 1 yard. 256 DECIMAL FRACTIONS. 11. I paid $.70 for 3.5 pounds of porterhouse steak. How much did I pay for one pound? 12. 4.8 yards of shirt-waist material cost me $1.44. How much was it per yard? 13. I paid $99 for 4.4 acres of land. How much did I pay for one acre? 14. Mr. Berry bought a farm for $1200 and sold it so as to gain 25^ of the cost. What was his gain? 15. Our neighbor receives a salary of $1500 per year. He pays 33^^ of this for house rent. How much money has he left after paying his rent? MULTIPLICATION OF DECIMALS. Note. — Introduce this work by means of one hundred one- inch squares, supplemented by United States money. 333. 1.2 times 1 row=? 2. 1 row is what part of the whole? 3. 2 times .1 = ? 4. 10times.l = ? 5. 5 times .2 = ? 6.5 times 4 rows=? 7. 5 times .4=? 8. 8 times .4=? 9. 9 times .5=? 10.2.5times.4=? 11.5.5times.8=? 12.2.2times.5=? MULTIPLICATION OF DECIMALS. 257 13. 4 times 1 small square = ? 14. 1 small square is what part of the whole? 15. .01X4=? 17. .10X10=? 19. 5.5X10=? 16. .02X4=? 18. 4.5X10=? 20. 6.5X10=? EXERCISE. 334. 1. 2 times 2.1 = ? 5. 4.8X3=? 2. 5.2X3 = ? 6. 2.5X2=? 3. 5.3X6 = ? 7. 4.2X10=? 4. 4.6X5=? 8. 8.1X5 = ? EXERCISE. 335, 1. Multiply .253 by .35. .253 Multiply as in whole numbers, and point off as 35 many decimal places in the product as there are in both multiplicand and multiplier. If there are not enough figures in the product to fill the decimal places, prefix as many ciphers as are necessary to 1265 759 .08855 make the required number. Multiply: 2. .386 by .47 6. 49.3 by .064 3. .231 by .36 7. 492 by 3.8 4. 48.2 by 25 8. 384.45 by .64 5. 48.2 by .25 9. 38.445 by .64 10. What is the cost of 26.25 pounds of sugar, at $.0625 per pound? 11. One acre of land produces 48.375 bushels of wheat. How many bushels will 7.25 acres produce? 12. At $6.37 a ton, what must be paid for 5.25 tons? 258 DECIMAL FRACTIONS, MENTAL EXERCISE. 336. 1. Make decimal fractions of the following: ■^\; TU'} 12] lo'WU': it' 2. State the numerator and the denominator of the fol- lowing: .03; 1.25; .075; .0075; .00205. 3. Change to common fractions in lowest terms: .4; .05; .6; .12; .75; .8; .24; .50; 2.5; .16|. 4. Compare the values of: .6 andt; .04 andi; .05 and I ; .875 and J. 5. Change to decimal fractions: il i'} 4; il i') J J A J il i'y i] ^s] /o. 6. What decimal of a pound do I buy in buying J of a pound of coffee? 7. A of a foot is what decimal of a foot? 8. Change J and f to decimals and add. 9. Add one hundredth and one tenth. 10. I paid f of a dollar for a knife and J of a dollar for a book. How many dollars and cents did I spend? 11. Add i and i as decimals. 12. I and J are equal to what decimal? 13. A boy spends .33^ of a day in sleep and ^ of a day in study. What fractional part of a day is left? 14. Subtract one hundredth from one tenth. 15. A man has 10 loads of hay and sells .05 of a load ; how much hay has he left? 16. From the sum of 4.4 and .08 take 4.1. 17. From 4.4 take the difference between .04 and 4.1. 18. 4X.05 is what common fraction? MULTIPLICATION OF DECIMALS. 259 19. One book costs 8 hundredths of a dollar. What will 9 such books cost? ($.08 X9 = .) 20. .3X.005-? 21. What is the shortest way of multiplying .48 by 100? 22. Change f to a decimal, take away .01, and multiply by 100. 23. From 1 take .001, and multiply by a thousand. 24. I own .1 of a farm and sell i of my share. What dec- imal part of the farm do I sell? 25. When apples are tV of a dollar a bushel, what must you pay for .7 of a bushel? 26. Divide $8 into 400 equal parts. 27. From 1 take .001 and multiply by 100. 28. Ten men can do a piece of work in 2.1 days. How long will it take one man to do it? 29. .lof .01 = ? 30. At $2 a yard, how much silk can be bought with $.40? 31. I buy a cord of wood for $5 and sell it for $5.50. The gain is what decimal part of the cost? 32. Multiply .75 by 10, and divide the product by .03. 33. A man gained 4 mills, or $0,004, on a quart of nuts. How many quarts must he sell to gain $.40? 34. What is the quotient of .1 divided by .01? 35. A grocer bought potatoes at $.40 a bushel and sold them so as to gain 10^, or .10, of the cost. What was the gain on the bushel? 36. My watch cost me $60. I sold it so as to lose 25^, or .25, of the cost. What did I lose? 260 DECIMAL FRACTIONS. MISCELLANEOUS PROBLEMS. 337. 1. The subtrahend is eight thousand and forty- eight thousandths, and the remainder is eight hundred seventy-three thousandths; what is the minuend? 2. There are 228.35 barrels of water in a cistern which will hold 410.5 barrels; how many barrels will be needed to fill the cistern? 3. At .085 of a dollar per dozen, what will lOf dozen steel pens cost? 4. From a barrel containing 43 gallons of vinegar, .125 gallons were drawn at one time, 3.5 at another, and .75 at another; now many gallons remained in the barrel? 5. Dry goods valued at $8000 were destroyed by fire; what would a man lose who owned .12 of the goods? 6. A gallon, liquid measure, contains 231 cubic inches; how many gallons are there in 13051.5 cubic inches? 7. At $6.80 an acre, how many acres of land can be bought for $4258? 8. Bought 17 chests of tea, each containing 59 pounds, at $0.67 a pound, and gave in exchange 118 bags of wheat, each containing 3.4 bushels; what was the value of the wheat per bushel? 9. When the dividend is .1 and the divisor 12.8, what is the quotient? 10. What is the quotient of 312.5 by 85? 11. If 38 yards of cloth cost $180.50, what will be the cost of 26 yards? 12. At $2.56 per yard, how many yards of cloth can be bought for $94.4? 13. From $62.40 take $7.37^ MISCELLANEOUS PROBLEMS, 261 14. A druggist sold 375 gallons of ink in bottles holding .375 of a gallon each; how many bottles of ink did he sell? 15. By selling a carriage for $195, I lost $34.50. For how much should I have sold it to gain an amount equal to .7 of what I lost? 16. From the sum of $15.75 and $1001.10 take the sum of $101,018 and $50,101. 17. Subtract $.50 from $1,005. 18. A man bought a coat for $16, a vest for $3.50, and a pair of trousers for $5.50; what two coins will exactly pay for them? 19. From the sum of $14.50 and $12.75 take 6 dimes 6 mills. 20. From $4.50 take 37i cents. 21. A grocer bought 3 barrels of apples for $6.75, a box of lemons for $2.50, and 5 barrels of flour for $30.00. He handed the merchant two gold pieces, and received $.75 in change. What were the two pieces of money? 22. At $.12^ a yard, how much muslin can be bought for $20.43? 23. If f of a yard of cloth cost $2.16, what will be the cost of 5i pieces, each containing 47 yards? 24. When rice is selling at $.075 a pound, how many pounds can be bought for $5.25? 25. How many days, of 9 hours each, must a man work in order to earn $576.72, at 18 cents an hour? 26. If a lady earns $15.00 a week, and spends an average amount of $11.37^, in how many weeks will she save $166.75? 27. 31.5 gallons of vinegar cost $11.81^; how much is that per gallon? CHAPTER IX. COMPOUND NUMBERS. Note. — Only so much of Compound Numbers should be used as seems to be adapted to the age and development of the pupils. 338. A Simple Quantity is expressed in units of one de- nomination; as 4 pecks. A Compound Quantity is expressed in units of different denominations which may be reduced to units of the same denomination; as 4 pecks, 3 quarts. 339. A Denominate Number is a number composed of denominate units. A Simple Denominate Nimaber is composed of units of one denomination; as 5 gal. or 13 bu. A Compoimd Denominate Number is composed of units of two or more denominations which may be reduced to units of the same denomination; as 4 lb. 9 oz. 340. Reduction is the process of changing the denomi- nation of a number without changing its value. In reducing denominate numbers, the increase or decrease in the number of units is irregular, instead of by ten as in simple numbers. DRY MEASURE. 341. Dry measure is used in measuring grain, fruit, seeds, vegetables, and other dry articles. DRY MEASURE. 263 The denominations are 'pintSj quarts, pecks, and bushels. 2 pints (pt.) = 1 quart (qt.). 8 quarts = 1 peck (pk.). 4 pecks = 1 bushel (bu.). 1 bu. = 32 qt. = 64 pt. The standard bushel is 18i inches in diameter and 8 inches deep, and contains 2150.42 cubic inches. 1. How many bushels in 24 pecks? In 25 pecks? In 35 pecks? 2. Reduce 5 bushels to pecks. To quarts. 3. Reduce 2 pecks to pints. 2 bushels to pints. 342. Reduce 16 bu. 3 pk. 1 pt. to pints. bu. pk. qt. pt. One bushel = 4 pecks. In 16 bushels there 16 3 1 are 4 times as many ones of pecks as ones of 4 bushels. 16 multiplied by 4 = 64. There are ~ , 64 pecks in 16 bushels. 64 pecks + 3 pecks =^ ^^P"^- 67 pecks. Z_ One peck = 8 quarts. In 67 pecks there are 536 qt. ^ times as many quarts. 8 times 67 = 536. 2 There are 536 quarts in 67 pecks. One quart = 2 pints. In 536 quarts there 1073 pt. are 2 times as many pints. 2 times 536 = 1072. 1072 pints + 1 pint = 1073 pints. 16 bu. 3 pk. 1 pt. = 1073 pints. 1. Reduce 8 bu. 3 pk. 1 pt. to pints. 2. Reduce 15 bu. 3 pk. to quarts. 3. Reduce 12 bu. 1 pk. 3 qt. to pints. 4. Reduce 3 pk. 6 qt. to pints. 264 COMPOUND NUMBERS. To reduce a compound denominate number to a lower denomination : Multiply the highest denomination by the number of ones of the next lower which make one of the higher, and add to the product the given number of the same denomination. Proceed in like manner with each successive result, until the number is reduced to the required denomination, 343. Reduce 689 pints to bushels. 2 ) 689 pt. There are in 689 pints as many quarts Q \^iAA + _L 1 + ^^ there are times 2 pints, which is 344, 8 )6^ qt. + 1 pt. ^^j^Ij ^ pjj^^ remaining undivided. 4 ) ^'^ pk There are in 344 quarts as many pecks in K 4- '-l It '^^ there are times 8 quarts, which is 43. ^ * There are in 43 pecks as many bushels as there are times 4 i)ecks, which is 10, with 3 pecks remaining. 689 pints -= 10 bu. 3 pk. 1 pt. 1. Reduce 817 pints to bushels. 168 quarts to bushels. 2. Reduce 682 pints to bushels. 95 pints to pecks. 3. Reduce 125 quarts to bushels. 87 pints to pecks. To reduce a compound denominate number to a higher denomination : Divide the given number by the number of ones that make one of the next higher denomination. Divide this quotient and each successive quotient in like manner, until the 7'equired denomination is reached. The last quotient, with the several remainders annexed in proper order, is the result required. LIQUID MEASURE. 265 LIQUID MEASURE. 344. Liquid Measure is used in measuring liquids. The denominations are gills, "pints, quarts, gallons, and barrels. 4 gills (gi.) - 1 pint (pt.). 2 pints = 1 quart (qt.). 4 quarts = 1 gallon (gal.). 31i gallons = 1 barrel (bbl). The gallon contains 231 cubic inches. l>j pints liquid me&sure equal 1 pint dry measure. The barrel contains 31i gallons ; the hogshead 63 gallons. 1. Reduce 15 gallons to pints. Reduce 18 gallons to gills. 2. Reduce 17 gal. 1 qt. 1 pt. 3 gi. to gills. Reduce 8 quarts to gills. 3. How many gallons in 47 quarts? How many gallons in 47 pints? 4. Reduce 86 gills to quarts. 98 gills to gallons. 5. Reduce 25 gal. 1 pt. to gills. Reduce 19 gallons to pints. AVOIRDUPOIS WEIGHT. 345. Avoirdupois weight is used in weighing all articles except gold, silver, and precious stones. The denomina- tions are ounces, pounds, hundredioeights , and tons. 16 ounces (oz.) -1 pound (lb.). 100 pounds -= 1 hundredweight (cwt.). 20 hundredweight, 2000 lb. = 1 ton (T.). 266 COMPOUND NUMBERS. 60 pounds of wheat =1 bushel. 56 corn or rye =1 " 32 oats =1 100 nails =1 cask or keg 196 flour =1 barrel. 200 beef or pork = 1 barrel. 1. Reduce 3 tons to pounds. Reduce 6 hundredweight to ounces. 2. Reduce 7 cwt. 48 lb. 9 oz. to ounces. Reduce 9 tons to ounces. 3. Reduce 54145 pounds to tons. Reduce 3684 ounces to pounds. 4. Reduce 36425 pounds to hundredweights. Reduce 32000 ounces to tons. 5. Reduce 5 T. 12 cwt. 36 lb. to pounds. EXERCISE. 346. 1. How many pint packages can a seedsman make from 4 bu. 2 pk. and 2 qt. of seeds? 2. What will IJ barrels of vinegar cost at 8 cents a quart? 3. In one season a market-gardener sold 12345 boxes of strawberries, averaging 1 quart each. How many bushels did he sell? 4. At 7 cents a pound, what will 2^ barrels of pork cost? 5. If a horse eats 1 pk. 6 qt. of oats in a day, how long will 7 bu. 2 pk. last? MEASURES OF LENGTH, 267 MEASURES OF LENGTH. 347. In measures of lengths and distances, the denomi- nations are inches^ feet, yards, rods, and miles, 12 inches (in.) =1 foot (ft.). 3 feet =1 yard (yd.). 5i yards, or 16J feet =1 rod (rd.). 320 rods =1 mile (mi.). 1760 yards, or 5280 feet = 1 mile. 1. Reduce 12 rods to feet. 2. Reduce 15 rd. 3 yd. 2 ft. to feet. 3. Reduce 136 rd. 4 yd. to inches. 4. Reduce 18 miles to rods. 5. Reduce 4 mi. 130 rd. to rods. 6. Reduce 5 mi. 20 rd. to inches. 7. Change to lowest denominations: 2^ miles; 16 rd. 25 ft.; 34 yd.; 16.8 rd. 32^ yd. 18^ ft. 8. Change to highest denominations: 16000 feet; 63360 inches; 3240 rd.: 7040 yd.; 47520 ft. 9. Measure one side of your school lot and give the length in rods. 10. How many rods are there in f of a mile? 11. 40 rods is what part of a mile? 12. If the large wheel of a wagon is 15 feet in circum- ference, how many times will it turn in going 5 mi. 182 rd. 4 yd.? 13. In a bundle of lath there are 100 pieces, each 4 feet long. If all the pieces of the 4 bundles were laid end to end, what would be the length in rods? 268 COMPOUND NUMBERS. 14. From A to 5 is 17 rods. One third of that distance is how many feet? 15. How long will it take George to walk a half-mile, if he walks at the rate of 20 rods a minute? SQUARE MEASURE. 348. In measures of surfaces, the denominations are square inches, square feet, square yards, square rods, and square miles. 144 square inches (sq. in.) =1 square foot (sq. ft.). 9 square feet =1 square yard (sq. yd.). 30i square yards = 1 square rod (sq. rd.). 160 square rods =1 acre (A.). 640 acres =1 square mile (sq. mi.), or section (sec). 349. A surface has two dimensions, length and breadth, A plane surface which has four square corners is called a Rectangle. A rectangle which has four equal sides is called a Square. The Area of a surface is the number of square units it con- tains. 350. Suppose the top of a table to be 4 feet long and 2 feet wide. There are two rows of 4 square feet each; that is, there are 2 times 4 square feet, or 8 square feet, in the surface of the table. The width of one end shows how many times 4 square units must be taken to give the whole area. SQUARE MEASURE. 269 To find the area of a rectangular surface, a certain num- ber of square units are taken a given number of times. What is the length of one side of a square yard? Measure off in your schoolyard a square rod. What is the length of each side? What is the area of a square that is 5^ yards on each side? 351. To find the area of a rectangular surface: The length and breadth being given in the same denomino, tion, multiply the length by the breadth. EXERCISE. 352. Reduce: 1. 140 square rods to square feet. 2. 18 acres to square rods. 3. 12 A. 50 sq. rd. 8 sq. yd. 1 sq. ft. to square feet. 4. 1 square mile to square inches. 5. 112 sq. rd. 5 sq. ft. to square feet. Reduce to higher denominations: 6. 1440 square rods to acres ; 4320 square rods to acres. 7. 23328 square inches to square yards. 8. 10890 square feet to square rods. 9. 102400 sq. rd. to square miles. 10. 5760 A. to square miles. EXERCISE. 353. 1. How many square inches of surface has a pane of glass 3 feet long and 2 feet wide? (Make a drawing to show the number of rows of square feet; the number of rows of square inches.) 270 COMPOUND NUMBERS. 2. How many .square inches are there in J of a square foot? In I of a square foot? (Drawing.) 3. How many square inches of surface are there in the top of a table which is 3 feet long and 2h feet wide? 4. Find the area of a floor which is 12 feet by 15 feet. 5. A floor has a surface of 180 square feet; if its length is 15 feet, what is its width? 6. How many acres are there in a field 18 rods long and 9 rods wide? 7. At $48 an acre, what will be the cost of a piece of land 160 rods long and 118 rods wide? CUBIC MEASURE. 354. Cubic Measure is used in measuring solids. Its denominations are cubic inrhes, cubic feet, cubic yards, and cords, 1728 cubic inches (cu. in.) =1 cubic foot (cu. ft.). 27 cubic feet =1 cubic yard (cu. yd.). 128 cubic feet =1 cord (cd.). In measuring wood, a pile 8 feet long, 4 feet wide, and 4 feet high is called a cord. 355. A Cube is a solid bounded by six equal squares. A Cubic Foot is a cube whose faces are each one foot square. . jo The Solid Contents of a body is the number of cubic \m$& it contains. :i>- CUBIC MEASURE. 271 The base of this cube is divided into square feet. There are 3 rows of 3 square feet each, making in all 3 times 3 square feet, which are 9 square feet. If upon each square foot we place 3 cubic feet, we shall have 9 times 3 cubic feet, which are 27 cubic feet. A solid which is 3 feet long, 3 feet wide, and 3 feet higli is a Cubic Yard, and contains 27 cubic feet. How many cubic feet are there in \ oi 2i cubic yard? How many cubic feet in i of a cubic yard? How many cubic feet in | of a 9 cu. ft. X 3 = 27 en. ft. cubic yard? 356. To find the solid contents of a rectangular solid: The length, breadth, and height being given in the same denomination, their product is the number of cubic units, of the same name as the linear units. 357. How many cubic feet of sand will be required to fill this box? How many cubic feet would there be in a layer of sand 1 foot high in this box? How many cubic inches are there in one cubic foot? How many square inches of sur- face has one of the faces? f >TfrV 272 COMPOUND NUMBERS. Build the cubic foot of 1-inch cubes. How many times must 144 cubic inches be taken to make one cubic foot? EXERCISE. 358. 1. How many cubic inches are there in 1 cubic yard? In J of a cubic yard? In | of a cubic yard? 2. Reduce 12 cubic feet to cubic inches. 3. Reduce 87 cubic yards to cubic feet; 62^ cubic yards to cubic feet. 4. Reduce 16 cords to cubic feet; 10} cords to cubic feet. 5. Reduce 20736 cubic inches to cubic feet. 6. Reduce 540 cubic feet to cubic yards. 7. Reduce 9 cu. yd. 7 cu. ft. to cubic inches. 8. Reduce 18 cu. yd. 12 cu. ft. 720 cu. in. to cubic inches. 9. Reduce 1152 cubic feet to cords; 6400 cubic feet to ' cords. 10. How many cubic feet are there in a rectangular block of stone 8 feet long, 5 feet wide, and 3 feet thick? (Make a drawing to show this.) 11. How many cubic feet are there in a pile of bricks 8 feet long, 4 feet wide, and 4 feet high? 12. A tank 6 feet long, 5 feet wide, and 3 feet deep con- tains how many cubic inches? 13. How many cubic feet of air does a room 18 feet long, 15 feet wide, and 10 feet high contain? 14. In digging a cellar 16 feet long, 12 feet wide, and 8 feet deep, how many cubic feet of earth must be removed? 15. A pile of wood 16 feet long, 5 feet high, and 4 feet wide contains how many cords? 16. At $.27 a cubic yard, what will it cost to dig a cellar 18 feet long, 14 feet wide, and 9 feet high? TIME MEASURE. 273 17. How many cubic feet are there in a stick of timber IS inches wide, 8 inches thick, and 12 feet long? 18. What is the value of a pile of wood 82 feet long, 4 feet wide, and 5 feet high, at $4.50 a cord? TIME MEASURE. 359. Time Measure is used in measuring time. The de- nominations are seconds ^ minutes, hours, days, weeks, months, years, and centuries. 60 seconds (sec.) = 1 minute (min.). 60 minutes = 1 hour (hr.). 24 hours = 1 day (d.). 7 days , = 1 week (wk.). 365 days = 1 year (yr.). 366 days = 1 leap year (1. yr.). 100 years = 1 century (C). February has 28 days, except in leap year, when it has 29. September, April, June, and November, each have 30 days; other months of the year (except February) each have 31 days. In business transactions, 12 months are considered a year and 30 days a month. 360. Reduce to lower denominations: 1. 12 hours to seconds; 5 days to minutes. 2. 8 d. 12 h. 40 min. to seconds. 3. How many minutes were there in the month of Feb- ruary, 1904 (1. yr.)? Reduce to higher denominations: 4. 1440 minutes to days; 86400 seconds to days. 5. 52560 hours to years; 4743856 minutes to years. 274 COMPOUND NUMBERS. MISCELLANEOUS PROBLEMS. 290 ft. 60 ft. 18 ft. 361. 1. Find from the dia- gram the number of square yards of tihng used for the fioor of a corridor. Divide into two rectangles and find the area of each. 2. Find the number of square yards in the floor of this room. Divide into two rectangles, one of which shall be 2 feet by 6 feet. 3. How many square yards are there in the ceiUng? 4. How many square yards are there in the w^alls of a room 20 feet by 16 feet, and 9^ feet high, if no allow- ance is made for doors and windows? The area of the four walls of a i»oom is equal to that of a rectangle whose length is equal to the sum of the four sides, and whose breadth is equal to the height of the room. 2 X 20 ft. + 2 X 16 ft. = 72 ft. 72 ft. x 9i ft. = 684 sq. ft. 684 sq. ft. -5- 9 = 76 sq. yd. Draw the rectangle which represents the area of the walls of the room. 30 ft. 5. How many square yards are there in walls, floor, and ceiling of this room? 6. How many square yards are there in a roof, the MISCELLANEOUS PROBLEMS. 275 rafters of which are 16 feet long and the ridge-pole 25 feet long? 7. If the height of a staircase is 15 feet, and that of each step is 9 inches, how many steps are there in the staircase? 8. How many cords of wood are there in a pile 40 feet long, 4 feet wide, and 5^ feet high? 9. How many cords of wood can be piled under a shed 24 feet long, 18 feet wide, and 12 feet high? 10. How many boxes, 4 inches long, 3 inches wide, and 2 inches deep, can be packed in a box 3 feet long, 3 feet wide, and 2 feet deep, measured on the inside? 11. How many fence boards, each 16 feet long, are re- quired to fence a field 80 rods long and 40 rods wide, the fence being 4 boards high? 12. Mr. A^s orchard covers 2^ acres. Allowing two square rods for each tree, how many trees are there? 13. How many loads of earth of 1 cubic yard each will be needed to fill in a lot, 45 feet front, 90 feet deep, to raise it 1^ feet? 14. The factors of a dividend are 16, 50, and 9; of the divisor, 15, 8, and 2. What is the quotient ? 15. A farmer gave 55 sheep for 11 young horses worth $60 each. What money value did he get for each sheep? 16. At 60 cents a cord, how many days will it take a man to earn $75.00, if he saws 2 cords of wood a day? 17. If a turkey weighing 10^ pounds costs $1.68, what is the cost, at the same rate, of one that weighs 15f pounds? 18. At 1^ dollars each, how many lamps can be bought for 65 dollars? 276 COMPOUND NUMBERS. 19. At iV of a dollar per yard, how many yards of rib- bon can be bought for 2j dollars? 20. A gentleman gave away I of the books in his library, lent i of the remainder, and sold i of what was left. He then had 360 books remaining How many had he at first ? 21. If a lady spends 4f dollars per month for carfare, in what time will she spend $27^? 22. The owner of a schooner sells .35 J of the vessel to the captain. What part does he still own? 23. The minuend is 67.081. What must the subtra- hend be to leave a remainder of 56.009? 24. A owns t\ of an iron foundry and sells .75 of his share for $2100. What is the value of the whole foundry? 25. A flour merchant bought 137 barrels of flour at $7,875 per barrel. He sold 89 barrels at $9,378 per bar- rel, and the remainder brought only $5.80 per barrel. What was his gain? 26. Two men start from the same place and travel in opposite directions. One travels 119.33 miles a day, and the other 123.75 miles a day. How far will they be apart at the end of six days? 27. Supposing that each child in a schoolroom ought to have 80 cubic feet of air, how many children should sit in a room which is 20 feet long, 18 feet wide, and 12 feet high? 28. The walk from our kitchen door to the stable is 75 feet long and 4.5 feet wide. How many bricks does it contain, each brick being 8 inches by 4 inches? 29. How many times is 4 cubic inches contained in a four-inch cube? YB 35810 I ■ ,- ^,V-;---«.. 541533 UNIVERSITY OF CALIFORNIA LIBRARY •' ^ J m^m^mm^mmmm^s^rmmm ^ Mi)t SCormal iEmtBt tit Nutnlirr ^ Cook and Cropsey ARITHMETICS INDIANA EDITIONS ^ a & ^ ''^^T^ Prices as fixed by Law New Elementary Arithmetic^ 35 cents New Advanced Arithmetic^ 45 cents AHY DEVIATION FROM THESE PRICES SHOULD BE IMMEDIATELY REPORTED TO Silver^ Burdett & Company Fuhiiskers^ Chicago^ Illinois ^ a