IN MEMORIAM FLOR1AN CAJORI SECOND LESSONS IN ARITHMETIC AN INTELLECTUAL WRITTEN ARITHMETIC UPON THE INDUCTIVE METHOD OF INSTRUCTION AS ILLUS- TRATED IN WARREN COLBURN'S FIRST LESSONS BT H. N. WHEELER HOUGHTON, MIFFLIN AND COMPANY Boston : 4 Park Street ; New York : 11 East Seventeenth Street >rw, Camfcri&ge 1892 Copyright, 1888, BY H. N. WHEELER. The Riverside Press, Cambridge, Mass., U. S. A. Electrotyped and Printed by H. O. Houghton & Company. PEEFAOE. SECOND LESSONS IN ARITHMETIC is the result of an attempt to prepare a text-book which, by its method of developing the mind of the learner, by the emphasis that it places on fundamental princi- ples, and by the omission of useless subjects and arithmetical terms known only in the school-room, will meet the wants of those teachers and busi- ness men throughout the United States who de- mand that the essentials of Arithmetic shall be better taught than heretofore, and that the non- essentials shall be omitted. I believe that there can be no better prepara- tion either for the business of life or for advanced intellectual work than a thorough understanding of the fundamental principles of Addition, Sub- traction, Multiplication, Division, Fractions, Deci- mals, and Percentage (including Interest), coupled with a self-reliant power of analysis sufficient to enable one to deduce what is required from what is given by the aid of his own understanding rather than by the aid of his memory of rules or iv Preface. methods. The subjects just mentioned I consider to be the essentials of Arithmetic ; everything else in this book has been brought in for the sake of illustration. Such illustrations l have been chosen, however, as will appeal to the intelligence of the pupil and give him a broader experience in those subjects to which in after life he will be called upon to apply his arithmetical powers. I have tried in all cases to follow the Inductive Method of Instruction as illustrated in Warren Colburn's First Lessons, a book which has con- fessedly done more for the cause of education than any other text-book that has ever been pub- lished. The Inductive-Colburn method, as I in- terpret it, consists in inducing the pupil to gain an experience of his own which will enable him to regard every definition as the result of his own personal observation and thought, every rule as a statement of the method by which he has done something, and every new word as only a labor- saving device for the expression of a familiar idea. This result is accomplished by a series of questions so framed as to lead the pupil to draw such con- clusions from what he already knows as will con- stitute new elements of knowledge on which in turn new questions may be based. These ques- tions at the beginning of each subject are made 1 See Stocks, Duties, Taxes, etc. Preface. v simple, and require the use of small numbers only, so that the mind of the learner may be occupied with principles rather than with their mechani- cal applications. More complicated relations and larger numbers are gradually introduced until finally the pupil finds it necessary to make a record of some of his work; his written work, however, is always intended to be a record of the results, not of mechanical processes, but of mental operations. In the application of this method the duty of the teacher is merely to provide material for work and to see that it is done ; the duty of the pupil is to work and to discover. In teaching we are strongly tempted to give the pupil the benefit of our own experience, hoping thus to shield him from the toilsome process of gaining for himself that experience which we have acquired only through hard work, but if we call to mind the pro- cess of learning to do that which we do best we shall find that it has consisted, not in learning by heart the results of the experience of others, but in individual, manly effort applied in the spirit of investigation and discovery. Although this book is intended as a continua- tion of Warren Colburn's First Lessons, I have drawn from the First Lessons, with permission, enough matter relating to Fractions to form a vi Preface. good review of the subject. I have also repeated here Chapters VI. and VII. which relate to writ- ten addition, subtraction, multiplication, and divi- sion ; these chapters did not form a part of the original edition of the First Lessons but were pre- pared by me for the revised edition of 1884. The matter taken from the First Lessons is carefully indicated wherever it occurs so that it may be readily omitted by pupils who are already suffi- ciently familiar with it. The Second Lessons, while complete in itself, can (as can any other book of the same scope) be used to the best advantage by pupils who have already studied Warren Colburn's First Lessons. H. N. W. CAMBRIDGE, Mass., Aug. 31, 1888. CONTENTS. SECTION I. PAGE Notation, Addition, and Subtraction .... 1-30 A. Questions on Notation, with Explanations . . 1-6 B. Addition, with Practical Illustrations . . . 6-14 C. Subtraction, with Practical Illustrations . . . 14-21 D. United States Money 21-24 E. Tables and Questions for Practice .... 25-30 SECTION II. Multiplication and Division 31-64 A. Multiplication: Examples and Problems, with re- marks and explanations ..... 31-45 B. Bills 46-51 C. Division: Examples and Problems, with remarks and explanations 51-62 D. Tables and Questions for Practice .... 62-64 SECTION III. Fractions 65-99 A. Fractional Notation and Fractional Terms . . 65-68 B. Common Denominator. Problems. Illustrations . 68-83 C. Multiplication of Fractions 83-90 D. Division of Fractions 90-95 E. Miscellaneous Questions 96-99 SECTION IV. Decimals : Introduction. Notation with simple il- lustrative examples in Addition, Subtraction, Multiplication, and Division .... 100-116 viii Contents. SECTION V. Multiplication of Decimals 117-128 A. Multiplication of a Decimal by a Whole Number. Examples and Problems, with remarks and expla- nations 117-122 B. Multiplication of a Decimal by a Decimal. Ex- amples and Problems, with remarks and expla- nations 122-128 SECTION VI. Division of Decimals 129-147 A. Division by a Whole Number. Examples and Problems, with remarks and explanations . 12!M.'I7 B. Division of a Decimal by a Decimal. Examples and Problems, with remarks and explanations . 137-143 C. Reduction of Common Fractions to Decimals. Circulating Decimals ..... 143-145 D. Miscellaneous Examples ..... 145-147 SECTION VII. Percentage 148-205 A. Interest 148-167 B. Compound Interest 167-171 ^ C. Partial Payments 172-176 D. Equation of Payments . . . . . . 176-179 E. Stocks 179-185 F. Taxes 185-188 G. Duties 189-193 H. Miscellaneous 194-205 SECTION VIII. Square and Solid Measures 206-225 A. Square Measure 206-215 B. Solid Measure 215-222 C. Board Measure 223-225 SECTION IX. Divisors, Factors, and Multiples .... 226-242 Contents. IX SECTION X. Cancellation and Analysis 243-258 APPENDIX. CHAPTER I. Roman Notation . 259-262 CHAPTER II. The Metric System of Measures A. Linear Measure .... B. Square Measure .... C. Solid Measure .... D. Capacity Measure .... E. Weight Measure .... F. Miscellaneous .... 262-275 268-267 267-270 270-271 271-272 272 273-275 CHAPTER III. Arithmetical Tables . United States Money English Money French Money German Money Length . Surveyor's Measure Surface . Solidity . Liquid Measure Dry Measure . Avoirdupois Weight Troy Weight Apothecaries' Weight Apothecaries' Measure Time . Miscellaneous 275-282 275 275 275 275 276 , 276 276 , 277 277 277 277 , 278 , 278 , 278 , 279 , 279 Contents. The Metric System. Linear Measure . . . 280 Square Measure 280 Cubic Measure 281 Capacity Measure 281 Weights 281 Compound Interest Table 282 SECTION L NOTATION, ADDITION, AND SUB- TRACTION. The portions of this section that are preceded by a star (*) have been drawn from Section VI. of the new edition of Colburn's First Lessons, and may at the discretion of the teacher be omitted by pupils who have studied them before. A. Questions on Notation, with Explanations. *1. What number is represented by 36 ? Answer: Thirty-six. What does the 3 stand for ? the 6 ? Answer : 36 = 3 tens + 6 units ; the 3, then, stands for the whole number of tens in thirty-six, and the 6 for the units left over. *2. What do the figures in 42 represent? in 69? in 58? in 17? in 23? *3. What does the right-hand figure of a num- ber denote ? what does the figure to the left of the right-hand figure denote ? Since the right-hand figure denotes units, and the figure to the left of ifc denotes tens, we say that the first figure from the right occupies the units 9 place, and that the second figure from the right occupies the tens 9 place. 2 Notation, Addition, and Subtraction. [ 1. *4. What is the number that has 8 in the ^ . units' place and 6 in the tens' place? (An- jj g swer : Sixty-eight.) 9 in the units' place and 6 8 7 in the tens' place ? 5 in the tens' place and 7 9 2 in the units' place? 4 in the tens' place 5 2 and in the units' place ? 40 *5. What number is represented by 857 ? Answer: Eight hundred and fifty-seven. What do the different figures stand for? Answer : 857 = 8 hundred + 5 tens + 7 units ; the 8, then, stands for the whole number of hun- dreds, the 5 for the whole number of tens left over, and the 7 for the units remaining after tak- ing away both the hundreds and the tens. *6. What do the different figures represent in 614? in 891? in 412? in 508? in 320? *7. What does the figure to the left of the tens' place denote ? *8. Which place, counting from the right, is the hundreds' place ? *9. What is the number that has 6 in rf the units' place, 8 in the tens' place, and I ^ 4 in the hundreds' place ? (Answer : Four Jj j| hundred and eighty-six.) 9 in the tens' 486 place, 7 in the hundreds' place, and 2 in 792 the units' place ? 6 in the units' place, 806 in the tens' place, and 8 in the hundreds' 600 place ? 6 in the hundreds' place, in the ; ens' place, and in the units' place ? *1O. The number 320 stands for 32 tens, but it is often convenient to call 10 tens a " hundred," A.] Questions on Notation, with Explanations. 3 so that 32 tens is 3 hundred and 2 tens, or three hundred and twenty. Similarly we call, for con- venience, 10 hundred (1000) a " thousand " ; 20 hundred (2000) 2 " thousand " ; 30 hundred (3000) 3 " thousand " ; | 3600, then, stands for 36 hundred, or 3 thousand and 6 hundred; 13200 for 132 hundred, or 13 thou- sand and 2 hundred; 168362 for 168 thousand 3 hundred and 62. Write in words each of the numbers in the adjacent column. First number : Three hundred and 982742 twenty-eight thousand nine hundred and sixty - eight. *11. What figures stand for thousands in each of the above numbers ? what figure stands for hun- dreds ? for tens ? for units ? *12. Counting from the right, what places do the figures that represent thousands occupy ? In order to make it easier to read a number, it is usual to separate by a comma the figures that denote thousands from the figure that denotes hun- dreds ; thus, in 198,642 we place a comma between the 8 and the 6, and read 198 thousand 6 hundred and 42. *13. Read : 986,432 ; 16,768 ; 270,432 ; 7,487 ; 100,708; 908,550; 2,008; 68,052; 111,684; 777,777. *14. Express by figures the numbers : Seventy-seven thousand six hundred and eighty- four. 4 Notation, Addition, and Subtraction. [ 1. Thirty-four thousand and fifty-six. Four hundred and seventeen thousand and six hundred. Seven hundred and fourteen thousand and six hundred. Six hundred thousand and two. Three hundred and ninety-one thousand six hun- dred and twelve. Fifty-seven thousand one hundred and nine. Nine hundred thousand four hundred and three. *15. 1,000,000, or 1 thousand thousand, is us- ually called a " million." 16,000,000, or 16 thou- sand thousand, may be called, then, 16 million ; 186,000,000, or 186 g | | thousand thousand, may be called | | n..s 186 million. J| j^AS 186 million, 186,000,000. 432 thousand, 432,000. 186 million 432 thousand and 792, 186,432,792. 799 million and 684, 799,000,684. 896 million and 479 thousand, 896,479,000. Counting from the right, what places do the figures that represent millions occupy? *16. What do we call the first place from the right ? the second place ? the third place ? the fourth, fifth, and sixth places ? the seventh, eighth, and ninth places ? *17. Read the numbers : 982,461,007 111,111,111 2,760,286 698,231,770 232,008,674 16,888,888 842,000,689 11,400,800 769,000,001. A.] Questions on Notation, with Explanations. 5 * 1 8. Express by figures the numbers : Six million seventy-five thousand and four. Three hundred and six million and forty thou- sand. Five million six hundred and seventeen thousand and forty-three. Four hundred and sixty million and twenty- seven. Seventy-three million forty-one thousand eight hundred and nineteen. *19. Express by figures the numbers : Nine hundred million one hundred and seven. Eighty-six million and seven hundred. Thirteen hundred and eighty-six million four hundred and ninety-seven thousand three hundred and sixty-two. 20. Light travels one hundred and eighty-six thousand three hundred and twenty-four miles in a second ; express this number of miles by figures. 21. Sound travels eleven hundred and twenty feet in a second ; express this number of feet by figures. 22. The distance around the earth at the equator is twenty-four thousand eight hundred and ninety-nine miles ; express this number of miles by figures. 23. The distance of the earth from the sun is ninety-five million miles : express this number of miles by figures. 24. The distance of the earth from the moon is two hundred thirty-eight thousand eight hundred 6 Notation, Addition, and Subtraction. [ 1. and forty-eight miles; express this number of miles by figures. 25. During the civil war of 1861 to 1865 two million eight hundred and fifty-nine thousand one hundred and thirty-two men were supplied to the Union army; express this number of men by figures. B. Addition, with Practical Illustrations. *1. How many complete rows of ten each can you make with 32 counters ? how many counters will be left over ? Answer : 3 complete rows of ten, and there will be two extra counters. *2. How many complete rows of ten each can you make with 24 counters ? how many counters will be left over ? *3. How many complete rows of ten counters each can you make with 49 counters? 83 counters? 61 counters? 17 counters? How many extra counters will there be in each of these cases ? *4. How many counters must you have in order to make 5 complete rows and have 3 count- ers over ? 7 rows and 2 counters over ? *5. How many complete rows of ten each can you make with 32 counters and 24 counters, and how many counters will be ** left over ? Answer : It is easy to see, by looking at J J the diagrams given above, that with 32 count- B.j Addition, with Practical Illustrations. 7 ers we should have 3 rows and 2 extra counters ; with 24 counters we should have 2 rows and 4 extra counters ; with 32 and 24 counters we should then have 3 + 2 rows and 2 + 4 extra counters, or 5 rows and 6 counters. *6. How many complete rows of ten counters each can you make with 42 counters + 53 counters? with 13 counters + 22 counters ? How many count- ers will be left over in each case ? *7. Mr. Smith paid 23 cents for a piece of cheese, and 36 cents for some butter ; how much did he pay in all? *8. Mrs. Jones paid 64 cents for veal, and 23 cents for vegetables ; how much did she pay for her dinner ? Let us solve this problem by the aid of counters. JJJJJJ* ( 64 counters = 6 rows of tens + 4 count- | 23 counters = 2 rows of tens + 3 count- ers. In all there are 8 rows of tens and 7 extra counters, or 87 counters ; therefore, 64 + 23 - 87, and Mrs. Jones paid 87 cents for her dinner. 8 Notation, Addition, and Subtraction. [ 1. *9. How many are 334-26 ? 44 + 16 ? } = 33 = 26 = 44 = 16 5 tens + 9 = 59. 5 tens + 10 = 6 tens = 60. *10. Add 35 41 97 23 84 16 80 41 to 43JL8_22561123^228 Answers: 78"591197995399269. *11. How many are 27 + 35? __ 07 In the two incomplete rows there are 7 + 5 or 12 counters, which make one row, with 2 counters left over : all together, then, there are (1 + 5) 6 rows of tens + 2 count- ers, or 62 counters. = 35 Therefore, 27 + 35 = 62. B.] Addition, with Practical Illustrations. 9 *12. How many are 36 + 28 ? }> =36 The extra counters make 1 row and 4 counters : all together, then, there are 6 rows and 4 counters. Therefore, 36 + 28 = 64. h =28 *13. Add 23 24 16 35 83 44 76 31 to 18 67 17 48 29 39 16 49 Answers: 41 ~91 33 83 112 83 92 80. WITHOUT COUNTERS. *14. How many are 74 + 68 ? 74= 7 tens+ 4 units. 68= 6 tens+ 8 units. 74 + 68 = 13 tens + 12 units = 14 tens + 2 units = 142. Or, more briefly, 74 8 and 4 are 12, or 1 ten + 2 over ; we set 68 down the 2 and save the 1 ten. 1 ten (the 142 one that was saved) and 6 are 7 and 7 are 14 tens ; we set down the 14 to the left of the 2, and have 142 for an answer. *15. Add 56 8 and 6 are 14 ; we set down to 28 the 4 and save the 1. 1 (the Answer : 84 1 that was saved) and 2 are 3, 10 Notation, Addition, and Subtraction. [ 1. and 5 are 8 ; we set down the 8 to the left of the 4, and get 84 for our answer. *16. Add to Answers : 25 65 89 98 57 75 84 29 76 56 90 187 132 113 132. *17. How many are 269 + 328? 269 = 2 hundred + 6 tens +9 units. 328 = 3 hundred + 2 tens + 8 units. 269 + 328 -5 hundred + 8 tens-f I 17 units ( or 1 ten 4- 7 units = 5 hundred + 9 tens + 7 units -597. *18. How many are 684 + 767? 684= 6 hundred + 8 tens + 4 units. 767= 7 hundred + 6 tens + 7 units. 684 + 767 = 13 hundred + 14 tens + 11 units = 13 hundred + 15 tens + 1 unit = 14 hundred + 5 tens + 1 unit = 1451. *19. Add 793 184 Write out your to 848 678 work. *20. Add 768 7 and 8 are 15 ; we set down to 857 the 5 and save the 1. 1 and 1625 ^ are 6, and 6 are 12 ; we set down the 2 and save the 1. 1 and 8 are 9, and 7 are 16 ; we set down the 16, and have for an answer 1625. *21. Add 689 972 439 139 to 763 684 698 984 Answers: 1452 1656 1137 1123. B.] Addition, ivith Practical Illustrations. 11 *22. Add 858 900 237 642 to 686 768 508 899 *23. Add 667 8 and 8 are 16, and 6 are 276 22, and 7 are 29 ; we set down 108 the 9 and save the 2. 2 (the 188 2 that was saved) and 8 are 1239 10, and 7 are 17, and 6 are 23 ; we set down the 3 and save the 2. 2 and 1 are 3, and 1 are 4, and 2 are 6, and 6 are 12 ; we set down the 12. We have for an answer 1239. *24. Add 284 326 522 683 355 719 799 793 123 268 655 573 618 400 321 498 1380 1713 2297 2547 687 322 623 60 864 921 888 798 730 777 760 666 231 815 298 208 419 23 402 476 *25. America was discovered by Columbus in 1492, and the War for Independence began 283 years afterwards ; in what year did this war begin ? *26. In July, 1776, the Declaration of Inde- pendence was made ; 87 years afterwards the Bat- tle of Gettysburg was fought . find the date of the Battle of Gettysburg. *27. A butcher bought 3 oxen : the first weighed 12 Notation, Addition, and Subtraction. [ 1. 1214 pounds, the second 1406 pounds, and the third 1384 pounds ; how much live-meat had he in all? 28. How many feet of fence will be required to surround a house-lot, the four sides of which are 126 feet, 233 feet, 126 feet, and 233 feet ? 29. A man paid 250 dollars for a carriage, 225 dollars for a horse, 3 dollars for a whip, and 6 dollars for a robe ; what did they all cost ? 30. How many public school children were there in New England in 1885 if in Maine there were 145,317, in New Hampshire 64,219, in Ver- mont 71,667, in Massachusetts 349,617, in Rhode Island 47,882, and in Connecticut 125,539 ? 31. The following is a summary of the students in Harvard University in October, 1887 : In the Undergraduate Department 1138, in the Divinity School 16, in the Law School 215, in the Scientific School 20, in the Medical School 263, in the Dental School 32, in the Bussey Institution 7, in the School of Veterinary Medicine 26, in the Graduate Department 96 ; how many students were there in all ? 32. The following is a summary of the students in Yale University in October, 1887 : In the Un- dergraduate Department 614, in the Divinity School 117, in the Law School 94, in the Scien- tific School 291, in the Medical School 26, in the School of Fine Arts 58, in the Graduate Depart- ment 69 ; how many students were there in all ? 33. George Washington was born in the year B.] Addition, with Practical Illustrations. 13 1732 ; he was elected President when 57 years old, and died 10 years afterwards ; in what year did he die ? 34. The difference of two numbers is 960,843, and the smaller number is 229,317 ; what is the larger number ? 35. The distance from Washington to Baltimore being 38 miles, thence to Philadelphia 99 miles, thence to New York 90 miles, thence to Worcester 175 miles, thence to Boston 44 miles ; how far is Boston from Washington ? 36. How many days are there in a leap year, there being 7 months of 31 days each, one month of 29 days, and 4 months of 30 days each. 37. How many strokes does a common clock strike in 24 hours? 38. When you go from Boston to San Francisco, if you take the Hoosac Tunnel Route through Buffalo and Detroit to Chicago, the Chicago, Bur- lington, and Quincy road from Chicago to Denver, the Denver and Rio Grande road from Denver to Ogden (Utah), and the Central Pacific road from Ogden to San Francisco, you will travel 150 miles in Massachusetts, 336 miles in New York, 236 miles in Canada, 222 miles in Michigan, 41 miles in Indiana, 227 miles in Illinois, 276 miles in Iowa, 371 miles in Nebraska, 623 miles in Colorado, 473 miles in Utah, 461 miles in Nevada, and 282 miles in California ; how many miles will you travel in all ? 39. When you go from Boston to the City of 14 Notation, Addition, and Subtraction. [ 1. Mexico, if you take the Hoosac Tunnel Route to Chicago, as indicated in the last example, the Chicago, Burlington, and Quincy road from Chi- cago to Kansas City, the Atchison, Topeka, and Sante Fe road to El Paso, and the Mexican Cen- tral road from El Paso to the City of Mexico, you will travel, after leaving Chicago, 263 miles in Illinois, 224 miles in Missouri, 485 miles in Kansas, 190 miles in Colorado, 498 miles in New Mexico, and 1225 miles in Mexico ; what will be the length of your entire journey ? C. Subtraction, with Practical Illustrations. *1. Take away 8 counters from 35 counters and how many will remain ? ^ =35 counters =3 rows + 5 counters. To take away 8 counters we first take away the 5 extra counters, and then going to the next row we keep on taking counters away until in all we have taken away 8 ; counting those that remain, we find that there are 2 rows + 7 counters or 27 counters. 35 less 8 are how many ? C.] Subtraction, with Practical Illustrations. 15 *2. Take away 18 counters from 35 counters and how many will remain ? The 35 counters are shown in the last question. 18 counters = 1 row + 8 counters. After taking away first the 8 counters, and then the 1 row, we find by counting that 17 counters remain. 35 less 18 are how many ? *3. From 82 71 95 76 take 23 45 37 59 Answers: 59 26 58 17 Illustrate by counters. WITHOUT COUNTERS. *4. How many are 48 less 32 ? 48 = 4 tens + 8 units. 32 = 3 tens + 2 units. 48 - 32 =JL ten + 6 units = 16. Answer. *5. From 68 99 84 74 take 42 76 jtt 32 Answers: 26 23 53 42 *6. How many are 53 less 18 ? 53 = 5 tens +3 units. 18 = 1 ten +8 units. We cannot take 8 units from 3 units ; we there- fore take one of the 5 tens and add it to the 3 units. 53 = 4 tens + 13 units. 18 = 1 ten + 8 units. 63 - 18 = 3 tens + 5 units = 35. Answer. 16 Notation, Addition, and Subtraction. [ 1. *7. How many are 97 less 68 ? 97 = 8 tens + 17 units. 68 = 6 tens+ 8 units. 97 - 68 = 2 tens 4- 9 units = 29. Answer. *8. From 77 97 73 36 take 38 ^9 58 28 Answers: 39 48 15 8 *9. A farmer raised 89 bushels of potatoes ; he kept 18 bushels for his family, and sold the rest : how many bushels did he sell ? *10. A man who owed a bill of 96 dollars, paid 20 dollars at one time, and 19 dollars at another ; how much did he then owe ? *11. Mr. Jackson raised 78 dollars' worth of hay above what he needed to feed to his stock ; he sold this hay to the grocer and received in part pay 29 dollars' worth of flour : how much money should he receive ? *12. Mr. Weston started to walk from Ports- mouth to Boston, a distance of 56 miles : he walked 34 miles on the first day ; how many miles had he left to go on the second day ? *13. Mr. Fowler, the mason, brought 73 bricks in his wheelbarrow to finish the arch he was build- ing : he used only 57 of them ; how many had he left? *14. Mr. Knapp's parlor is 97 inches high : the wainscoting above which the room is to be papered is 33 inches high. Into what lengths must the paper be cut ? C.] Subtraction, with Practical Illustrations. 17 *15. Mr. Lanman sent 67 books to be bound: the binder made mistakes in the lettering of 19 of them ; how many were lettered correctly ? *16. 683 less 421 are how many? From 683 = 6 hundred + 8 tens + 3 units, take 421 = 4 hundred + 2 tens + 1 unit. 683-421 = 2 hundred + 6 tens + 2 units = 262. Answer. *17. a. From 896 764 972 541 take 784 451 630 321 Answers: 112 JS13 342 ~220 6. From 468 532 769 497 take 351 412 347 182 *18. a. How many are 684 less 296 ? From 684 = 6 hundred + 8 tens + 4 units, take 296 = 2 hundred + 9 tens + 6 units. We cannot take 6 units from 4 units ; we there- fore take one of the 8 tens and add it to the 4 units, so that the problem will read : From 684 = 6 hundred + 7 tens + 14 units, take 296 = 2 hundred +9 tens-h 6 units. We cannot take 9 tens from 7 tens ; we there- fore take one of the 6 hundred and add it to the 7 tens ; and our problem is now as follows : From 684 = 5 hundred + 17 tens + 14 units, take 296 = 2 hundred + 9 tens + 6 units. 684-296 = 3 hundred + 8 tens + 8 units = 388. Answer, b. Show that 762 less 484 are 278. 18 Notation, Addition, and Subtraction. [ 1. >19. From take Answers : 811 422 623 237 762 478 436 184 389 386 284 252. 20. From take 431 292 973 584 763 184 842 754 *21. From 974 take 783. We may state briefly what we do thus : 3 from 4 gives 1 ; we set down the 1 in the units' 974 column. One of the 9 hundred added to 783 the 7 tens gives 17 tens, and 8 tens from 191 17 tens are 9 tens ; we set down the 9 in the tens' column. 7 hundred from the remaining 8 hun- dred is 1 hundred ; we set down the 1 in the hun- dreds' column. We have, then, 191 for an answer. *22. From 864 726 419 521 take 471 233 126 197 Answers: 393 493 293 324. *23. From 921 take 246. One of the 2 tens added to the 1 unit gives 11 units, and 6 from 11 are 5 ; we set 921 down the 5 in the units' column. One of the 9 hundred added to the remaining 1 675 ten gives 11 tens, and 4 tens from 11 tens are 7 tens ; we set down the 7 in the tens' column. 2 hundred from the remaining 8 hundred are 6 hundred ; we set down the 6 in the hundreds' column. We have, then, 675 for an answer. C.] Subtraction, with Practical Illustrations. 19 *24. From 684 555 911 722 take 296 166 223 333 Answers: ~388 ~389 ~688 389. *25. From 842 take 468. We may state our work very briefly thus : 8 from 12 are 4 ; 842 6 from 13 are 7 ; 468 4 from 7 are 3. 374 *26. From 633 6 from 13 are 7 take 216 l f rom 2 is 1 Answer : 417 2 from 6 are 4. *27. From 846 4 from 6 are 2 take 674 7 from 14 are 7 Answer: 172 6 from 7 is 1. *28. From 811 2 from 11 are 9 take 322 2 from 10 are 8 Answer : 489 3 from 7 are 4. *29. From 600 8 from 10 are 2 take 268 6 from 9 are 3 Answer : 332 2 from 5 are 3. *3O. America was discovered by Columbus in the year 1492 ; how many years ago was that ? *31. The Pilgrims landed at Plymouth in 1620 ; how many years ago was that? how many years after the discovery of America by Columbus ? *32. Harvard College was founded in the year 1636 ; how many years ago was that ? 33. In 1880 London had 3,832,441 inhabitants, and New York only 1,206,590 ; how man^ more inhabitants had London than New York ? 20 Notation, Addition^ and Subtraction. [ 1. 34. From 6942 7123 12111 take 3369 6876 8467 35. If I buy a house for 5875 dollars and sell it for 7100 dollars, do I gain or lose, and how much? 36. The population of the United States in 1840 was 17,069,453 ; in 1880, 49,369,595 ; what was the increase of population during these forty years ? 37. Subtract 1907 from 11442 until nothing re- mains. 38. Benjamin Franklin died in 1790, aged 84 years ; in what year was he born ? 39. If William is 23 years old, in what year was he born ? 40. From the sum of 4936 and 7208 take the sum of 1137, 2065, and 6820. 41. The distance from Boston to Albany by railroad is 200 miles. Suppose one locomotive to have gone 68 miles from Boston towards Albany, and another 95 miles from Albany towards Boston ; how far are they apart ? 42. The larger of two numbers is 987,564,321, and their difference is 14,097,738 ; what is the smaller number ? 43. The larger of two numbers is 842,260,084, and their difference is 179,742,986; what is the smaller number? 44. In 1870 there were in the United States 19,493,565 males and 19,064,806 females; how many more males were there than females ? D.] United States Money. 21 45. In 1880 there were in the United States 25,518,820 males and 24,636,963 females; how many more males were there than females ? 46. From the figures given in the last two ex- amples find how many more males there were in the United States in 1880 than in 1870; how many more females in 1880 than in 1870. D. United States Money. 10 mills = 1 cent. 10 cents = 1 dime. 10 dimes or 100 cents =1 dollar. *1. The sign $ is often used to stand for dollars ; thus, 6 dollars is usually written $6. Eead $12 ; 116 ; 124 ; 1196. *2. Write 48 dollars ; 96 dollars ; 132 dollars ; 2137 dollars ; using the dollar mark, $, in each case. *3. A man bought a sail-boat for $325, and paid $88 for a new set of sails, and $32 for having the boat painted ; how much did the whole cost ? *4. Mr. Kresus had $1000 in the Lydian Five- Cents Savings Bank, but he drew out $432 to pay for a lot of land ; how much was there left in the bank? *5. A dealer bought some flour for $642, and sold it at a gain of $97 ; what did he get for his flour? *6. A man bought a lot of land for $672, and paid $167 for having it graded : he then sold the 22 Notation, Addition, and Subtraction. [ 1. land at a profit of $125 ; how much did he get for the land ? *7. Write 37 dollars and 24 cents. In such a case as this it is customary to write the number denoting cents after the number denot- ing dollars and to separate the two numbers by a period, thus : 137.24. In the same way we write 48 dollars and 79 cents, $48.79 22 doUars and 53 cents, $22.53 22 dollars and 43 cents, $22.43 22 dollars and 33 cents, $22.33 22 dollars and 23 cents, $22.23 22 dollars and 13 cents, $22.13 22 dollars and 03 cents, $22.03. Notice that when the number of cents is less than 10, we fill out the empty tens' place by a zero. Thus we do not write 22 dollars and 3 cents $22.3, but $22.03. *8. Read $49.13 ; $117.88 ; $109.76 ; $104.70 ; $99.04; $98.10; $92.07; $100.01; $417.62; $3189.25 ; $4281.50. Notice carefully that $625 stands for 625 dollars, but that $6.25 stands for 6 dollars and 25 cents. *9. Write in figures : Four hundred and seven dollars. Fifty-six dollars and thirty-seven cents. Two hundred and forty-three dollars and five^cents. Five thousand six hundred and forty dollars and nine cents. Eight thousand seven hundred and eighty-three dollars. D.] United States Money. 23 Eighty-seven dollars and eighty-three cents. Six hundred dollars and eight cents. *1O. 85 cents may be written $0.85. Read 10.33; $0.07; $0.69; $0.38; $0.02. *1 1. Write in the same way 48 cents ; 83 cents ; 78 cents ; 50 cents ; 4 cents. *12. How many cents are there in $8.32 ? Answer : 832 cents. *13. How many cents are there in $6.19? in $41.31 ? in $64.09? in $3142.27 ? *14. Show that 183 cents are equivalent to $ 1.83. 192 cents are equivalent to $ 1.92. 7189 cents are equivalent to $71.89. 3787 cents are equivalent to $37.87. 3601 cents are equivalent to $36.01. *15. Add $37.13 to $43.81. Answer : $80.94. In doing such examples as this, write the num- bers in a column, taking care to write cents under cents and dollars under dollars ; the numbers may then be added as if the periods were not there. *16. Add $38.29 $26.18 $47.82 $32.51 $117.98 $ 21.08 $64.47 $80.33 $139.06 $316.47 $283.32 $4132.07 $ 8.17 $397.09 $ 71.93 $780.98 $ 0.18 $ 31.50 *17. Mrs. Wentworth paid $128 for a chamber- set, $46 for mattresses and bed linen, $48.50 for a carpet, $24 for curtains, and $32.38 for other 24 Notation, Addition, and Subtraction. [ 1. articles needed in her spare chamber ; what was the cost of furnishing the room? *18. A New England farmer wished to move out West, so he sold his farm and his stock and with the proceeds went to Kansas. He sold his two horses for $246, his oxen for $98, his sheep for $136, his pigs for $24.50, and his poultry for $16.48 ; how much did he get in all from the sale of his stock ? *19. On the morning of the 8th of May Mr. Brown had only $186.14 in the bank, but in the course of the day he deposited $112 in bills, and checks amounting to $347.32 ; how much had he then in the bank ? *20. Add A B C 14268.07 $138.06 $6842.16 $6842.16 $291.98 $6843.27 $7999.99 $236.72 $7234.82 $2316.11 $972.36 $6124.86 $7000.98 $842.16 $5555.55 $1119.72 $777.77 $3232.39 $2942.15 $666.66 $9998.99 $3715.35 $545.53 $6849.76 $1872.89 $379.23 $2332.29 $6792.91 $192.98 $6974.84 21. Take the first number in column B, of the last example, from each number in column A : also from each number in column C. Proceed in like manner with each of the remaining numbers of column B. E.] Tables and Questions for Practice. 25 E. Tables and Questions for Practice. TABLE 1. A B C D E F G H I 1 19 18 17 16 15 14 13 12 11 2 29 28 27 26 25 24 23 22 21 3 39 38 37 36 35 34 33 32 31 4 49 48 47 46 45 44 43 42 41 5 59 58 57 56 55 54 53 52 51 6 69 68 67 66 65 64 63 62 61 7 79 78 77 76 75 74 73 72 71 8 89 88 87 86 85 84 83 82 81 9 99 98 97 96 95 94 93 92 91 1. Add 2 to each number in column A of Table 1 ; add successively 3, 4, 5, 6, 7, 8, 9, 10, to the same numbers. [In this case the pupil may be asked : How many are 19 and 2 ? 29 and 2 ? etc. ; 19 and 3 ? 29 and 3 ? etc.] Proceed in like manner with each of the remain- ing columns. 2. Add 2 to each number in line 1 ; add suc- cessively 3, 4, 5, 6, 7, 8, 9, 10, to the same numbers. Proceed in like manner with each of the remain- ing lines. 3. Subtract 2 from each number in column A ; subtract successively 3, 4, 5, 6, 7, 8, 9, 10, from the same numbers. [In this case the pupil may be asked : How many are 19 less 2 ? 29 less 2 ? etc. ; 19 less 3 ? 29 less 3 ? etc.] Proceed in like manner with each of the remain- ing columns. 26 Notation, Addition, and Subtraction. [ 1. 4. Subtract 2 from each number in line 1 ; subtract successively 3, 4, 5, 6, 7, 8, 9, 10, from the same numbers. Proceed in like manner with each of the remain- ing lines. 5. Find the sum of the numbers in each column. 6. Find the sum of the numbers in each line. TABLE 2. A B C D E 1 426 6984 976 8432 19798336 2 684 7697 2864 6798 11604186 3 799 926 1984 1080 3222720 4 231 294 962 18006 55494492 5 700 7698 111 132 167376 6 119 8432 555 10832 178316384 7 294 9012 7006 69834 681649674 8 371 2984 1080 76497 219087408 9 187 6798 9807 984 9275184 10 679 832 16 16214 80453868 11 138 1981 484 1111 8552478 12 291 6462 108 5555 46839760 13 545 1268 132 7006 63201126 14 379 8006 806 9821 24395364 7. Find the sum of the numbers in each column of Table 2. 8. Find the sum of the numbers in each line. 9. Subtract the first number in column A from the first number in column B ; the second number in column A from the second number in column B, and so on to the ends of these columns. E.] Tables and Questions for Practice. 27 1O. Find the number of square miles, the number of inhabitants in 1820, and in 1880, in each of the two groups of states given below ; find also the number of United States soldiers furnished for the civil war of 1861-65. Group A. (New England States.) States. Sq. Miles. Population in 1820. Population in 1880. Soldiers furnished. Maine 29,890 298,335 648,936 72,114 New Hampshire 9,005 244,161 346,991 34,629 Vermont 9,135 235,764 332,286 35,262 Massachusetts 8,040 523,287 1,783,085 152,046 Rhode Island 1,085 83,059 276,531 23,699 Connecticut 4,845 275,248 622,700 57,379 Group B. (Middle States and the District of Columbia.) States. Sq. Miles. Population in 1820. Population in 1880. Soldiers furnished. New York 47,620 1,372,812 5,082,871 467,047 New Jersey 7,455 277,575 1,131,116 81,010 Pennsylvania 44,985 1,049,398 4,282,891 366,107 Delaware 1,960 72,749 146,608 13,670 Maryland 9,860 407,350 934,943 50,316 District of Co- lumbia 60 33,039 177,624 16,872 11. Find the number of square miles, and the number of inhabitants in 1880, in each of the fol- 28 Notation, Addition, and Subtraction. [ 1. lowing groups of states and territories ; find also the number of United States soldiers furnished for the civil war of 1861-65. Group C. States. Sq. Miles. Population in 1880. Soldiers furnished. Virginia 40,125 1,512,565 West Virginia 24,645 618,457 32,068 North Carolina 48,580 1,399,750 3,156 South Carolina 30,170 995,577 - Georgia 58,980 1,542,180 - Florida 54,240 269,493 - Alabama 51,540 1,262,505 2,576 Mississippi 46,340 1,131,597 545 Louisiana 48,420 939,946 5,224 Texas 262,290 1,591,749 1,965 Group D. States. Sq. Miles. Population in 1880. Soldiers furnished. Michigan 57,430 1,636,937 89,372 Wisconsin 54,450 1,315,497 96,424 Ohio 40,760 3,198,062 319,659 Indiana 35,910 1,978,301 192,147 Illinois 56,000 3,077,871 259,147 Kentucky 40,000 1,648,690 79,025 Tennessee 41,750 1,542,359 31,092 Minnesota 79,205 780,773 25,052 Iowa 55,475 1,624,615 76,309 Missouri 68,735 2,168,380 109,111 Arkansas 53,045 802,525 8,289 E.] Tables and Questions for Practice. 29 Group States. Sq. Miles. Population in 1880. Soldiers furnished. Nebraska 76,185 452,402 3,157 Kansas 81,700 996,096 20,151 Colorado 104,500 194,649 4,903 Nevada 112,090 62,265 1,080 California 155,980 864,694 15,725 Oregon 94,560 174,768 1,810 Group F. (Territories.) Territories. Sq. Miles. Population in 1880. Soldiers furnished. Alaska 577,399 35,000 Arizona 113,020 40,440 - Dakota 149,100 135,177 206 Idaho 84,800 32,610 Montana 146,080 39,159 - New Mexico 122,580 119,565 6,561 Utah 84,970 143,963 - Washington 69,180 75,116 964 Wyoming 97,890 20,789 12. What is the difference between the number of square miles in Texas, and the sum of the square miles in New York, Pennsylvania, Virginia, Wisconsin, and South Carolina ? 13. What was the whole number of inhabitants in the United States (including the District of Columbia and the Territories) in 1880 ? 30 Notation, Addition, and Subtraction. 14. In addition to the number of United States soldiers sent to the civil war of 1861-65, from the states and territories given above, there were 3,530 Indian troops and 93,441 colored troops. How many soldiers were furnished in all ? 15. How many more soldiers were sent by the Middle States than by the New England States ? 16. How many more inhabitants were there in the New England States in 1880 than in 1820 ? 17. How many more inhabitants were there in the Middle States and the District of Columbia in 1880 than in 1820 ? 18. How many more square miles are there in the largest state than in the smallest ? how many more inhabitants? 19. How many more square miles are there in Colorado than in Pennsylvania ? 20. How many more inhabitants are there in Pennsylvania than in Colorado ? 21. What is the difference between the number of square miles in Texas and the total number of square miles in the New England States ? SECTION II. MULTIPLICATION AND DIVISION. The portions of this section that are preceded by a star (*) have been drawn from Section VII. of the Revised Edition of Colburn's First Lessons, and may be omitted at the discretion of the teacher by pupils who have studied them before. A. Multiplication : Examples and Problems, with Remarks and Explanations. *1. What will four pounds of coffee cost at thirty-two cents a pound ? Four pounds of coffee will cost four times as much as one pound, or four times thirty-two cents. 32 = 3 tens + 2 units. 4 times 32 = 4 times 3 tens + 4 times 2 units = 12 tens -I- 8 units = 120 + 8 = 128. Four pounds of coffee, then, will cost 128 cents or 11.28. *2. Mr. Wood, the grocer, makes a profit of 21 cents on every bag of meal that he sells ; what will be his profit on 4 bags ? *3. How many are 5 times 41 ? *4. If beefsteak is 23 cents per pound, what will 3 pounds cost ? 32 Multiplication and Division. [ 2. *6. How many are 4 times 52 ? 52 = 5 tens + 2 units. 4 times 52 = 4 times 5 tens + 4 times 2 units = 20 tens + 8 units = 200 + 8 = 208. To save space we may write our work thus : Or, still more briefly : f 52 We may briefly de-J 4 scribe our work as fol- 208 lows : 4 times 2 are 8 ; 52 8 = 4x2 units. 200 = 4x5 tens. OQQ _ x co we write down the 8 : 4 times 5 are 20 ; we write down the 20 to the left of the 8 and get 208 for an answer. *6. Multiply 42 61 53 82 by JB _4 _3 _5 126 *7. When we have to perform an example in multiplication it is sometimes convenient to call the number which we are asked to multiply the MULTIPLICAND, the number by which we are to multiply the MULTIPLIER, and the result obtained by our work the PRODUCT. Thus, in example 5 the multiplicand is 52, the multiplier is 4, and the product is 208. Name the multiplicand, the multiplier, and the product in the examples of 6. *8. If one barrel of sugar costs 19 dollars, how much will 5 barrels cost? The answer will be 5 times 19 dollars, and we are to find out how many dollars this will make. 19 = 1 ten +9 units. A.] Multiplication. 33 5 times 19 = 5 times 1 ten + 5 times 9 units = 5 tens 4- 45 units = 5 tens + 4 tens + 5 units = 9 tens + 5 units -95. Five barrels of sugar, then, at 19 dollars a bar- rel will cost 95 dollars. *9. If a railway train goes 36 miles in an hour, how far will it go in 3 hours ? *1O. How many are 5 times 64 ? *11. Mr. Eeardon gets 25 cents an hour for his work ; how much does he get for 8 hours' work ? how much for 4 hours' work ? *12. How many are 6 times 78 ? 78 = 7 tens + 8 units. 6 times 78 = 6 times 7 tens + 6 times 8 units = 42 tens + 48 units = 42 tens + 4 tens + 8 units = 46 tens + 8 units = 460 + 8 = 468. Answer. To save space we may 78 write our work thus : 6 48 = 6x8 units. 420 = 6x7 tens. 468-6x78. *13. a. Multiply 29 by '_7 63 = 7x9 units. 140 = 7x2 tens. Answer: 203 = 7x29. 34 Multiplication and Division. [ 2. 6. How many are 36 x 5 ? 43 x 8 ? 57 x 6 ? *14. Multiply 84 ^ Instead of writing down by 6 the 24 and the 480 sepa- 24 v rately, we may add them 480 together in our heads as TT7 we go on and say: 6 times Answer: 504 ) A 4 are 24, or 2 tens and 4 units ; we set down the 4 and save the 2 tens to add in with other tens. 6 times 8 tens are 84 48 tens ; these with the 2 tens saved over 6 make 50 tens, and we write down the 50 to 504 the left of the 4. *15. Multiply 96^1 4 times 6 are 24; set by 4 I down the 4 and save the 2. 4 times 9 are 36 ; add in the 2 that were saved and set down the result, 38, to the left of the 4. *16. Multiply 85^ 7 times 5 are 35; set by 7 y down the 5 and save the 3. 7 times 8 are 56 ; add in the 3 and set down the result, 59, to the left of the 5. *17. Multiply 49 28 37 65 by _T_ _8 _5 _j> 343 "224 185 585. Name the multiplicand, the multiplier, and the product in each case. *18. What will six pounds of chocolate cost at 38 cents per pound ? *19. At 19 cents a pound, what will a roasting piece of beef weighing 8 pounds cost ? 595 I 3. ' ir A.] Multiplication. 35 *2O. If 7 men can dig a ditch in 24 days, how long would it take 1 man to do it ? *21. If one barrel of vinegar holds 32 gallons, how much vinegar can be put into 6 barrels ? how much into 8 barrels ? *22. If one barrel of flour contains 196 pounds, how many pounds will there be in four barrels ? 196 = 1 hundred + 9 tens + 6 units. 4x196 = 4x1 hundred + 4x9 tens + 4x6 units = 4 hundred + 36 tens + 24 units = 400 + 360 + 24 -784. Four barrels of flour, then, will contain 784 pounds. We may save space by arranging our work thus : 196 24 = 4x6 units. 360 = 4x9 tens. 400 = 4x1 hundred. ~784 = 4x196. *23. How many are 3 times 218 ? 218 _3 24 = 3x8 units. 30 = 3x1 ten. 600 = 3x2 hundred. Answer: "654 = 3x218. Multiply 341 208 121 610 by ' 6 3 8 6 36 Multiplication and Division. [ 2. *24. Mr. Blake paid three dollars and twenty- five cents apiece for six chairs ; how much did the whole cost him ? *25. If one barrel of flour is worth seven dollars and forty cents, how much are six barrels worth ? *26. Multiply 243 ^ Instead of writing down by 8 the 24, the 320, and the 24 I 1600, separately, we may 320 f ^^ them together as we 1600 g on with our work, and 1044] sav: ^ times 3 are 24; we set down the 4 and save the 2 tens. 8 times 4 tens are 32 tens, and these with the 2 tens saved over make 34 tens, or 3 hundred and 4 tens ; we set down the 4 tens and save the 3 hundred. 8 times 2 hundred are 243 16 hundred, and these with the 3 hundred 8 make 19 hundred, which we set down to the 1944 left of the figures that we already have. In -like manner Multiply 416 123 321 643 by 7452 *27. Multiply~134by7T~ After you have learned how to do such examples as those of No. 26 correctly and quickly, you may tell as briefly as possible what you actually do in multiplying two numbers together, thus : 134^1 7 times 4 are 28 ; we set down the 8 and 7 I carry the 2. 7 times 3 are 21, and 2 are 938 [ 23 ; we set down the 3 and carry the 2. 7 J times 1 are 7, and 2 are 9 ; we set down the 9. A.] Multiplication. 37 *28. Multiply 2196 1419 6021 8206 by 5 368 Answer: 10980 *29. 10 times 36 are how many ? We know that 10 times 36 are the same as 36 times 10. Therefore 10 times 36 = 36 tens or 360. 10 times 49 are how many ? Answer : 490. 10 times 64 are how many ? Answer : 640. 10 times 90 are how many ? Answer : 900. 10 times 16 are how many ? Answer : 160. 10 times 83 are how many ? Answer : 830. You have just multiplied different numbers by 10 ; writing down each of these numbers, and un- derneath it 10 times the number we have : 49 64 90 16 83 490 640 900 160 830 What do you do, then, when you multiply a number by 10 ? *30. 20 times 37 are how many ? 20x37 = 10x2x37 = 10x74 = 740. Answer. *31. Multiply 68 49 36 51 by ' 30 40 50 70 Answers: 2040 1960 1800 3570. -Notice that the first answer may be obtained by placing a zero after 3 times 68, the second by placing a zero after 4 times 49, the third by placing a zero after 5 times 36, and so on. *32. a. Multiply 46 46 46 46 by 100 200 300 700 Answers: 4600 9200 13800 32200. 88 Multiplication and Division. [ 2. Notice that the first answer may be obtained by writing two zeros after 1 x 46, the second by writ- ing two zeros after 2 x 46, etc., etc. 6. Multiply 64 82 76 98 by 600 500 800 400 Answers: 38400 41000 60800 39200. *33. How many are 24 times 36 ? 36 24 = 20+ 4. 144 = 36 x 4. 720 = 36x20. 864 = 36x24. *34. How many are 18 times 23 ? 23 _18 = 10+ 8. 184 = 23 x 8. 230 = 23x10. 414 = 23x18. *35. If butter is worth 28 cents a pound, what is the value of a tub containing 35 pounds ? Answer : 980 cents, or $9.80. *36. What is the cost of 24 hoes at 58 cents each ? *37. Mr. Jones bought 32 hens at 96 cents apiece ; how much did he pay for them all ? He then sold 21 of them at $1.10 apiece ; how much did he get for these ? *38. If a man, who is draining a field, can dig 48 feet of ditch in one day, how much can he dig in 22 days ? A.] Multiplication. 39 *39. I bought 42 boxes of soap each containing 18 bars ; how many bars of soap were there in all? *40. Multiply 98 69 73 46 by 47 J59 13 54 *41. After you have learned to multiply cor- rectly, you may shorten your written work as much as possible. Multiply 49 by 27. 49 49 27 or, more briefly, 27 343-49x 7. 343 _980 = 49x20. 98 1323 = 49x27. 1323. You may omit the zero after the 98, if you are careful to show that the 98 represents 98 tens, by moving the number one step to the left, so that the 8 may stand in the column of tens and not in the column of units. 2 times 6 are 12 ; we set down the 2 and carry the 1. 2 times 7 are 14, and the 1 make 15. 5 times 6 are 30 ; we set down the zero in the col- *42. Multiply 76 by 52 380 3952 umn of tens and carry the 3. 5 times 7 are 35, and the 3 make 38, etc., etc. *43. Multiply 71 48 64 49 83 by 19 12 T2 _38 25 Name the multiplicand, the multiplier, and the product, in each case. 196 48 or, more briefly, 1568 = 196 x 8. 7840-196x40. 40 Multiplication and Division. [ 2. *44. If a barrel of flour contains 196 pounds, how many pounds are there in 48 barrels ? 196 48 1568 784 9408-196x48. 9408. *45. If coal is worth four dollars and sixty-four cents a ton by the cargo, what will a cargo of three hundred and twenty-seven tons cost ? If one ton costs 464 cents, 327 tons will cost 327 times 464 cents. We are, then, to multiply 464 by 327. 327-300 + 20 + 7. 464 464 327 327 3248 = 464 x 7. or, 3248 9280 = 464 x 20. more 928 139200 = 464x300. briefly, 1392 151728 = 464x327. 151728 The whole cargo will cost, then, 151728 cents, or 1517 dollars and 28 cents. *46. a. Multiply 318 by 425 Multiply by Answer : J23 417 196 391 406 719 Answer : b. Multiply by 1590 636 1272 3654 406 2842 135150. 873 i 614 ] 291914. 803 516 A.] Multiplication. 41 *47. a. Multiply 456 by 204. 204 = 2 hundred + tens + 4 units. -200 + 4. 456 or, 456 204 more 204 1824- 456 x 4. briefly, 1824 91200-456x200. 912 . 93024-456x204. 93024. b. Multiply 471 822 583 649 by 306 507 109 408 Answer: 144126 *48. In multiplying one number by another, you have learned to write down the number to be multiplied (the " multiplicand ") over the multi- plier ; and, beginning at the right, to multiply the multiplicand by each figure of the multiplier sepa- rately ; and to write down your results one over another, so that the right-hand figure of each shall be directly under that figure of the multiplier which was used to obtain this result. By adding together the results thus arranged you get the product required. Thus : Multiply 8164^ In this case 8164 is the multiplicand and 4821 is by 4821 8164 16328 65312 32656 39358644 the multiplier. The re- . suit obtained by multiply- ing 8164 by the 1 is 8164, and is set down so that its right-hand figure 4 is di- rectly under the 1. In the same way the result, 16328, obtained by multi- 42 Multiplication and Division. [ 2. plying 8164 by the 2, is set down so that its right- hand figure is directly under the 2, etc., etc. Multiply 3142 7162 6824 5691 by 693 8237 9036 2007 49. Show that 93 multiplied by 587 is the same as 587 multiplied by 93. In this example and in others like it the product is the same, whether the first number is chosen as the multiplicand and the second as the multiplier, or the second is chosen as the multipli- cand and the first as the multiplier. 50. Find the product of 1728 and 132, first by using 1728 as the multiplicand and 132 as the multiplier, and second by using 132 as the multi- plicand and 1728 as the multiplier. Which number, the larger or the smaller, should you select for a multiplicand in order to get the product with the least labor ? 51. Find the product in each of the following cases, selecting for your multiplier in each case the number which enables you to get your result with the least labor : - 29x384 1142x38 17x684 497x36 79x11238 45x125 52. Find the product of 360 and 27. 360 = 10 x 36, therefore 360 x 27 = 10 x 36 x^27. The work may be arranged as follows : Either Or 27 360 360 27 162T 252 81 72 9720 9720 A.] Multiplication. 43 53. Find the product of ( 13600 and 64 and (6) of 96000 and 127. (a) 13600 (6) 127 64 96000 544~ ~762~~ 816 1143 870400. 12192000. 54. Find the product of (a) 386000 and 86. (c) 124000 and 42. (6) 96 and 121000. (d) 7900 and 132. 55. The railroad from Providence to Worcester is 44 miles long; there are 1760 yards in a mile; how many yards of track are there between Provi- dence and Worcester? 56. There are 480 sheets in a ream of paper ; how many sheets are there in 605 reams ? 57. A man owing $105,760 gives in payment 180 acres of land worth $65 an acre and $5000 in money. How much remains unpaid ? 58. How much more do 112 cords of wood at $5.25 a cord cost than 75 yards of cloth at $5 a yard? 59. How many miles has a railroad conductor traveled in 3 years who has gone over the road from Louisville to Nashville 185 miles once each day ? 60. A flour dealer bought 42 barrels of flour for $210. He sold of it at $4.75 a barrel, and the remainder at $6 a barrel. How much did he gain ? 44 Multiplication and Division. [ 2. 61. How much will 17 bales of cloth cost at 22 cents a yard if each bale contains 50 pieces and each piece contains 38 yards ? 62. I bought 350 acres of land at $17.60 an acre and sold the whole at $16 an acre. Did I gain, or lose, and how much ? 63. A man bought 3 tons of hay at $18.50 a ton, 12 barrels of apples at $3.50 a barrel, and a suit of clothes worth $50. In payment he gave 17 barrels of flour at $4.75 a barrel and the bal- ance in money. How much money did he give ? 64. How much less is 608 times 3607 than 6075 times 10476? 65. Find the cost of each of the following items and add the results together : 42 bushels of turnips at $0.37 per bushel. 213 pounds of butter at $0.23 per pound. 126 pounds of coffee at $0.33 per pound. 24 boxes soap at $4.25 per box. 12 pitchers at 42 cents apiece. 16 doz. lamp chimneys at 80 cents per doz. 450 Ibs. of rice at 9 cents per Ib. 66. Find the cost of the articles which Mr. Slade bought for his store when he was last in Boston. He bought : 60 yards of nun's veiling at 75 cents per yard. 4 doz. pair hose at 50 cents a pair. 72 yds. sateen at 37 cents a yard. 18 doz. towels at $3 per doz. 200 spools sewing silk at 8 cents per spooL A.] Multiplication. 45 67. The number of full pages in Hawthorne's Wonder- Book is 178 ; the average number of words on a page is 307. How many words are there in all ? 68. During the year ending June 30, 1887, the United States Post Office Department bought 25,500 leather mail bags at an average cost of $264 a hundred ; what was the entire cost ? 69. In 1880 the population of Cleveland, Ohio, was 160,142, and the debt of the city (amount of money owed by it) was such that all of it could have been paid if each inhabitant had contributed 125.45 ; what was the debt? 70. Below is given the population and debt per person * in 1880 of each of the nine largest cities of the United States ; what was the entire debt of each city ? Population. Debt per person. 1 New York, N. Y. 1,206,299 $90.69 2 Philadelphia. Pa. 847,170 19.18 3 Brooklyn, N. Y. 566,663 67.13 4 Chicago, 111. 503,185 25.42 5 Boston, Mass. 369,832 77.90 6 St. Louis, Mo. 350,518 65.18 7 Baltimore, Md. 332,313 81.55 8 Cincinnati, Ohio. 255,139 86.00 9 San Francisco, Cal. 233,959 13.12 * By debt per person is meant the amount that each inhabitant would have owed if the entire debt had been divided equally among all the inhabitants. 46 Multiplication and Division. [ 2. B. Bills. NOTE. On Dec. 11, 1887, John H. Brown bought of Jones, Smith & Co., of Chicago, 111., 24 copies of the Household Edition of Long- fellow's Poems at $1.17 a copy, and 36 copies of Colburn's Arithmetic at 29| cents a copy. He received with the books the following BILL : Mr. John H. Brown CHICAGO, ILL., Dec. 11, 1887. JONES, SMITH &, CO., Dr. 24 Longfellow's Poems, Hid Ed., @ $1.17 86 Colburn's Arithmetic, " 29\c. $28 10 08 71 $88 79 On Dec. 24 Mr. Brown took the bill, with the money that he owed, to Jones, Smith & Co.'s cashier, H. M. Twitchell, and received the bill back again, RECEIPTED, as shown below. CHICAGO, ILL., Dec. 11, 1887. Mr. John H. Brown &0 JONES, SMITH &, CO., Dr. $4 Longfellow's Poems, If'ld Ed., @ $1.17 86 Oolburn's Arithmetic, " %9\c. 08 7 JL 79 Received payment, JONES, SMITH & Co. Dec. 24, 1887. per H. M. TWITCHELL. $28 10 B.] Bills. 47 1. Copy and complete the following bills : A. BOSTON, MASS., DEC. 22, 1887. Mr. A. B. Cole DEE &, ELA, Dr. 10% Rolls of Paper, @ 50c. 3 " " Frieze, " 75c. 56 Feet of Moulding, 8c. Hanging 11 Rolls of Paper, " 37\c. " 3 " frieze, " 37\c. " 56 Ft. of Moulding, 3c. Taking off Paper . * 00 Car Fares ..... . . 48 Received Payment, DEE & ELA, B. CAMBRIDGE, MASS., Jan. 2, 1888. Mr. A. B. Jones &0 THE CAMBRIDGE GAS-LIGHT COMPANY, Dr. For Gas consumed during the quarter ending this day .... 21500 cubic feet at 20 cents per hundred, Discount if paid before Jan. 26, 1888, at 25 cents per thousand Received payment for the Company, ADOLPH VOOL. Date, Jan. 12, 1888. 48 Multiplication and Division. [ 2. Mr. C. L. Wood a BOSTON, MASS., Sept. 19, 1884. Of JOEL GOLDTHWAIT &, CO. 26 yds. Art Kidderminster, @ Sewing and Fitting, " We. 26 yds. Lining, " 12\c. Paid Oct. 1, 1884. JOEL GOLDTHWAIT & Co. per A. M. LOOMIS. D. CAMBBIDGEPORT, MASS., Oct. 16, 1886. Mr. O. F. Ambrose fto GEORGE F. RICKER & CO., Dr. Sept. Oct. 8 Taking up 25 yds. of Carpet, 2c. 8 Cleaning 25 yds. of Carpet, @ 4*. 8 " 1 Rug 8 Laying 28 yds. of Carpet, @ J^c. 8 " 25 " " " " 4c. 18 Renovating 8 pairs of Pillows, @ $1.00. 18 Washing 8 Pillow Ticks, @ 25c. 15 Credit by check received Balance due, B.] Bills. 49 E. BOSTON, Oct. 2, 1887. N. H. Cooler, Esq. Bought of L. T. PRICE &/ CO., IMPORTERS AND GROCERS. 25 Ibs. Coffee, @ 35c. 8 " Tea, " 75c. 1 doz. bottles Salad Oil, @ 75c. per bottle. 3 " boxes of Soapine, @ lie. per box. 9 " Sapolio, @ 96c. per doz. lt\ Ibs. Royal Baking Powder, @ lf.2c. 18 boxes Gelatine, 44 13c. 24 Ibs. Macaroni, " 13c. 16 " Spaghetti, " We. 8 doz. cans Tomatoes, @ $1.60 per doz. 17 Ibs. Rice, @ 8c. 17 boxes Silver White, u 8c. 16 Ibs. Am. Oatmeal, " 4\c. 24 " Rye Meal, " $& 29 " Granulated Sugar " 7ic. 16 " Pearl Tapioca, " Pl c - 12 gals. Dark Molasses, " ^c. 15 Ibs. Am. Chocolate, " We. 13 " Cream o j Tartar, " 62c. \ Ib. B. Pepper, We. 3 qts. S. S. P. Olives, u RQc j Ib. Ground Mustard, " 40c " Received payment, L. T. PRICE & Co. fy S. L. T. 50 Multiplication and Division. [ 2. F. BOSTON, MASS., Oct. 4, 1887. Messrs. Selwood $ Sons Ett ftcct. but!) BABK, STEELE & CO., Dr. 188 June 4 140 tons Steel Rails, @ $82.00 u 15 28 doz. Axes, 9.25 July 5 175 tons Zinc Dross, " 42-00 Aug. 20 487 Ibs. Zinc, " *05\ Sept. 80 814 cwt. Lead, 6.85 Cr. July 2 500 bbh. Flour, " 4M " 17 825 bu. Wheat, .95 Aug. 24 Draft on New York . 875 00 Sept. 8 117 SharesofR.R. Stock, @ $85.00 Balance due, . Received payment, BARR, STEELE & Co. by JOHN SMALL. 2. Make out a bill dated to-day against the City of Boston for 15 J hours of work done by yourself at 24 cents an hour. 3. Make out a bill for each of the following cases : a. Cambridge, Mass., Dec. 15, 1887. O. B. French made for Rufus Bullock 1056 ft. of board walk at 17i cents a foot. Paid Dec. 20, 1887. b. Boston, Mass., Apr. 27, 1885. Houghton, C.] Division. 51 Mifflin & Co. bound for B. O. Peirce 25 volumes of the Waverley novels at $1.25 a volume. Paid Apr. 27, 1885. c. Chicago, 111., June 1, 1888. James Brown owes John Smith, stable-keeper, for board of horse from May 1 to June 1, 1888, $26.52 ; for hack, May 2, $1.50 ; for hack, May 4, $3.00 ; for use of carryall, May 10, $1.50; for 1 halter, May 15, $1.25. Paid May 15, $5.00, May 25, $10.00, and the balance on June 10. d. Indianapolis, July 1, 1884. On June 24, 1884, A. S. Jones bought of Richardson & Bacon 8 tons of Excelsior coal at $6.00 a ton, and 12 tons of Draper Furnace coal at $5.75 a ton ; the charge for putting into the cellar was 25 cents a ton. C. Division: Examples and Problems, with Remarks and Explanations. *1. If one barrel of apples costs 3 dollars, how many barrels can you buy for 96 dollars ? We are to divide 96 by 3. 96 = 9 tens + 6. = 3 tens + 2 -32. You can, then, buy 32 barrels of apples, at 3 dollars a barrel, for 96 dollars. Divide 84 by 4 ; 48 by 2 ; and 63 by 3. *2. A man, who wished to pay a bill of 2 dollars and 48 cents in a neighboring town, bought the 52 Multiplication and Division. [ 2. money's worth of 2-cent stamps, and sent the stamps in a letter to his creditor; how many stamps did he buy? We are to divide 248 by 2. 248 = 2 hundreds + 4 tens + 8. 248-^2 = 2 hundreds -r 2 4-4 tens-f2 + 8-r2 = 1 hundred + 2 tens + 4 = 124. Answer. Divide 369 by 3 ; 484 by 4 ; and 624 by 2. *3. If you divide 92 counters among 4 boys, how many will each receive ? 92 counters =9 rows + 2 counters. There are evidently rows enough to give each boy 2 full rows ; now dividing the remaining 1 row and 2 counters, or 12 counters, among the 4 boys, each will receive 3 counters. Therefore, each boy will receive in all 2 rows + 3 counters, or 23 counters. *4. If you divide 87 counters among 3 girls, how many will each receive ? * I )> 87 counters =8 rows +7 counters. I C.] Division. 53 There are evidently rows enough to give each girl 2 full rows; now dividing the remaining 2 rows and 7 counters, or 27 counters, among the 3 girls, each will receive 9 counters. Therefore, each girl will receive in all 2 rows 4- 9 counters, or 29 counters. *5. Gertrude, Herbert, and Edith went out chestnutting together, agreeing to divide equally all that they should find: when they reached home they counted out their chestnuts into heaps of ten each, and found that they had 8 heaps and 4 chest- nuts over, that is, 84 in all; how many chestnuts should each receive? If you solve this problem by the aid of counters, using a counter to represent a chestnut, you will find that each should receive 28 chestnuts. *6. Mr. Smith, Mr. Jones, and Mr. Robinson cultivated together a vegetable garden : beside supplying their families with vegetables, they sold 57 dollars' worth and divided the money ; how much did each receive? *7. When we have to perform an example in division, it is sometimes convenient to call the number which we are asked to divide the DIVIDEND, the number by which we are to divide the DIVISOR, and the result obtained by our work the QUOTIENT. Thus, in Example 3 the dividend is 92, the divisor is 4, and the quotient is 23. Name the dividend, the. divisor, and the quotient in the Examples of 1 and 2 ; multiply the divisor by the quotient in each case and see if your result is the same as the dividend. 54 Multiplication and Division. [ 2. *8. Divide 96 by 4. 96 = 9 tens + 6 units. 4 goes into 9 tens 2 tens times and leaves an extra ten, which added to the 6 units makes 16 units. 4 goes into 16 units 4 times; therefore 4 goes into 96 2 tens times + 4 times or 24 times. We can save space by arranging our work thus : 4)96(20 + 4. 80 = 4x20. 16 remainder. 16 = 4x4. remainder. We may say 4 goes into 9 tens 2 tens times; we set down the 2 tens to the right of the dividend and subtract 2 tens times 4 or 80 from the dividend. The remainder is 16. 4 goes into 16 4 times ; we set down the 4 to the right of the dividend and subtract 4 times 4 from our 16. The remainder is nothing. Our quotient, then, is 20 + 4 or 24. In this example which is the dividend ? which the divisor ? which the quotient ? *9. Divide 78 by 3. 3)78(20 + 6. " 60 = 3x20. 18 = remainder. 18 = 3x6. Divide 52 by 4. 4)52(10 + 3. 40 " 12 12 = remainder. Answer : 26. Answer : 13. Divide 36 by 2 ; 54 by 3 ; 72 by 4 ; and 95 by 5. Name the dividend, the divisor, and the quotient, in each case ; multiply the divisor by the quotient and compare the result with the dividend. C.] Division. 55 *10. Divide 532 by 4. 4)532(100 + 30 + 3. 400-4x100. 4 goes into 5 hundred 1 hundred times ; we set down 100 in the quotient = remainder. , . -i i i i 190 ji OA land subtract 1 hundred f times 4 or 400 from 532 12 = remainder. 12 = 4x3. and get 132 for a re- mainder. 4 into 13 tens ) goes 3 tens times ; we set down 30 in the quotient and subtract 30 times 4 or 120 from 132 and get 12 for a remainder. 4 into 12 goes 3 times ; we set down the 3 in the quotient and subtract 3 times 4 or 12 from 12 and get nothing for a remainder. Our quotient, then, is 100 + 30 + 3, or 133. Divide 423 by 3 ; 738 by 6 ; 314 by 2 ; and 854 by 7. *11. Divide 156 by 4. 4)156(30 + 9. 120=7x80. 4 is not contained in 1 ; we therefore put the 1, which stands for 1 hun- 36 = remainder. \ , , . , - . . Q A Q I c * re d> with 5 tens making OO 4 X y. l-ir. 4*1 -ir, 15 tens. 4 into 15 tens J goes 3 tens or 30 times ; we write the 30 in the quotient and subtracting 30 times 4 or 120 from 156 we get 36 for a remainder. 4 into 36 goes 9 times ; we write the 9 in the quotient and subtracting 9 times 4 from 36 we get nothing for a remainder. Our quotient, then, is 30 + 9, or 39. Divide 432 by 6 ; 801 by 9 ; 336 by 4 ; and 525 by 7. 56 Multiplication and Division. [ 2. *12. If a bunch of fire-crackers costs 8 cents, how many bunches can you buy for $1.28? how many bunches for |3.36 ? *13. If a pound of brown sugar costs 9 cents, how many pounds can you buy for $2.34 ? how many pounds for |9.36 ? *14. A man bought 8 acres of land for $912; what was that an acre ? *15. How many miles a day must an Atlantic steamer make, if she is to cross the ocean, a dis- tance of 2870 miles, in 7 days? how many miles a day if she is to cross in 10 days ? *16. If 4 quarts make a gallon, how many gal- lons are there in 376 quarts ? how many gallons in 916 quarts ? *17. Sarah wants very much a fine writing-desk for which the dealer asks $3.87. If she gets 9 cents a quart for picking berries, how many quarts must she pick in order to earn enough to buy the desk? *18. Divide 792 by 6. 1 3 2 6)792 100 We may shorten our 6)792 600 30 work by omitting some 6 192 ^ the figures, thus : jg 180 18_ 12 12 12 12 Answer : 132. We may also write all the figures of our quo- tient on one line, as in the following : C.] Division. 57 Divide 833 by 7. 7)833(119. 7 13 7 Divide 968 by 4 4)968(242.* 8 16" 16 63 8 63 8 Divide 855 by 9 ; 784 by 8 ; and 795 by 5. *19. If a barrel of flour costs 8 dollars, how many barrels can be bought for 184 dollars? *2O. The 4.30 express from Boston to New York makes the distance of 234 miles in 6 hours ; how many miles an hour must the train cover? *21. How many yards of cotton cloth at 9 cents a yard can be bought for I486 ? *22. George gets 9 cents an hour for weeding onions; how long will it take him to earn $1.35? *23. Where the divisor is a small number, it is usual to do in one's head a large part of the work of dividing. a. Divide 725 by 5. 5)725 \ 5 into 7 goes once with a re- > mainder 2 ; we set down the 1 145 = quotient. \ Al - ,. ; under the 7 and put the re- mainder with the next figure of the dividend, mak- ing 22. 5 into 22 goes 4 times, with a remainder 2 ; we set down the 4 as the second figure of the quotient, and put the remainder with the next fig- ure of the dividend, making 25. 5 into 25 goes 5 times, with no remainder. 5, then, is the last fig- ure of the quotient, and 725 -r 5 = 145. 58 Multiplication and Division. [ 2. b. Show that 595-^-5 = 119; 552-.-4 = *24. a. Divide 632 by 4. 4)632 ) 4 into 6 goes once with 2 over. 4 into > 23 goes 5 times with 3 over. 4 into 32 ' goes 8 times with no remainder. 6. Divide 185 by 5. 5)185 } 5 into 1 will not go; 5 into 18 goes ( 3 times with 3 over. 5 into 35 goes 7 o7 \ ,. / times. c. Show that the work is correct in each of the following examples : 6)192 7)651 4)1264 5)7285 9)374504508 32 ~93 316 1457 41611612 *25. The distance from London to Edinburgh by the Great Northern Railway and its connec- tions is 405 miles ; how many miles an hour must the daily express, which covers the distance in 9 hours, make? *26. Where the divisor is large, it is necessary to write out in full most of the work. Divide 496 by 16. Divide 322 by 14. 16)496(31. 14)322(23. 48 28 ~~16 ~42 16 42 NOTE. Where the successive steps are set down, as in *26, the operation is called long division; where only the result is set down, as in *23 and *24, the operation is called short division. *27. Sometimes we can tell only by trial what the different figures of the quotient are. Divide 2001 by 29. C.] Division. 59 29)2001(69. 174 261 261 Here, in order to get the first figure of the quotient, we have to find, by trial, how many times 29 is contained in 200. If we try 7 we find that 7 x 29 = 203 ; therefore 7 is too large and we try 6. In the same way we have to find Ity trial how many times 29 is contained in 261. Show that 1862 -r 38 = 49, and that 4446 -r 78 = 57. 28. How long will it take a train of cars which goes at a rate of 29 miles an hour to go 2842 miles ? 29. I bought 127 bushels of potatoes for $95.25 ; how much was that a bushel ? 30. There are 12 inches in a foot, and 63360 inches in a mile ; how many feet are there in a mile ? 31. How many barrels of flour at 15.75 a bar- rel can I buy for $86.25 ? 32. If 29694 dollars were to be equally divided among 202 soldiers, how many dollars would each receive ? 33. A man bought a house for $1275, which he paid for at the rate of $15 a month ; how long was he in paying for the house ? 34. The expenses of a picnic party of 16 men and 14 ladies were $1.60 each. The men paid all the expenses. How much did each man pay ? 35. I bought 13 chests of tea, each chest con- taining 18 Ibs., for $152.10. How much was the tea per pound ? 60 Multiplication and Division. [ 2. 36. I bought 173 barrels of sugar for 13633, and sold it for $3935.75. How much did I make on each barrel? 37. The circumference of each fore wheel of a carriage is 8 feet, and of each hind wheel 12 feet ; how many more turns will the fore wheels make than the hind wheels in going 5280 feet or 1 mile? How many more turns in going from Providence to Worcester, a distance of 44 miles ? 38. The number of square miles in Siberia is 5,493,629, and in Great Britain is 120,832. How many countries the size of Great Britain could be made out of Siberia, and how many square miles would be left over ? 39. How many countries the size of Great Brit- ain could be made out of European Russia (2,261,- 657 square miles), and how many square miles would be left over ? 40. How many countries the size of Great Brit- tain could be made out of China and its depend- encies (3,924,627 square miles), and how many square miles would be left over ? 41. How many countries the size of Japan (156,644 square miles) could be made out of China and its dependencies, and how many square miles would be left over ? [For the sizes of the states referred to in the next five exam- ples, see pages 27-29. ] 42. Into how many states the size of Connect^ cut could Texas be divided, and how many square miles would be left over ? C.] Division. 61 43. Into how many states the size of New York could California be divided, and how many square miles would be left over? 44. Into how many states the size of New Jer- sey could New York be divided, and how many square miles would be left over ? 45. How many times is the smallest of the United States contained in the largest, and with what remainder? 46. The number of square miles in Denmark is 14,553. How many countries the size of Den- mark could be made out of Maine? how many out of North Carolina ? Georgia ? Mississippi ? and how many miles in each case woidd there be left over ? 47. A man took a 3 days' walking journey; on the first day he walked 16 miles, on the second day 25 miles, and on the third day 19 miles. How many miles a day would he have walked if he had made the journey in the same number of days but had gone the same distance each day? Ans. 20 miles a day, which is called the man's AVERAGE rate for the 3 days that he walked. 48. A man walked for five days, going 16 miles the first day, 19 miles the second day, 23 miles the third day, 21 miles the fourth day, and 26 miles the fifth day. What was his average rate per day ? [Get J of the entire distance.] 49. At the end of a fourteen days' journey a man found that he had spent $49 for traveling ex- penses. What was his average expense per day ? 62 Multiplication and Division. [ 2. at the same average how much would he need for a journey of 23 days ? 50. The following figures show how many pupils attended the Planktown Grammar School during the first five days of the first week in April : Monday 38 ; Tuesday 47 ; Wednesday 47 ; Thurs- day 42 ; Friday 46. What was the average attendance? 51. What is the average of the numbers 1, 2, 3, 4,5,6,7,8, 9? t D. Tables nml Questions for Practice. MULTIPLICATION TABLE TO 12 TIMES 12. 1 2 3 ^4 8 5 6 7 14 8 9 10 11 12 2 4 6 10 15 12 18 16 24 18 27 20 22 24 3 6 9 12 21 30 33 36 4 8 12 16 20 24 28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 77 72 84 7 14 21 28 35 42 .49 56 63 70 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 60 66 77 88 99 110 121 132 12 24 36 48 72 84 96 108 120 132 144 To find how many 3 times 9 are, put your finger on the 3 in the D.] Tables and Questions for Practice. 63 left-hand column ; then move the finger to the right till it comes under the 9 in the top line, and there you find 27, which is 3 times 9. In the same way the product of any two of the first twelve numbers may be found. 1. Repeat the multiplication table for each num- ber from 2 to 12 inclusive. 2. The product of any number multiplied by itself is called its SQUARE. Thus 2 times 2, or 4, is called the square of 2 ; 3 times 3, or 9, the square of 3; 12 times 12, or 144, the square of 12. , What is the square of 4 ? of 5 ? of 6 ? of 7 ? of 8? of 9? of 10? of 11? 3. Construct a table which shall give the square of each number from 13 to 25 inclusive. 4. Extend the following multiplication table to 10 times 20. 1 13 14 15 16 17 18 19 20 2 26 3 39 4 52 5 6 65 78 7 91 8 104 9 117 10 130 5. Repeat the multiplication table both for- wards and backwards from once 13 to 10 times 13 ; 64 Multiplication and Division. proceed in like manner with each number from 14 to 20, inclusive. NOTE. A multiplication table beyond 12 times 12 is not usually learned in school ; it is believed, however, that the pupil will be amply repaid in the future for the time spent in learning this table now. 6. Divide each number in the following table by each number less than 20 that is contained in it without a remainder. Thus 12 -r 2 = 6 ; 12 -r 3 = 4 ; 12^4 = 3; 12-^6 = 2, etc. 12 20 26 33 39 46 52 60 66 74 80 88 96 14 21 27 34 40 48 54 62 68 75 81 90 98 15 22 28 35 42 49 55 63 69 76 82 92 99 16 24 30 36 44 50 56 64 70 77 84 94 100 18 25 32 38 45 51 58 65 72 78 86 95 7. Each of the following numbers is the square of what number ? 25, 36, 16, 49, 9, 64, 4, 81, 144, 169, 100, 196, 121, 625, 225, 484, 256, 529, 324, 576, 441, 361, 400, 289. 8. Multiply the first number in column B (see p. 26) by the first number in column C ; the second number in column B by the second num- ber in column C, and so on to the ends of these columns. 9. Divide the first number in column E (see p. 26) by the first number in column D ; the second number in column E by the second num ber in column D, and so on to the ends of these columns. SECTION III. FRACTIONS. [This section, to Example 53, page 95, is drawn from the Re- vised Edition of Colburn's First Lessons. Most pupils will do well to review Fractions carefully as a preparation for the study of Decimals. In cases where a review is not deemed necessary, pages 65-95 may be omitted.] A. Fractional Notation and fractional Terms. Halves, thirds, fourths, fifths, etc., of a thing are called FRACTIONS. Fractions may be expressed by figures, thus : One third by . Three fourths by J. Two thirds by . Two fifths by f . One fourth by J. Four sevenths by %. 1. Eead the fractions : , J, f , ^ &, fa W, -ft, *, *, f 2. Express one fifth by figures ; three fifths ; one sixth ; four sixths ; five sixths ; three sevenths ; one eighth ; five eighths ; one ninth ; four ninths ; one thirteenth ; eleven thirteenths ; one twenty- first ; twenty twenty-firsts, 3. What is of 6 ? J of 8 ? f of 15 ? 4. What is f of 14? } of 32? of 18? 5. 14 is | of what number? 6. 18 is ^ of what number? 66 Fractions. [ 3. 7. 25 is $ of what number? 8. If I plant J of my garden with peas, \ with beans, J with potatoes, and i with tomatoes, how many eighths shall I have left for other vegetables ? How many eighths are l+J + i + i? Add together f , J, and $. 9. How many sevenths are } 4- ? 4- ? ? 10. How many ninths are $ + + }? As we have seen, it requires two numbers to ex- press a fraction ; one (which we write below the line) to show into how many equal parts the thing which we are talking about is divided ; the other to show how many of these parts are taken. For instance, ? of an apple is that portion of an apple which we get by cutting the apple into 7 equal parts, and then taking 3 of these parts. W f an orange represents that portion of an orange which we could get by cutting the orange into 12 equal parts, and taking 5 of these parts. 11. What is meant by | of an orange ? J of an apple ? T 7 ? of a melon ? The number which shows into how many parts the thing is to be divided is always written below the line, and is called the DENOMINATOR of the fraction. The number which shows how many of these equal parts are to be taken is written above the line, and is called the NUMERATOR. Thus, in J, 9 is the denominator and 5 is the numerator* A.] Fractional Notation. 67 12. a. In the fraction J, what does the 8 denote ? the 7 ? which is the numerator ? which the denom- inator ? 6. Which is the numerator and which is the de- nominator in each of the following fractions? -fa, f, T 3 TT> , if- Notice that when a thing is divided into 3 equal parts, the parts are called thirds ; when a thing is divided into 4 equal parts, the parts are called fourths; when a thing is divided into 19 equal parts, the parts are called nineteenths ; that is, the fraction takes its name from the denominator. 13. Read A, A, A, A, jf, Jif, 14. Mary had } of a melon, but gave away \ ; how much had she left? Whatisf-i? ft-|? -*? A- A? 15. Mr. Wood bought a yacht, and sold \ of it to Mr. Shellabarger ; what portion of the yacht did Mr. Wood then own ? 16. How much is A+A-A+H? Fractions like J, -, J, J, , in which the numera- tor is smaller than the denominator, are called PROPER fractions. Fractions like f , J, J, f , in which the numerator is larger than the denominator, are called IMPROPER fractions. 17. Which of the following are proper fractions and which improper fractions? |, f, $, J, 5, A? 68 Fractions. [3. 18. Change f to a whole number and a fraction. Answer : f^f + i^l + J, and this is usually writ- ten 1J. 19. Change ^ to a whole number and a fraction. Answer : 2 and f ; usually written 2 J. 20. A whole number and a fraction like li, 2|, 4J, is called a MIXED NUMBER. Change the mixed number 4 ^ to an improper fraction. Answer : 4i = 4 + = * + J = J 3 3 . 21. Change the mixed numbers 3, If, 2J, 4J, 3$ to improper fractions. 22. Change the improper fractions $, J, |, g, V, ^, to mixed numbers. B. Common Denominator. Problems. Illus- trations. 1. If you divide an apple into two equal parts, and then divide each one of these into two equal parts, how many pieces will you have? What part of the whole apple will each piece be ? Answer : There will be 4 equal pieces, so that each piece will be one fourth of the apple. J of an apple is equal to how many fourths of an apple ? 2. How many fourths of an orange are there in J of an orange ? Answer : 2 fourths of an orange. 3. If you give J of an orange to one boy and J B.] Common Denominator. 69 to another, how much more do you give the first than the second ? 4. If you give of an orange to one boy and J to another, how many times of an orange do you give away? how many fourths of the orange do you have left ? How many times J are J and J ? 5. A man gave to one laborer \ of a bushel of wheat, and to another J of a bushel ; how many times J of a bushel did he give to both ? how many bushels ? How many times \ are \ and f ? 6. If you divide an inch into halves, and then divide each of these halves into ^ ^ 3 equal parts, how many parts will you have ? what portion of an inch will each part be ? How many sixths of an inch are there in of an inch ? 7. A man gave i of a barrel of flour to one person, and of a barrel to another ; to which did he give the more ? 8. A man, paying some money to his laborers, gave each man \ of a dollar, and each boy i of a dollar ; how much more did he give to a man than to a boy ? how much did a man and a boy receive ? 9. What is the difference between J and \ ? what is the sum of J and J ? 10. If a man earns * of a dollar in a day, and a boy \ of a dollar, how much more does the man earn than the boy ? 1 1 . What is the difference between f and \ ? 70 Fractions. [3. 12. If you cut a loaf of cake into two equal parts, and then cut each part into quarters, how many pieces will you have in all ? what part of the whole loaf will each piece be ? 13. How many eighths of a loaf are there in \ of a loaf? In i of anything there are how many eighths ? 14. If you cut a line or loaf of cake into quar- ters, and then cut each one of these quarters into thirds, how many pieces will you have in all ? what part of the whole will each piece be? into how many twelfths of a cake can you cut one fourth of a cake ? 15. Into how many eighths of an apple can you cut $ of an apple? How many eighths are there in J ? 16. If you cut one quarter of a sheet of paper into 4 equal parts, what portion of the whole sheet will each part be ? [The pupil should find out the answer by actually cutting a sheet of paper.] If one cake of chocolate will make 16 cups, how many cups will \ of a cake make ? How many sixteenths are there in J ? 17. This figure is supposed to represent a sheet of paper divided into equal pieces. What part of the whole sheet is one of these pieces ? 2 pieces form what part of B.] Common Denominator. 71 the whole sheet ? How many twelfths are there in one sixth ? 3 pieces form what part of the whole sheet? 9 pieces ? How many twelfths are there in one fourth ? how many in 3 fourths ? 4 pieces form what part of the whole sheet ? 8 pieces ? How many twelfths are there in one third ? how many in 2 thirds ? 6 pieces form what part of the whole sheet ? How many twelfths are there in one half ? fy 18. Take a sheet of paper six inches long and 4 inches wide, and draw a line on it so as to divide it into halves ; now draw a second line so as to divide the sheet into quarters ; now draw 4 other lines, so as to divide the sheet into twelfths. Cut out with the scissors one twelfth of the sheet ; also one sixth. How long and how wide is each of these pieces ? Cut out one fourth of the sheet ; also one third. How long and how wide is each of these pieces ? How many twelfths of the sheet have you cut out in all ? How many twelfths, then, are 19. How many twelfths are equal to ? We may ask ourselves into how many twelfths of anything (for instance, a sheet of paper or a cake) we can cut \ of the thing, or we may reason thus: In anything there are 12 twelfths of that thing ; in j- of the thing, then, there must be \ of 12 twelfths, or 4 twelfths. Therefore = iV 72 Fractions. [ 3. 20. \ is equal to how many tenths ? Answer : In 1 there are 10 tenths ; in J of 1, then, there must be \ of 10 tenths, or 5 tenths. Therefore $ = &. 21. I is equal to how many thirtieths ? Answer : In 1 there are 30 thirtieths ; in J of 1, then, there must be of 30 thirtieths, or G thirtieths. Therefore J^^a- 22. \ is equal to how many twelfths ? four- teenths ? twenty-fourths ? 23. ^ is equal to how many ninths ? fifteenths ? twenty-fourths ? 24. is equal to how many twentieths ? twenty- fourths ? thirty-seconds ? 25. } is equal to how many fourteenths? twenty- eighths ? 26. How many thirty-sixths does it take to makei? i? 1? i? i? T 5 ? & ? 27. How many sixtieths does it take to make A? A? A? t? t? i? 4? i? 28. A man bought J of a bushel of wheat at one time, and f of a bushel at another ; at which time did he buy the more ? 29. An ounce of Mary's medicine makes 48 closes ; what part of an ounce is there in a dose ? how many doses are there in a quarter of an ounce? i is equal to how many forty-eighths ? 30. are equal to how many ninths ? Answer : One third equals 3 ninths, therefore two thirds must equal 2 times 3 ninths or 6 ninths. B.] Common Denominator. 73 31. ^ are equal to how many twenty -eighths ? Answer : One seventh equals 4 twenty-eighths, therefore five sevenths must equal 5 times 4 twenty-eighths or 20 twenty-eighths. 32. Show that $ = i$; | = ft; A = A; f = H- 33. tf are how many times ft ? f are how many times T ? f are how many times ^ ? 34. | are how many times ^ ? f are how many times 3^ ? ft are how many times ^ ? 35. How many twelfths are there in ? in | ? in J? 36. How many eighteenths are there in J? in J? in I? 37. How many forty-eighths are there in ? in |? in? in? in A? in ft? in JJ ? 38. A man bought of a yard of cloth at one time, and g of a yard at another ; how many sixths of a yard did he buy altogether ? at which time did he buy the more? 39. What shall we get if we multiply both numerator and denominator of \ by 2 ? First: If we multiply the denominator by 2 we get 1 1 2x2 " 4 Second : If we now multiply the numerator 1 also by 2 we get 2x1 _2 ^~ =4 ~ 74 Fractions. [ 3. But | is the same as J, which is the thing we started with. Therefore if we multiply both the numerator and the denominator of J by 2, we do not alter the value of the fraction. Shall we alter the value of the fraction \ if we multiply both numerator and denominator by 3 ? by 4? 40. What shall we get if we multiply both numerator and denominator of the fraction J by 4 ? Suppose our J to be J of a foot (12 inches). J of a foot = 4 inches. If we multiply the denominator of \ by 4 we get 1 or. 1 4x3 12 fv of a foot= 1 inch. If we now multiply the numerator by 4 we get A. i 4 * of a foot = 4 inches. But 4 inches is what we started with ; therefore we have not altered the value of J by multiplying both its numerator and its denominator by 4. Shall we alter the value of the fraction J if we multiply both numerator and denominator by 2 ? 41. If we multiply both numerator and denom- inator of f by 5 we shall get Jf ; let us compare this result with f : I -fs > therefore ^ = 3 times -f$ or ^f , and we see that we have not altered the value of the fraction ^ by multiplying both numerator and denominator by 5. Show that if you multiply both numerator and B.] Common Denominator. 75 denominator of f by 6 you will not alter the value of the fraction ; of / T by 4 ; of if by 2 ; of ^ by 6. In the same way you can take any fraction and show that its value will not be changed if you multiply both numerator and denominator by the same number, whatever that number may be. 42. The numerator and denominator of a frac- tion are sometimes called its TERMS. Thus the terms of the fraction ^ are 3 and 7. By what must you multiply both terms of the fraction J in order to get an equivalent fraction whose denom- inator is 12 ? 43. By what must you multiply both terms of the fraction in order to change it to eighths ? to sixteenths ? to twenty-fourths ? 44. Multiply both terms of each of the following fractions by something that will change the frac- tion to sixty-fourths : 1 ; i ; f ; -ft. 45. Change to forty-eighths J, j, j, g, J, H, H, H- 46. What shall we get if we divide both nu- merator and denominator of ^ by 3 ? Suppose our T 9 ^ to be ^ of a foot (12 inches). fs of a foot = 9 inches. If we divide the numerator of f y by 3 we get ^. ^ of a foot = 3 inches. If we now divide the denominator of -ft- by 3 we get f. | of a foot = 9 inches. But 9 inches is what we started with ; therefore we have not altered the value of -fy by dividing both its numerator and its denominator by 3. 76 Fractions [ 3. What will you get if you divide both numerator and denominator of ^ 2 by 4 ? 47. Divide both numerator and denominator of }g by 3 and compare the result with \$. 12-r3_4 15-3 5* J= ? 5 ; therefore $ = 4 times ^ or JJ: we have, then, not altered the value of \l by dividing both numerator and denominator by 3. Show that if you divide both numerator and denominator of JJ by 4 you will not alter the value of the fraction ; of |-g by 10 ; of if by 9 ; of }$ by 12 ; of ft by 7. In the same way you can take any fraction and show that its value will not be changed by dividing both numerator and denominator by the same number. 48. When, as in the last question, we divide both the numerator and denominator by the same number, we are said to reduce the fraction to lower terms ; and when there is no number that will go in both the numerator and denominator without a remainder, the fraction is said to have been reduced to its lowest terms. If, for example, we divide both terms of -ft- by 2, we reduce it to lower terms and get f ; if we now divide both terms of f by 2 we get , and have reduced our fraction T 9 ^ to its lowest terms, for there is no number (except 1) which will go in both 2 and 3 without a remainder. Reduce J$ to its lowest terms. Answer : f . B.] Common Denominator. 17 49. Reduce to lowest terms the fractions f, y 5 ^, I, A> , A, A, M> H, A> M, f i> H> M> M, 1!, M, fi, **, H, and f 5. 50. Where the denominators of two fractions are the same, the fractions are said to have a com- mon denominator. Thus, J and have the common denominator 8. When two fractions have a common denominator we may add them together by adding the numer- ators and writing the sum over the common de- nominator, thus : 4 ,3_ 4+3_7 9 + 9~~9 9' When we have to add together two fractions which have different denominators we may first change the fractions into equivalent ones which shall have a common denominator, and then add them, thus : + f = A + ft = H- J is how many twelfths ? is how many twelfths ? I and % are how many twelfths ? 51. J is how many twentieths? is how many twentieths ? J and J are how many twentieths ? 52. are how many fifteenths ? f are how many fifteenths ? and J are how many fifteenths ? 53. are how many sixths ? less are how many sixths ? 54. Change and to eighteenths and then add them. Answer : J = ft ; f = T V Therefore J + f = T |. 55. Change ^ and j to thirty-fifths and then add them. Which is the larger, and how much ? 78 Fractions. [ 3. 56. Change f and to forty-fifths and then add them. Which is the larger, and how much? 57. A man had a horse, a cow, and a sheep. The horse eat of a load of hay in the winter, the cow J, and the sheep . How many sixths of a load did each eat ? how many sixths did they all eat? 58. A boy, having a quart of nuts, wished to divide them so as to give one companion J, another J, and a third J of them ; but, in order to make a proper division, he first separated the whole into eight equal parts, and then he was able to divide them as he wished. How many eighths did he give to each ? how many eighths had he left for himself ? If we wish to replace two fractions by equivalent fractions having a common denominator we may choose for our common denominator any number which contains both denominators without a re- mainder. For instance, if our fractions are f and f we may choose 36 for our common denominator ; for 36 contains both 4 and 9 without a remainder. If our fractions are g and J we may choose 12, or 24, or 36, or 48 ; for each of these contains both 6 and 4 without a remainder, but it is convenient to take as small a number as we can, so in this case we should choose 12. We can always find a number which will contain both of our denominators without a remainder, by multiplying them together. B.] Common Denominator. 79 59. Reduce J and f to equivalent fractions hav- ing a common denominator. Solution : "7 x 9 = 63 contains both 7 and 9 without a remainder, and may be taken as our common denominator, i /^5 ? 15- 60. Reduce to a common denominator J and $ ; $ and f ; | and ; J. and f ; f and g. 61. Add together and f ; ^ and ^ ? ; f and T \; and and y 3 () . 62. Subtract J from J ; ^ from f ; f from ; from f . 63. A boy, distributing some nuts among his companions, gave J of a quart to one, and ^ of a quart to another; how much more did he give to one than to the other? 64. What is the difference between J and ? 65. A man, having two bushels of grain to dis- tribute among his laborers, wished to give J of a bushel to one, and of a bushel to another, and the rest to a third, but was at a loss to tell how to divide it ; at last he concluded to divide each bushel into six equal parts, or sixths, and then to distribute those parts. How many sixths did he give to each ? 66. is how many sixths ? 67. A man, who had a bushel of wheat, wished to give ^ of it to one man, and J of it to another ; but he could not tell how to divide it. Another man standing by advised him to divide the whole bushel into six equal parts first, and then take i of them for one, and of them for the other. How 80 Fractions. [ 3. many parts did he give to each? how many to both ? how many had he left ? 68. i is how many sixths? is how many sixths ? i and ^ are how many sixths ? 69. and J are how many times J? 70. J and \ are how many times J ? 71. | and i are how many times J? 72. | and f are how many times J ? 73. J and are how many times J ? 74. and are how many times ^ ? 75. J and and J are how many times J? 76. i and f and ^ are how many times T V ? 77. and J are how many times fa? 78. and J and j are how many times -fal 79. J and ^ and J are how many times ^ ? 80. \ and J and J and J and ^ are how many times 3^ ? 81. J and ^ are how many times -fal 82. g and J are how many times 83. and J- are how many times 84. less are how many times J ? 85. Which is the larger, } or f ? how much the larger ? 86. A boy, having a pound of almonds, said he intended to give J of them to his sister, and J to his brother, and the rest to his mamma. His mamma, smiling, said she did not think he could divide them so. " Oh, yes, I can," said he ; "I will first divide them into twelve equal parts, and then I can divide them well enough." Pray, how many twelfths did he give to each ? B.] Common Denominator. 81 87. i is how many times T ^? J is how many times A ? and \ are how many times -^ ? 88. Mr. Goodman, having a pound of raisins, said, " I will give Sarah J, and Mary , and James \ of them, and the rest shall go to Charles, if he can tell how to divide them." " Well," said Charles, " I would first divide the whole into twelve equal parts, and then I could take \ and \ and of them." How many twelfths would each have? 89. J and \ and \ are how many times T V ? 90. George bought a pine-apple, and said he would give -]- of it to his papa, and to his mamma, and f$ to his brother James, if James could di- vide it. James took it, and cut it into twenty equal pieces, and then distributed them as George had desired. How many twentieths did he give to each ? 91. and f less /., are how many times 92. g less $ are how many times 93. # less J are how many times 94. f less f are how many times 95. i and f and J and ^ less g, are how many times ^ ? 96. i and J and f and T V and ^V less J, are how many times ^ ? 97. | and J are how many times 98. f and ^ are how many times 99. ^ and f are how many times ^V? 100. Mr. Fuller said he would give of a pine- apple to Fanny, and f to George, and the rest to 82 Fractions. [ 3. the one that could tell how to divide it, and how much there would be left. But neither of them could tell ; so he kept it himself. Could you have told, if you had been there? How would you have divided it? how much would have been your share ? 101. A man sold 1| bushels of wheat to one man, and 4| to another ; how many did he sell to both? 102. A man bought 6i bushels of wheat at one time, and 2J at another ; how much did he buy in all? 103. A man bought 7 yards of one kind of cloth, and 6J of another kind ; how many yards in all? 104. A man bought J of a barrel of flour at one time, and 2^ barrels at another, and 6| at an- other ; how much did he buy in all ? 105. A man bought one sheep for 4 dollars, and another for 5$ dollars ; how much did he give for both? 106. There is a pole standing so that J of it is in the mud, and ? of it in the water, and the rest out of the water ; how much of it is out of the water ? 107. A man, having undertaken to do a piece of work, did of it the first day, J of it the sec- ond day, and of it the third day ; how much of it did he do in three days ? 108. A man having a piece of work to do, hired two men and a boy to do it. The first man could C.] Multiplication of Fractions. 83 do of the work in a day, and the other J of it, and the boy of it ; how much of it could they all do in a day ? C. Multiplication of Fractions. 1 . A boy, having of an apple, gave away J of what he had ; what part of the whole apple did he give away? Answer : J of an apple is the same as f of an apple therefore the boy gave away J of of an apple or \ of an apple. 2. What is of J? 3. If you cut an orange into three equal pieces, and then cut each of those pieces into two equal pieces, how many pieces will the whole orange be cut into ? What part of the whole orange will one of the pieces be ? 4. What is i of 4 ? 5. A boy had i of a pine-apple, and cut that half into three equal pieces, in order to give away J of it. What part of the whole apple did he give away? 6. What is 4 of J? 7. If an orange is cut into 4 parts, and then each of the parts is cut in two, how many pieces will the whole be cut into ? 8. What is of i ? 9. A man having of a barrel of flour, sold J of what he had ; how much did he sell? 84 Fractions. [ 3. 10. What is J of J ? 11. If an orange be cut into 4 equal parts, and each of these parts be cut into 3 equal parts, into how many parts will the whole orange be cut ? 12. What is i of J? 13. A boy, having of a quart of chestnuts, gave away J of what he had. What part of the whole quart did he give away ? 14. What is of J ? 15. What is of ? 16. A man, who owned J of a ship, sold J of his share ; what part of the ship did he sell, and what part did he then own ? [When we get J of J we are said to multiply the fractions together ; we can, then, use the mul- tiplication sign ( x ) instead of the word " of," thus x J.] 17. Whatis Jxj? 18. What is Jx J? 19. What is Jx J? 20. What is i of J? 21. What is J of j? 22. What is J of J? 23. What is of J ? 24. What is J of J ? 25. A boy, having of an orange (that is, 2 pieces), gave his sister of what he had ; how many thirds did he give her ? 26. What is i of ? 27. A boy, having j of a pine-apple, said he would give one half of what he had to his sister, C.] Multiplication of Fractions. 85 if she could tell how to divide it. His sister said, " You have f , or three pieces, if you cut them all in two, you can give me \ of each one of them." Flow much did his sister receive ? 28. What is \ of | ? 29. A man, who owned f of a share in a Boston bank, sold J of his part ; what part of a share did he sell ? 30. What is J of | ? 31. A man, who owned J of a ship, sold J of his share ; what part of the whole ship did he sell ? What part had he left ? 32. What is J of f? 33. What is J of $? 34. What is i of ^ ? 35. What is i of ?? Answer : \ of \ = ^ , therefore J of ^ will be 3 times as much, or ^. 36. A man, who owned f of a share in a bank, sold of his part ; what part of a whole share did he sell ? 37. What is J of ? 38. What is of f ? 39. A boy, having J of a watermelon, wished to divide his part equally among his sister, his brother, and himself, but was at a loss to know how to do it ; but his sister advised him to cut each of the fifths into 3 equal parts. How many pieces did each have ? and what part of the whole melon was piece ? 40. What is of f ? 86 'Fractions. [ 3. 41. What is J of f ? 42. What is J of J? 43. What is J of i ? 44. What is i of ? 45. What is f of ? Answer : We first get J of , Now * of J = 5^ ; i of f is, then, twice as much, or ^, which equals A- Therefore of , which is 3 times J of , is A- 46. What is 4 of ? 47. What is of f ? 48. What is J of f ? 49. What is of 4 ? 50. What is & of J? 51. What is Jof ?? 52. What is J of f ? 53. What is A of | ? 54. What is A of | ? 55. If a yard of cloth costs 2 J dollars, what will J of a yard cost ? Answer : If 1 yard costs $2^ or f , ^ a yard will cost \ as much or $|, which is $lj. 56. What is J of 2? 57. A boy had 2^ oranges, and wished to give J of them to his sister, and J to his brother, but he did not know how to divide them equally. His brother told him to cut the whole oranges into halves, and then cut each of the halves into 3 pieces. What part of a whole orange did each have ? 58. What is 4 of C.] Multiplication of Fractions. 87 59. A man bought 4 bushels of corn for 3 dol- lars ; what part of a dollar did 1 bushel cost ? Answer : $3 = f^. I f 4 bushels cost $Y> one bushel, of 4 bushels, will cost i of f V or ii: J- 60. What is of 3 ? 61. If 5 bushels of wheat cost 7f dollars, what is that a bushel ? 62. What is J of 7f ? 63. A man bought 6 gallons of alcohol for 8f dollars ; what was that a gallon ? 64. What is J of 8f ? 65. A man bought 7 bushels of potatoes for 8^ dollars ; how much was that a bushel ? 66. What is \ of 8f ? 67. A man bought 10 pieces of nankeen for 6 1 dollars ; how much was it a piece ? 68. What is ^ of 6 ? 69. If 9 bushels of rye cost 6? dollars, what is that a bushel ? 70. What is Jx6? 71. What is 4x5?? 72. What is Jx8|? 73. What is i x 6ft ? 74. What is x 9* ? 75. A man bought 7 yards of cloth for 18| dol- lars ; what was that a yard ? What would 3 yards cost? 76. What is \ of 18j ? What is f of 18 J ? 77. A man bought 5 barrels of vinegar for 27 1 dollars ; what was it a barrel ? What would 7 barrels cost at that rate ? 88 Fractions. [ 78. What is J of 27| ? What is J of 27| ? 79. If 6 barrels of flour cost 38$ dollars, what would 10 barrels cost at that rate ? 80. What is -V x 38| ? 81. Show that ^x _ = . 1 3 3 . 1 . 5 = 3-irrr5 and 3 Therefore Q X C == Q z o o x o 82. Show that s x w = - '15* 15' x f will be 4 times as much as J x J. ^ X g = JL=A. (See Example 81.) 4 15' To get J of a fraction, that is, to multiply it by , we may multiply its denominator 83. Show that V- * .!.! 3 4 3x4 12' l x l_ 1 _1. 3S/ /T X O r 32 3x2 6 ' 12 2 _2 . > ma 33 3x3 9 ' 1 V 2 2 2 by 35 3x5 15' 84. Show that 14 We know from Example 82 that ~ x - = - - o o o x o 4 J.D c.] Multiplication of Fractions. 89 Now f x f is 2 times as much as ^ x ; ^ ' 2 4 2x4 therefore ~ x - = - - = o 5 5 x 6 '15' 85. Show that 4 15 ; 2x2_ 4 3 3x3 9 ; 5_2x5_10 21' 2 2_2x2 3 X 5 3x5 O O 3 X 3"3^3 : 2 5_2x5 3 X "3x7 86. Show that To get f of a fraction, that I is, to multiply it by f , we | multiply the numerator by 2, and the denominator by 3. To multiply two fractions together, we multiply the nu- merators together to get a new numerator, and the de- nominators together to get a new denominator. 87. WhatisJxJ? Jxf? ixj? jxj? [Reduce your results to their lowest terms. See Ex. 48, p. 76.] 88. What is^xj? jxf? xj? |xj? A*|? 89. What irfxfMx|f fxA? fxff fKf? 90. What is Iix2f? 1JX3J? 3ix4j?5i l> 2 9 3 2 X 4 3 X 7 3 4 5 2 5 2 9 3 X X X X X fi 7 2 3 3 12 "I "35 5 4 27 5 9 2 4 X X 4 2 8 ' 8* 3 x f> 15' 91. A man bought a piece of cloth for 42^ dol- lars, and was obliged to sell it for J of what it cost him ; how much did he lose ? 92. A man bought a quantity of flour for 53? dollars, and sold it for f of what it cost him ; how much did he gain ? 90 Fractions. [ 3. 93. If 7 men can do a piece of work in 4$ days, how long will it take 1 man to do it ? How long will it take 3 men to do it? 94. If 4 men can do a piece of work in 9^ days, how long would it take 7 men to do it ? 95. There is a pole standing so that $ of it is in the water, and as much in the mud ; how much is in the mud ? 96. If a man can travel 13$ miles in 3 hours, how many miles can he travel in 8 hours ? 97. If 5 horses can eat 26 J loads of hay in a year, what will 8 horses eat in the same time ? 98. If 4 pipes can empty a cistern in 6$ hours, how long will it take 7 pipes of the same size to empty it? D. Division of Fractions. 1. A boy having 2 oranges, wished to give ^ of an orange apiece to his playmates ; how many could he give them to ? If he had given of an orange apiece, how many could he have given them to ? Answer: If he gives ^ of an orange to each playmate, he can give to as many playmates as there are thirds in 2, or to 6 playmates. Had he given of an orange to each, he could have given to only half as many, or to 3 playmates. Or, we may find the answer to the last question in this way : In two oranges there are 6 thirds ; if he gives 2 thirds to each, he can give to as many D.] Division of Fractions. 91 playmates as there are twos in 6, or to 3 play- mates. 2. How many times are there in 2 ? How many times are there in 2 ? 3. A man having 3 bushels of corn, distributed it among some poor persons, giving | of a bushel to each ; to how many did he give it ? Answer: In 3 bushels there are 12 fourths bushels. Had he given 1 fourth to each, he could have given to 12 persons ; if, then, he gives 3 fourths to each, he can give to only as many, or to 4 persons. Or, since in 3 bushels there are 12 fourths bush- els, he can give 3 fourths to as many persons as 3 is contained in 12, or to 4 persons. 4. In 3 are how many times i ? how many times i? 5. If f of a barrel of flour will last a family one month, how long will 4 barrels last the same family ? How long will 6 barrels last ? How long will 10 barrels last ? 6. How many times is contained in 4? how many times in 6 ? how many times in 10 ? 7. If | of a bushel of wheat will last a family one week, how many weeks will 6| bushels last the same family ? Answer : In 6| bushels there are \ 7 bushels ; at the rate of J in a week, the whole would last 27 weeks ; and at the rate of j in a week, the whole would last J of 27 weeks, or 9 weeks. Or, 6 bushels = V bushels ; at the rate of f in 92 Fractions. [ 3. a week, ^ will last as many weeks as 3 is con- tained in 27, or 9 weeks. 8. How many times is f contained in 6 J ? 9. There is a cistern having a pipe which will fill it in of an hour ; how many times would the pipe fill the cistern in 3 hours? Ann. 81 times. 10. How many times is contained in 3? 11. How much cloth at 1 J dollars (that is, f dol- lars) a yard can be bought for 4 dollars ? 12. How many times is l or f contained in 4 ? 13. How many barrels of potatoes at $lj a bar- rel can I buy for $8.1? 14. How many times is 1^ contained in 8i ? 15. If a soldier is allowed 1^ pounds (that is $ of a pound) of meat in a day, to how many soldiers would 6 pounds be allowed? 16. How many times is 1 : \ contained in 6? 17. If If tons of hay will keep a horse through the winter, how many horses will 10 tons keep ? 18. How many times is 1 contained in 10? 19. At 2J dollars a box, how many boxes of raisins can be bought for 10 dollars ? 20. How many times is 2 ; \ contained in 10 ? 21. At If dollars a pound, how many pounds of indigo can be bought for 9| dollars? 22. How many times is If contained in 9|? 23. At If dollars a barrel, how many barrels of raisins can be bought for 9y dollars ? 24. How many times is If contained in 9^ ? 25. At I of a dollar apiece, how many pieces of nankeen can be bought for 8 dollars ? D.] Division of Fractions. 93 26. How many times is J contained in 8| ? 27. At f of a dollar a pound, how many pounds of tea can be bought for 7f dollars ? 28. How many times is f contained in 7J? 29. How many times is 83 contained in 7? 30. How many times is 5^ contained in 17? 31. How many times is 4j contained in 9f? 32. How many times is 3 ; J contained in 33. At T V of a dollar a pound, how many pounds of meat can be bought for i of a dollar ? Note. Change \ to tenths. 34. How many times is ^ contained in J ? 35. If a man can do J of a piece of work in one hour, how many hours will it take him to do J of the work? Note. Change both fractions to twelfths ; that is, reduce them to a common denominator. 36. How many times is J contained in f ? 37. If a pound of almonds cost \ of a dollar, how many pounds can be bought for of a dollar ? Note. Reduce the fractions to a common de- nominator. 38. How many times is \ contained in ? 39. If a piece of nankeen costs of a dollar, how many pieces can be bought for 4| dollars, that is, for V dollars ? 40. How many times is f contained in 4| ? 41. If a bushel of barley costs f of a dollar, how many bushels can be bought for f of a dollar ? How many for If dollars? 94 Fractions. [ 3. contained in J ? How 42. How many times is many times in If ? 43. How many times is ? contained in -; ? 44. How many times is contained in ? 45. Show that J is contained in 1, 3 times ; in 2, 6 times; in 3, 9 times ; I To dl ^ a ^ hole n " mb ^ r b / in 4, 12 times ; * we multl P 1 y the mimber b ? 3 ' in 5, 15 times. 46. Show that in 1, 5 times ; in 2, 10 times ; in 3, 15 times ; in 4, 20 times. 47. Show that J in 1, f times ; * in 2, ^ or 5 times ; in 3, * times ; 3j or 10 times. 48. Show that in 4, is contained To divide a whole number by i we multiply the number by 5. is contained To divide a whole num- ber by % we may multiply the number by 5 and di- vide by 2. 49. Show that 1^S = t [ixl]^ To divide a whole number 2-^f= V [f x 2] ', by a fraction we may multiply 3-^J = 12 [f x 3] i by the denominator and divide 5 -=-$ = [|x5]J by the numerator, or, as is * | will be contained only J as many times as . D.] Division of Fractions. 95 sometimes said, we may invert the divisor and proceed as in multiplication. In the last example, for instance, 7x5 or 35 [the number of sevenths in 5] is the number of times that \ is contained in 5, and ^ will be con- tained i as many times or ^ times. 50. Whatis7^-? 4-f? 6-f? 2(Hf? 21^-J? 18-f T 3 T ? 16'-=- ft? 51. We have already shown that If now we divide i of 2 or f by the same divi- sors the results will be only J as large. Show, then, that T| = f x f A = 1 To DIVIDE ONE FRACTION BY ANOTHER WE MAY IN- VERT THE DIVISOR AND PRO- CEED AS IN MULTIPLICATION. [Reduce your results to their lowest terms.] 52. Whatisf-rV? ^? *^? H^l? 53. What is |-f A? if-i 7 ^? A-5-H? 54. Whatis2l-rl|? 2J-5-H? 6i^3i ? 6-U? 55. A man bought 6J- yards of cloth for |25 ; how much was that a yard ? 56. A man walked 12?- miles in 4f hours ; how many miles an hour was that ? 57. A man bought 2 pounds of butter for 60 cents ; how much a pound was that ? 58. At | of a dollar a pound, how many pounds of tea can I buy for 3 and f dollars ? 96 Fractions. [ 3. E. Miscellaneous Questions. 1 . If A of a ship's cargo is worth 114,000, what is the whole cargo worth ? 2. A owiis ft of a coal mine, and his share is worth $ 3,500. What is the whole mine worth ? 3. A man willed J of his property to his wife, i to a public library, to an orphan asylum, to his only brother, and the remainder in equal shares to his three children. What part was the share of each child ? If his wife's share amounted to $ 15,220, what did each of the other shares amount to? 4. A lady bought 3 pieces of cloth ; the first contained 8J yards, the second 21J yards, and the third 15| yards. How many yards did she buy in all? 5. Three men, C, D, and E, bought a house lot for $1488 ; C put in $248, D $744, and E $496. What part of the house lot did each man own ? 6. Four men, A, B, C, and D, bought a mill ; A. contributed i the cost, B f , C , and D $990. What did the mill cost ? 7. A person having sold and 4 of his farm bad 26 acres left. How many acres had he at first ? 8. Which of the following quantities is the greatest, and which is the least ? ^, ^ -f G ? 9. A ship is worth $16,000, and a person who owns YV f it sells f of his share ; what share has he remaining and what is it worth? E.] Miscellaneous Questions. 97 10. a. What is f x if ? 3 16 3x16 48 , , Note. | x = = (reducing to lower 16 4 terms) -op~q We reduced our result to lowest terms by divid- ing both numerator and denominator first by 3 (this gave J{j), and then by 4 (this gave f). Now we can just as well perform our divisions on the expressions 3 x 16 and 4 x 27 as on their equiva- lents 48 and 108. j = (dividing numerator and denominator by 16 4 3) " ( dividin s b y 4 )=- The following is a short way of expressing our work : 1 14 3 16_3xl6_gxl6_g _ ~ ( 919 1 4 316 j|x^ 4 1 9 In describing our work we may say : 3 divided by 3 is 1, we cancel * (draw a line through) the 3 and write the 1 above it ; 27 divided by 3 is 9, we cancel the 27 and write the 9 below it. 16 * This subject of cancellation is treated in detail in Section X. The little that is said about it here, however, will save the pupil time and labor in many of the examples that follow. 98 Fractions. [ 3. divided by 4 is 4, we cancel the 16 and write the 4 above it ; 4 divided by 4 is 1, we cancel the 4 1x4 and write the 1 below it. We now have ^ ^ or 1 x j a for our answer. b. What is A x i 8 5 ? [Find your answer by cancelling.] 7 11 84 121, 11. a. What is gg-^ L55-J20- 77 5 10 6. What is JSxJf? 12. WhatisxJ? x-H? JtxJl? 13. WhatisH-V? li-5-A? A* A? 14. What is JJ x f J x } ? 15. What is if X I X A? 16. How many quarts of berries at 11 cents a quart will it take to buy 2jJ yards of cloth at 16 cents a yard ? [2| x 16*^11 - 3 x ^ x ^ = ?] 17. How many barrels of apples at 2j dollars a barrel will it take to buy 4j tons of coal at 5i dol- lars a ton ? 18. A man failing in trade is able to pay only { of a dollar on a dollar ; how much can he pay on a debt of 832 dollars ? 19. At $7.86 per barrel, what will 18jj barrels of flour cost ? 20. If a horse will eat ^ of & ton of hay in a month, how much will 80 horses eat ? E.] Miscellaneous Questions. 99 21. How long will it take a stage to run 4 miles if in 1 hour it runs 9J miles ? if in 1 hour it runs 5 miles? 7J miles? 6 miles? 6 T 7 T miles? 22. If a family consume f of a barrel of flour in a month, how long will it take them to use 3 barrels? 4i barrels? 7 J barrels? 3J barrels? 12f barrels ? 23. If a man can build 15 rods of wall in 5j days, how much can he build in 1 day? in 7i days? 3| days? 6 days? 8 days? j of a day? of a day? 5i days? 13 ; \ days? 2g days? 24. If a roll of carpeting containing 75 yards is worth $132, what is | of a yard worth ? 25. If a man can perform a journey in 580 hours, how many days will it take him to perform it if he travel 8| hours in a day ? 26. If 2i bushels of oats will keep a horse 1 week, how long will 18$ bushels keep him ? 27. If If, that is |, of a yard of cloth will make a coat, how many coats may be made from a piece containing 6l yards? 28. If of a pound of fur is sufficient to make a hat, how many hats may be made of 4 T V pounds of fur ? 29. If lji yards of cloth is worth 11 \ dollars, what is a yard worth ? 30. A merchant bought a piece of cloth con- taining 21 1 yards and in exchange gave 23 J barrels of flour ; how much flour did one yard of the cloth come to ? How much cloth did 1 barrel of the flour come to ? \ SECTION IV. DECIMALS. INTRODUCTION. Notation with simple illustrative examples in Addition, Subtraction, Multiplication, and Di- vision. This section contains examples of fractions whose denominators are 10, 100, 1000, etc. , and points out a simple method of denot- ing such fractions without writing their denominators. (Ill The few numbers here given remind us that ev- ery time we move a figure one place to the left we multiply its value by 10 ; and that every time we move a figure one place to the right (until we reach the units' place) we divide its value by 10. Now suppose that, after reaching the units' place, we go a step further to the right and put the figure 6 for example in a column to the right of the units' column ; the principle 6 60 600 6000 6000 600 60 6 ffS3 6 06 6 x 10 60 x 10 600 x 10 6000-10 600-10 60-10 6 4- 10 or Introduction. 101 just referred to leads us to regard 6 in this posi- tion as denoting T V of 6 or ft. 1. On the same principle, what will the figure 6 denote, if put in the second column to the right of the units' column? 2. What will each of the figures 1, 2, 4, 8, 9, denote, if written in the first column to the right of the units' column ? what if written in the sec- ond column to the right of the units' column ? 3. What will be denoted by 1 in the first and 2 in the second column to the right of the units' column ? 4. What name would you give to the first col- umn to the right of the units' column ? 5. What name would you give to the second column to the right of the units' column ? In order to avoid writing the names of the columns at the top, the following- device has been adopted : Take the figure for example : When it belongs in the first place to the right of the units' place (that is when it denotes tenths) we put a point before it, thus .6 ; and when it belongs in the second place to the right of the units' place (that is when it denotes hundredths and there are no tenths) we put a zero before it to fill the vacant place ai*d a point before the zero, thus .06. From what precedes we know that we may de- note ft by .1 (sometimes written 0.1). ft by .2 (sometimes written 0.2). ft by .3 (sometimes written 0.3). }g by 1.0 or 1 li or 1 and ft by 1.1 H or 1 and ft by 1.2 }% or 1 and ft by 1.3 6. Write in the same way ft, ft, ft, } $, H, }g, 102 Decimal*. [ 4. 7. What is denoted by 0.7? 0.8? 0.9? 6.7? 4.6? 12.3? 3.8? 8. a. In 3.2 there are 32 tenths, or 3 units and 2 tenths left over ; what is the total number of tenths in each of the following numbers ? 1.7; 1.8; 1.9; 2.7; 4.6; 6.8; 9.2 ; 2 ; 4 ; 3. b. How many units are there in each of these numbers, and how many tenths are left over? 9. Write with a denominator and also without a denominator the sum of 0.7 and 0.8 (Answers \$, 1.5); the sum of 0.6 and 1.6; the sum of 0.3 and 0.6 ; the sum of 1.2 and 2.6. -10. Write in two ways, as in the last example, the answers to the following : . tors the answers to the following questions : a. 8x^ = ? (8xA = or 4.8.) 5. 9x0.5-? d. 7x0.6-? c. 8x0.4-? e. 10x0.5-? /. 2x6.9-? g. ft-5-3 = ? Glw*. A or 0.3.) A. 0.8-7-2 = ? j. 1.8-2-? i. 0.6-3-? L 3.6^3 = ? 22. Multiply each of these numbers by 10 : 0.1 (Ans. \% or 1) ; 0.3 (Ans. ?# or 3) ; 0.7 ; 0.9 ; 1.1 (Ans. W or 11) ; 1.5 ; 1.9 ; 3.8 ; 16.1 ; 18.1 ; 96.4 ; 184.6 ; 192.9. 23. What is the quotient of A^A? (A^A = AxJ^ = 4*) T V*-A? O 1 ^- 3) 0.4^-0.2? 0.2^- 0.2? 7.5-^0.3? 0.9-f 0.3? 2.7^-0.3? 98.4^32.8? 24. What is the quotient of 8^0.2? (8-r0.2 = 8^-^-8x^0^=40) 6^0.3? (^7^.20) 16^0.8? 27-^0.9? 144^-1.2? 2^-0.2? 4^0.2? 98.1^0.9? 5.7^-1.9? 25. Divide each of the following numbers by 10, expressing the answers both with and without denominators : 1 (1^-10 = ^ or 0.1) ; 64 (64-=-10 = ?J = 6.4) ; 3 ; 6 ; 123 ; 172 ; 968 ; 1728. * A g-ood knowledge of common fractions is here indispensa- ble ; the student should test the accuracy of each step for him- self, illustrating, if necessary, by simple examples of his own, and he should not be allowed to state a rule except as the result of his own fresh experience. 106 Decimals. [ 4. Now advancing one step further to the right, we may denote T fo by .01 or 0.01 T fa by .02 or 0.02 T fo by .03 or 0.03 T V& by .11 or 0.11 TV* by .12 or 0.12 m by 1.00 or 1 Hi by 1.01 m by 1.02 HJ by 1.11 m by 1.12 W% by 2.00 or 2 26. Write in the way indicated above yj^, r g^, T$U> T lV% T^ !*& H$ ?$ 27. What is denoted by 0.09? 0.23? 0.98? 1.82? 3.21? 0.90? 0.70. 28. In 1 \ J there are T W ; we may say, then, that 0.1 is equal to 0.10. How many hundredths are there in each of the following numbers ? Express the answers without denominators, 0.7 (Ans. 0.70) ; 0.9; 0.2; 0.6; 0.8. 29. What is the sum of 0.7 and 0.08? (0.7 is equal to 0.70 and this added to 0.08 gives 0.78) What is the sum of 0.6 and 0.09 ? (Ans. 0.69) of 0.5 and 0.13? of 0.4 and 0.09? of 0.8 and 0.62? of 1.2 and 0.32 ? 30. a. In 0.78 there are 7 tenths and 8 hun- dredths. How many tenths are there in each of the following numbers, and how many hundredths are left over ? 0.69 ; 0.60 ; 0.53 ; 0.48 ; 0.87. Introduction. 107 b. What is the total number of hundredths in each of the preceding numbers ? 31. a. How many units are there in each of the following numbers, and how many hundredths are left over ? 1.86 (Ans. 1 unit and 86 hundredths over); 2.03; 1.68; 17.64; 9.10; 38.42; 6; 1; 4. b. What is the total number of hundredths in each of the preceding numbers ? [In the first (1.86) there are 186 hundredths.] 32. Express with a denominator and also with- out a denominator the answers to the following questions : a. 0.64 + 0.29 = ? Answers fl&; 0.93 b. 0.67 + 0.11 = ? /. 0.5 + 0.91 = ? c. 0.32 + 0.16 = ? g. 0.4 + 0.2 + 0.18=? d. 0.30 + 0.12=? h. 0.6 + 0.66 + 0.53 = ? e. 0.6 + 0.17=? i. 4 + 0.4 + 0.04 = ? 33. Add the numbers in each of the following columns : a. 6. c. d. e. / 0.12 6.82 0.01 1.61 8.4 1.11 0.68 1.61 1.01 .80 6.84 1.23 0.73 0.05 0.10 .94 7.32 4.56 0.96 .70 8.76 1.79 9.99 7.89 34. From 3.68 1.83 6.07 1.08 0.99 take 1.72 .76 1.02 .18 .02 Answer : 1.96 35. What is the product of 8 x T fo ? (Ans. ffo or 0.48) 7 x ^ ? 5 x 0.06 ? 9 x 0.16 ? 4 x 0.02 ? 25x0.25? 108 Decimals. [ 4. 36. Multiply each of the following numbers by 100: 0.06 (Ans. 0. 06 = T -|fo, therefore 0.06x100 = T foxlOO = 6); 0.09; 0.16; 0.89; 1.16; 9.84; 11.84 ; 69.42. 37. What is the product of A x ^ ? (Ans. ffo or 0.32) Ax A? 0.4x0.9? 1.6x1.6? 2.5x2.5? 0.9x8.1? 38. a. Give three new examples, like those of No. 37, of tenths multiplied by tenths. b. When we multiply tenths by tenths, what is the denominator of the product ? c. When the product is expressed without a denominator, how many figures are there to the right of the point ? 39. Multiply 0.6 by 0.9 ; 0.8 by 1.6 ; 0.7 by 1.7 ; 2.3 by 7.6 ; 1.2 by 0.4. 40. Multiply 0.8 by 0.6 ; 0.9 by 1.7 ; 0.4 by 0.3 ; 1.4 by 1.6 ; 8:5 by 6.2. 41. What is the quotient of T f^2? (Ans. T fo or 0.04) TVfc-s-8? y^-7? 42. What is the quotient of 0.06-^3? (Ans. 0.02) 0.08-J-4? 1.44 -M2? 43. What is the quotient of 0.8^10? (0.8 is the same as 0.80 and 0.80^-10-0.08) 0.9^-10? 1.2-5-10? 14.4^10? 172.8-rlO? 44. What is the quotient of 2-flOO? (2 is the same at 2.00 or f gg, and this divided by 100 is T o or 0.02) 12-^100? 168-rlOO? 1123^-100? 45. What is the quotient of T ^ -r A ? (TTF "^ T%-TSo x -V- ::= A or 0.3) What is the quotient Introduction. 109 of 0.08-0.4? (Ans. & or 0.2) 0.16-K8? 0.18- 0.9? 0.27-^0.3? 0.24^0.8? 0.09^0.3? 0.22^0.2? 0.27-0.9? 0.12*0.6? 46. a. Give three new examples, like those of No. 45, of hundredths divided by tenths. 6. When we divide hundredths by tenths, what is the denominator of the quotient ? c. When the quotient is expressed without a denominator, how many figures are there to the right of the point ? 47. Divide 0.33 by 1.1 (Ans. 0.3) ; 0.24 by 1.2 (Ans. 0.2) ; 0.48 by 1.2 ; 0.36 by 1.2 ; 0.28 by 1.4 ; 0.48 by 2.4 ; 0.32 by 1.6 ; 0.22 by 1.1. 48. What is the quotient of T^-y^? (Ans. 4) TVfe-M*? 0.21-f0.07? 0.84-0.21? 16.38-f8.19? 49. What is the quotient of 8 -f T D ? (^Us. 400) 6^0.03? (Ans. 200) 16^-0.08? 27^-0.09? 18-0.03 ? Advancing another step to the right, we may denote by -001 or 0.001 by .002 or 0.002 by .011 or 0.011 by .060 or 0.060 by .684 or 0.684 HU by 1.111 by 2.000 or 2 by 16.213 50. Write without the denominators 110 Decimals. [ 4. 51. Write without the denominators T i8^ TiSiT> iMtfi TfffflH T$$ff5 T& the number just below it and o!o6 * s wnat P art f the number 0.006 just above it? 0.0006 0.00006 NOTE. We know that the value of a figure is multiplied by TEN by moving it one place to the left (with reference to the point) ; and that this value is consequently divided by TEN by moving the figure one place to the right ; it is on this account that the system of numbers we have used is called the ten or decimal * system of numbers ; the point or period placed between the unit's place and the tenth's place is called the decimal point, and the fraction to the right of the point is called a decimal fraction. * The word decimal is from the Latin decem, which means ten. SECTION V. MULTIPLICATION OF DECIMALS. A. Multiplication of a Decimal by a Whole Number. Examples and Problems with He- marks and Explanations. 1. a. Multiply 9.6 by 4. Applying the method used in the multiplication of whole numbers we say 4 QQ Multiplicand, times o tenths are 24 tenths, 4 Multiplier. or 2.4 ; write the .4 vertically T^~ ' J, 38.4 Product, under the .0 and save the 2 [units]. 4 times 9 are 36 ; add the 2 that were saved and write the result 38 to the left of the point. I). Show that the product of 9.6 and 4 is 38.4, by first reducing the 9.6 to a common fraction and then proceeding as in the multiplication of a frac- tion by a whole number. 2. Multiply 1.842 by 4, in each of the two ways just indicated. Ans. 7.368 3. Multiply 1.842 by 6, in two ways. Ans. 11.052 4. Multiply 0.1842 by 9, in two ways. Ans. 1.6578 5. a. How many decimal places are there in the 118 Decimals. [ 5. multiplicand, and how many in the product in each of the last four examples ? b. Find the product of 1.842 and 12. 6. a. Find the answer to each of the following examples, and see if you have the same number of decimal places in the product as in the multipli- cand. b. Take any number you please containing one decimal place and multiply it by 7. c. Take any number you please containing two decimal places and multiply it by 8. d. Take any number you please containing three decimal places and multiply it by 11. e. Take any number you please containing four decimal places and multiply it by 6. NOTE. From what precedes we see that when the multiplier is a whole number we may multiply exactly as in whole numbers, pointing off the same number of decimal places in the product as there are in the multiplicand. 7. Multiply 184.2 by 144. 8. Multiply 1.842 by 144. 9. Multiply 8.42316 by 32. 10. Multiply 16.82 by 39. 11. Multiply 0.693 by 842. 12. Multiply 0.0396 by 97. 13. Multiply 0.0064 by 4. 14. How many gills are there in 0.35 pints? Ans. 1.4 gi. 15. How many pints are there in 0.125 quarts? (Ans. 0.25 pts.) how many gills? (Ans. 1 gi.) 16. How many quarts are there in 0.0125 gal- lons ? how many pints ? how many gills ? A.] Multiplication. 119 17. How many gills are there in 0.6875 gallons ? 18. Show that 0.9375 gallons = 3 qts. 1 pt. 2 gi. First: Reducing to quarts, we get 0.9375 gal- lons = 4 x 0.9375 qts. = 3.75 qts. Second: Redu- cing the .75 qts. to pints, we get .75 qts. = 2x .75 pts. = 1.5 pts. Third : Reducing the .5 pts. to gills, we get .5 pts. = 4x.5 gi. = 2 gi. .-. 0.9375 gallons = 3 qts. 1 pt. 2 gi. This is called reducing the decimal part of a gallon to quarts, pints, and gills. The work may be more briefly arranged as fol- lows: 0.9375 gallons. 4 3.7500 qts. _2 1.50 pts. 4 19. Reduce 0.175 gallons to quarts, pints, and gills. Ans. qts, 1 pt. 1.6 gi. 20. Reduce 0.435 gallons to quarts, pints, and gills. 21. Reduce 0.2175 gallons to quarts, pints, and gills. 22. What is the cost of 2 quarts of alcohol at 11 cents a gill ? 23. What is the cost of 24 gallons of vinegar at 7 cents a quart ? 24. What is the cost of 8 two-gallon cans of milk at 6 cents a quart ? 120 Decimals. [ 5. 25. What is the cost of 3 gallons of ice-cream at 15 cents a quart? 26. What is the cost of 3 gallons of astral oil at 10.0625 a quart? 27. How many pecks are there in 0.35 bushels? how many quarts ? how many pints ? 28. How many pints are there in 1.45 bushels ? 29. Reduce 0.109375 bushels to pecks, quarts, and pints. 30. Reduce 0.809375 bushels to pecks, quarts, and pints. 31. Reduce 0.984375 bushels to pecks, quarts, and pints. 32. How many pints are there in 0.02 bushels ? 33. At 12 cents a quart, what will 3 pecks of berries cost ? 34. At $0.145 a quart, what will a bushel of grass seed cost ? 35. At $0.1875 a peck, what will 2 bushels of potatoes cost ? 36. At $0.285 a peck, what will 3 bushels of apples cost ? 37. How many pounds are there in 0.001 tons? how many ounces ? 38. How many ounces are there in 0.0025 tons? 39. Reduce 0.00125 tons to pounds and ounces. 40. Reduce 0.0006 tons to pounds and ounces. 41. Reduce 0.21512 tons to pounds and ounces. 42. At $0.345 a pound, what will 14 pounds of coffee cost? 43. At $0.00275 a pound, what will a ton of range coal cost ? A.] Multiplication. 121 44. At $ 0.006 a pound, what will a ton of Amer- ican Cannel coal cost? 45. At $0.011 a pound, what will a ton of hay cost? 46. At 10.115 a pound, what will tons of lead cost? 47. How many feet are there in 0.001 miles ? how many inches? 48. How many rods are there in 0.12525 miles ? how many yards ? how many feet ? how many inches ? 49. Reduce 0.002 miles to rods, yards, feet, and inches. 50. Reduce 0.5135 miles to rods, yards, and feet. 5 1 . What would it cost to build 38 rods of fence at 10.305 a foot? 52. What would 2 miles of telegraph wire cost at $0.003 a foot? 53. What would 108 fathoms of rope cost at $0.0125 a foot ? [A fathom = 6 f t.] 54. What would be the cost of enough steel rails for a mile of a single track railroad at $0.055 a foot ? 55. If sound travels a foot in 0.0008 seconds, how long will it take it to travel 10 miles? 56 Reduce 0.615 miles to rods, yards, feet, and inches. 57. Reduce 0.115 miles to rods, yards, feet, and inches. 58. Reduce 0.115125 miles to rods, yards, feet, and inches. 122 Decimals. [ 5. 59. Reduce 0.002125 miles to rods, yards, feet, and inches. 60. Reduce 0.5 rods to hands. [A hand = 4 in.] 61. Reduce 0.0025 miles to hands. 62. How many seconds are there in 0.01 hours ? 63. How many minutes are there in 0.05 days? 64. Reduce 0.01 years to days, hours, minutes, and seconds. 65. Reduce 0.0005 years to days, hours, min- utes, and seconds. 66. Reduce 0.0004 years to days, hours, min- utes, and seconds. B. Multiplication of a Decimal by a Decimal. Examples and Problems, with Remarks and Explanations. 1. Multiply 18.42 by 1.2 Here the multiplier 1.2 is the same as \$ ; there- fore 18.42 x 1.2 - 18.42 x ^ = 2 -^ Q4 = 22.104 Ans. 2. Multiply 6.82 by 0.2 Ans.. 1.364 3. Multiply 36.7 by 0.6 Ans. 22.02 4. Multiply 0.067 by 0.4 Ans. 0.0268 5. a. In Example 1 we first multiplied by 12 and then divided by 10 ; now the effect of dividing by 10 is merely to move the point one place to the left. We may say, therefore, that in order to multiply 18.42 by 1.2, we may first multiply by 12 and then move the point one place to the left in the product. We may arrange our work as fol- lows : B.] Multiplication. 123 18.42 18.42 Multiplicand. 12 1.2 Multiplier. 3684 or more briefly 3684 1842 1842 221.04 22.104 Product. 22.104 Ans. b. Multiply 1.732 by 1.4 1.732 c. Multiply 267.3 by 1.6 1.4 Ans. 427.68 6928 1732 2.4248 Ans. 6. Multiply 184.2 by 0.4. Ans. 73.68 7. Multiply 1.842 by .6 8. Multiply 18.42 by 0.8 9. Multiply 0.695 by 13.2 10. Multiply 0.00763 by 3.8 11. Multiply 7.614 by 38.2 12. When there is one decimal place in the multiplier (as in each of the last six examples), how many more decimal places are there in the product than in the multiplicand, and why? How many decimal places will there be in the product in each of the following examples, and why ? a. 18.42x12.3 c. 2.135x1.3 b. 69.1x1.1 c?. 0.1684x1.4 13. Find the product in each of the cases just given. 14. a. Multiply 18.42 by 0.06 124 Decimals. [ 5. Here the multiplier 0.06 is the same as jg^ ; therefore 18.42 x 0.06 = 18.42 x ^ = Uftp = 1.1052 Ans. b. Multiply 1.842 by 0.12 Ans. 0.22104 15. Multiply 1.842 by 0.08 Ans. \A~CM 16. Multiply 18.432 by 0.09 Ans. 1.65888 17. Turning back to Example 14 a. we see that the last step in the process consists in dividing by 100, the effect of which is merely to move the point two places to the left. We may say, then, that in order to multiply 18.42 by 0.06 we may first multiply by 6 and then move the point two places to the left in the product. We may ar- range the work briefly, as follows : 18.42 Multiplicand. 0.06 Multiplier. 1.1052 Product. a. Multiply 18.42 by 1.23 18.42 b. Multiply 0.2268 by 0.16 1.23 Ans. 3.06288 5526 3684 1842 22.6566 Ans. 18. Multiply 16.8 by 0.03 Ans. 0.504 19. Multiply 144 by 0.12 20. Multiply 111.234 by 0.09 21. When there are two decimal places in the multiplier (as in each of the last eight examples), how many more decimal places are there in the B.] Multiplication. 125 product than in the multiplicand, and why ? How many decimal places will there be in the product in each of the following cases, and why ? a. 1.842x1.23 c. 69.1x0.11 b. 1.842 x 0.32 d. 0.6842 x 1.21 22. Find the product in each of the cases just given. 23. Multiply 18.42 by 0.006 Ans. 0.11052 24. Multiply 2.634 by 0.004 Ans. 0.010536 25. Multiply 111.32 by 0.015 26. Why is it that when there are three decimal places in the multiplier the number of decimal places in the product is three more than in the multiplicand ? 27. How many decimal places will there be in the product in each of the following cases, and why ? a. 184.2x0.012 c. 1.234x1.112 b. 6.91x0.006 d. 162x0.016 28. Find the product in each of the cases just given. Below is given the work for the first case : a. 184.2 0.012 3684 1842 2.2104 Ans. 29. Multiply 1164.1 by 0.0006 Ans. 0.69846 30. Multiply 1762 by 0.0012 Ans. 2.1144 31. Multiply 236.8 by 0.1112 Ans. 26.33216 32. Why is it that when there are four decimal 126 Decimals. [ 5, places in the multiplier the number of decimal places in the product is four more than in the multiplicand ? 33. How many decimal places are there in the product in each of the following cases, and why ? a. 113.46x0.1111 c. 2137.1x0.0123 b. 23.684 x 1.6842 d. 1.6931 x 1.2311 34. Find the product in each of the cases just given. NOTE. When we multiply by a number which contains decimal places, the last step in the process always consists in a division, the effect of which is to move the point as many places to the left as there are decimal places in the multiplier ; we may say, then, that the number of decimal places in the product is the sum of the number of decimal places in the multiplicand and in the multiplier. 35. How many decimal places are there in the product in each of the following cases, and why ? a. 68.2 x. 4 j. 186.6 x. 66 b. 13.45 x. 6 k. 1215 x. 012 c. 86.2 x. 8 I. 26.943xl.09 d. 13691 x .004 m. .9642 x .009 e. .2841 x. 18 n. 233.1x98 /. .1234 x. 11 o. 1684 x. Ill g. 1.7236 x. 01 p. .068x68 h. 11.123 x. 002 q. 7.123x9.8 /. .2783 x. 0123 r. 1.672x1.31 36. Find the product in each of the cases just given. B.] Multiplication. 127 37. At $0.667 a foot, what is the cost of enough steel rails for a rod (16.5 ft.), of railroad track ? 38. If the circumference of a wheel is 3.14 times the diameter, what is the circumference when the diameter is 3.87 feet ? 39. If a man can walk 3.75 miles in an hour, how far, by walking 6.3 hours a day, can he walk in 3.8 days ? 40. If it takes a train, which travels at an aver- age rate of 26.22 miles an hour, 10.25 hours to go from A to B, what is the distance between these places ? 41. At $8.26 an acre, what is the cost of .06 of an acre of land ? of .07 of an acre ? of .08 of an acre? Ans. 10.4956; 10.5782; $0.6608 NOTE. Since a cent is the smallest coin we have in circulation, the amounts given above can- not be paid exactly. In cases of this kind the number of cents required is that which is nearest the exact amount ; in paying these amounts, then, there would be required $0.50 for the first, $0.58 for the second, and $0.66 for the third. 42. What must I pay for 6.3 square feet of leather at $0.37 a square foot? what for 8.7 square feet? Ans. $2.33; $3.22 43. What must I pay for 9.5 square feet of leather at $0.37 a square foot ? Ans. The exact amount is $3.515 or $3.5l, which is equally near $3.51 and $3.52 In a case of this kind the man who sells generally adds a half cent to his bill. 128 Decimals. In this case, then, I should probably be called upon to pay $3.52 * 44. What must a carpenter pay for the follow- ing bill of lumber: 4500 shingles at $4.70 per thousand ; 13,842 feet of boards at $28.35 per thousand; 4849 feet of planks at $42.75 per thou- sand ; 18,382 laths at $0.38 per hundred ? 45. What will 18,763 bricks cost at $7.75 per thousand ? 46. What will be the freight charge from Boston (a) to Chicago, on 1263 Ibs. at 75c. a hundred ? (6) to New York, on 1878 Ibs. at 35c. a hundred? (c) to Albany, on 2034 Ibs. at 30c. a hundred ? (d) to New Haven, on 689 Ibs. at 28c. a hundred ? (e) to Buffalo, on 568 Ibs. at 44c. a hundred? * Where a half cent comes into the exact amount, an old busi- ness custom still sometimes followed gives the half cent to the man who makes the change. If, for example, A owes B $0.405 and offers him exactly $0.4(3, B should not object; but if A offers B $0.50 then B need give back only 3 cents in change. SECTION VI. DIVISION OF DECIMALS. A* Division by a Whole Number. Examples and Problems, with Remarks and Explanations. 1. a. Divide 9.7 by 4. Divisor 4)9.7 Dividend. 2.4J Quotient. Applying the method used in the division of whole numbers, we say i of 9 units is 2 units, with a remainder of 1 unit ; we set down the 2 in the units' place and save the 1 [unit]. 1 unit added to 7 tenths makes 17 tenths ; J of 17 tenths is 4J tenths [.4J] ; we set down the .4 to the right of the 2. 6. Show that 9.6 divided by 4 is 2.4, by first re- ducing the 9.6 to a common fraction and then proceeding as in the division of a fraction by a whole number. 2. Divide 9.48 by 6, in each of the two ways just indicated. A ns. 1.58 3. Divide 1.728 by 12, in two ways. Ans. 0.14^ 4. Divide 48.6912 by 24, in two ways. 5. a. How many places are there in the divi- dend, and how many in the quotient in each of the last 4 examples ? 130 Decimals. [ 6. b. Give a new example of each of the following cases : With a divisor a whole number, and a divi- dend containing one decimal place ; a dividend with two decimal places ; three decimal places ; four decimal places. How many decimal places will there be in the quotient of each ? c. How many decimal places will there be in the quotient of 9.68479 divided by 4 ? Find the quotient. Ans. 2.42119| 6. Give a new example of each of the following cases before answering the question : With the divisor a whole number, how many decimal places will there be in the quotient when the dividend contains one decimal place ? two decimal places? three decimal places 1 four deci- mal places ? NOTE. From what precedes we see that when the divisor is a whole number we may divide ex- actly as in whole numbers, pointing off the same number of decimal places in the quotient as are pointed off in the dividend. 7. Divide 80.4144 by 4. 8. Divide 864.144 by 6. 9. Divide 8641.44 by 9. 10. Divide 4.216 by 2. 11. Divide 0.6824 by 8. 12. Divide 0.0132 by 9. 13. Divide 6.55 by 7. 14. Divide 4.18 by 4. 15. Divide 0.167 by 11. 16. Divide 3.264 by 24. Divide 78.428 by 36. A.] Division. 131 17. How many hundredths are there in f (that is, in 3 divided by 4) ? We know that in 3 there are 300 hundredths (?8#), which may be written 3.00 Now, dividing as in previous examples, we have 4 ) , therefore | -.75 .75 How many hundredths are there in 2 \ ? 18. How many thousandths are there in ? [5 may be written 5.000] Ans. .625 How many thousandths are there in T | ^ ? 19. Change J to thousandths. Ans. .875 20. How many tenths are there in J ? how many hundredths with how many thousandths left over ? 21. Reduce ^ to hundredths. 22. Reduce f to thousandths. 23. Reduce T | 5 to thousandths. 24. Reduce J>J to hundredths. Ans. .68 Since .68 is nearer to .7 than to .6, we may say that, expressed to the nearest tenth, JJ is .7 25. What number of tenths comes nearest the true value of ? Ans. .8 26. What number of tenths comes nearest the true value of J ? (Ans. .3) What number of hun- dredths? (Ans. .33) 27. Express to the nearest hundredth the value of , of f , of ft, of T \. 28. a. How many hundred-thousandths are there in , ? 7)2.00000 ' 0.28571+ Ans. [The + is used here to denote that there is still a remainder after using five places.] 132 Decimals. [ 6. 1). How many In u id red-thousandths are there in | ? & = . 77777, etc. Using only 4 decimal places we should call our answer .7778,* because this is nearer the true value of J than .7777 In this case we may say, if we like, ^ = .7778 29. How many hundredths are there in { ? Since j =.625 is just half way between .63 and .62, we may here choose for our answer either .62 or .63 Where there is any choice, most com- puters, for the sake of uniformity, choose an even number for the last figure. They would therefore choose .62 rather than .63 30. How many hundredths are there in Jg ? .I//*. .82 31. Divide 11 by 13, carrying out the result to 4 decimal places only. The work may be written as indicated below on the left ; or, since we can easily tell how many zeros are to be used without actually writing them all in the dividend, it may be written more briefly as indicated on the right. 13)11.0000(.8461+ 13)11.0(.8461 + 104 104 60 ~~60 52 52 80 80 78 78 ~20 ~20 13 13 * The result should always be made as near the true value as possible. A.] Division. 133 32. Divide 11 by 14, carrying out the result to 3 places of decimals. Ans. .786 33. Divide (a) 23 by 32, and (6) 87 by 64, continuing the process of division in each case un- til there is no remainder. 34. Divide 18 by 12. Ans. 1.5 35. Divide 25 by 8. Ans. 3.125 36. Divide 27 by 24. 37. Divide 11 by 8. 38. Divide 8 by 11, carrying out the result to three places of decimals. Ans. 0.727 39. Divide 2 by 9, carrying out the result to 3 places of decimals. 40. Divide 9 by 24. 41. An Englishman on reaching America sold 3 English pounds for $ 13.58 ; how much was that a pound ? 42. Change 1.3 gills to the decimal of a pint. [1 gill = i of a pint, therefore 1.3 gills = l f- pts. = .325 pts.] 43. Change 1 gill to the decimal of a quart. 44. Change 3 gills to the decimal of a gallon. Ans. .09375 galls. 45. Change 2 qts. 1 pt. to the decimal of a gallon. Solution: First step, 1 pt. = .5 qts. .*. 2 qts. 1 pt. = 2.5 qts. Second step, 2.5 qts. = .625 galls. Ans. The work may be arranged as follows : 2 1.0 pt. 2.5 qts. .625 galls. Ans. 134 Decimals. [ 6. 46. Change 3 qts. 1 pt. 2 gi. to the decimal of a gallon. Solution: 4 2.0 gi. 1.50 pts. 3.75 qts. .9375 galls. Ans. 47. Change 1 pt. 1.6 gi. to the decimal of a gallon. 48. Change 1 qt. 1 pt. 3.52 gi. to the decimal of a gallon. 49. Change 1 pt. 2.96 gi. to the decimal of a gallon. 50. If 2 quarts of alcohol cost $2.16, what will be the cost of a gill ? 51. If 24 gallons of vinegar cost 16.72, what will be the cost of a pint ? 52. If 8 two-gallon cans of milk cost #6.72, what will be the cost of a pint ? 53. If 3 gallons of ice cream cost $ 4.80, what will be the cost of a pint ? 54. If 4 gallons of molasses cost $ 2.00, what will be the cost of a quart ? 55. If 3 gallons of astral oil cost 75 cents, what will be the cost of a quart ? 56. Change 1 pk. 3 qts. 0.4 pts. to the decimal of a bushel. Ans. .35 bu. 57. What is the number of bushels expressed decimally in 1 bu. 1 pk. 6 qts. .8 pts. ? 58. Change 3 qts. 1 pt. to the decimal of a bushel. 59. Change 3 pks. 7 qts. 1 pt. to the decimal of a bushel. A.] Division. 135 60. Change 1.28 pts. to the decimal of a bushel. 61. If 3 pecks of berries cost 12.88, what will a quart cost ? 62. If a bushel of grass seed costs $4.64, what will a quart cost ? 63. If 2 bushels of potatoes cost 11.50, what will a peck cost ? 64. If 3 bushels of apples cost 13.42, what will a peck cost? 65. Change 32 ounces to the decimal of a ton. 66. Change 5 pounds to the decimal of a ton. 67. Change 1624 Ibs. 12 oz. to the decimal of a ton. 68. Change 1 Ib. 3.2 oz. to the decimal of a ton. 69. Change 430 Ibs. 3.84 oz. to the decimal of a ton. 70. If 14 pounds of coffee cost $5.25, how much will a pound cost ? 71. If a ton of range coal costs $5.50, what will a pound cost ? 72. If a ton of American Cannel coal costs $12, what will a pound cost ? 73. If a ton of hay costs $22, what will a pound cost? 74. If 3 tons of lead cost $690, what will a pound cost ? 75. Change 63.36 inches to the decimal of a mile. Solution : We may change successively to feet, yards, rods, and miles, and may arrange our work as follows : 136 Decimals. [ 6. 12)03.86 in. 3)5.28 ft. here dm^" by H and *i_ ix 2 ~ *~ .16 multiply by 2. 320)^20 rds. .001 mi. Ans. 76. Change 40 rds. 1 ft. 3.84 in. to the decimal of a mile. Ans. .12525 miles. 77. Change 3 yds. 1 ft. 6.72 in. to the decimal of a mile. 78. Change 36 rds. 1 yd. 2 ft. 3.36 in. to the decimal of a mile. 79. If it costs $11.59 to build 38 rods of fence, what will it cost to build a rod ? 80. If 108 fathoms of rope cost 18.10, what will a fathom cost ? [A fathom = 6 ft.] 81. How much is railroad iron a foot when rails enough to lay a mile of track cost f 580.80 ? 82. If it takes sound 42.24 seconds to travel 10 miles, how long will it be in going 1 foot ? How' many feet will it travel in a second ? 83. Change 66 rds. 4 yds. 1 ft. 2.4 in. to the decimal of a mile. 84. Change 36 rds. 4 yds. 1 ft. 2.4 in. to the decimal of a mile. 85. Change 3 rds. 3 yds. 2 ft. 3.432 in. to the decimal of a mile. 86. Change 1 rd. 1 ft. 3.84 in. to the decimal of a mile. 87. Change to the decimal of a mile 33 hands; 52.8 hands. [A hand = 4 in.] I B.] Division. 13' 88. Change 36 seconds to the decimal of an hour. 89. Change 1 h. 12 min. to the decimal of a day. 90. Change 15 h. 39 min. 36 sec. to the decimal of a day. Ans. .6525 dys. 91. Change 4 h. 22 min. 49.08 sec. to the deci- mal of a day. Ans. .1825125 dys. 92. Change 3 h. 30 min. 23.04 sec. to the deci- mal of a day. 93. Change 2 h. 15 m. to the decimal of a week. B. Division of a Decimal by a Decimal. Exam- ples and Problems, with Hemarks and Expla- nations. 1. Divide 86.4144 by 1.2 Since 1.2 is the same as \\ we may say that 86.4144 -M.2- 86.4144^ \\. Now to divide by 1$ we may first dmde^by 12 and then multiply by 10 ; we know therefore that , 86.4144-7-13 -= ^\\ 4 4 x 10 : = 7.20^2 x 10- 72.012 2. Show that 86.4144 -f. 6 = 144.024^ 3. Show that 86.4144^-2.4 = 36.006 4. Show that 86.4144-f-.9-96.016 5. Divide 86.4144 by .8 Here, as in the last four examples, two steps are to be taken : First, we are to divide by 8, and, second, we are to multiply by 10. Now multiply- ing by 10 moves the point one place to the right 138 Decimals. [ 6. (see p. Ill), and therefore gives only 3 decimal places in the quotient, or one less than there are in the dividend. We may, then, arrange our work as follows, first dividing by 8 as in whole numbers, and then pointing off only 3 decimal places in the quotient. Divisor 0.8)86.4144 Dividend. 108.018 Quotient. 6. How many decimal places are there in the quotient in each of the following examples, and why? a. 4.263^.3 c. 6.1224^-.^. 5.064^-2.4 6. 91.26-^.9 d. 8.8-5- .8 / 0.01728^14.4 7. Find the quotient in 6 a. 8. Find the quotient in 6 b. 9. Find the quotient in 6 c. 10. Find the quotient in 6 d. 11. Find the quotient in 6 e. 12. Find the quotient in 6/1 13. Divide 86.4144 by .06 Since .06 is the same as T $Q, we may say that 86.4144 * .06 = 86.4144 * T - = 14.4024 x 100 - 1440.24 Ans. How does this quotient differ from that of Ex- ample 2 ? 14. Show that 86.4144^0.12 = 720.12 Compare with Example 1. 15. Show that 86.4144-K24- 360.06 Compare with Example 3. 16. Show that 86.4144 -r. 09 = 960.16 Compare with Example 4. B.] Division. 139 17. Divide 86.4144 by .08 Here the two steps in our work are : (1) to di- vide by 8 and (2) to multiply by 100. Now mul- tiplying by 100 moves the point two places to the right (see p. 112), and therefore gives only two decimal places in the quotient, or two less than there are in the dividend. We may, then, arrange the work as follows, first dividing by 8 as in whole numbers, and then pointing off only 2 deci- mal places in the quotient. .08)86.4144 1080.18 Compare with Example 5. 18. How many decimal places are there in the quotient in each of the following examples, and why? a. 4.263^.03 c. 6.1224^.04 e. 5.064 -r. 24 b. 91.26-K09 d. .88-r.OS /. .01728-^1.44 19. Find the quotient in 18 a. 20. Find the quotient in 18 b. 21. Find the quotient in 18 c. 22. Find the quotient in 18 d. 23. Find the quotient in 18 e. 24. Find the quotient in 18 f. 25. Divide 86.4144 by .006 Since .006 is the same as T ^-Q we may say that 86.4144-^.006 = 86.4144^^ = 86.4144 x 1000 ^ 14.4024x1000 = 14402.4. How does this quotient differ from those of Examples 2 and 13 ? 26. Show that 86.4144 -r. 012 = 7201.2 140 Decimals. [ 6. Compare with Examples 1 and 14. 27. Show that 86.4144 ^.024 -3600.6 Compare with Examples 3 and 15. 28. Show that 86.4144-- .009 = 9601.6 Compare with Examples 4 and 16. 29. How many decimal places are there in the quotient of 86.4144 divided by .008, and why ? Find the quotient by arranging the work as in Examples 5 and 17. 30. How many decimal places are there in the quotient in each of the following examples, and why ? a. 4.263-^.003 d. .00088^.008 6. .9126^.009 e. 5.064 -r. 024 c. 6.1224^.004 /. .01728-T.144 31. Find the quotient in 30 a. 32. Find the quotient in 30 6. 33. Find the quotient in 30 c. 34. Find the quotient in 30 d. 35. Find the quotient in 30 e. 36. Find the quotient in 30 /. 37. When the divisor contains 3 decimal places (thousandths) the number of decimal places in the quotient will be how many less than in the divi- dend, and why? NOTE. In each of the last 37 examples (where the divisor contains not more than three decimal places) the number of decimal places in the quo- tient was found to be equal to the difference be- tween the number of decimal places in the divi- dend and in the divisor. That this statement is B.] Division. 141 true, for any example where the divisor has more than three decimal places can easily be shown by reducing the divisor to a common fraction, as in Example 1. 38. How many decimal places are there in the quotient of 0.643214 divided by 0.00011? Find the quotient. 39. How many decimal places are there in the quotient of 0.001233 divided by .0003 ? Find the quotient, arranging the work as in Example 17. 40. Why are there no decimal places in the quotient of .0066 divided by .0011? Find the quotient. [The difference between the number of decimal places in the dividend and in the divisor being zero, the statement contained in the note following Ex- ample 37 holds good as well here as in previous examples.] 41. a. Divide 5.7102 by .1842 .1842)5.7102(31. Answer. 5526 1842 1842 6. Divide 0.01664 by .0064. Ans. 2.6 42. Divide .697368 by .000168 Ans. 4151 43. Divide 6.316125 by 1.6843 Ans. 3.75 44. Divide 16.7376 by 1.32; 8.552478 by 1.111 45. Divid^.199584 by 2.31; .625 by .25 46. Divide .6 by .06 The statement contained in the note following 142 Decimals. [ 6. Example 37 does not appear to hold good here be- cause the number of decimal places in the divi- dend is less than in the divisor ; we will therefore first solve this example by the method of Exam- ples 1, 13, and 25, where the divisor was in each case reduced to a common fraction : .6-K06 = .6-J- T fa = .6x 100 = .1x100 = 10. 6 47. Divide .044 by .0011 48. Divide 12.33 by .009 49. Divide 14.4 by .012 50. Although the method given above offers no difficulty, we may find the quotients by a shorter process, as follows : a. In Example 46 the dividend .6 or -f$ is the same as T \ ^, and may be written .60 We can now divide .60 by .06 as in Example 41 a, thus : .06).6Q 10. b. In Example 47 the dividend .044 may be written .0440 ; may, then, divide as indicated on the ' ^~ right, and place the decimal point in the quotient in accordance with the state- ment contained in the note on page 140. c. Solve in the same way Example 48. 51. Solve in the same way Example 49. 52. From the last six examples we learn that the statement contained in the notion page 140 can be made to apply to cases where there are fewer decimal places in the dividend than in the C.] Reduction of Common Fractions. 143 divisor, if we first annex zeros to the dividend until it contains as many decimal places as there are in the divisor. Divide 172.8 by .144 Answer 1200 53. Divide 1.8 by .006 54. Divide .144 by .0004 55. Divide 86 by .43 56. Divide 2.8 by .007 57. Divide 1 by .25 58. Divide 10 by .02 59. Divide 50 by .001 C. Reduction of Common Fractions to Deci- mals. Circulating Decimals. Some common fractions can be exactly expressed by decimals, and others cannot be so expressed. In the following cases, for example, each fraction can be exactly expressed by a decimal : | =.75 $=.625 ^=.35 T |^ = -072 2 ' r , = .16 $=.875 | =.375 _ 7 ^ = .008 But if we try to reduce ^ to a decimal, each suc- cessive division gives a quotient Q^ QQQOOO 3 and a remainder 1, and this - OOOOOQI _ i will evidently continue to be the case no matter how far the process of division be continued. There can be, therefore, no decimal fraction which is the exact equivalent of ^. The decimal .3 expresses the value of J to within ^; .33 to within T ^ ; .3333 to within T oiroT, etc ' 144 Decimals. [ G. 1. Find the decimal of three places which ex- presses to within -^-^ the value of |. Ans. .666 [Expressed to the nearest thousandth the answer would be .667] What figure would be repeated if the process of division were continued ? 2. Find the decimals which express to within TtfFoT) the va l ues of the following fractions : () 4; (*)|; 001; 001; 00 I ; (/) 5- What figure would be repeated in each case if the process of division were continued ? 3. Find the decimals which express to within nnfWlF the fractions (a) ^ ; (&) T^ ; 0) 1 .1- What figure would be repeated in each case if the process of division were continued ? 4. Find the decimals which express to within Ttfowoo the fractions, (a) ff ; (6) f f ; (c) ,; ! , r [Answer to (a) .696969 + ] What ^o figures would be repeated in each case if the process of division were continued ? 5. Find the decimals which express to within TOO^OOIT () HI ; (6) III- ^n. (a) .684684 + ; (6) .362362 + What three figures would be repeated in each case if the process of division were continued ? 6. Divide 1 by 7, carrying out the result to 12 places of decimals. Ans. .142857142857 + What six figures would be repeated if the pro- cess of division were repeated ? 7. Divide 6 by 13, carrying out the result to 12 places of decimals. Ans. .461538461538 + D.] Miscellaneous Examples. 145 What six figures would be repeated if the pro- cess of division were continued ? NOTE. In the last seven examples we have dealt with decimals in which, if the process of division were continued, the same figure or series of figures would be repeated, over and over again ; such deci- mals are called CIRCULATING DECIMALS, and are more commonly written by placing a dot over the first and last of the series of repeated figures. Thus, in Example 4, J| = .696969 + may be more briefly written ff =.69 .1428571428574- (see Example 6) may be writ- ten .142857 .362362 4- (see Example 5 6) may be written .362 .58333 -f (see Example 3 6) may be written .583; .666+ (see Example 1) may be written .6 Express in the same manner the remaining an- swers of the last seven examples. D. Miscellaneous Examples. 1. If a man walks at the rate of a mile in .4 of an hour, how far can he walk in 3.432 hours ? 2. In one rod there are 5.5 yards ; how many rods are there in 27.225 yards? 3. The quart of liquid measure contains 57.75 cubic inches ; how many quarts of liquid measure are there in a cubic foot (1728 cubic inches) ? Carry the result out to two places of decimals. 146 Decimals. [6. 4. The quart of dry measure contains 67.2 cu- bic inches ; how many quarts of dry measure are there in a cubic foot? Carry the result out to two places of decimals. 5. At $7.75 per thousand, how many bricks can be bought for $26.35 ? 6. At 35 cents a hundred, how many laths can be bought for 183.86 ? 7. If in 34 lines there are 331 words, what is the average number of words per line ? Carry out the result to two places of decimals. Ans. 9.74 8. The figures given below are taken from the Census of 1880. Compute to two places of deci- mals the average size in acres and the average value of the farms of each state. Number of farms. No. of acres in farms. Value of farms. Alabama 135,864 6,375,706 $ 78,954,648 California 35,934 10,669,698 262,051,282 Colorado 4,506 616,169 25,109,223 Massachusetts 38,406 2,128,311 146,197,415 Michigan 154,008 8,296,862 499,103,181 Ohio 247,18918,081,091 1,127,497,353 Pennsylvania 213,542 13,423,007 975,689,410 South Carolina 93,864 4,132,050 68,677,482 Texas 174,18412,650,314 170,468,886 Virginia 118,517 8,510,113 216,028,107 9. In 1887 the Chicago, Burlington, and Quincy Railroad Company sold D.] Miscellaneous Examples. 147 a. In Nebraska 7,079.48 acres of land for 151,739.58 b. In Iowa 7,357.32 acres of land for 151,724.11 What was the average price per acre in each case ? Carry out the result to two places of deci- mals. SECTION VII. PERCENTAGE. A. Interest. 1. If I must pay 6 cents for the use of $1 for 1 Money year, what must I pay for the use of $ 6 for theYse'of the same time ? what for the use of $ 16 ? money. of $8 ? of * 10.50 ? of 118.75 ? of $25 ? 2. If I must pay 6 cents for the use of $1 for 1 year, what must I pay for the use of $ 1 for 2 years? what for 6 years? for 2 years and 6 months?* for 4 years and 4 months ? for 10 years and 4 months ? for 3 years and 8 months ? for 2 months ? for 4 months ? for 6 months ? for 8 months? for 10 months? for 1 month? for 3 months? for 9 months? 3. a. If I must pay 8 cents for the use of $1 for 1 year, how much must I pay for the use of f 3 for 4 years? Solution : If I must pay 8 cents for the use of f 1 for 1 year, for the use of $3 for 1 year I must pay 3 times 8 cents, or 24 cents. If I must pay 24 cents for the use of $3 for 1 * In cases of this kind, business men, for the sake of conven- ience, usually regard a month as ^ of a year: this, though very nearly true, is not exactly so, since no month contains a number of days which is exactly ^ of 365. A.] Interest. 149 year, for the use of $3 for 4 years I must pay 4 times 24 cents, or 96 cents. b. If I must pay 8 cents for the use of 11 for 1 year, how much must I pay for the use of $6 for 7 years? Ans. $3.36. How much for the use of f3.50 for 3 years ? Ans. 84 cents. How much for the use of $6.50 for 5 years and 6 months? Ans. $2.86. How much for the use of $1 for 6 months? for 3 months ? 9 months ? 1 month ? 4. If I must pay 4 cents for the use of $1 for 1 year, what must I pay for the use of $8 for 3 years ? what for the use of $7 for 6 years and 3 months ? what for the use of $10.50 for 5 years and 6 months ? what for the use of $2 for 12 years and 9 months ? 5. If I must pay $6 for the use of $100 for 1 year, how much must I pay for the use of $300 for 4 years ? $400 for 2 years ? $200 for 3 years and 6 months ? $500 for 3 years and 2 months ? $600 for 1 year and 9 months ? $100 for 2 years and 8 months ? 6. If an Englishman has to pay T |^ of IX for the use of 1 for 1 year, how many shillings must he pay for the use of 3 for 4 years? (Ans. 12s.) 5 for 3 years ? 4X for 2 years and 6 months ? 10< for 6 months ? 9X for 4 months ? 7. If I must pay $6 for the use of $100 for 1 year, what must I pay for the use of $100 for 1 month? for 15 days? * for 6 days? for 24 days? * In cases of this kind, business men usually regard 1 MONTH AS 30 DAYS, and therefore I DAY AS ^ OF A MONTH, 15 days as io or i * a montn - 150 Percentage. [ 7. for 18 days? for 12 days? for 3 days? for 1 day? for 8 days ? 8. If I must pay $8 for the use of $100 for 1 year, what must I pay for its use for 1 month ? for 15 days? for 20 days? for 10 days? for 16 days? Money paid by a borrower to a lender for the interest, use of money is called INTEREST. 9. If the interest on $ 1 for 1 year is 6 cents, what is the interest on $3.50 for 4 years and 2 months ? Ans. $0.875 10. If the interest on $3 for 4 years is 72 cents, what is the interest on $1 for 1 year ? Ans. $0.06 (6 cents). 11. If the interest on $4 for 3 yrs. 4 m. is 80 cents, what is the interest on $1 for 1 yr. ? Solution : If the interest on $4 for 3 yrs. 4 m. is 80 cents, the interest on $1 for the same time is | of 80 cents, or 20 cents. 3 yrs. 4 m. is the same as 3^ or -^- yrs. ; now if the interest on $1 for JgQ- yrs. is 20 cents, then for 1 of a year it is T ^ of 20 cents, or 2 cents, and for a whole year it is 3 times 2 cents, or 6 cents. 12. If the interest on $1 for 1 yr. is 6 cents, what is the interest on $0.50 for 10 yrs. ? for 24 days ? 13. If the interest on $100 for 1 yr. is $6, what is the interest on $300 for 3 yrs. 6m.? 14. If the interest on $1000 for 1 yr. is $60, what is the interest on $100 for the same time? on $1? The usual interest for 1 yr. is 6 cents on each Per cent, dollar, 6 dollars on each hundred dollars, A.] Interest. 151 or Y^Q- of the sum borrowed ; the rate of interest is here said to be 6 PER CENT * a year ; 3 per cent, 4 per cent, etc., signify, then, T | T , T ^, etc., of the sum borrowed. 15. At 6 per cent a year, what is the interest on |3 for 4 yrs. ? (Ans. 72 cents) on |2 for 6 yrs. ? on $10 for 2 yrs. ? on $50 for 4 yrs. 8 m. 6 d. ? (Ans. $14.05) 16. At 4 per cent a year, what is the interest on 10< for 3 yrs. ? (Ans. 1 4s.) on 160 for 5 yrs. 9 m. ? (Ans. 36 16s.) 17. At 8 per cent a year, what is the interest on $4 for 2 yrs. 3m.? 18. At 3 per cent a year, what is the interest on $6 for 8 yrs. 8m.?' 19. At 6 per cent a year, what is the interest on $12.25 for 3 yrs. 8m.? Instead of the words per cent, the symbol /o is often used ; instead, then, of 6 per cent, 4 per cent, 8 per cent, etc., we may write /& 6$fc, 4>, 8J6, etc. 20. At 6fi a year, what is the interest on $1 for 2 months ? for 4 months ? for 6 months ? for 8 months ? for 10 months ? for 1 month ? for of a month ? for 18 days ? 21. At 8fi a year, what is the interest on $1 for 3m.? for 6m.? for 9m.? for 15 d. ? 22. At 6/0 a year, how long will it take $1 to earn 12 cents ? how long to earn 3 cents ? 2 cents ? * Per cent is from the Latin per centum, which means by the hundred. 152 Percentage. [ 7. 4 cents ? 1 cent ? 5 mills or |- of a cent ? 100 cents or $1 ? 23. At 8^> a year, how long will it take 1100 to earn $8 ? how long to earn $2 ? 14 ? $1 ? $24 ? $100? 24. At 6 a year, how long will it take $100 to earn $1 ? how long to earn $2.50 ? $4.50 ? $1.50 ? $5.50 ? 50 cents ? 10 cents ? 5 cents? NOTE. Since interest is usually reckoned by the year, 6jfc, 4^b, etc., when not followed by any specified time are understood to mean 6jfc a year, 4fy a year, etc. 26. If the interest on $2 for 5 yrs. is 90 cents, what is the rate per cent. ? Ans. 9$fc. 27. What is the rate per cent when the interest on $1 for 3 yrs. is 18 cents ? what when the inter- est on $1 for 2 in. is 1 cent ? when the interest on $1 for 3 yrs. 9 m. is 30 cents ? what when the in- terest on $1 for 2 yrs. 6 m. is 10 cents ? what when the interest on $1 for 6 days is 1 mill ? 28. What is the rate per cent when the interest on $1000 for 1 yr. is $40 ? what when the interest on $1000 for 1 yr. is $60 ? what when the interest on $1000 for 1 yr. is $70 ? 29. What is the rate per cent when the interest on !< for 1 yr. is Is. ? when the interest on 3X for 1 m. is 3d. ? 30. What is the rate per cent when the interest 011 $300 for 2 yrs. 6 m. is $30 ? A.] Interest. 153 31. What is the rate per cent when the interest on $200 for 3 yrs. 6 m. is $49 ? when the interest on the same sum for the same time is f 42 ? ' 32. What is the rate per cent when the interest on $12 for 3 m. is 12 cents ? 33. If I borrow $100 for 6 m. 3 d. at 6^, how much must I pay back, including the $100 and in- terest, at the end of the 6 months and 3 days ? 34. If I borrow $20 for 3 yrs. 6 m. at 8^, how much must I pay back ? The sum borrowed is called the PRINCIPAL, and the total sum paid back, includ- principal. ing the Principal and Interest, is Amount. called the AMOUNT. 35. If the principal is $50, what will be the amount at the end of 6 years, when the rate of interest is 5j& ? 36. If the principal is $7000 what will be the amount at the end of 10 yrs. 3 m., when the rate of interest is 4^ ? 37. What principal, if put at interest at 8*fe, will earn $16 in two years ? 38. What principal, if put at interest at 6^, will earn $1.50 in 8 yrs. 4 m. Solution : $1 will earn 50 cents in 8 yrs. 4 m. at Qfi ; therefore, to earn $1.50 will take as many times one dollar as 50 cents is contained in $1.50, or 3 times one dollar. The answer, then, is $3. 39. What principal, if put at interest at 6^, will earn $25 in 8 yrs. 4m.? [Find first how much $1 will earn in the given time.] 154 Percentage. [ 7. 40. What principal, if put at interest at 4j6, will earn 60 cents in 7 yrs. 6m.? 41. What is the principal when the amount at the end of 1 yr., at 6^, is 1106 ? 42. What is the principal when the amount at the end of 2 yrs., at 4^, is $108 ? 43. What is the principal when the amount at the end of 2 yrs., at 8^, is $348 ? Solution : $1 will amount to $1.16 in 2 yrs. at 8^> ; therefore, it will take as many times 1 dollar to amount to $348 as 1.16 is contained in 348, or 300 times $1. The answer, then, is $300. What will be the amount of this principal at the end of 1 yr. ? what at the end of 6 in. ? what at the end of 4 yrs. ? 44. What is the principal when the amount at the end of 3 yrs., at 4^), is $448 ? What would this principal amount to at the end of 1 yr. ? at the end of 2 yrs. ? 45. What is the principal when the amount at the end of 1 yr. 6 m., at 6^, is $218 ? What would this principal amount to at the end of 1 yr. ? at the end of 2 yrs. ? at the end of 3 yrs. 3 d. ? 46. What is the principal when the amount at the end of 2 yrs., at 10^, is $60 ? What would this principal amount to at the end of 1 yr ? at the end of 5 yrs. ? 47. What is the principal when the amount at the end of 2 yrs. 6 m. 18 d., at 6^, is $1000? A.] Interest. 155 Solution : The interest on $1 for 2 yrs. is . . $0.12 The interest on $1 for 6 m. is . . 0.03 The interest on $1 for 1 m. is $0.005 The interest on $1 for 6 d. (% of 1 m.) is $0.001 The interest on $1 for 18 d. (f of 1 m.) is 0.003 The interest on $1 for 2 yrs. 6 m. 18 d. is 10.153 The amount of $1 for 2 yrs. 6 m. 18 d. is 11.153 If $1 amounts to $1.153, it will take as many times 11 to amount to 11000 as $1.153 is contained in $1000. 1.153)1000.000(867.302 Ans. $867.302, which 9224 expressed to the 7760 nearest cent is 6918 $867.30 8420 8071 3490 3459 3100 2306 48. What is the principal when the amount at the end of 4 yrs. 3 m. 12 d., at 6^, is $400 ? Ans. (to the nearest cent) $318.22 49. What is the principal when the amount at the end of 4 yrs. 2 m., at 6/0, is $180. Ans. $144. 50. What is the principal when the amount at the end of 8 yrs. 2 m., at 6^, is $1024. Ans. (to the nearest cent) $687.25 51. What is the principal when the amount at 156 Percentage. [ 7. the end of 3 yrs. 4 m., at 6^, is 13668 ? Ans. (to the nearest cent) $3056.67 Promissory A written promise to pay money is called Note - a PROMISSORY NOTE, or simply a NOTE. 52. J. G. Bland of Augusta, Maine, who bor- rowed from A. B. Smith on Jan. 5, 1886, $300, payable in 3 years from date (that is from Jan. 5, 1886), with interest at 6^, to be paid annually, gave Mr. Smith the following note : $300.00 AUGUSTA, ME., Jan. 5, 1886. For Value Received, I promise to pay A. B. Smith, or order, three hundred -j-^ dollars in three years from this date, with interest at 6^, to be paid annually. J. G. BLAND. The words " to be paid annually " required Mr. Bland to pay at the end of each year (Jan. 5, 1887, Jan. 5, 1888, and Jan. 5, 1889) the interest then due. How much interest was due at the end of each year? The words " or order " after Mr. Smith's name gave him the power to order Mr. Bland to pay the money to some one else. Desiring him to pay the money to T. F. Brown, he would write on the back of the note, " Pay to T. F. BROWN. A. B. SMITH." It is not necessary to word a note just like the A.] Interest. 157 foregoing ; all that is required is that the whole promise shall be distinctly stated. It is advisable, though not absolutely necessary, that the person who makes the promise (the maker of the note, as he is called) should acknowledge by some such expression as " For Value Received " that he has received either the money or something else that he considers an equivalent. For greater distinct- ness the sum named in a note should be expressed both in words and by figures ; this sum is called the FACE of the note. 53. Write a note that you would give your next-door neighbor, if you were to borrow from him $430.60, payable in 3 years from now, with interest, to be paid annually, at 6jfc. How much interest would you have to pay him at the end of each year ? 54. Write a note for each one of the following cases : A. On Feb. 1, 1885, H. M. TwitcheU bought of J. P. True a tricycle, for which he promised to pay $150.50 on March 16, 1885, with interest at 6/0. [Business men would regard the time here as 1 m. 15 d., or 1J months ; national governments, however, would count the exact number of days (42), and would consider the interest to be ^g- of the interest for a year.] B. On July 15, 1885, Edward Gilchrist bor- rowed of Frank Allen $450, which he promised to pay on Jan. 21, 1886, with interest at 856. [The 158 Percentage. [ 7. time here is to be regarded as 6 m. 6 d., or 6 months.] C. On Dec. 24, 1885, Willard Small bought of Iloughton, Mifflin & Co. books to the value of 1206.34, which he promised to pay on Feb. 17, 188G, without interest. D. On Feb. 1, 1885, Clarke & Carruth bought of Roberts Brothers books to the value of 1000, to be paid for in 60 days, with interest at 6jfc. [When the time is specified in days the exact number of days must be counted from the date of the note to find when the money is due. In this case, for instance, 27 days in February, plus 31 days in March, plus 2 days in April, make up the 60 days. The money was due, then, on April 2.] E. On Feb. 15, 1885, Richard Roe bought of John Doe a pair of horses for $600, to be paid for in 90 days, with interest at 6^>. F. On July 31, 1885, Amos Ames bought of Byron Burns a barn for $1500, to be paid for in 60 days, with interest at 8^fc. G. On Sept. 1, 1888, John Jones bought of J. C. Ayer 11 acres of wood-land for 400, to be paid for in 90 days, with interest at 5J^. 55. Compute the interest in each of the preced- ing cases, excepting C. 56. This morning James Pike said to Irving Blake, " If you will give me 1106 for my bay mare Dolly you need not pay for her until one year from date." Mr. Blake accepted the offer, and gave Mr. Pike the following note : A.] Interest. 159 *TTTO" CAMBRIDGE, MASS.,* For Value Received, I promise to pay James Pike, or order, one hun- dred six and -f^ dollars one year from date, with- out interest. IRVING BLAKE. A few minutes after the note was given, Mr. Pike found that he needed some money at once, and therefore offered to sell the note to M. M. Sawin ; now supposing that Mr. Sawin has money that he is willing to invest at 6^, what can he afford to pay for the note ? [Compare with Ex- ample 41.] Ans. $100 ; because the payment of $100 now will insure his getting $106 from Mr. Blake a year hence ; that is, in one year he will get back his 1100 and in addition $6 (6^) as interest. The $100 that Mr. Sawin can afford to pay for the note now is called the PRESENT WORTH of the note. The PRESENT WORTH of a debt due at some future time, without interest, is the sum Present which, put at interest, will amount to Wort b. the debt when it becomes due. 57. Find the present worth on Jan. 1, 1885, of each of the following notes, regarding the rate of interest in getting the present worth of A to be 4fo ; of B to be 8^ ; of C to be 4fi ; of D to be 6^; and of E to be Wjb. * The pupil should fill in the date of the day on which he per- forms this example. 160 Percentage. [ 7. A. $108^^. SAN FRANCISCO, CAL., Jan. 1, 1885. For Value Received, I promise to pay B. M. Snow, or order, one hundred eight and y^ dollars in two years from this date.* A. B. COOK. [Compare with Example 42.] B. $348^. NEW YORK, N. Y., Jan. 1, 1885. For Value Received, I promise to pay D. E. Faunce, or order, three hundred forty-eight and -ffa dollars in two years from this date. B. C. DOWD. [Compare with Example 43.] C. $448 T ^j. CHICAGO, ILL., Jan. 1, 1885. For Value Received, I promise to pay E. F. Griffin, or order, four hundred forty-eight and -^fa dollars in three years from this date. C. D. EVARTS. [Compare with Example 44.] D. $218^. NEW ORLEANS, LA., Jan. 1, 1885. For Value Received, I promise to pay F. G. Hale, or order, two hundred eighteen and -j^ dollars in one year and six months from this date. D. E. FALES. [Compare with Example 45.] * Where nothing is said ahoiit interest, no interest can be claimed if the sum promised is paid when due. A.] Interest. 161 E. $^lTO* INDIANAPOLIS, IND., Jan. 1, 1885. For Value Received, I promise to pay G. H. Ives, or order, sixty and y^fo dollars in two years from this date. E. F. GATES. [Compare with Example 46.] 58. Find the present worth on July 1, 1885, of each of the notes of Example 57, regarding the rate of interest in each case the same as before. [The fact that July 1, 1885, is not the date of each note has nothing to do with the question. All that we are required to find is the sum which, put at interest on July 1, 1885, will amount in each case to the sum promised at the time it is due.] The sum which, subtracted from the True Dis- face of a note, would give its present count - worth, is called the TRUE DISCOUNT. The True Discount, then, is the interest on the present worth. 59. Find the true discount on Jan. 1, 1885, on each of the notes of Example 57. 60. Write a note for each of the following cases, and find the true discount at Gfi of each note on Jan. 1, 1885 : A. On Jan. 1, 1885, A. M. Brown bought 11500 worth of groceries of C. B. Smith, promising to pay for them in 6 months from date. B. On Jan. 1, 1885, G. H. Irwin bought a house 162 Percentage. [ 7. of H. I. Knight for $2000, promising to pay for it in 2 yrs. 6 m. from date. C. On Jan. 1, 1885, 1. K. Lyons bought a horse of K. L. Minot for $248, promising to pay for him in one year from date. D. On Jan. 1, 1885, L. M. Nason bought $ 861.50 worth of boots of M. N. Ogdeu, promising to pay for them in 6 months from date. 61. The interest on the face of the note of Example 60 A. for 6 months, at 6^, is $45, whereas the true discount is $43.69, therefore the interest on the face of the note exceeds the true discount by $1.31. Find by how much the interest on the face of each of the other notes of Example 60 exceeds the true discount. 62. $1060. OMAHA, NEB., Jan. 16, 1886. For Value Received, I promise to pay James Brown, or order, one thousand sixty and T ^ dollars in one year from date. J. B. SMITH. What is the face of this note ? What was the true discount on Jan. 16, 1886, at 6^ ? what was the present worth ? [The man who paid $1000 (the present worth) for this note on Jan. 16, 1886, is said to have discounted the note because he paid for it the face value ($1060) less the true discount ($60).] In case a man does not pay his debts when A.] Interest. 163 they become due, an appeal to the law may be made to make him pay them. The maker of a note, however, excepting a note payable on de- mand, need not pay it until three days after the time of payment mentioned in the note : that is to say, a note is not legally due until three days after it is nominally due. These three j) ays O f additional days allowed by law are called Grace. DAYS OF GRACE. The maker of a note, payable at a given time with interest, may be compelled legally to pay in- terest for the three days of grace ; but if he pays his note when it becomes nominally due he is not usually asked to pay interest for the days of grace, except when he deals with a bank. When a bank discounts a note it deducts a dis- count larger than the true discount, in order that it may receive pay for its trouble, as well as inter- est on the money that it advances. The BANK DISCOUNT, as it is called, on a note payable at a specified time, without in- Bank terest, is the interest on the face for the Discount, time that is to elapse before the note is legally due.* 63. What is the bank discount on the note of Example 62 ? Ans. 164.13. [The true discount, as we have seen, is only f 60.] How much will a man receive who gets this note discounted at a bank ? Ans. $995.87. * As has been stated, it is not legally due until the third day uiter the time of payment mentioned in the note. 164 Percentage. [ 7. 64. How much would each of the following notes have brought on Jan. 1, 1886, if discounted at 6^, bank discount ? Note A, for $1632.12, dated July 1, 1885 ; pay- able March 1, 1887. Note B, for $56.28, dated Dec. 1, 1885 ; pay- able June 1, 1886. Note C, for $2361.18, dated Nov. 1, 1885 ; pay- able Dec. 1, 1886. Note D, for $1500, dated Jan. 1, 1886 ; pay- able Sept. 1, 1886. 65. $100^. CAMBRIDGE, MASS., Sept. 1, 1887. For Value Received, I promise to pay the Cambridge National Bank, or order, one hundred ^^ dollars in 6 months from date, with interest at 6^>. F. D. JONES. What was the amount of this note at the time it was due ? A?is. $103.05. What is the face of a note of the same date and payable at the same time, in which no mention is made of interest,* that will give the bank the same amount of money ? Ans. $103.05. 66. Make a note equivalent to the following, of the same date and payable at the same time, but without interest : $ 566 lTo- BOSTON, MASS., Sept. 1, 1887. For Value Received, I promise to pay the Suffolk National Bank, or * It will be remembered that where no mention is made of interest, none can be collected if the note be paid when due. A.] Interest. 165 order, five hundred sixty-six -f^ dollars in 9 months from date, with interest at 4j6. 67. In each of the following cases find the face of a note, payable at the end of the time given, but without interest, allowing as before for the three days of grace : A. $684.50, due in 1J years, at 5%. B. $3984.00, due in 6 months, at 6 ft. C. $1000.00, due in 9 months, at 4/o. 68. How much would a bank, whose rate of dis- count is 6^fc, have given me on Sept. 1, 1887, for a note -for $500, dated Sept. 1, 1887, due in 6 months, with interest at 4^>? [Suggestion: First find the face of an equivalent note due Sept. 1, 1887, without interest, and then compute the dis- count on this face.] 69. How much would a bank, whose rate of dis- count is 8/0, have given me on Jan. 1, 1888, for a note for $1000, dated Jan. 1, 1888, and due in 9 months, at 6yfo ? 70. How much would a bank, whose rate of dis- count is 6^b, have given me on Jan. 1, 1886, for a note for $750, dated Jan. 1, 1886, and due in 18 months, at 4^fc ? 71. How much would a bank, whose rate of dis- count is 6^fe, have given me on June 1, 1888, for a note for $1500, dated June 1, 1888, and due in 1 year, at 5^> ? how much would it have given me on Jan. 1, 1888, for a note for $800, dated Jan. 1, 1888, and due in 6 m., at Sfi ? 72. a. How much would a bank, which discounts 166 Percentage. [ 7. at 6'/>, have given me on Jan. 1, 1886, for a note for $1.00, dated Jan. 1, 1886, and payable, without interest, April 28, 1886 ? Ans. $0.98 b. How large a note should I have had to give the bank in order to get $1.96 ? [We have just seen that a note for $1 would have brought me $0.98; therefore, to get $1.96 my note would have had to be for as many dollars as $0.98 is contained in $1.96, that is for $2.] c. How large a note should I have had to give a bank in order to get $2.94? $3.82? $9.80? $19.60? $29.40? $39.20? $98.00? $150? $500? $1200? 73. If I want a bank, whose rate of discount is 6^f>, to pay me $979.50 to-day, for how much must I make my note payable to the bank, without interest, in 4 m. from now ? [A note for $1.00 will bring now $0.9795] 74. If I want a bank, whose rate of discount is 8^), to pay me $500 to-day, for how much must I make my note payable to the bank, without inter- est, in 8 m. from now ? [A note for $1.00 will bring now $0.946] 75. If I want a bank, whose rate of discount is 6^>, to pay me $2400 to-day, for how much must I make my note payable to the bank, without in- terest, in 6 m. from now ? 76. If I want a bank, whose rate of discount is 6^fc, to pay me $175 to-day, for how much must I make my note payable to the bank, without inter- est, in 2 yrs. from now ? B.] Compound Interest. 167 Business Customs in regard to Computing Interest. A month is regarded as ^ of a year. (See foot-note on page 148.) In computing the interest for a given number of days less than a month, a day is regarded as ^ of a month. (See foot-note on page 149.) In computing the time between Jan. 1, 1885, and July 20, 1885, we count the months from Jan. 1 to July 1, and then the days from July 1 to July 20, and thus get 6 m. 19 d. In a similar way the time between any two dates is computed. A note dated Jan. 31, and payable in one month, is nominally due on the last day of February, that is, on February 28, except in a leap year, and then on Feb. 29 ; and a note dated Feb. 28, and pay- able in 1 month, is due March 28. These same principles are applied in other similar cases. When a note falls due on Sunday, or on a legal holiday, it is payable the day previous. B. Compound Interest. 1. a. On Jan. 1, 1880, I deposited 11000 in the Bonanza Savings Bank. The rules of the bank allow depositors 4 , for 2 yrs. 6m.? $200. .06 $12.00 Interest for the first year. $200. $212.00 New Principal at the end of first year. .06 $12.72 Interest for the second year. $212.00 $224.72 New Principal at the end of second year. .03 $6.7416 Interest for 6 months. $224.72 $231.46 Amount at the end of 2 yrs. 6 m. 3. At compound interest, compounded semi- annually, what is the amount of $150, at 4^>, for 1 yr. 9 m. ? 4. At compound interest, compounded semi- annually, what is the amount of $3000, at 6^fc, for 2 yrs. 2m.? 5. a. At compound interest, what is the amount of $1, at 6jfc, for 1 yr. ? for 2 yrs. ? for 3 yrs., etc., up to 8 yrs. ? B.] Compound Interest. 171 Arrange the results, carried out to six places of decimals, in a table, as indicated below. Years. Amount. 1 1.060000 2 3 4 5 6 7 8 b. Knowing the amount of $1 for 5 years at 6^ compound interest, how can we find the amount of $3 for the same time ? c. What is the amount of 17000 for 8 years? [Refer to the table that you have just formed for the amount of $1 for 8 years.] 6. At compound interest, what is the amount of $200, at 6^, for 2 yrs. 6m.? [Take the amount of $1 for 2 yrs. from the table of the preceding example, and from this find the amount of $200 for the same time, then compute and add in the interest for 6 m. Compare the result with that ob- tained in Ex. 2.] 7. At compound interest, what is the amount of $1800 at 6^b for 5 yrs. 8m.? 8. At compound interest, what is the amount of $2500 at 6/0 for 7 yrs. 4 m. ? NOTE. A table showing the amount of $1 at compound interest from 1 year to 50 years, inclu- sive, at 3, 3, 4, 4j, 5, 6, and 7 per cent is given on page 282. 172 Percentage. [ 7. (7. Partial Payments. 1. I owed $100 at 6^ : At the end of 8 months I paid $54; how much did I then owe? At the end of 18 months I paid in full ; how much did I then have to pay ? Solution : $100.00 Interest on $100 for 8 m. . . $4.00 Amount at the end of 8 m. . . . $104.00 Paid at the end of 8 m. . . . $54.00 Unpaid at end of 8 m $50.00 Interest on $50 for 10 m. . . $2.50 Due at end of 18 m $52.50 2. I owed $1000 at 6J& : At the end of 6 m. I paid $400, and at the end of 18 m. $300. How much did I owe at the end of 18 m. ? Ans. $367.80 3. $500 T ^. BOSTON, MASS., July 1, 1885. For Value Received, I promise to pay A. B. Coes, or order, five hun- dred fa dollars on demand, with interest at 6jfc. B. C. DOLE. Payments $50 Jan. 1, 1886; $150 Sept. 1, 1886 ; $100 Nov. 1, 1886. How much was due Jan. 1,1887? 4. $1000 T ^-. ALBANY, N. Y., Jan. 1, 1884. For Value Received, I promise to pay C. D. Earle one thousand -ffi dollars on demand with interest at 4/o. C.] Partial Payments. 173 Payments $400 Dec. 1, 1884 ; $200 March 1, 1885 ; $300 Jan. 1, 1886. How much was due July 1, 1886 ? 5. I owed $10000 at 6fo. At the end of 4 m. I paid $300 ; at the end of 7 m. $100 ; and at the end of 12 m. $800. How much did I owe at the end of 16 m. ? $10000.00 Interest on $10000 for 4m. . . $200.00 Amount of $10000 at end of 4 m. . $10200.00 Paid at end of 4 m $300.00 Unpaid at end of 4 m. . . . $9900.00 Interest on $9900 for 3 m. $148.50 Amount of $9900 for 3 m. $10048.50 Paid at end of 7 m. $100.00 Here the payment ($100) is less than the interest ($148.50) earned since the last payment. The Supreme Court of the United States has decided that such a payment must not be subtracted from the amount, but must be added to the next payment, and be treated as if paid at the same time with the next payment. In this case, then, we add the $100 to the next payment ($800), and proceed as if $900 were paid at the end of 12 m. and nothing at the end of 7 m. Unpaid at end of 4 m. . . . $9900.00 Interest on $9900 for 8 m. . . . $396.00 Amount of $9900 for 8 m. . . . $10296.00 Paid at end of 12 m $900.00 Unpaid at end of 12 m. . . . $9396.00 Interest on $9396.00 for 4 m. . . $187.92 Due at end of 16 m $9583.92 6. I owed $10000 at 6fi. At the end of 4 m. I paid $800, at the end of 8 m. $100, at the end of 174 Percentage. [ 7. 12 ra. $100, and at the end of 16 m. $1000 ; how much did I owe at the end of 18 m. ? [If, after adding a payment to the next, as in the last example, we find that the sum of the two payments is less than the interest earned at the time of this next payment, we add this sum to the following payment, and so on until we get a sum that equals or exceeds the interest.] Ans. $8851.64 7. $346.36 CAMBRIDGE, MASS., March 26, 1880. For Value Received, I promise to pay A. B. Clark, or order, three hundred forty-six and -j 3 ^ dollars on demand, with interest at QJb. B. C. DOLE. Payments July 20, 1880, $54.75 ; April 8, 1881, $10 ; Sept. 26, 1881, $5.50; Jan. 6, 1882, $150.46. What was due May 2, 1882 ? Ans. $161.43 NOTE. The method just indicated of treating part or PARTIAL PAYMENTS called the United States method has been adopted by most states. Where, however, a settlement is made in a year or less, the Merchants' method, indicated in the next example, is often used. 8. A owed B $100, payable in 1 yr. at 6^> : at the end of 4 m. A paid B $50 ; how much did he owe him at the end of the year? By the Merchants' method a payment is regarded as a loan, to be accounted for at the time of settle- ment. In this example, then, at the time of settle- ment we regard B as having loaned A $100 for 1 yr. and A as having loaned B $50 for 8m.; there- fore, at the time of settlement, A owes B the C.] Partial Payments. 175 amount of $100 for 1 yr., or $106.00 and B owes A the amount of $50 for 8 m., or $52.00 Balance to be paid by A to B, $54.00 9. On July 1, 1884, Amos Brown bought of Byron Coe $1000 worth of boots, for which he gave his note at 6^), payable in 1 year. Brown made payments as follows : Sept. 1, $200 ; Dec. 1, $300; Feb. 1, 1885, $100. How much did he owe Mr. Coe on July 1, 1885 ? By the Merchants' method no balance is found until the time of settlement, at which time we say that on the one hand Brown owed Coe the amount of $1000 for 1 yr., or $1060.00 and on the other hand Coe owed Brown the amount of $200 for 10 m., or $210.00 $300 for 7 m., or $310.50 $100 for 5 m., or $102.50 Total $623.00 Balance owed by Brown on July 1, 1885, $437.00 1O. $387.75 BOSTON, MASS., May 15, 1880. For Value Received, I promise to pay Charles Doe three hundred eighty-seven -$$ dollars on demand with interest at 6/0. DAVID EMERY. Payments July 21, 1880, $75 ; Oct. 10, 1880, $125 ; Feb. 24, 1881, $50. The account was set- tled on May 15, 1881, what was then due ? Ans. $152.19 176 Percentage. [ 7. 11. On a 6% note of $1263 dated Jan. 1, 1886, the following payments were made : Feb. 1, 1886, 1100 ; March 16, 1886, $150 ; April 30, 1886, $200. How much was due July 1, 1886 ? D. Equation of Payments. 1. In how long a time will 1 dollar gain as much interest as $15 will gain in a month ? 2. In how long a time will 1 dollar gain as much interest as $8 will gain in 3 months ? 3. In how long a time will 1 dollar gain as much interest as $24 will gain in 5 months ? 4. In how long a time will 1 dollar gain as much interest as $158 will gain in 11 months? 5. In how long a time will $3 gain as much interest as 1 dollar will gain in 24 months ? 6. In how long a time will $28 gain as much interest as 1 dollar will gain in 157 months ? 7. A lent B $8, which was paid in 2 months ; afterwards B lent A 1 dollar. How long should A keep the 1 dollar in order to compensate him- self for his loan to B ? 8. C lent D 1 dollar, which was paid in 15 months ; afterwards D lent C $5. How long should C keep the $5 in order to compensate him- self for his loan to D ? 9. A borrowed of B $17 for 11 months, prom- ising him a like favor. Afterwards B lent A $25 ; how long ought he to keep it to balance the favor ? D.] Equation of Payments. 177 Suggestion: Find how long he should keep 1 dollar, and then from this how long he should keep 25 dollars. 10. I lent a friend $257, which he kept 15 months, promising to do me a like favor, but he was not able to loan me more than $100 ; how long could I keep it ? 11. A owes B notes to be paid as follows : $7 to be paid in 3 months, and $5 to be paid in 8 months ; but he wishes to pay the whole at once. In what time should he pay it ? Solution : $7 for 3 months is the same as 1 dollar for 21 months ; and $5 for 8 months is the same as 1 dollar for 40 months. He may have 1 dollar 40 + 21, or 61 months; the question now is how long may he keep 7 + 5, or 12 dollars. It is evi- dent he may keep it -^ of 61 months, or 5 months and 2 days. 12. C owes D $380, to be paid as follows: $100 in 6 months ; $120 in 7 months ; and $160 in 10 months. He wishes to pay the whole at once. In how many months is it due ? 13. A merchant has due him 300X, to be paid as follows : 50< in 2 months ; 100< in 5 months ; and the rest in 8 months. It is agreed to make one payment of the whole. In how many months is it due ? 14. F owes H $1000, of which $200 is now due, $400 is due in 5 months, and the rest in 15 months. F wishes to make one payment of the whole. Required the time ? 178 Percentage. [ 7. 15. A merchant has due him a certain sum of money, of which is to be paid in 2 months, -J- in 3 months, and the rest in 6 months. In what time is the whole due ? 16. a. I owe 1200 to be paid in 3 months, '$500 to be paid in 4 months, and $100 to be paid in 6 months. In how many months must I make one payment of the whole in order to cancel my indebtedness ? Arts. In 4 months. 6. How much should I have to pay now in order to cancel my indebtedness if 6 a year on my money, how much per share can I afford to pay for stock which pays a dividend of 8fi a year ? 10. I bought some mining stock at 182, which pays a dividend of 10^) a year. What per cent a year do I get on my investment? NOTE. Below is given a partial list of the sales of stock at Boston on April 24, 1886. The figures on the left indicate the number of shares and those on the right the number of dollars received for each share. 184 Percentage, [7. 11. Find how much was received for each lot of shares and the brokers' commission on each lot. Boston Stock Exchange. April 24. BAILKOADS. 100 Atch., T., & St. Fe, 87 70 Union Pacific, 50J 325 Union Pacific. 511 150 u 50J 181 61* 470 " 51 215 51$ 325 51 100 51* 25 51i 260 61J 15 Atch., T., & St. Fe*, 86* 219 61* 1140 86} 430 52 110 " B6{ 100 52 3 86 108 N. Y. & N. E., 37J 5 Fl. & P. Mar., 96^ 100 37* 4 Chi., Bur., & Q., 134 550 374 85 134 110 Clev. & Can., 22 50 M., II., & 0., 34i 100 Cin., S., & Clev., 15 15 Mex. Central, 7} 7 Bost. & Lowell, 127J MINING COMPANIES. 60 127 19 Cal. & Hecla, 227* 5 Bost. & Albany, 189 10 227 2 Eastern, so* 100 Bost. & Montana, oi 3 Old Colony, 165* 5 165* LAND COMPANIES. 47 Metropolitan H., 80 400 Bost. W. Power, 81 5 Chi., Bur., & Q., 134 200 " 8A 148 " 133g 25 Brookline, 3f 400 N. Y. & N. E., 37 10 Cambridge H., 83* MISCELLANEOUS. 300 Atch., T., & St. F<$, 86J 3 Am. Bell Tel., 162* 3 87 50 N. E. Teleph., 32? 350 " 86J 50 33 1000 " 87 15 Am. Bell Tel., 162J NOTE. A large number of the best examples of stocks can be found in any daily paper that makes a specialty of reporting the stock exchange. F.] Taxes. 185 12. Define the terms Capital Stock ; Stockhold- ers or Shareholders ; Dividend ; Stock Broker ; Commission ; Bears ; Bulls. 13. What is meant by the expression "Union Pacific Eailroad Stock sells at 50j " ? F. Taxes. A town needs money for various purposes, such as Building and repairing school-houses, and paying the salaries of teachers. Making and repairing roads and bridges. Supporting a fire department. Supporting a police department. Building and repairing public buildings a town-house, a library building, etc. A town must also pay its share of the expenses of the county and state in which it is situated. Money needed for the purposes just mentioned and for other similar purposes is obtained from the inhabitants, each of whom is required to pay what is considered to be his share of the money to be spent for the good of all. Money obtained in this way is called a TAX* Each person's share is determined, in most states, somewhat as follows : First. Every male citizen who has reached the age of 21 years is required to pay what is called a poll tax. In Massachusetts this tax is 12.00, one 186 Percentage. [ 7. half of which is paid to the county treasurer and the other half to the state treasurer. Second. Each property owner is required to pay an amount proportional to the value of what he owns.* 1. In the town of X, Mass., the amount of the entire tax to be raised is $ 10,000 ; the number of persons who must pay a poll tax ($2.00) is 500 ; the value of all the property owned in the town is $600,000. How large a tax must be paid by a male citizen over 21 years old, the value of whose taxable property is $12,000 ? [The amount of all the poll taxes is 500 x $2, or $1000. There remain, then, $9000 to be raised from property valued at $600,000 ; therefore, from property valued at $1 there would be raised ^ow8l7 dollars, or $0.015. The rate of taxation in this case may be expressed as 1JJ6, or 15 mills on a dollar, or $15 on a thousand dollars.] 2. The citizens of the town of Draco have voted to raise a tax of $17,400 ; the number of inhabit- ants who are subject to a poll tax of $2 apiece is 450 ; the real estate f owned in the town is valued at $850,000, and the personal property at $250,000. What is the total tax of B, who pays one poll tax, and whose property is valued at $15,800 ? What is the rate of taxation ? * Certain kinds of property, such as government bonds, etc., are exempt from taxation. t Immovable property, such as land and building's, is called real estate ; movable property, such as horses, cows, money, etc., is called personal property. F.] Taxes. 187 3. The tax to be raised in a town is $ 10,600. The property is valued at 11,250,000, and there are 300 inhabitants who are subject to a $2 poll tax. What is the total tax of a man who has to pay one poll tax and whose property is valued at $7,500 ? What is the rate of taxation ? 4. In a town where the rate of taxation is 11 mills on a dollar, what is the total tax of a man who pays one $2 poll tax, and whose property is valued at 112,800? 5. In a town where the property is valued at $3,300,000 the total tax to be raised is $41,400 ; the number of polls at $2 each is 900. What is the total tax of Mr. Brown, who pays one poll tax, and whose property is valued at $24,000 ? What is the rate of taxation ? How much more would Mr. Brown's tax have been if it had been voted to raise $5,800 more for a new school-house ? What would the rate of tax- ation have been in this case ? 6. A's entire tax is $306 ; he pays one poll tax of $2, and the rate of taxation is 8 mills on a dol- lar. What is the value of his property ? 7. What is the value of the property in a town where a tax of $30,937.50 is to be raised, at a rate of 7^ mills on a dollar? 8. The richest man in a city where the rate of taxation is 15 mills on a dollar pays, in addition to his poll tax of $2, a tax of $22,500. What is the value of his property, and how much could he 188 Percentftye. [ 7. save each year in taxes by moving to a town where the rate of taxation is only 12 mills on a dollar? 9. If the property of a city be valued at $250,- 000,000, and a property tax of 14,000,000 is to be raised, what tax (including a $2 poll tax) must a man pay whose property is valued at $15,000? 10. How much will a tax collector get for col- lecting a tax of $20,000 if he is paid a commission of 2^> on what he collects ? 11. How much will a tax collector get for col- lecting a tax of $95,000 if he is paid a commission of lJf> on what he collects ? 12. If a tax collector whose commission is lg^> receives $640 for his services, how much does he collect ? What per cent of the whole tax remains after he has taken his commission ? 13. If a tax collector whose commission is 2^> receives $1200 for his services, how much does he collect ? What per cent of the whole tax remains after he has taken his commission ? 14. If a tax collector whose commission is l^^fc receives $1000 for his services, how much does he collect ? What per cent of the whole tax remains after he has taken his commission ? 15. What sum must be raised in order that $19,600 may remain after paying a commission of 2^ for collection ? 16. What sum must be raised in order that $44,325 may remain after paying a commission of for collection ? G.] Duties. 189 G. Duties. The United States Government needs money for The salaries of the president, senators, congress- men, judges, and other officers. Pensions to men disabled in fighting for the preservation of the Union. The support of an army and navy, and the building and maintenance of arsenals and forts. The improvement of rivers and harbors, so that transportation by water may be easy and safe. Interest on the public debt. And for such other objects as may tend to pro- mote the welfare of the whole people. The largest part of the money needed by the United States Government is obtained by imposing a tax on property that is brought into the United States from other countries. Such a tax is called a DUTY. 1 . The duty on books is 25J& of the cost. How much duty will a man have to pay on a collection of books which will cost him in London $1270 ? 2. The duty on kid gloves is 50^o of the cost. How much duty will an importer have to pay on 10 dozen pairs of kid gloves, the cost of which in Paris is 40 cents a pair ? 3. In 1887 watches to the value of $1,198,109 were imported to the United States ; the duty was of the cost. What was the total duty ? 4. In 1887 30,027,670 yards of cotton cloth 190 Percentage. [ 7. were imported into the United States at an aver- age cost of 12 cents a yard ; the average duty was of the cost. What was the total duty ? NOTE. A duty based on the cost, as in the last four examples, is called an AD VALOREM * DUTY. 5. The duty on silk umbrellas is 50^? ad va- lorem. How much duty must a man pay on 25 silk umbrellas, the cost of which in Paris is $1.75 apiece ? 6. The duty on fine salt is 8 cents on every hundred pounds. How much duty must a man pay who imports 24,840 pounds ? 7. The duty on the best molasses is 8 cents a gallon. How much duty must a man pay who im- ports 80 hogsheads containing 63 gallons each ? 8. In 1887, 972,570 pounds of oat meal were imported into the United States ; the duty was ^ cent per pound. What was the total duty ? 9. In 1887, 11,207,548 pounds of filberts and walnuts were imported into the United States ; the duty was 3 cents a pound and the cost was 5|- cents a pound. What was the total duty and also the total cost ? 1O. During the year 1887 there were imported into the United States 33,731,463 pounds of rice at a cost of 2 cents a pound ; the duty was 2 J cents a pound. What was the total duty and also the total cost ? * Ad valorem means according to value. G.] Duties. 191 NOTE. A duty based on quantity, as in each of the last five examples, is called a SPECIFIC DUTY. 11. On spirit varnishes there is a specific duty of $1.32 a gallon and an ad valorem duty of 40jfc. How much duty must a man pay on 85 gallons of varnish, the cost of which in London is $ 2.38 a gallon ? 12. During the year 1887 there were imported into the United States 15,671 gallons of cologne water at a cost of $15.09 a gallon ; there was a specific duty of $2 a gallon and an ad valorem duty of 50^. What was the total duty ? 13. During the year 1887 there were imported into the United States 50,315 gross of lead pencils at a cost of $1.91 a gross ; there was a specific duty of 50 cents a gross and an ad valorem duty of 30^. What was the total duty ? 14. During the year 1887 a family in Boston bought 600 pounds of imported sugar at an average price of 7 cents a pound ; this sugar cost in the countries that it came from 2.6 cents a pound, and the duty was 82jfc. How much less would the year's supply have cost the family if there had been no duty? 15. During the year 1887, 500,085 pounds of hemp yarn were imported into the United States, at an average cost of 12 J cents a pound ; the duty was 35^). What was the total duty ? 35^ is how many cents a pound ? [This is the same as asking what specific duty per pound is equivalent to an ad valorem duty of 192 Percentage. [ 7. 16. What specific duty on buckwheat is equiva- lent to an ad valorem duty of 10^>, when the cost is 38 cents a bushel ? 17. In 1887 there was a specific duty on barley of 10 cents a bushel ; the cost price was 60 cents a bushel. What ad valorem duty would have been equivalent to the specific duty ? [10 cents on 60 cents (the price of a bushel) is $ of the 18. What ad valorem duty on bituminous coal is equivalent to a specific duty of 75 cents a ton when the cost is $3.08 a ton ? 19. What ad valorem duty on raisins is equiva- lent to a specific duty of 2 cents a pound when the cost is 5 cents a pound ? 20. What ad valorem duty on lead pencils which cost $1.91 a gross is equivalent to a specific duty of 50 cents a gross and an ad valorem duty of 30^ ? Ans. 56^ nearly. 21. What ad valorem duty on cologne water which costs $15.09 a gallon is equivalent to a spe- cific duty of $2 a gallon and an ad valorem duty of The United States Congress is now (July, 1888) considering the advisability of reducing the duties on imported goods. The following table shows the values of different kinds of dutiable property brought into the United States during the year 1887 ; the present average ad valorem duties ; and the duties contained in a proposition now before Congress, called the Mills bill. G.] Duties. 193 Schedule. Values of im- portations in 1887. Ad valorem duties. Present %. Proposed %. Chemicals $ 5,050,325 40 22 Earthenware and glassware 10,492,067 66 49 Metals 16,152,789 52 43 Wood and wooden- ware 889,558 35 29 Sugar 68,897,102 82 66 Tobacco 26,441 82 38 Provisions 3,235,987 53 43 Cotton and cotton goods 2,423,585 51 40 Hemp, jute, and flax goods 17,434,514 36 24 Wool and woollens 42,448,127 69 40 Books, papers, etc. 57,298 24 18 Sundries 11,221,253 44 35 Wood, salt, etc. 61,672,120 27 00 Wool 18,206,987 30 00 22. a. Find from the table given above the amount of the duties collected in 1887 on each kind of property ; find also the total amount. 5. Find the amount that would have been col- lected on each kind of property with the duties as proposed ; find also the total amount. c. How much less would the duties on sugar have amounted to in 1887 if the rate of duty had been as now proposed ? d. How much less would all the duties have amounted to in 1887 if the rates had been as now proposed ? 194 Percentage. [ 7. H. Miscellaneous. 1. If I can buy the following books at a dis- count or reduction of 10$fc ( T a ^ or ^) from the list prices, what must I pay for each ? LIST PRICES. I. The Lamplighter. By Maria S. Cummins .... $0.50 II. Frederick the Great. By T. B. Macaulay 60 III. Handbook of American Authors. By O. F. Adams ... .75 IV. Tom Brown at Rugby. By Thomas Hughes 1.00 V. The American Statesmen Series, per volume . . . .1.25 VI. The Story of a Bad Boy. By T. B. Aldrich 1.50 VII. Longfellow's Poems. Household Edition 1.75 VIII. The Autocrat of the Breakfast-Ta- ble. By O. W. Holmes . . 2.00 IX. Holmes' Poems. Household Edi- tion, in full gilt . . . .2.25 X. The Children's Book. By H. E. Scudder 2.50 XI. Whittier's Poems. Household Edi- tion in half calf . . . . 3.00 XII. Yesterdays with Authors. By J. T. Fields. In half calf . . 3.25 XIII. Agassiz. By Elizabeth C. Agassiz 4.00 H.] Miscellaneous. 195 XIV. History of Our Country. By Abby S. Richardson .... 4.50 2. What will each book in the preceding list cost at a discount of 15^6 ? NOTE. To get the cost of VII., for instance, we may say I5fi of $1.75 = $1.75 x .15 = $0.2625. Therefore, the cost is $1.75 -$0.2625 = $1.4875 or $1.48|. The work may be done mentally if we note that 15^b or -^-fa = -fa + T ^ iV ~*~ 2" ^ lV T V of $1.75 = $0.175 or $0.17 J. J of ^ of $1.75 = .0875 or .08|. Therefore 15^ = $0.26 J; and the cost is $1.75- $0.26 or $1.48|. 3. What will each of the books cost at a dis- count of 20^ ? [20^> = T 2 oo = ^ or .] 4. What will each of the books cost at a dis- count of 25$fc ? [25^ = |.] 5. What will each of the books cost at a dis- count of 30 f /) ? [30^ = ^.] 6. What will each of the books cost at a dis- count of 33$ or 1 ? Suggestion: After taking off J we shall have ^ left. Therefore f of each price will give the cost. 7. What will each book cost at a discount of 40/o ? [The cost will be 60^ or -^ or |.] 8. a. What will be the cost of 10 copies of I., 10 of II., 15 of IV., 25 of VI., and 15 of VII. at a discount of ? Ans. $59.83. b. If, by paying cash, I can get a discount of 5^o from the charge for the above, what must I pay ? Ans. $56.84. 196 Percentage. [ 7. NOTE. In this case I am said to get a discount of ^ and 5J&, which means that I may take off 5jfc of the result found by taking off J. 9. a. What will each book cost at a discount of \ and lOJfc ? 6. What at a discount of 40^> ? 10. Show that a discount of J and 10$fc is the same as a discount of 40^. 1 1. a. If, on an order for 1000 copies of VII. (see Example 1), 800 of XIII., and 400 of V., I am allowed a discount of 50^, what must I pay ? 6. If I am allowed a discount of 40$fc and 10^, what must I pay ? 12. What Jb is a discount of 40^fc and 10# ? 13. a. How much does a bookseller make on a $3.00 book that he buys at a discount of ^ and sells to you at a discount of 20*fc ? What per cent does he make on his investment ? [He makes 40 cents on $2.00, or 20 cents on $1.00, or 20^.] 6. If he buys the book at a discount of 40^ and sells it to you at a discount of ^, what is his profit ? what per cent ? Solution: On the $1.80 (180 cents) that he pays for the book he makes a profit of 20 cents ; a profit of 1 cent would be T ^ of the cost ; his profit, then, is -^fa or ^ of the cost ; and this, reckoned by per cent (that is, by the hundred), is 1 of 100 or ll^Jfc. We get the same result if we reduce -J- to hundredths, as in decimals. Thus = c. Find HJJfe of $1.80. H.] Miscellaneous. 197 14. A bookseller sold a shop- worn book for 85 cents that cost him $1.00 ; what per cent, did he lose ? Ans. 15^. 15. A man sold a horse for $ 130 that cost him $150 ; what per cent did he lose ? Ans. 13j6. 16. A merchant sold goods at auction for $3250 that cost him $5000 ; what per cent did he lose ? 17. A merchant sold a quantity of goods for $273.00, by which he gained 10 per cent on the cost. What was the cost ? Suggestion: 10 per cent, is y 1 ^ of the cost. Consequently $273.00 must be {$ of the cost. 18. A merchant sold a quantity of goods for $135.00, by which he gained 13 per cent. How much did the goods cost, and how much did he gain? 19. A merchant sold a quantity of goods for $3875, by which he gained 65 per cent. How many dollars did he gain ? 20. A merchant sold a quantity of goods for $983.00, by which he lost 12 per cent. How much did the goods cost, and how much did he lose ? NOTE. If he lost 12 per cent., that is T ^, he must have sold it for -ffa of what it cost him. 21. A farmer sold 3 cows for $248.37, by which he* lost 25 per cent. How much did the cows cost him, and how much did he lose ? 22. A merchant sold a quantity of goods for $87.00 more than he gave for them, by which he gained 13 per cent of the cost. What did the goods cost him, and how much did he sell them for? 198 Percentage. [ 7. NOTE. Since 13 per cent, is T ^, |87 must be ^^ of the first cost. 23. A merchant sold a quantity of goods for $4$. 00 more than they cost, and by so doing gained 20 per cent. How much did the goods cost him ? 24. A merchant sold a quantity of goods for $137.00 less than they cost him, and by doing so lost 23 per cent. How much did the goods cost, and how much did he sell them for ? 25. A man having put a sum of money at inter- est at 6 per cent, at the end of one year received 13 dollars for interest. What was the principal ? NOTE. Since 6 per cent is -j ^ of the whole, 13 dollars must be ^077 f the principal. 26. What sum of money put at interest for 1 year will gain $57, at 6 per cent ? 27. A man put a sum of money at interest for 1 year, at 6 per cent, and at the end of the year he received for principal and interest 237 dollars. What was the principal ? NOTE. Since 6 per cent is y-^, if this be added to the principal it will make |f , therefore $237 must be \^ of the principal. 28. What sum of money put at interest at 6 per cent will gain $53 in two years ? 29. What sum of money put at interest at 6 per cent will gain $97 in one year and 6 months ? 30. What sum of money put at interest at 6 per cent will amount to $394 in 1 year and 8 months ? 31. What sum of money put at interest at 7 per cent will amount to $183 in one year ? H.] Miscellaneous. 199 32. What sum of money put at interest at 8 per cent will amount to $137 in 2 years and 6 months ? 33. Suppose I owe a man $287 to be paid in one year without interest, and I wish to pay it now; how much ought I to pay him, when the usual rate is 6 per cent. ? 34. A man owes $847 to be paid in 6 months, without interest ; what ought he to pay if he pays the debt now, allowing money to be worth 6 per cent a year ? 35. A merchant being in want of money sells a note of $100, payable in 8 months, without inter- est. How much ready money ought he to receive, when the yearly interest of money is 6 per cent. ? 36. According to the above principle, what is the difference between the interest of $100 for 1 year at 6 per cent and the discount of it for the same time ? 37. What is the difference between the interest of $500 for 4 years at 6 per cent and the discount of the same sum for the same time ? 38. At an arithmetic examination a boy did correctly only | of the work required ; what per cent of the work did he do ? 39. At a civil service examination a man got 58 marks out of 80 ; what was his per cent ? 40. On Jan. 1, 1887, a horse-car conductor who had been receiving $50 a month had his wages lowered 10J6. On Feb. 1 his request for 10^ more pay than he was then receiving was granted. How much a month did he receive after these two changes ? 200 Percentage. [ 7. 41. A man who had 11000 lost 20^ of it in a trade ; he then invested what he had left in flour ; what per cent did he make on the flour if he sold it for $1000 ? 42. Just before Christmas a jeweler increased by lOjfc the selling prices of the following articles : Selling prices. Waltham silver watches . . . $25.00 Gold watch chains .... 10.00 Small clocks . . . . . 2.50 What did the selling prices then become ? 43. After Christmas he [see Example 42] low- ered the new selling prices by lOyfc ; what did the selling prices then become ? 44. A superintendent of schools called for writ- ten answers to Example 40 from the pupils in each of 6 schools under his charge. The following table shows the number of pupils in each school, and the number who answered the question cor- rectly. Complete the table by entering in the fourth column the per cent of the pupils in each school who gave correct answers. School. No. 1 Whole No. 58 No. answering correctly. 14 No. 2 64 12 No. 3 38 16 No. 4 42 12 No. 5 75 13 No. 6 80 20 answering correctly. H.] Miscellaneous. 201 45. A man's net * income from his property is $5000 ; how much property has he if it earns 5^> a year, subject to no expenses or charges except a tax of l/o ? 46. A man bought a house for $1200 ; he spent $400 for repairs, paid a tax of \ffl of the cost, and at the end of a year sold it for $1650. Would he have made more or less, and how much, if he had loaned his money at 5j#, and had paid a tax of ? 47. A merchant buys furniture at a discount of and 10^ from the list prices, and sells it at a discount of 10^ and 5fi. What per cent of the cost is his profit ? 48. What is the difference on a bill of $425 between a discount of 50^fc and a discount of 40yfc and 10# ? 49. 80 marks were to be given for perfect an- swers to all the questions on an examination paper in arithmetic. The following are the marks of the different pupils in a class of 6 : 80, 65, 58, 42, 38, and 25. What per cent did each get ? 50. By selling a house for $2340 a man lost 10$6 of the cost ; what should he have sold it for to make 10# ? 51. A collector deducts his commission of 2^? from a bill ; the balance is $1960. What was the biU? 52. A boy buys chestnuts at $2.50 a bushel, and sells them at 5 cents a pint. What per cent does he make ? * What remains after deducting all charges and expenses. 202 Percentage. [ 7. 53. A stationer marks his writing paper at above cost. What discount must he allow a cus- tomer in order to make a profit of lOjfc ? Solution : Paper that costs $1.00 is marked $1.25; 15 cents or T \^ from this leaves $1.10, which gives a profit of lOjfc on the cost. There- fore from the marked price he must make a dis- count of VA = i$ = lffi> = l2 fi A > 54. The marked prices of a jeweler's articles are 40^) above the cost prices ; how much discount can he make from the marked prices and make a profit of 20*fc on the cost ? 55. What Jo of the cost will his profit be if he makes a discount of 20 over, it is itself the g. c. d. of 8 and 16, and therefore of the numbers we ~Q~ started with. IV. We may now say that to get the g. c. d. of two numbers, we may divide the larger by the smaller, and then the smaller by the remainder , if there be any, and then continue dividing the last divisor by the last remainder until nothing remains. TJie last divisor will be the g. c. d. Find the g. c. d. of 22176 and 23328. 22176)23328(1 22176 1152)22176(19 1152 10656 10368 288)1152(4 Ans. 288. 1152 65. Find the g. c. d. of 11385 and 16335. 66. Reduce -* to its lowest terms. V I 2 V 67. Reduce $$$$ to its lowest terms. 68. Reduce to its lowest terms 236 Divisors, Factors, and Multiples. [ 9. 69. Reduce to its lowest terms 4M o 1 J. o 70. Reduce to its lowest terms |r;;|^. 71. Reduce to its lowest terms \\^ r 72. Reduce to its lowest terms ffify* 73. Reduce to its lowest terms JjiilJ". 74. Reduce to its lowest terms [: 75. Reduce to its lowest terms ^||.]* 76. a. Add together \, -J-, and \. 6. Subtract \ from \. Ans. \ - ' = . 1.Z NOTE. In examples like 76 a and 76 6 we have to reduce fractions to a common denominator. Any number that will contain each of the denominators will serve for a common denominator, but the smallest number that will do this is, of course, the easiest to manage. The next few examples lead up to a method of finding the smallest number that will contain two or more different numbers as divisors. 77. a. 15 is the product of what two numbers ? Ans. 3 and 5. b. 18 is the product of what numbers ? A?IK. 2 and 9, or 2, 3, and 3. . NOTE. Those numbers which when multiplied together will produce a given number are called its FACTORS. 78. What are the factors of 30 ? 79. What are the factors of 39 ? Factors and Multiples. 237 80. What are the prime factors of 154 ? NOTE. From what precedes we see that the factors of a number are also divisors of the num- ber, and that the prime factors are the same as the prime divisors. 81. What is the number whose factors are 3 and 5 ? Ans. 15. 82. What is the number whose factors are 2, 7, and 11 ? 83. What is the number whose factors are 2, 2, 3, 3, and 3 ? NOTE. Any number which contains another number as a factor or divisor is called a MULTIPLE of that other number. Thus 15 is a multiple of 3 and also of 5 ; 18 is a multiple of 2 and also of 3 and also of 9. 84. Of what numbers is 30 a multiple ? 85. Of what numbers is 48 a multiple ? 86. What is the least number which is a multi- ple of 5 and at the same time of 3 ? Ans. 5 x 3, or 15. 87. What is the least number that is a multiple of 3, 5, and 2 ? Ans. 3 x 5 x 2, or 30. NOTE. The least number that is a multiple of two or more numbers is called their LEAST COM- MON MULTIPLE (I. c. m.). 88. What is the I. c. m. of 15 and 21 ? Solution : Separating each into its prime fac- 15 = 3 x 5 tors, we get 91 _ o x 7- In order to contain 15, the required number must contain 3 and 5 ; and in or- 238 Divisors, Factors, and Multiples. [ 9. tier to contain 21 it must contain 3 and 7. That number, then, which contains only 3, 5, and 7 must be the required number ; therefore 3x5x7 = 105 is the I. c. m. 89. a. What is the I. c. m. of 18 and 21 ? Solution : Separating each number into its r 18 = 2x3x3 prime factors, we get 91-3x7 We see from the above that the least number which will contain 18 and 21 must contain 2, 3 twice, and 7 ; therefore the 1. c. m. = 2x3x3x7 = 126. b. What is the L c. m. of 36, 48, and 60 ? Ans. 720. 90. What is the L c. m. of 6, 20, and 45 ? Solution : Separating each number into its prime factors we get : 6=2x3 ) 20 = 2x2x5 [' th e/. c- - 45 = 3x3x5) -2x2x3x3x5 = 180. NOTE. From the preceding illustrations we see that the L c. m. of two or more numbers is the product of all the different prime factors of all the numbers, each prime factor being used the greatest number of times that it occurs in any one of the numbers. In the last example, for instance, 2 occurs twice as a factor in 20, and 3 occurs twice in 45 : the only remaining prime factor is 5, which occurs only once in the same number ; therefore the L c. m. = 2x2x3x3x5 = 180. Factors and Multiples. 239 The following is another method of finding the L c. m., although the principle is the same. 6 20 45 3 10 45 1 10 15 2 3 L c. m. = 2x 3x5x2x3 = 180. This method consists only in a briefer arrange- ment of the following work which we should do in finding the prime factors of each number sepa- rately : 2|6_ 3 20 45 15 6 = 2x3 20 = 2x5x2 45 = 3x5x3 91. Find the 7. c. m. of 21, 33, and 28. 21 33 7 11 28 "28 ~ 4 1 11 4 Ans. 3x7x11x4. 92. Find the L c. m. of 100, 400, and 1000. In this case the number 100 100 400 JLOOO "To" 100 is a common divisor of all the numbers, we can therefore just as well divide by it at once as divide by its prime factors one after the other. 93. Find the L c. m. of 8 and 12. 94. Find the L c. m. of 8 and 14. 95. Find the I. c. m. of 9 and 15. 240 Divisors, Factors, and Multiples. [ 9. 96. Find the /. c. m. of 15 and 18. 97. Find the I. c. m. of 10, 14, and 15. 98. Find the L c. m. of 15, 24, and 35. 99. Find the L c. m. of 30, 48, and 56. 100. Find the L c. m. of 32, 72, and 120. 101. Find the /. c. m. of 42, 60, and 125. 102. Find the L c. m. of 250, 180, and 540. 103. Reduce f and | to the least common de- nominator and then add them together. 104. Reduce | and T 5 ? to the least common de- ^r 1 O nominator and then add them together. 105. Reduce f, ^, ^5, and ^ to their least common denominator. 106. Find the greatest common divisor of 48 and 130. 107. Reduce ^, f , ^, and -J-J to their least com- mon denominator. 108. Subtract 15| from 18f . 109. Reduce | and -| to the least common de- nominator and then add them. 110. Reduce f and T 5 ? to the least common de- nominator and then add them. 111. Reduce -f% and -fy to the least common denominator and then add them. 112. Reduce ^ and -fa to the least common denominator and then add them. 113. Reduce fa and -^ to the least common denominator and then add them. 114. Reduce T |, / y , and \\ to the least common denominator and then add them. 115. Reduce |, f, -f^, and ^y to the least com- mon denominator and then add them. Factors and Multiples. 241 *[116. a. Find the I. c. ra. of 407 and 481. In a case of this kind, where none of the prime factors of either number can be found by inspection, it is best to find first the g. c. d. In this example we shall find the g. c. d. to be 37, which is contained 11 times in 407 and 13 times in 481 .-. ) . . _. and the I. c. m. is 11 x 13 x 37 = 5291, b. Find the I. c. m. of 731 and 817. 117. Find the I. c. m. of 451 and 943. 118. Find the I. c. m. of 217 and 341. 119. Find the I. c. m. of 203 and 319.]* 120. Find the I. c. m. of 17 and 31 and 2. 121. Find the I. c. m. of 7, 13, and 3. NOTE. Where, as in the last two examples, two or more numbers have no common factors, their I. c. m. is evidently their product. 122. Find the I. c. m. of 3, 13, and 31. 123. What is the I. c. m. of 20, 24, and 36 ? 124. Add f, |, 2 T %, and 8&. *[125. What is the g. c. d. of 1181 and 2741?]* 126. Reduce |, T %, and T 7 y to a common denom- inator. 127. Name all the prime numbers in the series of numbers from 1 to 29 ; resolve all the composite numbers into their prime factors ; and name all the perfect squares. 128. Add together f , Jf, and T 4 ^, and from their sum subtract -. 242 Divisors, Factors, and Multiples. 129. a. Find the g. c. d. of 12, 30, and 45. [The g. c. d. of 12 and 30 is 6 ; and the g. c. d. of 6 and 45 is 3. Therefore the answer is 3.] 6. Find the g. c. d. of 720, 336, and 1736. 130. Reduce -J| $- to its lowest terms. 131. Reduce T ^, {, ^, -fa, and ^ to their least common denominator, add them and reduce the sum to its simplest form. 132. Find the g. c. d. and the I. c. m. of 630, 840, and 2772. 133. Find the g. c. d. and /. c. m. of 144 and 780. 134. Reduce J, f, j 3 ^, and || to their least common denominator. 135. Subtract 15J from 18 J. 136. Reduce |f $$ to its lowest terms. 137. What is the g. c. d. of the two numbers 4760 and 3432 ? 138. What is the I. c. m. of 48, 98, 21, and 27 ? 139. What is the g. c. d. of 1872 and 432 ? [Obtain the answer by factoring.] 140. Find the g. c. d. of 187 and 153 ; also their I. c. m. SECTION X. CANCELLATION AND ANALYSIS. 1. How many tons of coal at $6 a ton can be bought for 15 tons of hay at $18 a ton? Solution : 15 tons of hay at 118 a ton is worth 15 x 18 dollars, or $270. As many tons of coal at $6 a ton can be bought for $270 as 6 is contained in 270, or 45 tons. The answer just found was got by dividing 15 times 18 by 6. We may say, then, that the an- swer is equal to ~ tons. Now, dividing both the dividend and the divisor first by 3, and then 5 9 . VM ^ j'jjA by 2, we get - - = 45, and thus save ourselves * some time and labor. Striking out the common factors from the divisor and dividend is called CANCELLATION. 2. If 6 men can build a stone wall 40 ft. long, 5 ft. high, and 2 ft. thick in 6 days, how long will it take them to build a wall 80 ft. long, 5 ft. high, and 3 ft. thick? 244 Cancellation and Analysis. [ 10. Solution : 6 men build 40 x 5 x 2 cu. ft. in 6 dys. , .,,40x5x2 o men build - ~ - cu. ft. in 1 day. 6 men, to build 80 x 5 x 3 cu. ft., will require * OA . 40 x 5 x 2 , x x 3 x 6 _ 80xox3-l-- - days, or - --- days, b 40 x f x g or 18 days. Ans. NOTE. If, in the solution of any problem, there is to be a series of multiplications and divisions it is always well first to express them all, and then to cancel if possible. 3. Divide $ of f of 2 by -J,. Solution : First expressing the successive steps and then 31 ----------- & , .._ a ~w _ f --- 2 - O I ~ I X ? X 2 * if 4. Divide 1 by 1 J. Multiply 1J by 1|. 5. Multiply 48 by ^. Divide ^ by ^. 6. Reduce ^ ^-^ i to its simplest form. 7. From | of f take J of f . 8. Divide || x 721 by | of f of 9|. 91 4 9. Reduce ^rl to a simple fraction. Reduce Sf to a simple fraction. Cancellation and Analysis. 245 10. What is the product of | of ^ of 15 and 15 Of 11 1? 11. Divide 100 by 4. 12. Multiply || by ^ of 2J. 4 3" 13. From | of % subtract ^ of 14. Divide & of ^ of 3J by 15. Multiply f of tf of 41 by 16. From fa of If subtract $ of 17. Divide V- of ^ of If by o 18. From |- of f f subtract ^ of 2J. 19. Divide J | of W of IS? by ^. ^2 9 20. Divide if 1 by 42. 21. Divide | of || by ^ T of f f. 18 2 22. Reduce ^ . Q 7 f r to its simplest form I of I of I 23. Add S, 1, and ^ of f . $ 24. What part of 6 is 2 ? 25. What part of T \ is ? 26. What part of f is f ? 27. Divide J of f of 2 J by -. 28. Divide T V of T ^ of 81 by - . 29. From 3 subtract (^ of -^ of 1|) -r 246 Cancellation and Analysis. [ 10. 30. Subtract J of f from f of -jf ; add to the 5 remainder -^g ; divide the result by 6 j. 31. Add -jj-|- and -=| ; divide the result by 7f . "To ' 8 3 32. From \ of If take -^-, add to the remain- ^2" der |, and divide the result by 6f . 33. From ^ of 2| subtract the product of 0.075 and 1^, and divide the remainder by 12. 34. Divide 10 times ( of -^ of 9&) by -||. 35. From 5 subtract $& + of of 71 315 36. From the sum of and -- subtract aud divide the result by the product of 3^ and 37. Divide (2}xJL) by (2J-lf). 38. Divide- by | of (-1^1 ). - 39. Add to 40. Add to 41. Add W of -^ to if. -ii 1 9 42. Divide 0.75 by -||x 0.081. *o. TY iictt 10 tuc . 44. How many tiles 8 inches square will cover a hearth 12 ft. wide and 16 ft. long? Cancellation and Analysis. 247 45. If 12 tailors can make 13 suits of clothes in 7 days, how many tailors will it take to make the clothes of a regiment consisting of 494 soldiers in 19 days. Solution : 12 tailors make 13 suits in 7 days. 1 tailor makes -if suits in 7 days ( T ^ as many as 12 tailors). 13 1 tailor makes ^ =, suits in 1 day. 13x19 1 tailor makes ^ suits in 19 days. 494 To do 494 suits in 19 days will take JQ TQ tailors. -Lo x j. y 12x7 2 n 494 13x19 Ans. 12 x 7 46. If a family of 9 persons spend $306 in 4 months, how many dollars would a faro ly of 15 persons spend, at the same rate, in 8 months ? Ans. 11020. 47. If 20 bushels of wheat are sufficient for a family of 15 persons 3 months, how much will be sufficient for 4 persons 9 months ? Ans. 16. 48. If 7 men can build 36 rods of wall in 3 days, how many rods can 20 men build in 14 days ? 49. If 7 men can reap 84 acres of wheat in 12 days, how many men can reap 100 acres in 5 days ? 248 Cancellation and Analysis* [ 10. 50. If 18 men can build a wall 40 rods long, 5 ft. high, and 4 ft. thick in 15 days, in what time will 20 men build a wall 87 rods long, 8 ft. high, and 5 ft. thick ? 51. How many yards of flannel that is 1^ yards wide will line a cloak containing 9 yards of cloth that is | of a yard wide. Ans. 4| yds. 52. A regiment of soldiers, consisting of 1000 men, is to be supplied with new coats ; each coat is to contain 2| yards of cloth 1J yards wide, and is to be lined with flannel f of a yard wide. How many yards of flannel will be required ? 53. A ship's crew of 18 men has enough pro- visions to last the voyage, if each man is allowed . 20 oz. per day. If a shipwrecked crew of 6 per- sons is picked up, what must then be the daily allowance of each person ? Suggestion : 18 x 20 oz. per day is to be divided among 24 men. 54. If 8 boarders will eat a quantity of flour in 15 days, how long will it last if 4 more boarders join them ? [How long would it last one man ? how long would it last 12 men ?] 55. Suppose 650 men are in a garrison, and have provisions enough to last them two months ; how many men must leave the garrison in order to have the provisions last those who remain five months? Ans. 390. 56. If a staff 4 feet long cast a shadow on level ground 6 ft. 8 in. long, what is the height of a steeple which casts a shadow 175 feet long at the same time? Ans. 105 feet. Cancellation and Analysis. 249 57. If a man travels 64 rods in .05 of an hour, how many minutes will it take him to travel a mile? 58. A man receives $ 18 for six days' work of 8 hours each; what should he receive for 5 days' work of 9 hours each ? 59. If 6 men can build 20 feet of a stone wall in 10 days, how many men can build 360 feet of the same wall in 90 days ? 60. If 3 men can build a wall 60 feet long, 8 feet high, and 3 feet thick in 64 days of 9 hours each, how many days of 8 hours each will 20 men require to build a wall 400 feet long, 9 feet high, and 5 feet thick ? 61. If 6 men can build a wall 80 feet long, 10 feet high, and 9 feet thick, in 100 days of 9 hours each, how many days of 10 hours each will be re- quired by 15 men to build a wall 200 feet long, 9 feet high, and 5 feet thick ? 62. If 4 men dig a trench 84 feet long and 5 feet wide in 3 days of 8 hours each, how many men can dig a trench of the same depth 420 feet long and 3 feet wide in 4 days of 9 hours each ? 63. If 496 men, in 5 days of 12 h. 6m. each, dig a trench of 9 degrees of hardness, 465 feet long, 3 1 feet wide, and 4| feet deep, how many men will be required to dig a trench 2 degrees of hardness, 168| feet long, 7^ feet wide, and 2| feet deep in 22 days of 9 hours each ? Aiis. 15 men. 64. A man has a bin 7 ft. long, 2^ ft. wide, 250 Cancellation and Analysis. [ 10. and 2 ft. deep, which contains 28 bushels of corn. How deep must he build another bin, which is to be 18 ft. long, 1 ft. 10| in. wide, in order to con- tain 120 bushels? Ans. 4 ft. 5 in. 65. Two men, A and B, traded in company ; A furnished f of the stock and B ; they gained $864.00 ; what was each man's share of the gain ? 66. Three men, A, B, and C, traded in com- pany; A furnished |J of the capital, B J|, and C the rest. They gained 18,453.28 ; what was each one's share of the gain ? 67. Two men, B and C, bought a barrel of flour together. B paid $5.00 and C 13.00 ; what part of the whole price did each pay ? What part of the flour ought each to have ? 68. Three men, C, D, and E, traded in com- pany; C put in $855, D $945, and E $1179; how many dollars did they all put in ? What part of the whole did each put in? They gained $1340.55; what was each man's share of the gain? 69. Five men, A, B, C, D, and E, freighted a vessel: A put on board goods to the amount of $4000, B $15,000, C $11,000, D $7500, and E $850. During a storm the captain was obliged to throw overboard goods to the amount of $13,039 ; what was each man's share of the loss ? 70. Three men hired a pasture for $42.00 ; the first put in 4 horses ; the second 6 ; and the third 8. What ought each to pay ? Cancellation and Analysis. 251 71. A man failing in trade owes A 12700, B $1800, C $1500 ; and he has only $2100 in prop- erty, which he agrees to divide among his creditors in proportion to the several debts. What will each receive ? 72. During a storm a master of a vessel was obliged to throw overboard -^ of the whole cargo. What part of the whole cargo did a man lose who owned f of it ? o 73. A man owned -f^ of the capital of a cotton manufactory, and sold T \ of his share. What part of the whole capital did he sell ? What part did he then own ? 74. How many bushels of apples, at \ of a dol- lar per bushel, may be bought for f of a dollar ? How many at f of a dollar per bushel ? 75. Two men bought a barrel of flour : one gave 2 1 dollars and the other 3f dollars; what did they give for the whole barrel? What part of the whole value did each pay ? What part of the flour should each have ? 76. Two men hired a pasture for 21 dollars. One kept his horse in it 5^ weeks, and the other 7 :t weeks ; what ought each to pay ? 77. A man being asked his age answered, that he was 24 years old when he was married, and that he had lived with his wife f of his whole life. What was his age ? 78. A person having | of a vessel sold f of his share for $8,400.00 ; what part of the whole vessel did he sell ? What was the whole vessel worth ? 252 Cancellation and Analysis. [ 10. 79. If | of a ship is worth ^ of her cargo, and the cargo is valued at 2100c, what is the value of the ship ? 80. There is a pole standing so tnat |- of it is in the water, | as much in the mud as in the water, and 7 feet above the water. What is the whole length of the pole ? 81. Two men, A and B, having found a bag of money, disputed who should have it. A said .] , .1 , and I of the money made 130 dollars, and if B could tell him how much there was in the bag he should have all the money, otherwise, he should have nothing. How much was there in the bag ? 82. A merchant sold a quantity of goods for $3,846, by which bargain he gained of the first cost. What was the first cost, ancWiow-rnuch did he gain ? 83. A merchant sold a bale of cloth for 1351, by which he gained -^ of what it cost him. How much did it cost him, and how much did he gain ? 84. A merchant sold a quantity of flour for 1143.00, by which he gained | o f the cost. How much did it cost, and how much did he gain ? 85. A merchant sold a quantity of goods for 1187.00, by which he lost | of the first cost. How much did it cost, and how much did he lose ? 86. A merchant sold a quantity of molasses for $259.00, by which he lost | of the cost. How much did it cost, and how much did he lose ? 87. A merchant sold a quantity of goods for $946, by which he lost T \ of the cost. How much did he lose ? Cancellation and Analysis. 253 88. A farmer mixed 15 bushels of rye, at 64 cents per bushel ; 18 bushels of corn at 55 cents per bushel ; and 21 bushels of oats, at 28 cents per bushel. How many bushels were there of the mixture ? What was the whole worth ? What was it worth per bushel ? 89. A grocer mixed 123 Ibs. of sugar that was worth 8 cents per pound ; 87 Ibs. that was worth 11 cents per pound ; and 15 Ibs. that was worth 13 cents per pound. What was the mixture worth per pound ? 90. Three merchants, A, B, and C, freight a ship with coal. A puts on board 500 tons, B 340, and C 94 ; in a storm they are obliged to cast 150 tons overboard. What loss does each sus- tain? 91. Two men hired a pasture for 136. A put in 3 horses for 4 months, and B 5 horses for 3 months. What ought each to pay? Suggestion : 3 horses for 4 months is the same as 4 times 3 or 12 horses for 1 month ; and 5 horses for 3 months is the same as 3 times 5 or 15 horses for 1 month. 92. Four men jointly hired a pasture for f 1.00. A turned in 7 oxen for 12 days, B 9 oxen for 14 days, C 11 oxen for 25 days, and D 15 oxen for 37 days. How much ought each to pay ? 93. Three men entered into partnership, and traded as follows : A put in $150, and at the end of 7 months took out f 50 ; 5 months after he put in $170 ; B put in $205, and at the end of 5 months 254 Cancellation and Analysis. [10. he put in $110 more, but 4 months after took out $150 ; C put in $300, and when 8 months had elapsed he drew out $150.00 ; but 5 months after he put in $500.00. Their partnership continued 18 months, at the end of which time they had gained $660. Required each person's share of the gain. 94. A owes B a sum of money, of which J is to be paid in 2 months, J in 3 months, and the rest in 6 months. If A prefers to pay the whole sum at the same time, when should he pay it? [A method of solution may perhaps be suggested by first solving the problem on the supposition that the entire sum is $6.] 95. Two men were talking of their ages ; one said, " | of my age is equal to \ of yours ; and the sum of our ages is 95." What were their ages ? Suggestion : -^ 5 of the second equals \ g of the first ; therefore the second = -^ of first, therefore first + second = -1-jj- of first = 95. 96. If a man can do f of a piece of work in one day, in what part of a day can he do | of it? How long will it take him to do the whole ? 97. A farmer hired two men to mow a field ; one of them could mow ^ of it in a day, and the other I of it. What part of it would they both together do in a day ? How long would it take them both to mow it ? 98. A gentleman hired 3 men to build a wall ; the first could build it alone in 8 days, the second in 10 days, and the third in 12 days. What part of it could each build in a day ? How long would it take them all together to build it ? Cancellation and Analysis. 255 99. A can do a certain piece of work in 10 days, working 8 hours a day. B can do the same work in 9 days, working 12 hours a day. They decide to work together, and to finish the work in 6 days. How many hours a day must they work ? 100. A man and his wife found that when they were together a bushel of corn would last 15 days, but when the man was absent, it would last the woman alone 27 days. What part of it did both together consume in a day ? What part did the woman alone consume ? What part did the man alone consume ? How long would it last the man alone ? 101. A cistern has 3 cocks to fill it, and one to empty it. The first cock will fill it alone in 3 hours, the second in 5 hours, and the third in 9 hours. The other will empty it in 7 hours. If all the cocks are allowed to run together, in what time will it be filled ? 102. Divide 25 apples between two persons so as to give one seven more than the other. Suggestion : Give one of them 7, and then di- vide the rest equally. 103. A gentleman divided an estate of 115,000 between his two sons, giving the elder $2500 more than the younger. What was the share of each ? 104. A gentleman bequeathed an estate of $ 40,000 to his wife, son, and daughter: to his wife he gave $1500 more than to the son, and to the son 13500 more than to the daughter. What was the share of each ? 256 Cancellation and Analyst*. [ 10. 105. A, B, and C built a house which cost $35,000 ; A paid 1500 more than B, and C $300 less than B. What did each pay ? 106. A man bought a sheep, a cow, and an ox for $82 ; for the cow he gave $10 more than for the sheep ; and for the ox $10 more than for both. What did he give for each ? 107. A man sold some calves and some sheep for $216 ; the calves at $10, and the sheep at $16 apiece. There were twice as many calves as sheep. What was the number of each sort ? Suggestion: There were two calves and one sheep for every $36. 108. A farmer drove to market some oxen, some cows, and some sheep, which he sold for $1498 ; the oxen at $56, the cows at $34, and the sheep at $15. There were twice as many cows as oxen, and three times as many sheep as cows. How many were there of each kind ? 109. Said A to B, my horse and saddle together are worth $150 ; but my horse is worth 9 times as much as the saddle. What was the value of each ? 110. A man driving some sheep and some cat- tle, being asked how many he had of each kind, said he had 174 in all, and there were ^ as many cattle as sheep. Required the number of each kind. 111. A gentleman left an estate of $13,000 to his four sons, in such a manner that the third was to have 1| as much as the fourth, the second as much as the third and fourth, and the first as Cancellation and Analysis. 257 much as the other three. What was the share of each ? 112. Three persons, A, B, and C, traded in company. A put in $75, B $ 40. They gained $64, of which C took $18 for his share. What did C put in ? 113. A person buys 12 apples and 6 pears for 17 cents, and afterwards 3 apples and 12 pears for 20 ceiits. What is the price of an apple, and of a pear ? Suggestion : At the second time he bought 3 apples and 12 pears for 20 cents. 4 times all this will make 12 apples and 48 pears for 80 cents : the price of 12 apples and 6 pears being taken from this, will leave 63 cents 'for 42 pears, which is 1J cent apiece. 114. Two persons were talking of their ages ; one said, " | of mine is equal to | of yours, and the difference between our ages is 10 years." What were their ages ? 115. A man having $100 spent a part of it ; he afterwards received five times as much as he spent, and then his money was double what it was at first. How much did he spend ? 116. A, B, and C hire a pasture for $92. A pastures 6 horses for 8 weeks, B 12 oxen for 10 weeks, and C 50 cows for 12 weeks. Now, if 5 cows are reckoned as 3 oxen, and 3 oxen as 2 horses, how much should each man pay ? 117. By a pipe of a certain capacity a cistern can be emptied in 3y^ hours ; in what time can it 258 Cancellation and Analyxix. be emptied by a pipe the capacity of which is | greater ? 118. A and B, 44 miles apart, travel towards each other. A travels -^ of the whole distance, v;hile B travels | of the remainder. How far are they then apart ? 119. A can do a piece of work in 10 days, A and C can do it in 7 days, A and B can do it in 6 days : in how many days can B and C together do it? APPENDIX. CHAPTER I. Roman Notation. The figures that are ordinarily used to represent numbers are called Arabic figures, because they were first introduced into Europe by the Arabs, who had derived them, however, from Hindostan. Numbers are also sometimes represented by Roman letters, as indi- cated in the following table : One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Ten, Eleven, Twelve, Thirteen, Fourteen, Fifteen, I. Sixteen, XVI. II. Seventeen, XVII. III. Eighteen, XVIII. IV. Nineteen, XIX. V. Twenty, XX. VI. Twenty-one, XXI. VII. Twenty-two, XXII. VIII. Twenty-three, XXIII. IX. Twenty-four, XXIV. X. Twenty-five, XXV. XL Twenty-six, XXVI. XII. Twenty-seven, XXVII. XIII. Twenty-eight, XXVIII. XIV. Twenty -nine, XXIX. XV. Thirty, XXX. 260 Appendix. [Ch. 1. Thirty-one, XXXI. Two hundred, CC. Thirty-two, XXXII. Three hundred, ccc. Forty, XL. Four hundred, cccc. Fifty, L. Five hundred, D. Sixty, LX. Six hundred, DC. Seventy, LXX. Seven hundred, DCC. Eighty, LXXX. Eight hundred, DCCC. Ninety, XC. Nine hundred, DCCCC. One hundred, C. One thousand, M. One thousand eight hundred and twenty-six, MDCCCXXVI. Note 1. The following description of the development of the Roman Notation, taken from Warren Colburn's Arithmetic, A Sequel to Intellectual Arithmetic (see page 109), may be found in- teresting, and will tend to fix the notation in mind. One was written with a single mark, thus : / Two was written with two marks, // Three was written /// Four was written //// Five was written ///// Six was written II I III Seven was written II Hill Eight was written II I Hill Nine was written ///////// Ten, instead of being written with ten marks, was expressed by two marks crossing each other, thus, X Two tens, or twenty, were written XX Three tens, or thirty, were written XXX And so on to ten tens, which were written with ten crosses. But as it was found inconvenient to express numbers so large as seven or eight with marks as represented in the foregoing, the X was cut in two, thus, X, and the upper part V was used to express one half of ten, or five, and the numbers from five to ten were ex- pressed by writing marks after the V, to express the number of units added to five. Roman Notation. 261 Six was written ]/f Seven was written ]/H Eight was written J7// Nine was written /)( Eleven was written XI Twelve was written XII Twenty-seven was written XXVII To express ten X's, or ten tens, or one hundred, three marks were used, thus, C ; and to avoid the inconvenience of writing seven or eight X's, the C was divided, thus, H, and the lower part, L, used to express five X's, or fifty. To express ten hundreds, four dashes were used, thus, I VI. This last was afterwards written in this form CD, and sometimes C ID, and was then divided, and ID was used to express five hundreds. These dashes resemble some of the letters of the alphabet, which letters were afterwards substituted for them. I resembles I ; V resembles V ; X resembles X ; L resembles L ; C was sub- stituted for C; ID resembles D; and I VI resembles M. The numbers four, nine, forty, and ninety were afterwards denoted more briefly, as shown below. IIII by IV XXXX by XL VIIII by IX LXXXX by XC Writing an I before a V or an X decreases the value of the V or X by I ; and writing an X before an L or a C decreases the value of the L or C by X. Wherever these numbers (four, nine, forty, and ninety) occurred they were abbreviated in the same way. Thus XIIII became XIV, XVIIII became XIX, LVIIII became LIX, CLXXXX became CXC, etc. Note 2. The explanation, substantially as given in Note 1, of the origin of the characters of the Roman Notation, was advo- cated by Sir John Leslie in his Philosophy of Arithmetic (1820), and by some writers of the 16th and 17th centuries. On this point, however, the Encyclopaedia Britannica (Ninth Edition), says : " This explanation is perhaps too ingenious. . . One does not readily see how the C could be formed from the X or the M from the C ; and it appears far more likely that the signs for 100 and 1000 are merely the initial letters of Centum and Mille. ... In any case the V, L, and D appear to be respectively the halves of X, the angular C (C) and the rounded M (po) [fre- quently written C ID]." 262 Appendix. [Ch. 2. EXAMPLES. 1. Read the numbers XLIV, XXXIX, CXIX, CLIII, LXXXVI, LX, XXI, Mil, DXCIX, MDCCCLXXXVIL 2. Denote by Roman letters the numbers thirty-eight, eighty-four, thirteen, four hundred and thirty-two, one thousand and seventy -two, eighteen hundred and seventy-five. APPENDIX. CHAPTER II. THE METRIC* SYSTEM OF MEASURES. In 1799 the French Legislature adopted a fixed length, which it agreed to call a metre (a word derived from the Greek metron, a measure), as the standard unit of linear measure. According to the best measure- ments that could then he obtained, this fixed length (called in English a meter f) was the ten-millionth part of the distance from the equator to the north pole ; it was chosen because it appeared to meet the demand for an invariable unit of length which could be re-determined from the definition at any future time. There was then constructed, with the meter as a basis, the so-called Metric System of Measures, which will be described in the following pages. The metric system of measures has also been adopted in Germany, Spain, Portugal, Belgium, Holland, Switzer- land, Italy, Austria, Sweden, Denmark, Greece, British India, Turkey, Mexico, Brazil, and other States of South America, and is in use to some extent in this and most other countries. * Pronounced me'tric. t Pronounced mee'-ter. A.] The Metric System. 263 A. Linear Measure. In the metric system of measures the principal meas- ure or unit of length is the meter, a measure very nearly 39.37 inches long.* The smaller measures of length are one tenth, one hundredth, and one thousandth of a meter; and the larger measures of length are ten, one hundred, one thousand, and ten thousand meters. The names of these measures are : MILLIMETER f (mm) for T73 ^ ^ of a meter. The prefix milli denotes thousandth. CENTIMETER | (cm) for T J^ of a meter. The prefix centi denotes hundredth. DECIMETER f (dm) for T ^ of a meter. The prefix deci denotes tenth. METER (m) for the principal unit. DEKAMETER (Dm) for 10 meters. The prefix deka denotes ten. HECTOMETER (Hm) for 100 meters. The prefix hecto denotes hundred. KILOMETER (Km) for 1000 meters. The prefix kilo denotes thousand. MYRIAMETER (Mm) for 10000 meters. The prefix myria denotes ten thousand. 1. Complete the following table by filling in the * The standard meter is a platinum bar carefully preserved at Paris ; and every meter-stick anywhere in use should be of the same length as the standard. The length of the standard meter, expressed in inches, is 39.3707904 ; it is sufficiently accurate for most practical purposes to say that a meter contains 39.37 inches. t Pronounced millimeter, centimeter, decimeter. All the metric names are accented on the first syllable. 264 Appendix. [Ch. 2. proper numbers on the right, and the proper abbreviations within the parentheses : 1 millimeter (mm) = .001 meter 1 centimeter ( ) = 1 decimeter ( ) = 1 meter ( m ) = 1 meter 1 dekameter ( ) = 1 hectometer ( ) = r-1 1 kilometer ( ) = 1 myriameter ( ) = 2. Complete the following table by filling in the proper numbers on the left and the proper abbreviations within the parentheses : 10 millimeters =1 centimeter (cm) centimeters = 1 decimeter ( ) decimeters =1 meter ( ) meters = 1 dekameter ( ) dekameters = 1 hectometer ( ) hectometers = 1 kilometer ( ) kilometers =1 myriameter ( ) 3. Make a decimeter rule or measure, by drawing with a fine pen on the edge of a piece of cardboard the figure given on the right. 4. By the aid of the measure just made find (a) the nearest number of decimeters in the length of the cover of this book (Z>), the nearest number of centimeters. Find (a) the nearest number of centimeters in the thickness of this book (&), the nearest number of millimeters. 5. Find the nearest number of centimeters in the length of your lead pencil. .4 A.] The, Metric System. 265 6. Find (a) the nearest number of decimeters in the width of the nearest window ; (b) the nearest number of centimeters. 7. What is the nearest number of decimeters in the distance of the window-sill from the floor ? 8. Measure the length and width of a window- pane and find the nearest number of centimeters in each. 9. Find the nearest number of decimeters in the width of your desk. 10. Find the nearest number of decimeters in the distance of the highest part of your chair from the floor. 1 1. What is the nearest number of centimeters in the distance from the point of your elbow to the tip of your middle finger ? 1 2. With a stick or a string make, by the aid of your decimeter rule, a meter rule. 1 3. a. How long is the room that you are now in ? [Find first how many whole meters there are and then the nearest number of centimeters that are left over.] b. How many centimeters are there in the length of the room ? c. How many steps do you take in walking from one end of the room to the other ? d. What is the nearest number of centimeters in the length of each step if all the steps are equally long ? 14. a. How many steps do you take in walking across the room ? b. How wide then is the room if the length of each step is the same as before ? c. Measure the width of the room with your rule and see how much the result differs from that just found. 15. Is the length of the room more or less than a dekameter ? how much ? [Give the answer in centi- meters.] 16. Is the width of the room more or less than a dekameter ? how much ? 266 Appendix. [Ch. 2. 17. a. How many steps do you take in walking the length of the building that you are now in ? how many meters long is the building if the length of each step is the same as found in Example 13 d ? b. Find in the same way the approximate width of the building. 18. How many millimeters are there in a centimeter ? how many centimeters in a decimeter ? how many decimeters in a meter ? how many meters in a deka- meter ? 19. How many decimeters are there in 3 Dm, 2 m, and 6 dm? (Ans. 326) how many meters? (Ans. 32.6) how many dekameters ? (Ans. 3.26) 20. How many centimeters are there in 4 m, 3 dm, and 8 cm ? how many decimeters ? how many meters ? how many dekameters ? 21. How many hectometers are there in 1.6046 Km ? how many dekameters ? how many meters ? how many decimeters ? 22. How many decimeters are there in 1632 cm? how many meters ? how many dekameters ? 23. How many centimeters are there in 736.8 mm ? how many decimeters ? how many meters ? 24. If a building is 123 m long, what measure is used when its length is denoted by 12.3 ? by 1.23 ? by 0.123 ? by 1230 ? by 12300 ? 25. a. How many meters are there in 6 Km ? in 3 Hm ? in 8 Dm ? b. How many meters are there in 8 mm ? in 9 cm ? in 4 dm ? 26. How many millimeters are there in 2 dm, 3 cm, and 6 mm ? How many meters ? 27. How many meters are there in 6 Hm and 2 B.] The Metric System. 267 28. If there are 39.37 inches in a meter how many inches are there in a decimeter ? how many in a centi- meter ? 29. How many yards are there in a meter ? Ans. 1.09 4- yds. 30. How many rods are there in a dekameter ? 31. How many miles are there in a kilometer ? Ans. .62 miles. 32. How many centimeters are there in an inch ? 33. How many decimeters are there in a foot ? 34. How many meters are there in a yard ? 35. How many dekameters are there in a rod ? 36. How many kilometers are there in a mile ? Ans. 1.60 + Km. 37. How long a strip of paper as wide as this page would be required for 200 pages of this book ? [Find the answer first in centimeters and then change it to meters.] 38. At $1.75 a meter what will a kilometer of silk cost? 39. How far must I walk in going from A to B, a distance of 6 Hm 3 m, and then from B to C, a dis- tance of 9 Dm 8 m ? Ans. 1 Hm 1 m. B. Square Measure. 1. The figure in the margin is a square centimeter (sq. cm). How many square mil- limeters does it contain ? 2. How many square centimeters are there in a figure 4 centimeters long and 2 centimeters wide ? how many in a figure twice as long and twice as wide ? 3. How many square centimeters are there in a figure 4 centimeters long and 4 centimeters wide ? how 268 Appendix. [Ch. 2. many in a figure 8 centimeters long and 2 centimeters wide ? How much more string is required to surround the second figure than the first ? 4. Draw on paper as accurately as you can a square decimeter (sq. dm). How many square centimeters does it contain ? 5. How many square decimeters are there in a square meter (sq. m) ? 6. How many square meters are there in a square dekameter (sq. Dm) ? [The square dekameter, called also an ar* (a), is a convenient measure to use in measuring land. The square meter is also called a centar (ra).] 7. How many square dekameters or ars are there in a square hectometer ? [The square hectometer is also called a hectar (Aa).] 8. How many square hectometers or hectars are there in a square kilometer (sq. km) ? 9. Complete the following table hy filling in the proper numbers on the left, and the proper abbreviations within the parentheses : 100 square millimeters = 1 square centimeter (sq. cm) square centimeters = 1 square decimeter ( ) square decimeters = 1 square meter ( ) or centar ( ) square meters or =1 square dekameter ( ) centars or ar ( ) square dekameters = 1 square hectometer ( ) or ars or hectar ( ) square hectometers = 1 square kilometer ( ) or hectars * Pronounced like the word are. This and our word area are derived from the Latin area, a broad, level piece of ground. B.] The Metric System. 269 10. How many square meters or centars are there in. 3 ha 4 a 2 ca ? 11. Change 684.2 sq. m to ars ; to square decimeters. 12. Since, in square measure, each measure or unit is 100 times the next smaller, how many places must we move the decimal point when we change from one meas- ure to the next ? 13. Reduce 398420 ars to hectars ; to square kilome- ters ; to centars ; to square centimeters. 14. Reduce 2.246 square kilometers to centars. 15. How many ars are there in a barn-yard 89 dm long and 7 m wide ? 16. How many square kilometers are there in a field 3100 m wide and 100 Hm long ? 17. How many square meters of carpet would he re- quired to carpet the room that you are now in ? 18. How many square centimeters are there in the cover of this book ? 19. How many square meters of paper are needed for 300 pages of this book ? 20. How many meters of carpet .6 m wide are needed to carpet a room 10.8 m long and 4.6 m wide ? 21. Complete the following table by filling in the proper numbers on the right, and the proper abbrevia- tions within the parentheses : 1 sq. millimeter (sq. mm) = .000001 sq. meter. 1 sq. centimeter ( ) sq. meter. 1 sq. decimeter ( ) sq. meter. 1 centar ( ) or sq. meter ( ) = sq. meter. 1 ar ( ) or sq. dekameter ( ) = sq. meters. 1 hectar ( ) or sq. hectometer sq. meters. 1 sq. kilometer ( ) sq. meters. 270 Appendix. [Ch. 2. C. Solid Measure. A cubic centimeter (cu. cm) is a solid, each of whose edges is a centimeter. 1. How many cubic centimeters are there in a solid 2 cm long, 1 cm wide, and 1 cm high ? how many in a solid 4 cm long, 1 cm wide, and 1 cm high ? 2. How many cubic centimeters are there in the solids whose dimensions (length, breadth, and thick- ness) are as follows : a. 10 cm, 2 cm, and 1 cm b. 10 cm, 4 cm, and 1 cm c. 10 cm, 10 cm, and 1 cm d. 10 cm, 10 cm, and 2 cm e. 10 cm, 10 cm, and 10 cm. 3. How many cubic centimeters are there in a cubic decimeter ? 4. How many cubic decimeters (cu. dm) are there in a cubic meter ? [The cubic meter (cu. m) is also called a ster * ().] 5. How many cubic millimeters (cu. mm) are there in a cubic centimeter ? 6. Complete the following table by filling in the proper numbers on the left : cubic millimeters = 1 cubic centimeter cubic centimeters = 1 cubic decimeter, cubic decimeters = 1 cubic meter or ster. * Pronounced like the word stair. D.] The Metric System. 271 7. Reduce 16900 cubic decimeters to sters; to cubic centimeters. 8. Reduce 3 5, 4 cu. dm, 3 cu. cm to cubic centime- ters. 9. In cubic measure each measure or unit is 1000 times the next smaller ; how many places, then, must we move the decimal point when we change from one measure to the next smaller or larger ? 10. Change 164300000 cu. mm to cubic centimeters ; change the result to cubic decimeters ; change the last result to cubic meters. 11. In 3.1647 sters how many cubic decimeters are there ? how many cubic centimeters ? 12. How many cubic meters are there in a box whose dimensions are 4 m, 5 dm, and 5 dm ? 13. How many sters are there in a pile of wood 8 m long, 1 in wide, and 2 m high ? 14. How many cubic meters are there in the room that you are now in ? 15. How many cubic decimeters of water will a tank hold that is 2 m long, .5 m wide, and .7 m deep ? 1U/ W D. Capacity Measure. 1. Draw on a piece of card- board a figure like that given here, making each of the lines 1 decimeter long ; cut the figure out with a sharp knife ; cut the card-board half through under- neath the dotted lines ; fold up the outer portions so as to form a box. This box will hold a cubic decimeter or liter. 272 Appendix. [Ch. 2. NOTE. The liter is the principal measure of capacity ; the smaller measures of capacity are one tenth, one hundredth, and one thousandth of a liter ; and the larger measures of capa- city are ten, one hundred, and one thousand liters. The names of the different measures of capacity are formed by prefixing to the word liter the same names that are used as prefixes in linear meas- ure. 2. Form a table of the measures of capacity like that of the measures of length. 3. How many liters will a tank hold that is 2 dm wide, 5 dm long, and 6 cm high ? E. Weight Measure. The weight of the water* that can be put into a tight box of the size of the one just mentioned, that is, the weight of a liter or cubic decimeter of water is called a kilogram (kg) or, more briefly, a kilo. 1. How many kilograms will 5 liters of water weigh ? how many kilos will a hectoliter of water weigh ? 2. How many grams (g) will a cubic centimeter of water weigh ? how many kilograms will a cubic meter of water weigh ? 3. Form a table of the measures of weight like that of the measures of length. 4. The weight of a cubic meter of water (1000 kg) is called a metric ton. How many metric tons of water will a tank hold that is 3 m long, 1.5 m wide, and .5 m deep ? * Pure water taken at its greatest density, that is, a little above the freezing point. F.] The Metric System. 273 F. Miscellaneous. Examples. 1. Reduce 2.15 km to centimeters. 2. Reduce 2.15 miles to inches. 3. How many hectars are there in a rectangular park 1 km 26 m long, and 675 m wide ? 4. How many acres are there in a rectangular park 1 mile 26 yds. long and 675 yds. wide ? 5. Reduce 6 Dm to decimeters. 6. Reduce 6 gallons to gills. 7. Reduce 6 metric tons to grams. 8. Reduce 6 tons to ounces. 9. Examples 1, 3, 5, and 7 are like Examples 2, 4, 6, and 8, except that in the former the French or metric system of measures is used, and in the latter the Eng- lish system of measures is used. Which set of exam- ples can be solved in the least time and with the least amount of figuring ? As we have seen, there are the following simple rela- tions between the metric measures of length, weight, and capacity : A CUBIC DECIMETER is A LITER, AND A LITER OF WATER WEIGHS A KILOGRAM ; whereas in the common system of measures there are no simple re- lations between the corresponding measures, the foot (or inch), the pound, and the quart. In the entire metric system the only words required to designate the different measures are meter, liter, gram, metric ton, ar, ster, and the prefixes milli, centi, deci, deka, Tiecto, kilo, and myria ; whereas in the com- mon system, in addition to numerous words occasionally required, the following 21 are in common use : inch, 274 Appendix. [Ch. 2. foot, yard, rod, mile, acre, cord foot, cord, gill, pint, quart, gallon, peck, bushel, grain, scruple, dram, ounce, pennyweight, pound, ton. The name of a metric measure tells at once how many principal units it contains ; the name kilometer, for instance, means 1000 meters ; whereas the name mile is no indication of the number of feet. From what has been said, we see that if the metric system of measures were everywhere substituted for the common system, a great saving of time and labor ivould be made both in the education of children and in the necessary computations of every-day life : 1. Because, as it is a decimal system, a change from one denomination to another can be made by a mere change in the position of the decimal point. 2. Because of the simple and easily remembered re- lations which exist between the measures of length, capacity, and weight. 3. Because of the small number of words required to designate all the different measures. 4. Because almost every word used has a meaning that tells the size of the measure to which it belongs. Professor LEONE LEVI, in his Metric System, says : ' ' Here is a tool which offers facilities for saving one half the time in arithmetical education, and one fourth, or one third of the time spent in all the transactions which include calculations of weights and measures." The following are extracts from a paper by Edward Wiggles- worth, M. D. : "John Quincy Adams, even in his day, spoke of the metric system as ' the greatest invention of human ingenuity since that of printing.' It has been calculated by large committees of our ablest teachers that the complete introduction of the metric system will save a full year of the school-life of every child . . . simpler than others as our money is simpler than pounds, shillings, and Arithmetical Tables. 275 pence, . . . competent authorities computed that the London and Northwestern Railway alone would annually save 10,000 sterling by the use in all its computations of the metric instead of the old system. How vast, then, would be the saving in the entire busi- ness of the country ! In 1860 the foreign business of the United States equalled $762,000,000. Of this $700,000,000 was with nations using the metric system, and that, too, before Germany had adopted it." APPENDIX. CHAPTER III. ARITHMETICAL TABLES. [Measures less frequently used are printed in smaller type.] UNITED STATES MONEY. 10 mills = 1 cent. 10 cents = 1 dime. 10 dimes = 1 dollar ($). $10 = 1 eagle. ENGLISH MONEY. 4 farthings (f) = 1 penny (d.). 12 pence =1 shilling (s.). 20 shillings = 1 pound sterling (). A florin = 2 s. A sovereign = 20 s. A crown = 5 s. A guinea = 21 s. 1 = $4.866 J. FRENCH MONEY. 100 centimes = 1 franc ($0.193). GERMAN MONEY. 100 pfennigs = 1 mark ($0.238). 276 Appendix. [Ch. 3. LENGTH. 12 inches (in.)^l foot (ft.). 3 feet = 1 yard (yd.). 16 feet or) 5* yards I = 320 rods or) 5280 feet j 12 lines = 1 inch. 18 inches = 1 cubit. 4 inches = 1 hand. 40 rods = 1 furlong. 9 inches = 1 span. S furlongs 1 mile. 6 feet = 1 fathom. SURVEYORS' MEASURE OF LENGTH. 7.92 inches =1 link (Ink.). 100 links =1 chain (ch.). 80 chains = 1 mile. 25 Inks = 1 rd. The Surveyors' Measure was obtained by first calling one tenth of an acre * a square chain and then divid- ing a linear chain (4 rds or 792 in.) into 100 equal parts which were called links. SURFACE. 144 square inches =:1 square foot (sq. ft.). 9 square feet = 1 square yard (sq. yd.). 272J square feet ) * , \ \ square rod (sq. rd.). or 30J square yards ) 160 square rods =1 acre (a.). 640 acres =1 square mile (sq. m.). 40 sq. rds. = 1 rood. 1 sq. m. =1 section. 4 roods = 1 acre. 36 sections = 1 township. * One tenth of an acre, or 16 sq. rds., is equivalent to a square each side of which is 4 rds. If, then, 16 sq. rds. are called a square chain, a linear chain would be 4 rds. or 792 in. Arithmetical Tables. 277 SOLIDITY. 1728 cubic inches (cu. in.) = l cubic foot (cu. ft.). 27 cubic feet =1 cubic yard (cu. yd.). 16 cubic feet = 1 cord foot. 8 cord feet or ) HOO i = 1 cor d (cd.). 128 cubic feet ) MEASURES OF CAPACITY. LIQUID MEASURE. 4 gills =1 pint (pt.). 2 pints =1 quart (qt.). 4 quarts = 1 gallon (gal.). DRY MEASURE. 2 pints =1 quart (qt.). 8 quarts = 1 peck (pk.). 4 pecks =1 bushel (bu.). 4 quarts liquid measure = 231 cubic inches. 4 quarts dry measure = 26&| cubic inches. AVOIRDUPOIS WEIGHT. FOR ALL GOODS EXCEPTING GOLD, SILVER, AND PRECIOUS STONES. 16 drams (dr.) = l ounce (oz.). 16 ounces = 1 pound (lb.) 2000 pounds = 1 ton (t). 14 pounds = 1 stone. 100 pounds = 1 hundred-weight (cwt.). 2240 pounds == 1 long ton.. The long ton is used in the United States custom- house and in wholesale transactions in coal and iron. 278 Aj,jx'nflix. [Ch. 3. TROY WEIGHT. . FOB GOLD, SILVER, AND PRECIOUS STONES. 24 grains (grs.) =1 pennyweight (dwt.). 20 penny weights = 1 ounce (oz.). 12 ounces =1 pound (lb.). 1 lb. Troy = 5700 grs. 1 lb. Avoirdupois = 7000 grs. 1 carat of diamond = 4 grs. 1 carat of gold or silver = 240 grs. 24 carats of gold or silver = 1 lb. Pure gold is too soft for practical use ; when coins, jewels, etc., are said to be made of gold they really consist of a mixture of pure gold and some other metal. "Jewelers' gold," a pound ('J4 carats) of which contains only 18 carats of pure gold, is said to be 18 carats fine. " Standard gold " is 22 carats fine ; that is, in every 24 parts there are 22 parts of pure gold and 2 parts of some other metal. On the inside of a gold watch case or of a gold ring we usually find a stamp to indicate the fineness of the gold, such as 18 k or merely 18. APOTHECARIES' WEIGHT. 20 grains (grs.) = l scruple (3). 3 scruples =1 dram (3). 8 drams = 1 ounce (oz. or |). 12 ounces = 1 pound (lb.). The grain, ounce, and pound are the same as in Troy weight. APOTHECARIES' MEASURE. 60 minims (rn.) =1 dram 8 drams =1 ounce (fl. drm. viij.). 16 ounces =1 pint (fl. oz. xvj.). HI lx. means Ix. (sixty) minims ; fl. drm. viij. means viii. fluid drams ; and fl. oz. xvj. means xvi. fluid ounces. The j at the end is used instead of an i. Arithmetical Tables. 279 TIME. 60 seconds (sec.) 1 minute (min.) 60 minutes =1 hour (hr.J. 24 hours =1 day (dy.). 7 days =1 week (wk.). 365 days or ) 52 wk 8 . i dy. } common 366 days = 1 leap year. 100 years = 1 century. Thirty days hath September, April, June, and November. All the rest have thirty-one, Excepting February alone, To which we twenty-eight assign, Till leap year gives it twenty-nine. A solar year is 365 dys. 5 hrs. 48 min. 50 sec., that is, nearly 365 days. A common year of 365 days is, therefore, nearly one fourth of a day shorter than a solar year ; to make up for this defect some years (leap years) are reckoned as 366 days. When- ever the number representing the year is divisible by 4 and not by 100, or whenever it is divisible by 400, then the year is a leap year. Thus 1888 is a leap year ; 1889 is not a leap year ; 1900 is not a leap year ; 2000 is a leap year. MISCELLANEOUS. NUMBERS. PAPER. 12 units =1 dozen. 24 sheets =1 quire. 12 dozen = 1 gross. 20 quires = 1 ream. 12 gross = 1 great gross. 2 reams = 1 bundle. 20 units = 1 score. 5 bundles = 1 bale. A barrel of flour = 196 Ibs. A cask of lime = 240 Ibs. Hr A barrel of pork = 200 Ibs. A quintal of fish = 100 Ibs. or beef 280 Appendix. THE METRIC SYSTEM OF MEASURES. LINEAR MEASURE. 10 millimeters (mm) = 1 centimeter (cm). 10 centimeters = 1 decimeter (dm). 10 decimeters = 1 meter (m). 10 meters = 1 dekameter (Dm). 10 dekameters = 1 hectometer (Hm). 10 hectometers =1 kilometer (Km). 10 kilometers =1 myriameter (Mm). 1 meter = 39.3707904 in. = 3.28090 ft. = 1.09363 yds. 1 kilometer = 0.62138 miles. SQUARE MEASURE. 100 square millimeters (sq. mm) = 1 square centimeter (sq. cm). 100 square centimeters = 1 square decimeter (sq. dm). 100 square decimeters = 1 square meter (sq. m) or centar (ca). 100 square meters = 1 square dekameter (sq. Dm) or ar (a). 100 square dekameters = 1 square hectometer (sq. Hm) or hectar (Ha). 100 square hectometers = 1 square kilometer (sq. Km). 1 sq. m = 1.19603 sq. yds. = 10.76430 sq. ft. 1 ar = 3.95383 sq. rds. CUBIC MEASURE. 1000 cubic millimeters (cu. mm) = 1 cubic centimeter (cu. cm), 1000 cubic centimeters =1 cubic decimeter (cu. dm). 1000 cubic decimeters = 1 cubic meter (cu. m) or ster (s.). 1 cu. m. = 35.31658 cu. ft. = 1.30802 cu. yds. Arithmetical Tables. 281 CAPACITY MEASURE. 10 milliliters (ml) = 1 centiliter (cl). 10 centiliters = 1 deciliter (dl). 10 deciliters =1 liter (I). 10 liters =1 dekaliter (Dl). 10 dekaliters =1 hectoliter (HI). 10 hectoliters =1 kiloliter (Kl). 1 / = 1 cu. dm = = 1.0567 quarts liquid measure. = .90792 quarts dry measure. WEIGHTS. 10 milligrams (ing) = 1 centigram (eg). 10 centigrams =1 decigram (dg). 10 decigrams =1 gram (g). 10 grams =1 dekagram (Dg). 10 dekagrams = 1 hectogram (ffg) 10 hectograms =1 kilogram (K). 1000 kilograms =1 metric ton (T). 1 I or 1 CM. dm of water weighs 1 K. 1 K. = 2.20462 Ibs. Avoirdupois. = 2.67923 Ibs. Troy. Of the coins of the United States The silver half-dollar weighs 12J grams. " " quarter-dollar weighs 6 grams. " " twenty-cent piece weighs 5 grams, ten-cent piece weighs 2i grams. " nickel five-cent piece weighs 5 grams. 282 Compound Interest Table. COMPOUND INTEREST TABLE. SHOWING THE AMOUNT OF $1.00, AT COMPOUND INTEREST, FROM 1 YEAR TO 50. Year. 3 p. cent. 3 p.cent. 4 p. cent. 4Jp.cent. 5 p. cent. 6 p. cent. 7 p. cent. 1 2 3 4 5 1.030000 l.OGOOOO 1.092727 1.125509 1.159274 1.035000 1.071225 1.108718 1.147523 1.187686 1.040000 1.081600 1.124864 1.169859 1.216653 1.045000 1.092025 1.141166 1.192519 1.246182 1.050000 1.102500 1.157625 1.215506 1.276282 1.060000 1.123600 1.191016 1.262477 1.338226 1.070000 1.144900 1.225043 1.3107% 1.402552 6 7 8 9 10 1.194052 1.229874 1.266770 1.304773 1.343916 1.229255 1.272279 1.316809 1.362897 1.410599 1.265319 1.315932 1.368569 1.423312 1.480244 1.302260 1.360862 1.422101 1.486095 1. 5521)60 1.3400% 1.407100 1.477455 .551328 .628895 1.418519 1.503630 1.593848 1.689479 1.790848 1.500730 1.605781 1.718186 1.838459 1.967151 11 12 13 14 15 1.384234 1.425761 1.468534 1.512590 1.557967 1.459970 1.511069 1.5631)56 1.618694 1.675349 1.539454 1.601032 1.665073 1.731676 1.800043 1.622853 1.695881 1.7721% 1.851945 1.935282 .710339 .795856 .885649 1.979932 2.078928 1.898299 2.0121% 2.132928 2.260904 2.39655S 2.104862 2.252192 2.409845 2.578534 2.759031 16 17 18 19 20 1.604706 1.652848 1.702433 1.753506 1.806111 1.733986 1.794675 1.857480 1.1)22501 1.989789 1.872981 1.947900 2.025816 2.106849 2.191123 2.022370 2.113377 2.208479 2.307860 2.411714 2.182875 2.292018 2.406619 2.526050 2.653298 2.540352 2.692773 2.854:53') 3.02551*0 3.207135 2.952164 3.158815 3.379931 3.616526 3.800683 21 22 23 24 25 1.860295 1.916103 1.973586 2.032794 2.093778 2.059431 2.131512 2.206114 2.283328 2.363245 2.278768 2.369919 2.464715 2.563304 2.665836 2.520241 2. (33652 2.752166 2.876014 3.005434 2.785963 2.025261 3.071524 3.225100 3.386355 3.399564 3.603537 3.819750 4.048935 4.291871 4.140561 4.430400 4.740528 5.072365 5.427431 26 27 28 29 30 2.156591 2.221289 2.287928 2.356565 2.427262 2.445959 2.531567 2.620177 2.711878 2.806794 2.772470 2.883300 2.998703 3.118651 3.243397 3.140679 3.282009 3.429700 3.584030 3.745318 3.555673 3.733456 3.920129 4.116136 4.321942 4.549383 4.822346 5.111687 5.418388 5.743491 5.807351 6.213866 6.648836 7.114255 7.612253 31 32 33 34 35 2.500080 2.575083 2.652335 2.731905 2.813862 2.905031 3.006708 3.111942 3.220860 3.333590 3.373133 3.508059 3.648381 3.794316 3.946089 3.913857 4.089981 4.274030 4.466361 4.667348 4.538039 4.764941 5.003188 5.253348 5.516015 6.088101 6.453387 6.840590 7.251025 7.686087 8.145110 8.715268 9.325337 9.978110 10.676578 36 37 38 39 40 2.898278 2.985227 3.074783 3.167027 3.262038 3.450266 3.571025 3.696011 3.825372 3.959260 4.103932 4.268090 4.438813 4.616366 4.801021 4.877378 5.096860 5.326219 5.565899 5.816364 5.791816 6.081407 6.385477 6.704751 7.039989 8.147252 8.636087 9.154252 9.703507 10.285718 11.423939 12.223614 13.079277 13.994827 14.974465 41 42 43 44 45 3.359899 3.4606% 3.564517 3.671452 3.781596 4.097834 4.241258 4.389702 4.543342 4.702358 4.993061 5.192784 5.400495 5.616515 5.841176 6.078101 6.351615 6.637438 6.936123 7.248248 7.391988 7.761587 8.149667 8.557150 8.985008 10.902861 11.557033 12.250455 12.985482 13.764611 16.022677 17.144265 18.344363 19.628469 21.002461 46 47 48 49 50 3.895044 4.011895 4.132252 4.256219 4.383906 4.866941 5.037284 5.213589 5.396065 5.584927 6.074823 6.317816 6.570528 6.833349 7.106683 7.574420 7.915268 8.271455 8.643671 9.0326361 9.434258 9.905971 10.401270 10.921333 11.467400 14.590487 15.465917 16.393872 17.377504 18.420154 22.472634 24.045718 25.728918 27.529943 29.457039 ARITHMETIC in TWO BOOKS. *Warren Colburn's First Lessons, - 35 cents. *H. 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The numbers already issued have been extensively used for the study of Language, for the study of Literature, for Supplementary Reading, and as substitutes for the graded Readers. In whatever way they may be used, the principal benefit to be derived from them will be the forma- tion of a taste in the reader for the best and most enduring literature ; this taste the pupil will carry with him when he leave: school, and it will remain through life a powerful means of self-education. An inspection of the titles of the different numbers of the series will show at it con- tains a pleasing variety of reading matter in Biography, History, Poetry, and Mythology. While each number has been issued in paper covers, several combinar tions of two and three numbers in board covers have also been made, in response to a demand for a larger amount of material in a single volume with a more permanent binding. The Publishers take pleasure in announcing that several new num- berscontaining some of the best and purest literature will be added to the Riverside Literature Series during each school year ; the wide-spread popularity among teachers and pupils of the numbers a'.oady published is a sufficient guarantee that future numbers wil meet with favor. GRADING. Numbers 47, 48, 49, and 50 are suitable for pupils of the Second and Third Reader grades. The following: numbers, given in the order of their simplicity, have been found well adapted to tKe tastes and capabilities of pupils of the Fourth Reader grades 29, 10, 7, 8, 9, 17, 18, 22, 23, 46, 11, 21, 44, 28, 36, 24, 19, 20 f 32, 37, 31, F, G, and H. The other numbers of the series are suita- ble for pupils of the Fifth and Sixth Reader grades and for the study of literature. 1 There are in the entire series perhaps half a dozen cases where a sentence has been very flight]/ changed in order to adapt it for use in the schoolroom ; and in to tvN, lor similar reasons- three page0f the original have been omitted. * C^e IBtoersilie literature Aeries. With Introductions, Notes, Historical Sketches, and Biographical Sketches. Each single number in paper covers, 15 cents. 1. Longfellow's Evangeline. 2. Longfellow's Courtship of Miles Standish. 3. Longfellow's Courtship of Miles Standish. DRAMATIZED for private theatricals in schools and families. 4. Whittier's Snow-Bound, Among the Hills, and Songs of Labor. 5. Whittier's Mabel Martin, Cobbler Keezar, Maud Muller r and Other Poems. . Holmes's Grandmother's Story of Bunker Hill Battle, and Other Poems. 7, 8, 9. Hawthorne's True Stories from New England His- tory. 1620-1803. In three parts.f 10. Hawthorne's Biographical Stories. Sir Isaac Newton, Samuel Johnson, Oliver Cromwell, Benjamin Franklin, Queen Christina. With Questions. [29 and 10 also in one volume, board covers, 40 cents.] 11. Longfellow's Children's Hour, and other Selections. 12. Studies in Longfellow. Containing Thirty-Two Topics for Study, with Questions and References relating to each Topic. 13, 14. Longfellow's Song of Hiawatha. In two parts.f 15. Lowell's Under the Old Elm, and Other Poems. 16. Bayard Taylor's Lars; a Pastoral of Norway. 17, 18. Hawthorne's Wonder-Book. In two parts.J 19, 20. Benjamin Franklin's Autobiography. With a chaptei completing the Life. In two parts.! 21. Benjamin Franklin's Poor Richard's Almanac, and othec Papers. 22, 23. Hawthorne's Tanglewood Tales. In two parts.f 24. Washington's Rules of Conduct, Letters and Addresses. 25, 26. Longfellow's Golden Legend. In two parts.:}: 27. Thoreau's Succession of Forest Trees, and Wild Apples With a Biographical Sketch by R. W. EMERSON. 28. John Burroughs^ Birds and Bees. [28 and 36 also in one volume, board covers, 40 cents.] 29. Hawthorne's Little Daffydowndilly, and other Stories. [29 and 10 also in one volume, board covers, 40 cents.] 30. Loweirs Vision of Sir Launf al and Other Pieces. 31. Holmes's My Hunt after the Captain and Other Papers. 32. Abraham Lincoln's Gettysburg Speech, and Other Papers, 33. 34, 35. Longfellow's Tales of a Wayside Inn. In three parts. [The three parts also in one volume, board covers, 50 cents.] 36. John Burroughs's Sharp Eyes and otter Papers. [28 and 36 also in one volume, board covers, 40 cents.] 37. Charles Dudley Warner's A-Hunting of the Deer, and Other Papers. t Also in one volume, board covers, 45 cents, i Also in one volume, board covers, 40 cents. Continued on the inside of this cover. HOUGHTON, MIFFLIN AND COMPANY, 4 PARK STREET, BOSTON, MASS. Literature [.4. list of the first thirty-seven numbers is given on the back cover."] 38. Longfellow's Building of the Ship, Masque of Pandora, and Other Poems. 39. Lowell's Books and Libraries, and Other Papers. 40. Hawthorne's Tales of the White Hills, and Sketches. 41. Whittier's Tent on the Beach. 12. Emerson's Fortune of the Republic, and Other American Essays. 43. Ulysses among the Phaeacians. From W. C. BRYANT'S Trans- lation of Homer's Odyssey. 44. Maria Edgeworth's Waste Not, "Want Not, and Barring Out. 45. Macaulay's Lays of Ancient Rome. 46. Old Testament Stories in Scripture Language. From the Dispersion at Babel to the Conquest of Canaan. 47, 48. Fables and Folk Stories. Riverside Second Reader. Phrased by HORACE E. SCUDDER. In two parts.* 49, 50. Hans Andersen's Stories. JSewly Translated. Riverside Second Reader. In two parts. $ 51, 52. "Washington Irving : Essays from the Sketch Book. [51.] Rip Van Winkle and other American Essays. [52.] The Voyage and other English Essays. In two parts. J 53. Scott's Lady of the Lake. Edited by W. J. ROLFE. With copious notes and numerous illustrations. (Double number, 30 cents.) 54. Bryant's Sella, Thanatopsis, and Other Foems. $ Also in one volume, board covers, 40 cents. EXTRA NUMBERS. A American Authors and their Birthdays. Programmes ana Suggestions for the Celebration of the Birthdays of Authors. With a Record of Four Years' Work in the Study of American Authors. By ALFRED S. ROB, Principal of the High School, Worcester, Mass. S Portraits and Biographical Sketches of Twenty American C A Longfellow Night. A Short Sketch of the Poet's Life, witl? songs and recitations from his works. For the Use of Catholic Schools and Catholic Literary Societies. By KATHERINE A. O'KEEFFE. J> Literature in School; The Place of Literature in Common School Education ; Nursery Classics in School ; American Classics in School. By HORACE E. SCUDDER. E Harriet Beecher Stowe. Dialogues and Scenes from Mrs. Stowe's Writings. Arranged by EMILY WEAVER. F Longfellow Leaflets. (Each, a Double Number, 30 cents.) G Whittier Leaflets. Poems and Prose Passages from H Holmes Leaflets. the Works of Longfellow, Whittier, and Holmes. For Reading and Recitation. Compiled by JOSEPHINE E. HODGDON. Illustrated, with Introductions and Biographical Sketches. t The Riverside Manual for Teachers, containing Suggestions and Illustrative Lessons leading up to Primary Reading. By I. F. HALL, Super- intendent of Schools at Arlington, Mass. HOUGHTON, MIFFLIN AND COMPANY, 4 PARK STREET, BOSTON, MASS. 4je JBitocm&e literature In my opinion nothing cultivates the reading habit like putting fresh and interesting books into the hands of the children. I think, however, that I have made the mistake of selecting too many information books. ... I intend to try the experiment of introducing more of the classics of our noble literature. E. A. GASTMAN, Supt. of Schools, Decatur, III. If reading is carefully taught, we do not need the Fifth Reader in the ungraded schools. GEO. R. SHAWHAN, Supt. of Schools, Champaign Co., III. If a Fifth Reader is dispensed with, aa some have advised, some, thing as good or better mast take its place, such as supplementary reading of the proper grade, consisting of good selections taken from the best American and English authors. State Course of Study for the Common Schools of Illinois. The Superintendent of Public Instruction for the State of Illinois re- commends for reading in the Higher Course of Study for the Common Schools of Illinois the following masterpieces of American Authors, published in the Riverside Literature Series : No. 2. The Courtship of Miles Standish . . . Longfellow. No. 28. Birds and Bees Burroughs. No. 4. Snow-Bound Whktier. No. 31. My Hunt after the Captain .... Holmes. No. 40. The Great Stone Face Hawthorne. No. 32. Essay on Abraham Lincoln .... LowelL The following numbers of the Riverside Literature Series have been adopted for use in the graded and ungraded schools of Knox County, Illinois, as a part of the course in Literature for 1890-91 ; Nos. 2, 4, 14, 17, and 81. Riverside Literature Series, by mail, paper covers, post-paid. Single numbers, each . 15c. 10 or more at one time, each 14c. 100 or more books selected from the Series at one time, each . . 13c. Sample sets at the " 100 or more " rate, post-paid. These prices are net HOUGHTON, MIFFLIN AND COMPANY, i PARK STREET, BOSTON; 11 EAS$ I?TH STREET, NEW YORK; 28 LAKESIDE BUILDING, CHICAGO. YB THE UNIVERSITY OF CALIFORNIA LIBRARY