I RATIONAL ARITHMETIC COMPLETE BY GEORGE P. LORD n • ■>_ > ;« J > O 5 . , » ° I ' ' THE GREGG PUBLISHING COMPANY NEW YORK CHICAGO BOSTON SAN FRANCISCO LIVERPOOL 1 .copyright, 192 0, by the '•/:.•': *:geegg publishing company ' •'. 1 • • •» • • A5a PREFACE Rational Arithmetic is intended for use in business colleges^ and in commercial high schools, by pupils who have com- pleted the equivalent of the eighth or ninth grade in the public school system. While deficiencies of early training may be remedied by its use, it is not intended as a textbook for those who are approaching the subject for the first time. Neither is it intended to take the place of any of the many excellent works now in use in the grades for the purpose of develop- ing a general understanding of mathematical principles. Such books, while they have satisfactorily discharged this function, have failed to develop the accuracy and facility so vitally essential in commercial calculations. Other commercial arithmetics have tried to overcome this weakness by following similar plans of instruction in abridged form. Rational Arithmetic follows a very different plan. It is purely a vocational work and aims to teach the "how'* rather than the "why." It is a reference book of com- mercial operations, rather than a method of presentation, and should be so used. Part One is a collection of practice exercises arranged along the lines of the generally accepted order of presentation. Part Two contains illustrated solutions covering the entire range of commercial arithmetic as generally understood. The methods used are those of business. The explanations are expressed in language which may be understood easily, rather than in the more scholarly language usually employed. iii 460956 iv PREFACE References throughout the book are by paragraph num- bers, which will allow the pupil to ascertain for himself the best method of solving any desired problem. The aim has been to produce a book so elastic that the teacher may arrange a course of study to suit himself. The author has found it advisable, however, to start pupils with the subject of balancing accounts which arouses their in- terest and gives them something new and practical. Drill on decimals should immediately follow this, for the purpose of developing accuracy in locating the decimal point. The advisability of work on the subject of fractions de- pends entirely upon the attainments of the individual pupil. The writer has found that fully 75 per cent of his pupils are greatly benefited by taking up this subject before beginning the strictly commercial work which commences with the subject of aliquot parts. It is suggested that the problems in addition at the be- ginning of Part One be used as drill problems throughout the course. Teachers will not find it necessary to use all the problems provided for each subject. The aim has been to give enough problems to meet any demand that may arise. No claim is made for originality in any of the methods presented. Every method that appears in this book may be found, in some form, elsewhere. To give credit to the sources from which the author has obtained assistance in the compilation of this book would be to name all the text- books consulted by him in an experience of nearly thirty years as arithmetic teacher. George P. Lord Salem, Mass. CONTENTS PART ONE PAGE Preliminary Problems 1 Addition 1 Subtraction ^ Decimals 1^ Multiplication 1^ Division ....•••••• 1^ Fractions ^^ Addition of Fractions 16 Subtraction of Fractions 17 Multiplication of Fractions 17 Division of Fractions 18 Practice Problems — Fractions and Decimals . . .18 Denominate Numbers 2^ Aliquot Parts ^"^ Exercises in Billing 30 Percentage ....•••••• ^^ General Problems in Percentage 38 Profit and Loss ....••••• "^^ General Problems in Profit and Loss . . . .47 Trade Discount ^^ General Problems in Trade Discount . . . .55 Commission ^^ General Problems in Commission 66 Time 69 V vi CONTENTS PAGE Interest 73 Ordinary Interest ... .o ... 73 Accurate Interest .77 To Find Time 77 To Find Rate 78 To Find Principal 79 General Problems in Interest . . . . . .81 Partial Payments . 83 Bank Discount 86 Compound Interest ....... 89 Periodic Interest ........ 90 Averaging Accounts 92 Taxes 99 Customs and Duties . . . . . . .100 Insurance 102 Life Insurance . . . . . , . .103 Exchange 105 Domestic Exchange . . . . . . .105 To Find the Value of a Sight Draft . . . .105 To Find the Value of a Time Draft . . . .105 To Find the Face of a Draft 106 Foreign Exchange . . . . . . , .107 Stocks and Bonds 109 PART TWO Definitions « . o o 1 Notation 2 Common Processes 5 Addition — Integers ....... 5 Subtraction — Integers ....... 7 Multiplication — Integers ...... 9 Division — Integers . , . . . . .11 CONTENTS Vll PAGE Decimals 13 Addition — Decimals . . . . . . .13 Subtraction — Decimals . . . . . . .13 Multiplication — Decimals . . . . . . 14 Division — Decimals . . . . , . . 1 .5 Factoring . . . . . . . . .17 Least Common Multiple . . . . . . .18 Greatest Common Divisor . . . . . .19 Fractions 22 Changing to Lower Terms ...... 24 Changing to Higher Terms . . . . . .26 Changing an Improper Fraction to a Mixed Number . 27 Changing a Mixed Number to an Improper Fraction . 28 Changing a Decimal Fraction to a Common Fraction . 29 Changing a Common Fraction to a Decimal Fraction . 30 Addition of Fractions . . . . . . .31 Subtraction of Fractions ....... 33 Multiplication of Fractions ...... 34 Division of Fractions ....... 37 Denominate Numbers 40 Reduction of Denominate Numbers .... 40 Addition of Compound Numbers ..... 43 Subtraction of Compound Numbers .... 43 Multiplication of Compound Numbers .... 44 Division of Compound Numbers ..... 44 Computing Time . . . . . . . . .45 Aliquot Parts ......... 46 Percentage 49 Profits and Losses 57 Discount .......... 63 Trade Discount ........ 63 Commission and Brokerage 70 viii CONTENTS PAGE Interest .... ..o ... 76 Accurate Interest . . ... . . .77 Ordinary Interest ........ 79 . Explanation ......... 79 Sixty-Day Method — Ordinary Interest Rule ... 80 Sixty-Day Method — Accurate Interest .... 83 Commercial Papers 94 Partial Payments 96 The United States Rule for Partial Payments . . 96 Merchants' Rule 98 Bank Discount 100 Compound Interest 104 Periodic or Annual Interest . . . . . .105 Averaging Accounts . . . . . . . .108 General Principles of Average . . . . . .108 Taxes 116 Duties and Customs . . . . . . .117 Insurance .......... 120 Life Insurance . . . . . . . .122 Exchange .......... 124 Domestic Exchange . . . . . . .124 Foreign Exchange .125 Stocks and Bonds 127 Tables 130 Index 147 , , 5 J 3 3 1 3' 3 3 V> 3 1 3 -, -, , , ' •■ '^ 3' » ' *•> '3 ' ' RATIONAL ARITHMETIC PART ONE The following problems are intended to afford sufficient practice to develop a thorough working knowledge of practical business arithmetic. An effort has been made to confine the problems as far as possible to actual business conditions and to present only problems similar to those met in actual business experience. References are by paragraphs to Part Two and are sufficiently copious to allow ready solution by the pupil of all problems. PRELIMINARY PROBLEMS ADDITION The following exercises in addition afford opportunity for frequent drills. The pupil should practice upon them and similar problems provided by the teacher throughout the course, or until he is able to add in the time specified, or in less time. Study carefully paragraphs 96 to 101 inclusive. 1 ^'•/RATIONAL ARITHMETIC Practice' -tintil you ; are able to add each of the following in 15 seconds or • less : 1 2 3 4 5 6 324 196 596 287 812 285 436 289 321 422 263 635 243 781 284 389 426 149 429 423 675 674 529 728 182 317 329 263 198 463 327 262 918 721 984 824 148 425 786 416 623 296 283 348 465 129 467 179 2. Practice until you are able to add each of the following in 25 seconds or less : 7 8 9 10 11 12 324 463 247 472 289 521 642 721 962 749 394 347 ^85 567 721 638 672 625 763 289 463 236 416 262 297 143 265 429 781 729 425 264 789 642 186 453 642 721 496 187 237 642 193 168 721 346 421 287 721 459 453 721 563 472 438 672 624 254 464 296 267 284 289 689 789 563 193 596 198 746 462 189 3. Practice until you are able to add each of the following :n 45 seconds or less : RATIONAL ARITHMETIC 3 13 14 15 16 17 2864 3829 4962 8426 5479 4233 6471 2794 7195 6294 7185 1687 6171 3824 1781 1679 2762 2437 6271 3326 2763 4463 8263 2617 7182 4638 9174 2819 5409 4963 9162 2896 6279 6276 7126 2746 4789 2854 2830 4413 8427 6217 1962 3418 7824 1679 2834 7148 9016 1671 2634 9753 6523 2468 5607 4791 2891 3764 1695 2896 4. Practice until vou are able following in 60 seconds or less : to add each of the 18 264118 428307 711695 386472 369143 642785 617192 548237 167589 294462 162781 146229 382716 19 528563 742896 478132 264389 1462^27 584296 817529 428127 362419 780962 278438 261971 446236 20 428137 298461 541672 832744 167182 322907 541891 851693 395816 724594 280790 642031 451682 21 252763 376329 167251 146327 421791 573619 287513 324671 271293 348162 912872 268047 634918 22 427183 284562 711456 378275 462871 146265 551681 287354 851762 718319 440892 632757 642819 4 RATIONAL ARITHMETIC 5. Practice until you are able to add each of the following in 75 seconds or less : 23 24 25 26 27 416342 487902 284062 614385 812716 913457 226439 713345 422716 341675 296731 914362 167182 312814 472386 284562 167943 421671 567583 611743 811706 208209 284371 642217 296342 273468 613317 176327 551638 542138 296329 672438 420416 281954 617516 284672 719243 798296 371621 271642 542983 264738 146329 560932 182133 718296 194513 817043 174837 162904 287981 382761 241671 165329 816238 273468 280642 146329 241671 271609 28 29 30 31 32 24627 93281 27682 14632 91387 48231 44638 38225 71136 26785 62783 71289 16781 28483 54321 24167 45642 91483 43729 28654 85262 71483 82675 52847 43832 71843 19721 46721 37625 71819 29636 54163 53482 54783 44623 71083 27386 27624 29654 48729 27642 16721 14729 18729 17453 29827 28294 61453 47387 37529 16429 26783 27185 26475 62745 54540 54296 16291 83267 71258 68296 78287 54385 29453 54183 RATIONAL ARITHMETIC 33 34 35 36 $6743.76 $3462.78 $4527.82 $7345.60 2846.75 5287.95 3675.18 2847.29 8421.62 6379.86 2643.89 5640.36 7329.44 7429.80 1796.97 7281.28 3780.50 5463.29 4238.54 1267.43 2894.62 7128.42 2879.36 3629.75 7481.13 3864.19 7481.29 4678.37 2563.27 7133.64 3279.81 8126.42 6247.16 4461.72 4782.63 3726.42 3729.42 9562.70 2871.54 4671.38 8427.16 4683.49 7627.18 6275.29 2945.71 7216.30 2375.62 5463.72 4453.75 3200.50 4671.16 2963.47 37 38 39 40 $3678.44 $5617,81 $2896.75 $8297.50 2917.53 2976.37 4429.36 6384.96 6279.45 3728.44 1785.29 7185.16 1468.71 1671.38 6271.38 4183.94 2783.94 5418.75 1862.45 5467.28 7162.85 2763.48 5483.92 4392.40 1671.36 6378.27 2861.57 5671.39 2768.29 5423.75 4675.38 2716.42 5419.75 1627.30 9863.75 3895.16 1683.27 4862.19 5728.50 5482.95 4462.91 7143.84 6275.81 7436.81 2789.65 3627.52 3829.62 2918.15 3601.82 5462.79 7426.85 4183.95 4618.79 2183.96 5387.09 6343.19 4844.60 4528.17 6425.70 2874.67 6 RATIONAL ARITHMETIC 6. Practice until you are able to add each of the following in 90 seconds or less : 41 42 43 44 45 283 487 275 438 219 347 365 381 622 736 462 791 456 563 432 728 384 179 729 275 529 429 527 247 863 633 186 623 384 179 387 795 819 862 327 472 324 942 721 453 694 432 163 903 618 458 175 209 415 721 182 293 725 541 153 791 764 483 287 286 453 379 342 453 723 645 453 671 628 429 286 186 538 143 186 729 791 827 517 791 452 428 113 386 917 628 384 452 295 283 139 292 938 672 418 255 574 286 286 726 386 387 721 295 814 675 198 453 721 428 341 421 618 193 297 296 502 721 382 548 416 218 453 453 353 209 721 618 182 287 381 347 721 763 186 RATIONAL ARITHMETIC 7 7. Pupils who have acquired proper facihty in the preceding problems will have no difficulty in per- forming the following with sufficient rapidity : $1283.64 $ 284.16 $ 94.43 $3728.54 785.30 1728.32 2168.75 6241.57 2721.83 4278.19 1413.80 287.63 1763.20 2674.19 287.60 729.42 487.32 905.16 782.19 98.17 9.05 728.40 3187.42 46.54 87.62 7.14 4163.27 19.72 44.33 287.64 973.29 83.71 1286.75 4239.17 4871.30 129.40 343.06 2476.28 2972.43 987.60 428.06 287.19 187.62 2763.42 71.37 458.16 98.43 1486.71 289.70 6724.13 642.19 4938.27 2729.40 8.60 3894.16 9132.38 2876.42 278.40 721.32 764.20 29.00 4291.62 193.64 381.65 427.40 71.85 4287.16 28.53 9.90 13 63 1328.72 9.19 727.62 2842.75 46.84 2.85 3478.24 4871.20 13.16 642.38 9288.75 389.45 9.68 94.73 2471.05 642.16 171.24 2652.81 287.63 71.00 283.95 453.17 274.28 9 80 90.60 287.29 297.62 2621.13 468.13 4182.19 48.39 38.79 71.24 78.50 184.19 178. '3 457.62 1246.53 8 RATIONAL ARITHMETIC 8. Complete the following statement by finding the totals of columns and the totals from left to right : Mar. 1 2 3 4 5 7 8 9 10 11 13 14 15 16 17 18 20 21 22 23 24 25 27 28 29 30 31 Totals Corn 284.65 78.39 164.70 348.62 175.29 98.40 467.28 298.70 587.64 298.63 728.45 687.44 285.90 429.18 697.65 597.67 738.42 914.16 576.80 894.48 1098.27 384.62 725.30 456.82 689.63 521.16 782.73 Wheat 487.62 629.34 587.19 984.63 729.40 1180.68 960.27 450.18 829.30 486.90 752.83 647.92 473.85 738.24 287.16 587.90 699.48 826.16 487.28 745.60 952.75 397.26 678.40 1238.50 927.24 627.42 795.38 Oats 125.30 326.45 217.16 98.43 454.20 90.12 72.15 248.00 316.90 109.09 402.14 48.30 178.85 148.16 242.90 371.42 416.78 212.13 98.42 248.16 500.00 187.19 238.16 71.45 122.58 108.12 314.60 Apples 285.64 171.80 219.17 287.60 58.09 171.23 264.12 175.11 147.16 209.04 348.17 250.60 658.12 347.60 795.80 721.80 424.70 368.40 519.67 295.48 487.29 368.12 563.27 448.12 219.60 348.75 218.90 Beans 697.60 721.80 450.67 385.90 287.40 458.62 721.30 318.90 427.65 386.37 516.80 428.54 719.80 579.60 387.40 473.20 517.68 429.90 343.80 624.39 387.26 419.72 611.41 516.75 218.19 365.40 409.08 Flour 48.60 75.60 308.00 9.50 24.72 317.80 240.16 421.75 48.90 97.50 105.12 215.70 84.95 198.60 385.00 95.16 171.20 384.60 142.80 218.70 140.68 214.09 190.54 138.20 97.42 164.83 120.95 Totals SUBTRACTION While no practice seems necessary on simple sub- traction of integers, the pupil should read carefully 102 to 109 inclusive, and then balance the following RATIONAL ARITHMETIC 9 accounts according to the method explained in 108 and 109. 9. Balance the following accounts : 1. Dr. QUAKER OATS CO. Cr. <2A--e-^<- $76.50 16.75 39.75 2. Dr. D. T. AMES & CO. Cr. $185.25 $603.75 8. 215.50 2.10 73.94 121.50 - 24. 3. Dr. INTEREST Cr. $1. $ .68 7.70 9.21 2.52 10. 3.70 1.40 1.12 10 RATIONAL ARITHMETIC 4. Dr. F. B. SMITH Cr, $ 6.75 $33.45 3.75 6. 14.50 10.25 12.25 5.40 13.25 5. Dr. SALES Cr. $12.20 $ 6.60 4.65 1143.75 5.10 843.19 4.85 6. 8. 5.50 6. Dr. PARK & STEWART Cr. $1021.65 $589.05 964.41 56.40 558.72 236.25 280. 661.50 RATIONAL ARITHMETIC 11 7. Dr. JAMES CARTER Cr. $ 1.91 $278.60 716. 956.20 733.13 8.20 588.03 9.16 436. 833.70 47.31 8. Dr. BLANCHARD & CO, Cr. $ 16.50 $ 3.60 7.64 10.80 18.50 7.75 219.21 19.60 12.50 4.23 9. Dr. F. G. MILLER Ci\. $63.60 $ 4.17 25. .60 14.50 140. 25. 207.50 21.15 55. 16.67 9. 61.50 12 10. Dr. RATIONAL ARITHMETIC CHARLES SMITH CV. $ 76. $ 51.01 98.55 60.03 83.02 171. 59.20 80.61 105.08 2.10 43.20 210.33 13.10 29.76 Dr. H. A. THOMAS Cr. $1240.50 $ 500. 876. 750. 453.35 1250. 96.73 2575. 1000. 354.56 Dr. SAWLER & HASKINS Cr. $1246.34 878.14 2543.65 3746.32 $2354.56 3143.42 1245.15 1873.94 DECIMALS MULTIPLICATION The object of these exercises is to acquire accuracy in locating the decimal line. References 133, 134, 135. 10. 1. .974X.35 = 2. 8.2X9.6 = 3. 284X.75 = 4. 346.5X10.02 = 5. 27X.38 = 6. .1655X18 = 7. .4355X16.66 = 8. 844.5X7.404 = 9. 1263X8.04 = 10. .44X8.05 = 11. 65.410X.585 = 12. 75000X.0098 = 13. 5.9X26.7362 = 14. 75X6.0053 = 15. 3.926X464 = 16. 4.872X.386 = 17. 84672X8.4 = 18. 2.8973X. 80806 19. .94X100.82 = 20. 446.8X3.044 = 21. 287X9.0104 = 22. .9634X58 = 23. 283.86X.396 = 24. 94.652X4.87 = 25. 84X. 000238 = DIVISION Study 136 to 141 inclusive. In solving the following problems make strict applica- tion of the rules given in 137 and 138. 13 14 RATIONAL ARITHMETIC Paragraphs 139, 140, method of sokition. 11. 1. 14.875^3.5 = 2. 338.52^8.4 = 3. 1385.128-^21.6 = 4. 3.456^12 = 5. 654.5 H- 11 = 6. 2.464-7-1100 = 7. 43.4172^74.6 = 8. .01581 -^ .255 = 9. 6.305 -=-3.25 = 10. 8.63-^3.84 = 11. 79.896 H- .53264 = 12. 372.012-^58 = 13. 14.157-^2.6 = 14. 14.157-^.26 = 15. 1284.7^.3875 = 16. 246.9^23 = 17. 1640.625^1875: 18. 286.996 -^ .914 = 19. 17.408-^12.8 = 20. 6264-^.348 = 21. 14.825^-8.29 = 22. 286.327 -^ 156 = 23. 16.38^.284 = 24. 284.62^84 = 25. 29.728-^8.4 = and 141 fully illustrate the 26. 1.75 -^ 23.5765 = 27. 437.8675^23.8 = = 28. 23.183^19.7463 = 29. 246.-^7.875 = 30. 75-^125 = 31. 1.8675^5.75 = 32. 124.56^15.8 = 33. 48567.75^4.875 = 34. 76.50^1250 = 35. .5863^12.52 = 36. .9875 -^. 584 = 37. 23.45675 -^ 1.375 = 38. 23.-^27 = 39. 125.^56 = 40. 324. -T- 678 = 41. 12.875-^4.25 = 42. 125.6^2^7.75 = 43. 415.875^.1275 = 44. 234. 15 -r- 5.875 = 45. 56.625^-128.50 = 46. 153.8756^53.962 = 47. 346.4278^8.4695 = 48. 16.4584-^3.4565 = 49. 125.4632-7-18.4965 = 50. 4356.4589-^27.4875 = FRACTIONS Study carefully IGO to 180 inclusive. References 181, 18^2, 183. 12. Reduce the following to lowest terms : 1. 7 20 12 00 6. 2. 6 30 2 8 35 7. 3. 9 1 104 8. 4. 105 2 3 1 9. 5. 2 1 A 5 5 10. 35 45 16 08 _6JL5_ 12 00 LJ.5. '7 8 2 2 3 2 11. 12. 13. 5 5 1 u 9 12 Q2 8 71 14. ^ 25 15. ^ 18 References 184, 185, 1 80. 13. Change the following fractions to the denomina- tions designated : 1. ttol25ths. 6. tto88ths. 2. I to 56ths. 7. tV to 1524ths. 3. i to 72ds. 8. f to 126ths. 4. A to 195ths. 9. A to 352ds. 5. f to 84ths. 10. i to 27ths. References 187, 188, 189. 14. Change the following to mixed numbers : 1 2J7 3. 1-6 9 2. 1 23 5 A. 124 5. 6. 15 5 1 5 16 3 2 3 1 5 7. 8. 4 27 9 ] 2 5 16 RATIONAL ARITHMETIC References 190, 191, 192. 15. Change the following to improper fractions : 1. If 4. 9t 7. 81i 10. 214f 2. 4i 5. 23f 8. 371 11. 34 A 3. 8| 6. 46| 9. 123* 12. 43| References 193, 194, 195. 16. Change the following to common fractions or mixed numbers : 1. .375 ' 5. .15625 9. .66f 2. .875 6. 8.25 10. .272t\ 3. .0625 7. 17.875 11. 1.77J 4. .125 8. 9.28125 12. .384t'^ References 196, 197, 198, 199, 200. 17. Change the following to decimal equivalents : 1. 3 4 5. 4 9 9. 1 1 17 13. 8i 2. 4 5- 6. 5 6^ 10. 9 2^3" 14. 9i3 3. 5 8 7. 7 8 11. 3f 16. 12| 4. 1 1 1 3 8. 9 1 3 12. 4f ADDITION OF FRACTIONS References 201, 202, 203, 204, 205. 18. 1. 3 1 5 _ 4 1 8 "" 2. 7 14 1 _ 8 1 5 2 — 3. 1 3 5 _ 13 2 4 1 8 ~~ 4. 12S+8f+27| = RATIONAL ARITHMETIC 17 6. 21t+46M+29t = 7. 628A+56f+16i+30A = 8. 34i+26f +91x^0 + 631 = 9. 4f+27oV+33i+12iJ = 10. 7|+9i+15A+5i = SUBTRACTION OF FRACTIONS References 206, 207. 19. 1. 1-1= 5. 146^-791 = 2. 12f-7|= 6. 1246^^-9831 = 3. 214H-185i= 7. 25f-14f = 20. 4. ^5i-5i= 8. 3461 -217 A MULTIPLICATION OF FRACTIONS References 208 to 218 inclusive, and 135. 5 1. 4 v 5 — 11. 319tXl8f = 2. 12iX8 = 12. 2.15iX24 = 3. 124X161 = 13. 24X.12i = 4. 218x29f = 14. 246X.16f = 5. 289fXl26 = 15. 814X.44i = 6. 1^2"X03" = 16. 31.62iX8f = 7. 124iX27f = 17. 312x24.08i = 8. 128fVX23i = 18. 459X3121 = 9. 5122^X137 = 19. 12461X34.52 = 10. 461X43 = 20. 4.16fX.12i = 18 RATIONAL ARITHMETIC DIVISION OF FRACTIONS References 219 to 225 inclusive, and 139, 140, 141. 21. 1. 4-5 5 • 8 ~ 11. 246f^l3f = 2. 1 2 • 8 _ 1 9 • 9 ~ 12. 12461 : 41 = 3. 7 • 5 _ 8*6 13. 4181-151 = 4. 125^1 = 14. 2161-^19 = 5. 24 : 1 = 15. 1248^ : 27^ = 6. 346 --1 = 16. 456fV-M25 = 7. 1246^151 = 17. 5. 141 --23 = 8. 482 --171 = 18. 4246 : .171 = 9. 8461 -^ 26 = 19. 128.571 : .12^ = 10. 5321 ^ 18 1^ = 20. 43.55f^.l6f = PRACTICE PROBLEMS INVOLVING THE USE OF FRACTIONS AND DECIMALS The following problems are intended to show the application of the general principles of common frac- tions. Their proper solution involves a knowledge of paragraphs 160 to 225 inclusive. 22. 1. Four pieces of cloth measure respectively 311 yd., 43i^ yd., 5Q^ yd., and 44i yd. What is the total length.^ 2. What is the sum of 23.8t%, 32.35f, 56|, 194, and i? Carry to the fourth decimal place. 3. I am about to ship a box containing 20f lb. coffee, 3^6 lb. tea, 23^ lb. ham and 16f lb. bacon to my camp. If the box weighs 2f lb., what is the total weight of the shipment? RATIONAL ARITHMETIC 19 4. A grocer bought six bags of coffee, weighing respectively 132f lb., 128^ lb., 127f lb., 136| lb., 134f lb., and 128| lb. Allowing H lb. for the weight of each bag, what would be the total net weight of the coffee ? 5. I bought 5 barrels of sugar. The net weight of each respectively was 275t lb., 283i lb., 2711 lb., 293f lb., and 2851 lb. Find the total net weight. 6. I bought a f interest in a bowling alley, and sold my brother a -re interest. How^ much do I own ? 7. An automobilist on a tour completes -^e of the trip on the first day, i on the second day, and i on the third day. He then finds himself 350 miles from his destination. What is the total length of the trip and how far has he alreadv advanced ? 8. I bought a house for $7500. I paid i of the pur- chase price in cash, f of the remainder was paid in six months, and I am now ready to make the final pay- ment. For what sum must mv check be WTitten .^ 9. I can do a piece of work in 5 da vs. Mv brother requires 7 days to do the same thing. If we work together how long will it take to complete the job ? 10. If coffee loses yq of its weight in roasting, how many pounds of green coffee must be roasted to pro- duce 375 lb. ? 11. A farmer bought a cow for $52f and a ton of hay for $29f . How much change would he receive out of Si one-hundred-dollar bill ? 12. A bookkeeper's pay envelope contains three $10's, one $5, one $2, and one $1. He paid for board 20 RATIONAL ARITHMETIC $1H, a bill amounting to $8f , and bought a hat for $3, a pair of gloves for $lf , and two pairs of stockings at $.50 a pair. What part of his week's pay did he have left ? 13. What will \l-r2 dozen eggs cost at $.58f per dozen ? 14. If 71 tons of hay cost $182^, what will llf tons cost ? 15. I bought 4375f bushels of corn at $.80f a bushel, and 2350^ bushels of oats at $.61f a bushel. What was the entire investment.'^ 16. Find the total cost of the following: 350 lb. Rio coffee at $.47| ; 450 lb. Mocha coffee at $.41f ; 900 lb. white sugar at $.10f ; 900 lb. brown sugar at $.09f; 970 lb. granulated sugar at $.08f; 172 lb. butter at $.56|. 17. A merchant sold 80 lb. of butter at $.57f ; 43 dozen of eggs at $.61f per dozen ; 32^ gallons of milk at $.60 a gallon. What was the total amount of sales ? 18. A piece of cloth containing 47f yd. was sold for $9. 94 J. What was the price per yard ? 19. One- third of a firm's capital is invested in merchandise, three-eighths in real estate, and the rest, $18,200, is cash. What is the capital of the firm.^ How much is invested in merchandise, and how much in real estate ? 20. A farm yields 96.08 bushels of potatoes to the acre, 36.625 bushels of oats per acre, 15.52 bushels of wheat per acre. 156 acres were planted in potatoes, 214 acres in oats, and 19.3 acres in wheat. What is the total number of bushels harvested ? Rx\TIONAL ARITHMETIC 21 21. I have withdrawn J of my money from the bank and have $376.40 remaining. How much did I with- draw? 22. A partnership consists of three members who in- vest respectively i, i, i of the capital, and agree to share losses and gains in the same proportion. How much will be each partner's share, if there is a profit of $13,416.75 ? 23. A business man finds himself unable to meet his entire obligations. He owes $12,360 and has $10,300 available with which to pay. What part of his lia- bilities can he meet? How many cents on the dollar is this ? 24. A invested i of the capital of a firm, B i, C i, and D the remainder. D's share is $1690. What was A's, B's, and C's investment? 25. A house and lot cost $6600. The house costs i more than the land. What was the cost of each? 26. How manv bushels is .75 of 640 bushels? 27. A merchant sold 162 barrels of flour which is f of his stock of flour. How much flour had he at first ? 28. A merchant sold 480 barrels of flour which is .625 of his entire stock. Hov/ many barrels had he at first ? 29. A man, at his death, left $30,000 to his wife, son, and daughter ; .5 of this sum went to his wife, .375 to his daughter, and .125 to his son. How much did each receive ? 30. I have just learned that one of my customers has failed and is able to pay only $.525 on the dollar. My claim against him amounted to $134.40. How much will I receive? DENOMINATE NUMBERS Reference ^29. 23. 1. Reduce £34, 8^, 7d to pence. 2. Reduce 4 T., 5 cwt., 85 lb. to pounds. 3. Reduce 14 gal., 3 qt., 1 pt. to pints. 4. Reduce 1 cwt., 24 lb., 3 oz. to ounces. 5. Reduce 1 da., 3 hr., 25 min. to seconds. 6. Reduce 14 yr., 5 mo., 3 wk. to days. 7. Reduce 1 m. 25 rd., 4 yd., 2^ ft. to inches. 8. Reduce 2 hhd., 14 gal., 3 qt. to pints. 9. Reduce 14 bu., 3 pk. to pints. 10. Reduce 3 A., 2 sq. rd., 10 sq. yd. to sq. ft. Reference 230. 24. 1. Reduce 3462 sq, in. to higher denominations. 2. Reduce 14>6Sd to higher denominations. 3. Reduce 17696 lb. to higher denominations. 4. Reduce 32625 gr. to higher denominations. 5. Reduce 12760 in. to higher denominations. 6. Reduce 18428 sq. in. to higher denominations. 7. Reduce 3896 cu. ft. to higher denominations. 8. Reduce 4843c? to higher denominations. 9. Reduce 120615 sec. to higher denominations. 10. Reduce 633 pt. to higher denominations. 22 RATIONAL ARITHMETIC 23 Reference 231. 25. 1. Reduce .327 m. to lower denominations. 2. Reduce .35 hr. to lower denominations. 3. Reduce .875 yd. to lower denominations. 4. Reduce .135 yr. to lower denominations. 5. Reduce f mo. to lower denominations. 6. Reduce .125 m. to lower denominations. 7. Reduce £2.3456 to lower denominations. 8. Reduce £12.456 to lower denominations. 9. Reduce f m. to lower denominations. 10. Reduce tt yr. to lower denominations. Reference 232. 26. 1. Reduce 4 yd., 2 ft. to a decimal of a rod. 2. Reduce 3 gal., 2 qt., 1 pt. to gallons. 3. Reduce 2 pk., 3 qt., 1 pt. to a decimal of a bushel. 4. Reduce 35 min., 18 sec. to a decimal of a day. 5. Reduce 18 rd., 4 yd., 2 ft. to rods. 6. Reduce 4 cwt., 85 lb. to a decimal of a ton. 7. Reduce Ss, lOd, 2/ to a decimal of a pound. 8. Reduce 18 sq. rd., 4 sq. yd. to a decimal of an acre. 9. Reduce 14 hr., 35 min., 10 sec. to a decimal of a day. 10. Reduce 185 lb., 12 oz. to a decimal of a ton. ALIQUOT PARTS Study paragraphs 240 and 241, memorizing the table and noting appHcation as explained in note a. Reference 241. 27. Find the cost of : 1. 720 1b. at 50^; at 33J^ ; at 25^. 2. 120 lb. at 33i^; at 25^; at 20^; at 12i^. 3. 360 lb. at 6U \ at 6f ^ ; at 10^ ; at 12i^. 4. 840 yd. at 10^; at 12^^; at 14f ^ ; at 25^. 5. 4800 lb. at 8^^ ; at 6i^; at 12^^; at IGf^; at 10^. 6. 240 yd. at 8i^ ; at 6f ^ ; at 10^ ; at 12^^. 7. 2480 yd. at 25^; at 50^; at 33i^ ; at 20^. 8. 480 yd. at 6i^ ; at 8^^ ; at 6f ^ ; at 10^ ; at 12i^. 9. 560 yd. at 8J^ ; at 6i^ ; at 6f ^ ; at 10^; at 12i^. 10. 204 yd. at 50^; at 33^^; at 25^; at 11. 4200 yd. at 10^; at 12^^; at 14f ^ ; at 16f^; at 25^. 24 RATIONAL ARITHMETIC 25 12. 1800 lb. at nW. at 16f^; at 20^; at 25^; at 33i^. 13. 1500 yd. at $1 ; at 12J^ ; at 14f ^ ; at 16f^; at 25^. 14. 490 doz. at 12i^ ; atlOf^; at 20^; at 6f ^. 15. 960 yd. at 8i^ ; at 6i^; at 10^; at UW, at 6f ^. Reference 241. 28. Find the total cost of : 1. 38 lb. at 25^ 2. 63 yd. at 28^^ 84 lb. at 37ijzi 81 yd. at 33i^ 72 lb. at 75^ 18f gr. at 52^ 48 lb. at 41|^ 28 doz. at 50^ 96 lb. at 33i^ 58 yd. at 14f ^ 24 lb. at 12i^ 235 yd. at 40^ 3. 61 lb. at 48^ 4. 25 bu. at 96<^ 480 lb. at 16f ^ 20 bu. at 88jZ^ 25 lb. at 44^ 31i bu. at $2 161 lb. at 48^ 50 bu. at $11.50 240 lb. at 18f ^ 12i bu. at $3.60 72 lb. at 37i^ 25 bu. at $1.64 5. 25 yd. at 76^^ 6. 37i bu. at 72^ 37^ yd. at 96^ 75 bu. at $3.20 750 yd. at 12^^ 62^ bu. at $1.36 168 doz. at 12i^ 14f bu. at $1.54 420 yd. at 33i^ 50 bu. at $5.85 176 yd. at 31i^ 12i bu. at $.64 26 RATIONAL ARITHMETIC Reference 241. 29. Find the cost of : 1. 6i A. land at $192. 2. 125 lb. tea at 48^. 3. 34 lb. tea at 50^. 4. 25 lb. coffee at 44^. 5. 25 T. coal at $10.80. 6. 72 pieces lace at $1.25. 7. 44 yd. velvet at $2.50. 8. 2i bu. potatoes at $1.48. 9. 12i bu. turnips at 74^. 10. 12i yd. silk at $1.04. 11. 84 tables at $12.50. 12. 36 sets chairs at $125, 13. 12i yd. linen at 56^. 14. 25 pieces lace at $6.60. 15. 62i T. coal at $9.50. 16. 375 T. coal at $11.50. 17. 264 A. land at $37.50. 18. 320 bu. potatoes at $2.12^. 19. 810 T. coal at $12.50. 20. 1250 bbl. pork at $24. 21. 1280 lb. rice at 12^^. 22. 366 yd. silk at $1.66f. 23. Hi yd. duck at 36^. 24. 474 gal. cider at 33i^. 25. 1680 qt. vinegar at 16f ^. 26. 648 lb. sugar at 10^. 27. 208 yd. tape at 2i^. 28. 176 lb. tea at 50^. 29. 1742 yd. silk at $1.50. 30. 560 gal. oil at 12i^. Reference 241. 30. Find the total cost of the following : RATIONAL ARITHMETIC 27 1. 180 \h. at SSU 2. 138 lb. at 33i^ 760 lb. at 25^ 7^28 lb. at 62^^ 54 lb. at 37i^ 224 lb. at 25^ 144 lb. at 33i^ 960 lb. at 66f ^ 72 lb. at 37i^ 72 lb. at 12^^ 150 lb. at 66f f!^ 904 lb. at 87^^ 3. 196 yd. at 16f ^ 4. 147 yd. at 55|^ 180 yd. at 66|^ 24 yd. at 66f ^ 288 yd. at 33i^ 28 yd. at 75^ 459 yd. at IH^ 84 yd. at 25^ 72 yd. at 25^ 56 yd. at 12^^ 48 yd. at 16S^ 48 yd. at 75^ 183 yd. at 33^^ 246 yd. at 25^ 5. 66 gal. at 33i^ 6. 441 gal. at 55U 64 gal. at 87i^ 3248 gal. at 6U 63 gal. at llU H^ gal- at 22f ^ 144 gal. at 83^^ 266 gal. at 28^^ 91 gal. at 71f ^ 384 gal. at 62^^ 96 gal. at 62^^ 368 gal. at 31i6 16 gal. at 87i^ 248 gal. at 87^^ 945 gal. at 55U ^^^ gal. at 83ij^ 7. 750 yd. at 33^^ 8. 648 yd. at QU 427 yd. at 42f ^ 684 yd. at 33i^ 87i yd. at 50^ 496 yd. at 75^ 52 yd. at 62^^ 186 yd. at 83i^ 450 yd. at 6f ^ 125 yd. at 18^ 2112 yd. at SU 144 yd. at 37i^ 240 yd. at SU 297 yd. at 444^ 174 yd. at 16f ^ 287 yd. at UU 249 yd. at 25^ 918 yd. at 33i^ 28 RATIONAL ARITHMETIC 9. 144 lb. at 83i^ 10. 144 lb. at 16f ^ 480 lb. at ^lU 216 lb. at m^ 282 lb. at 83i^ 872 lb. at 12i^ 312 lb. at 33i^ 348 lb. at 25^ 427 lb. at 71f ^ 72 lb. at SU 184 lb. at 12i^ 186 lb. at 87^^ 940 lb. at IH 138 lb. at 334^ 462 lb. at 66f ^ 96 lb. at QU 342 lb. at 16f ^ 384 lb. at 50^ 11. 84 yd. at 91f^ 12. 84 lb. at 58^^ 288*^ yd. at 12^^ 960 lb. at 16f ^ 345 yd. at 66|^ 728 lb. at 62i^ 192 yd. at 37^^ 36 lb. at 43f ^ 423 yd. at 33i^ 64 lb. at 41f ^ 280 yd. at 12i^ 96 lb. at 12^^ 324 yd. at 411^ 72 lb. at 41f ^ 284 yd. at 25^ 348 lb. at 75^ 396 yd. at 33^^ 246 lb. at 33i^ 64 yd. at 5Q\i 344 lb. at 37i^ 13. 87i yd. at $2.48 14. 176 yd. at $1.12i 192 yd. at 87^^ 75 yd. at 16^ 28 yd. at 75^ 27 yd. at 75^ Hi yd. at 18^ 5Q yd. at 83^^ 144 yd. at lli^ 17 yd. at 25^ 25 yd. at 44^ 12^ yd. at 39^ 75 yd. at 24^ 72 yd. at 41|^ 87i yd. at $2.88 344 yd. at 37^^ 270 yd, at \\U ^4 yd. at 8^^ 24 vd. at 75^ 87i yd. at 88^ RATIONAL ARITHMETIC 29 15. 11511 lb. at 20^ 16. 156 lb. at 66f^ 960 lb. at 16f ^ 284 lb. at 25^ 728 lb. at 62i^ 396 lb. at SSU 32 lb. at 43f ^ 64 lb. at 5^^ 64 lb. at ^U 384 lb. at SlU 96 lb. at 12i^ 84 lb. at 58^^ 72 lb. at 41f ^- 960 lb. at 16f ^ 348 lb. at 75^ 728 lb. at 62^^ 246 lb. at 33i^ 96 lb. at 43|^ 344 lb. at 37i^ 98 lb. at 37^^ 17. 96 yd. at SU 18. 594 lb. at 66f^ 96 yd. at 12^^ 963 lb. at 83^^ 348 yd. at 75^ 312 lb. at 37i^ 72 yd. at 41f ^ 251 lb. at 50^ 246 yd. at 33^^ 603 lb. at 11^^ 344 yd. at 37^^ 552 lb. at 66f ^ 156 yd. at 66f ^ 133 lb. at 14f ^ 132 yd. at 91f ^ 528 lb. at 61^ 84 yd. at 50^ 273 lb. at 83^^ 328 yd. at 25^ 368 lb. at 31i^ • 19. 146 gal. at 37^^ 20. 200 yd. at 37i^ 245 gal. at 42f ^ 384 yd. at 18f ^ 672 gal. at 16f ^ 288 yd. at 83i^ 18 gal. at 87i^ 294 yd. at mU 162 gal. at 16f ^ 918 yd. at 44^^ 332 gal. at 18 J^ 459 yd. at 11^^ 369 gal. at 33i^ 18 yd. at 12i^ 828 gal. at 311^ 111 yd. at 33i^ 693 gal. at 16f ^ 164 yd. at 62i^ 918 gal. at 44|c^ 8 yd. at 87^^ 30 RATIONAL ARITHMETIC EXERCISES IN BILLING Reference 241. 31. 1. Copy and extend the following bill : Salem, Mass., July 20, 1919 Mr. J. A. Brown, 3 Leach St., City. To S. S. Pierce & Co., Dr. Terms Cash 8 bu. beans @ $3.75 108 lb. butter @ .50 84 lb. cheese @ .33i 129 doz. eggs @ .50 150 lb. lard @ .42f 25 bu. potatoes @ 2.16 72 lb. rice @ .161 12 lb. Japan tea @ .5Q\ 360 lb. granulated sugar @ .10 128 lb. coffee @ .43f 2. Feb. 1, A. W. Smith & Co., Boston, Mass., sold to Jones & French, Marblehead, Mass., on 30 days' credit : 36 boxes oranges at $3.66| ; 12 chests T. H. tea, 840 lb., at 50e; 14 chests Japan tea, 980 lb., at 37^^; 12 bbl. St. Louis flour at $9.10 ; 4 bags coffee, 576 lb., at 33^^ ; 54 boxes lemons at $4.60 ; 14 bbl. pineapples at $7.50 ; 44 bunches bananas at $5.45. Write the bill. 3. R. C. Adams, Danvers, Mass., bought of Davis & Bicknell, Salem, Mass., on account 60 days: 25 bbl. RATIONAL ARITHMETIC 31 St. Louis flour at $11.48 ; 35 boxes apricots, 25 lb. each, at 23^ ; 20 boxes apples, 25 lb. each, at 114^ ; 10 boxes peaches, 25 lb. each, at 374|Z^ ; 6 boxes raisins, 25 lb. each, at 25^; 8 boxes prunes, 25 lb. each, at 16f^; 13 boxes currants, 25 lb. each, at 16^ ; 15 cases Quaker Oats at $3.75 ; 12 cases canned corn, 24 doz., at $2.12| ; 18 cases canned tomatoes, 36 doz., at $1.87^; 15 chests Japan tea, 70 lb. each, at 54^; 1600 lb. gunpowder tea at 43i^. Write the bill. 4. Jan. 31, K. R. Good & Co. sold to Lewis W. Sears, Middleton, Mass., on 60 days' time : 6 pairs men's kid gloves at $2.48 ; 5 doz. napkins at $5.50 ; 4 doz. children's hose at $2.75 ; 9 pr. blankets at $6.50 ; 2 pieces jeans, 80 yd., at 25f^; 2 pieces point, 80 yd., at 12^^ ; 5 doz. towels at $3.50 ; 15 doz. spools thread at 87i^ ; 7 doz. ladies' collars at $2.12^; 7 doz. ladies' cuffs at $3.25 ; 14 robes at $2.33i ; 4 pieces Irish linen, 156 yd., at 66f ^. Write the bill. 5. March 1, James K. Broderick & Co., Boston, Mass., sold to Henry T. Lewis, Peabody, Mass., on 30 days' time: 2 pieces gingham, 63 yd., at 37^^; 1 piece blue denim, 31 yd., at 28^; 1 piece brown denim, 30 yd., at 28^^ ; 1 piece duck, 31 yd., at 50^%. 38. $271.35 for 4 mo. 24 da. at 7%. 39. $361.75 for 44 da. at 7%. 40. $285.60 for 2 yr. 8 mo. 24 da. at 7i%. 41. $397.80 for 7 mo. 12 da. at 5%. 42. $184.25 for 8 mo. 21 da. at 4^%. 43. $1495.60 for 1 yr. 2 mo. 13 da. at 6%. 44. $372.75 for 11 mo. 8 da. at 7%. 45. $175.43 for 2 yr. 7 mo. 14 da. at 7i%. 46. $295.60 for 9 mo. 24 da. at 6%. 47. $362.70 for 8 mo. 21 da. at 6%. RATIONAL ARITHMETIC 75 48. $467.80 for 1 yr. 3 mo. 5 da. at 3%. 49. $284.60 for 9 mo. 13 da. at 8%. 50. $575.80 for 6 mo. 18 da. at 4^%. References 335, 336. 55. Find the interest on : 1. $600 from Jmie 1, 1916 to Aug. 13, 1916 at 6%. 2. $360 from Oct. 3, 1914 to June 3, 1915 at 5%. 3. $180 from Dec. 6, 1915 to July 13, 1916 at 8%. 4. $840.75 from May 12, 1912 to Dec. 8, 1914 at 7%. 5. $454.54 from Jan. 16, 1916 to Oct. 28, 1916 at 5%. 6. $544.44 from April 5, 1916 to Jan. 1, 1917 at 1\%, 7. $850 from July 18, 1914 to Dec. 31, 1916 at 9%. 8. $809 from Sept. 13, 1912 to June 11, 1915 at 5%. 9. $256 from Nov. 28, 1913 to Mar. 12, 1914 at 1\%. 10. $660.80 from Aug. 17, 1914 to Jan. 3, 1916 at 7%. 11. $840 from April 1, 1914 to Jan. 3, 1916 at 7%. 12. $629 from Nov. 13, 1908 to Aug. 4, 1910 at \\%, 13. $548 from Jan. 30, 1912 to Sept. 28, 1914 at 7%. 14. $465.10 from Oct. 2, 1913 to Sept. 12, 1914 at 8%. 15. $654 from Feb. 12, 1914 to Dec. 21, 1916 at 6%. 16. $360 from June 16, 1909 to Sept. 30, 1914 at 4^%. 17. $126 from Aug. 28, 1907 to Feb. 16, 1912 at 4%. 18. $480 from April 6, 1910 to June 30, 1914 at 5%. 19. $1000 from Oct. 13, 1914 to Nov. 28, 1915 at 6%. 20. $975 from May 12, 1916 to Nov. 11, 1916 at 9%. 21. $649.24 from Jan. 1, 1914 to April 3, 1916 at 6%. 22. $100 from Oct. 2, 1915 to Feb. 12, 1916 at 5%. 76 RATIONAL ARITHMETIC 23. $654 from Dec. 12, 1915 to Aug. 7, 1916 at 5%. 24. $962 from March 8, 1915 to Aug. 7, 1916 at 4^%. 25. $269.05 from Dec. 3, 1909 to Sept. 8, 1915 at 4%. 26. $680 from April 21, 1914 to Nov. 2, 1914 at 6%. 27. $500 from Aug. 19, 1913 to May 28, 1915 at 8%. 28. $85 from Nov. 12, 1914 to April 3, 1915 at 4%. 29. $450 from Feb. 9, 1916 to Dec. 21, 1916 at 5%. 30. $240 from Jan. 8, 1913 to June 9, 1915 at 7%. 31. $400 from Sept. 12, 1909 to Oct. 9, 1914 at 7^%. 32. $560 from July 4, 1910 to Sept. 8, 1914 at 4i%. 33. $200 from March 2, 1913 to Sept. 27, 1913 at 9%. 34. $460 from Dec. 8, 1916 to Feb. 3, 1917 at 3%. 35. $296.50 from May 8, 1904 to June 11, 1913 at 8%. 36. $320.60 from Oct. 28, 1913 to July 7, 1915 at 4%. 37. $576 from Jan. 23, 1909 to Aug. 13, 1912 at 6%. 38. $320.60 from Dec. 6, 1914 to June 28, 1916 at 4>i%. 39. $720.14 from July 11, 1912 to Aug. 29, 1916 at 8%. 40. $365.40 from Sept. 14, 1911 to Feb. 18, 1915 at 7%. 41. $428.60 from May 11, 1916 to Nov. 8, 1916 at 12%. 42. $576.80fromJune20,1909toDec.31,1909atl0%. 43. $162.38 from Oct. 27, 1914 to Dec. 3, 1916 at 8%. 44. $316.20 from July 26, 1913 to Oct. 19, 1914 at 5%. 45. $483.90 from Jan. 8, 1915 to July 4, 1916 at 4i%. 46. $265.70 from Sept. 4, 1915 to May 30, 1916 at 9%. 47. $456.75 from Dec. 9, 1914 to June 30, 1916 at U%. 48. $195 from Nov. 19, 1913 to April 19, 1914 at 4%. 49. $362.80 from Sept. 12, 1914 to Aug. 8, 1915 at 6%. 50. $195.64 from Aug. 14, 1915 to May 3, 1916 at 3%. RATIONAL ARITHMETIC 77 ACCURATE INTEREST References S'2G, 327, 328 ; 337 to 343 inclusive. While both methods are used in business, the one explained in 340 is better because of the infrequency with which accurate interest is used. 56. Find the accurate interest of : 1. $1436 for '295 days at 6% ; at 8%. 2. $484.50 for 193 days at 6% ; at 7i%. 3. $956.35 for 1 year 214 days at 6% ; at 4^%. 4. $632 for 462 days at 6% ; at 4^%. 5. $1284.50 from Aug. 8, 1916 to Jan. 12, 1917 at 8%. 6. $543.32 from Apr. 7, 1916 to Feb. 25, 1917 at 5%. 7. $246.50 from Jan. 5, 1915 to May 27, 1916 at 4i%. 8. $3432.40 from June 13, 1914 to Feb. 29, 1916 at Si%. 9. £ 120 9s Sd for 214 days at 8%. 10. £253 ll5 10^ for 313 days at 5%. 11. £586 Us 4>d for 1 year 246 days at 4i%. 12. £732 Ids 9d for 2 years 97 days at 3%. Note : Change English money to pounds ; see 232, and then apply 343. Reduce resulting decimal to lower denomination, 231. TO FIND TIME References 346, 347, 348. 57. 1. In what time will $417.40 produce $7.43 at 3i% interest ? 2. How long will it take $325.80 on interest at 10% to produce $30.86 ? 78 RATIONAL ARITHMETIC 3. $895.80 earned $16.50 at 7% interest. How long was the money at interest ? 4. How long will it take $9500 to earn $1524.75 at 9%? 5. I loaned $592.25 to my brother at 5% interest. He paid me $13.74 interest. How long did he have the money ? 6. $182.40 drawing interest at 4% earned $9.48. How long was it invested ? 7. $4150.30 was invested at 8% long enough to earn $295.58. How long was it invested ? 8. $318.60 at 3% would require how long to produce $7.38 interest.^ 9. I loaned $7500 at 7%. When the loan was paid I received $7702.71. For what time was the loan made ? 10. I received a check for $533.85 to cancel a loan of $519.75 effected at 4^%. How long had the loan been standing ? TO FIND RATE References 349, 350. 58. 1. At what rate will $1836 earn $23,46 in 115 days ? 2. The interest on $852 for 18 days is $2.13. What is the rate ? 3. In 93 days $75 increases at interest $1.55. What is the rate ^ RATIONAL ARITHMETIC 79 4. In 144 days $375 amounts to $381.75. What is the rate ? 5. At what rate would $982 be placed at interest for 2 mo. 12 da. to earn $9.82 ? 6. A loaned $2160 for 5 yr. 9 mo. 1 da. It amounted to $3091.95. What rate did he charge? 7. $588 on interest for 2 yr. 3 mo. and 18 da. earns $67.62. What is the rate ? 8. At what rate must $3500 be placed at interest for 4 mo. and 15 da. to amount to $3605 ? 9. $296 produced $3.70 in 45 days. At what rate was it invested ? 10. $810 amounts to $829.44 in 6 months and 12 days. What was the rate ? 11. $750 on interest from March 1, 1916 to August 7, 1916 earns $26.50. What is the rate.^ 12. From April 7, 1915 to October 2, 1915, $360 earns $16.02. What is the rate ? TO FIND PRINCIPAL References 351 to 356. 59. 1. What principal will be required to earn $13.18 in 11 mo. 11 da. .^ 2. W'hat principal will be required to earn $39.92 in 192 days at 3^% ? 3. How much money invested at 5^% for 86 days will earn $24.62? 80 RATIONAL ARITHMETIC 4. The interest is $16, time 5 months and 18 days, rate 8%. What is the principal ? 5. At 9% the interest for 3 mo. 27 da. is $150.07. What is the principal ? 6. W^hat principal will in 6 mo. 13 da. at 6% amount to $720.55 .? 7. What principal will in 8 mo. 25 da. at 5% earn $15.73? 8. In 4 mo. 9 da. at 4% what principal will amount to $591.61.^ 9. At 6% what principal will in 1 yr. 9 mo. 28 da. yield $74.99? 10. Money invested for 1 yr. 7 mo. 14 da. at 4^% amounts to $531.30. What was the principal? 11. A certain sum of money on interest at 3% from May 19, 1910 to July 15, 1915 earns $392.35. What is the investment ? 12. W^hat sum was loaned on April 5, 1910 at 7% if it were paid by check for $942.66 on Jan. 9, 1916? 13. What principal on interest from October 25, 1905 to May 21, 1912 at 12% would amount to $566.44 ? 14. What sum on interest from July 15, 1915 to October 19, 1915 at 7% will earn $14.90? 15. What sum on interest from May 13, 1910 to August 25, 1919 would amount to $11,446.47? RATIONAL ARITHMETIC 81 GENERAL PROBLEMS IN INTEREST Before attempting to solve the following problems, the student should be thoroughly familiar with the entire subject of interest as presented in paragraphs 317 to 356 inclusive. The following prob- lems comprise a series of tests on the subject of interest and are not graded according to difficulty, but are arranged as such problems might present themselves in business. Solve each in the simplest possible way. 60. 1. A note for $852 dated May 2, 1915 with interest at 5% was paid on Feb. 14, 1916 by certified check. What was the amount of the check, time being computed in exact days ? 2. I have just received a legacy of $5000. I have an obligation of $395 which will be due in 1 yr. 6 mo. and 14 da. from to-day. I have decided to set aside enough of my legacy at 4% interest to pay the obliga- tion when it is due. How much will I have to set aside ? 3. Brown borrowed of me $400 at 7%, Jones $647 at 5%, and Smith $398 at 4^%. Brown's loan ran 1 year 8 months and 11 days, Jones' ran 6 months and 19 days, and Smith's ran 297 days. What were my total re- ceipts for interest ? 4. After a loss by a fire the insurance company has agreed to pay me $4340 in full settlement of the claim. They will pay this amount in full at the end of 60 days or will make a cash settlement, deducting 2%. Which proposition should I accept and what will I gain by so doing, money being worth 6% ? 5. On May 26, 1915 I gave my note for $1000 for 6 months at 5%. November 26, 1915 I paid the note 82 RATIONAL ARITHMETIC and accumulated interest, reckoned on the basis of exact days. How much did I pay? 6. A bill of $780.14 due on March 3, 1909, was not paid until October 15, 1909, when it w^as settled with interest at 6%. What was the amount paid, computing time in exact days ? 7. A house costing $7500 rents for $60 a month. What rate of interest does the investment pay if the annual expenses, including repairs, taxes, etc., amount to $250 ? 8. What will be the difference in the amount of interest involved on a claim for $85, running from May 12, 1909 to October 19, 1909 at 8%, between the amount due by computing the time in exact da^^s and computing the time in months and da^^s ? 9. On May 1, 1915, I bought a bill of hides at $15,300, on 60 days' credit, 2% off for cash, and bor- rowed the money at 6% on the hides as security to accept the cash price, giving my note for 60 days. At the expiration of the note I had tanned the hides at an expense of $1500 and sold them at an advance of 25% on the price paid for them. After paying my note, what was my net profit ? 10. I bought a bill of $1450 subject to a discount of 20%, 10%, and 5%, with an additional discount of ^Z% for cash, and borrowed the money to pay for it at 5%. After 45 days I sold the goods at the same list price, subject to a discount of 25%, 2% extra for cash, re- ceiving cash settlement, and paid my loan. What was my profit ? PARTIAL PAYMENTS Study carefully 357 to 370 inclusive, before attempting to work on partial payments. References 371, 372, 373. Use the United States Rule. 61. 1. What is the balance due on July 1, 1916 on a note for $1500 dated February 1, 1913, upon which the following payments were made : July 24, 1913, $250; Aug. 7, 1913, $100; March 9, 1915, $50; Jan. 1, 1916, $300; interest at the rate of 5%? 2. What is the balance due on December 31, 1917 on a note for $1400 dated May 2, 1915, bearing in- terest at 6% and having the following indorsements : July 1, 1915, $200; September 25, 1915, $90; Febru- ary 28, 1916, $175; May 19, 1917, $475? 3. On April 21, 1913, I gave my note for $3550, payable in three years, interest at 4 5%. I paid as follows: Oct. 15, 1913, $125; March 3, 1914, $125; July 20, 1914, $875. How much will be required to settle the note at maturitv ? 4. W. A. Jones gave a note on June 4, 1914 to Frank Brown for $1285.50 with interest at 7%. He made payments as follows: Dec. 15, 1914, $340; Feb. 17, 1915, $330; March 11, 1915, $400. What was due on Mav 15, 1915 ? 83 84 RATIONAL ARITHMETIC 5. Find the value of a note for $2500, given Oct. 10, 1915, with interest at 4^%, and on which the following payments have been indorsed : Jan. 5, 1916, $815 ; June 18, 1916, $350; Oct. 10, 1916, $250; Jan. 17, 1917, $150. Settlement was made Sept. 6, 1917. 6. On a note of $3080, dated Oct. 1, 1915, the follow- ing payments have been made, interest being at 6% : Dec. 31, 1915, $300; Feb. 29, 1916, $50; June 5, 1916, $500 ; Oct. 2, 1916, $700. What will be due on Dec. 31, 1916? 7. Find the amount due Feb. 24, 1917 on a note for $3000, dated March 12, 1915, with interest at 7%, upon which the following payments were made : Aug. 18, 1915, $235; April 9, 1916, $80; July 3, 1916, $400; Dec. 5, 1916, $175. 8. What is the balance due on Jan. 1, 1917 on a note for $500, dated July 15, 1916, bearing interest at 5% and having the following indorsements : Aug. 20, 1916, $27.50; Oct. 8, 1916, $125; Nov. 12, 1916, $110; Dec. 11, 1916, $65? References 374-375. Use the Merchants' Rule. 62. 1. A note of $850 was dated May 25, 1914, interest at 6%. It was indorsed Aug. 13, 1914, $50 ; Nov. 7, 1914, $324.95. What was due March 25, 1915 ? 2. A note for $1250, dated Jan. 25, 1916, interest at 8%, bears the following indorsements : March 10, 1916, $462.50; Aug. 4, 1916, $100; May 22, 1917, $556, What was due on Jan. 1, 1918 to settle the note in full ? RATIONAL ARITHMETIC 85 3. On a note for $550, dated Feb. 5, 1913, interest at 6%, the following payments were made : Oct. 17, 1913, $66.10; March 5, 1914, $140. What was due Nov. 11, 1914.^ 4. A note for $2000 was dated Dec. 12, 1915, interest at 7%. It was indorsed as follows : June 19, 1916, $200; Dec. 6, 1916, $338; Aug. 21, 1917, $276.50; Sept. 12, 1917, $60. What was due Oct. 15, 1917.^ 5. I gave my note for $1080 with interest at 5% on Jan. 25, 1914. I made the following payments : Mar. 1, 1914, $364.40 ; May 13, 1914, $341.50 ; Sept. 1, 1914, $205. What was due on settlement, Jan. 25, 1915? 6. A note for $1500 was dated May 11, 1913, bearing interest at 7.2%. The following payments were made : Feb. 14, 1914, $150; Sept. 23, 1914, $300; July 8, 1915, $100; May 29, 1916, $200. What was due Sep- tember 4, 1916? 7. On a note for $1120, dated August 7, 1914, interest at 7%, payments were made as follows : Sept. 13, 1914, $80; Nov. 7, 1914, $200; Sept. 15, 1915, $450. What was due Aug. 7, 1916? 8. The following pa^mients were made on a note for $580, dated Oct. 17, 1915, bearing interest at 5% : Aug. 5, 1916, $52.50; April 17, 1917, $49.30; Aug. 5, 1917, $250. What was due Sept. 9, 1917? BANK DISCOUNT Study 376 to 381 inclusive. Reference 38*2. 63. Find the bank discount and net proceeds Face 1. $^40 2. $300 3. $1000 4. $400 5. $250 6. $350 7. $500 8. $850 9. $600 10. $375 11. $460 12. $2500 13. $36500 14. $845 15. $280 16. $430 17. $375 18. $3000 19. $575 20. $490 21. $450 22. $340 23. $500 24. $1500 25. $475 Date Time Jan. 3, 1916 60 da. Sept. 8, 1916 2 mo. June 1, 1915 90 da. May 1, 1916 3 mo. June 1, 1916 3 mo. Dec. 30, 1914 4 mo. Apr. 3, 1915 6 mo. June 1, 1916 6 mo. Mar. 3, 1915 90 da. Feb. 3, 1916 60 da. May 9, 1914 3 mo. July 1, 1915 6 mo. Mar. 3, 1916 4 mo. Jan. 3, 1916 90 da. Sept. 8, 1914 30 da. Nov. 9, 1915 3 mo. Oct. 12, 1916 2 mo. Jan. 10, 1916 4 mo. Dec. 4, 1915 3 mo. Aug. 8, 1915 60 da. Mar. 3, 1916 90 da. May 2, 1915 6 mo. July 5, 1916 60 da. Dec. 3, 1914 4 mo. Jan. 4, 1916 3 mo. 86 Date of Disc. Jan. 4, 1916 Oct. 1, 1916 June 7, 1915 June 1, 1916 Aug. 3, 1916 Jan. 2, 1915 May 15, 1915 Oct. 14, 1916 Apr. 4, 1915 Feb. 20, 1916 May 12, 1914 Aug. 30, 1915 June 30, 1916 Feb. 8, 1916 Sept. 10, 1914 Nov. 20, 1915 Nov. 13, 1916 Jan. 12, 1916 Jan. 29, 1916 Sept. 18, 1915 Apr. 5, 1916 Aug. 3, 1915 July 10, 1916 Feb. 9, 1915 Jan. 20, 1916 Rate op Disc. 6%. 7%. 5%. 6%. 0- ^0- 4%. n%. 41%. 5%. 6%. '0- 710/ '2/0- 0- 7%. 7%. /o- 7i%. 4%. 5%. 0- RATIONAL ARITHMETIC 87 Reference 383. 64. Find the bank discount and proceeds of the following interest-bearing notes : Face 1. $800 2. $400 3. $2240 4. $480 5. $1530 6. $285 7. $390 8. $2500 9. $460 10. $1400 11. $2150 12. $580 13. $490 14. $3400 15. $1560 16. $780 17. $?40 18. $1375 19. $650 20. $425 21. $730 22. $260 23. $475 24. $390 25. $1575 Date Time Sept. 1, 1915 3 mo. June 15, 1916 2 mo. Jan. 10, 1916 90 da. Jmie 1, 1916 6 mo. Mav4, 1916 60 da. Sept. 3, 1916 4 mo. Feb. 6, 1916 30 da. Mar. 10, 1916 3 mo. June 12, 1916 4 mo. Aug. 3, 1916 90 da. July 5, 1916 60 da. Mavl2, 1916 6 mo. June 6, 1916 30 da. Dec. 2, 1916 3 mo. Mar. 3, 1916 5 mo. Aug. 1, 1915 4 mo. July 11, 1916 90 da. Mar. 18, 1916 30 da. June 19. 1916 5 mo. Jan. 31, 1916 1 mo. Oct. 6, 1916 2 mo. Jan. 14, 1916 5 mo. May 10, 1916 90 da. Apr. 5, 1916 30 da. July 1, 1916 1 mo. '0 5% 6% 5% Date Date of Disc. 6% Sept. 11, 1915 6% June 30, 1916 7% Feb. 8, 1916 5% Aug. 3, 1916 May 4, 1916 Sept. 20, 1916 Feb. 6, 1916 Mar. 15, 1916 Aug. 2, 1916 Sept. 1, 1916 Julv 7, 1916 May 12, 1916 Junes, 1916 Jan. 3, 1917 Mar. 15, 1916 Aug. 11, 1915 Aug. 1, 1916 Mar. 24, 1916 Aug. 1, 1916 Feb. 4, 1916 Oct. 6, 1916 Mar. 3, 1916 May 11, 1916 /o Apr. 12, 1916 7i% July 5, 1916 '0 6% 5% 4% 5% 6% 4% o% 7% ^0 10% Rate of Disc. '0- 5%. 0- 0- 4%. 5%. '0- 7%. ^0- 6%. 5%. 6%. 7%. 5%. 7%. 7%. 6%. 5%. 6%. References 384-385. 65. For what sum must I write my note in order to yield the following proceeds if discounted on the date of the note ? 88 RATIONAL ARITHMETIC Proceeds Time Rate 1. $385 90 da. 6% 2. $450 3 mo. 5% 3. $1285 60 da. 8% 4. $370 4 mo. 7% 5. $260 90 da. 8% 6. $580 2 mo. 4i% 7. $290 6 mo. 8% 8. $365 4 mo. 7% 9. $290 90 da. 6% 10. $460 30 da. 8% 11. $1340 2 mo. 7% 12. $360 3 mo. 6% 13. $1500 1 mo. 4% 14. $2560 5 mo. 7i% 15. $775 60 da. 6% 16. $550 30 da. 7% 17. $1250 2 mo. 8% 18. $875 4 mo. 8% 19. $1360 90 da. 6% 20. $728 60 da. 5% 21. $450 5 mo. 7% 22. $1385 30 da. 8% 23. $3760 60 da. 6% 24. $1485 3 mo. 5% 25. $960 90 da. 8% RATIONAL ARITHMETIC 89 COMPOUND INTEREST Study 386 to 388 inclusive. Reference 388. 66. Find the compound interest '. Principal Rate Time Compounded 1. $7800 6% 2 yr. Annually 2. $4600 5% 2yr. Semi-annually 3. $8400 8% Syr. Annually 4. $9000 4% Syr. Quarterly 5. $3500 6% 4 yr. 5 mo. Semi-annually 6. $4650 6% 1 yr. 8 mo. Quarterly 7. $3865 4% 1 yr. 2 mo. 15 da. Quarterly 8. A note for $495.60, dated June 10, 1912, and drawing interest at 6% per annum, compounded semi- annually, was paid March 22, 1916. What was the amount due, if no payments of either interest or prin- cipal had been made ? 9. What amount will, on June 30, 1917, discharge a note of $3560, dated Dec. 1, 1914, and drawing in- terest at 8% per annum, compounded quarterly, no previous payments having been made ? 10. What is the amount due April 1, 1916, upon a note for $480.50, dated May 10, 1911, and drawing interest at 8% per annum, compounded semi-annually, no previous payments having been made ? 11. A young man deposited $200 in a savings bank which paid 4% per annum, compounded quarterly. If nothing was withdrawn, what amount was to his credit at the end of the third year ? 90 RATIONAL ARITHMETIC 12. For the benefit of his son who is 12 years old, Mr. A deposited in a savings bank $1000 at 4%, in- terest compounded semi-annually. How much should the son receive when he becomes 21 years old? PERIODIC INTEREST Study 389 to 390 inclusive. 67. Find the periodic interest : 1. 2. 3. 4. 5. 6. 7. Principal $5500 $450 $3000 $2850 $4650 $956 $380 Rate per Annum 6% '0 7% 4% Time 4 yr. 3yr. 4 yr. 5 yr. 3 mo. 1 yr. 2 yr. 10 mo. Interest Due Annually Annually Semi-annually Annually Quarterly Quarterly Semi-annually 4 yr. 1 mo. 8. What amount will be due Feb. 1, 1922, on a note of $3000, dated Jan. 1, 1920, and drawing interest at 6% per annum, payable semi-annually, if. the first four interest payments are paid when due, and no subse- quent payments made ? 9. What amount was due July 15, 1917, on a note of $4600, dated March 13, 1913, drawing interest at 5% per annum, payable semi-annually, no previous pay- ments having been made ? 10. No interest having been previously paid, what was the amount of a note of $1400 at 6%, interest pay- able quarterly, dated Jan. 1, 1914, and paid Feb. 1, 1916? RATIONAL ARITHMETIC 91 11. What sum was due Jan. 28, 1917, on a note of $4000, dated May 18, 1913, and drawing interest at 5% per annum, payable semi-annually ; no payments hav- ing been made previous to that time ? 12. A merchant bought a store building for $9000, giving his note without interest, payable 2 years from date, and 8 separate non-interest-bearing notes for the quarterly interest at 6% per annum. If nothing was paid until the maturity of the note, what was the amount then due ? 13. I purchased a $1250 mortgage on which interest at 6% was due semi-annually on Jan. 15 and July 15. Owing to the fact that no interest had been paid since Jan. 15, 1917, I secured the mortgage at less than its face value. On Oct. 27, 1919, I made arrangements with the mortgagor whereby he paid the interest in full and $500 on the face of the mortgage. How much did I receive in all ? 14. What interest is due Jan. 7, 1920, on $875 from Nov. 13, 1917, at 6%, interest due quarterly and none having been paid ? 15. What is the total interest on $386.40 from Dec. 31, 1914, to Sept. 1, 1919, interest due annually and none having been paid ? Rate 4^%. 16. What amount was due Jan. 8, 1920, on a note of $2340 dated Sept. 1, 1917, drawing interest at 6%, in- terest payable semi-annually, if the first two payments were made when due and no subsequent payments made ? AVERAGE ACCOUNTS Study 391 to 396 inclusive. Reference 397. 68. Average the following : 1. Dr. Harold Chute 1916 Oct. 12 Dec. 20 1917 Jan. 5 Mar. 2 $ 67.85 71.15 143.50 116.20 Cr. 1917 Jan. 10 $316.20 Feb. 19 415.23 Mar. 24 99. May 10 271. 2. Dr. Fred Ellis Cr. 1917 Apr. 18 $367.40 May 6 572. May 23 923. June 2 134.50 3. Dr, Benjamin Jones Cr. 92 4. Dr. RATIONAL ARITHMETIC Howard Colson 93 Cr, 1916 Dec. 1 $540. Dec. 15 236.10 1917 Jan. 2 200. Jan. 31 150. 5. Dr, James Mullaney Cr. 1917 Feb. 5 $1050.10 Mar. 10 826. May 1 924. May 31 186. Reference 398 69. Average the following : 1. Find cash balance on April 18, 1915. Dr. Paul Duncanson Cr, 1915 Jan. 27 30 da. $420. Feb. 17 10 da. 300. Mar. 1 20 da. 540. Apr. 12 30 da. 600. 94 RATIONAL ARITHMETIC 2. Find cash balance on Jan. 1, 1917. Dr. Arthur Bennett Cr. 1916 Oct. 17 15 da. $432. Nov. 20 2 mo. 864. Nov. 30 30 da. 286. Dec. 19 10 da. 627. 3. Find cash balance on Dec. 31, 1916. Dr C. D. Adams 4. Find cash balance on Sept. 7, 1916. Cr, 1916 Aug. 9 30 da. $234. Sept. 15 60 da. 562. Nov. 29 10 da. 52.96 Dec. 21 15 da. 715. Dr. George Duncan Cr, 1916 May 6 30 da. $128. June 30 20 da. 126. July 19 10 da. 213.20 Sept. 3 30 da. 185. RATIONAL ARITHMETIC 95 5. Find cash balance on May 1, 1915. Dr. Paul Jones 70. 1. Dr. References 399-400. A. C. Davis When is the above due by average ? What was the cash balance Apr. 15, 1917.^ CV. 1915 • Jan. 5 10 da. $400. Jan. 31 30 da. 90.60 Mar. 8 10 da. 150. Apr. 25 2 mo. 86.12 Cr. 1917 1917 Jan. 1 2 mo. $600. Feb. 1 Cash $200. Feb. 2 30 da. 240. Mar. 18 Cash 150. Apr. 6 10 da. 360. Apr. 3 Cash 75. 2. Dr. J. F. Howard Cr, 1916 1916 May 18 60 da. $209.70 June 1 Cash $100. June 3 30 da. 180. June 30 Cash 50. July 10 15 da. 750. July 19 Cash 300. Aug. 1 10 da. 280.50 When is the above due by average ? What was the cash balance Aug. 21, 1916? 96 RATIONAL ARITHMETIC 3. Dr, George Stevens When is the above due by average ? What was the cash balance July 1, 1915.'^ 4. Dr. Fred Ellis When is the above due by average .^ What was the cash balance May 5, 1917.'^ Cr. 1915 1915 Jan. 20 2 mo. $219.50 Feb. 25 Cash $ 50. Feb. 25 30 da. 218.75 Mar. 31 Cash 75. Mar. 28 10 da. 413. Apr. 30 Cash 200. June 30 10 da. 216. Cr. 1917 1917 Jan. 1 10 da. $600. Feb. 28 Cash $400. Feb. 2 10 da. 200. Mar. 31 Cash 100. Mar. 3 10 da. 350. Apr. 30 Cash 150. 5. Dr. William Walker Cr. 1915 1915 Jan. 31 2 mo. $540. Feb. 15 Cash $225. Feb. 15 60 da. 450. Mar. 1 Cash 345. Mar. 30 10 da. 306.50 Mar. 10 Cash 295. When is the above due by average ? What is the cash balance Julv 3, 1915 ? RATIONAL ARITHMETIC 97 6. D r. Benjamin Brown Cr. 1916 1916 Apr. 2 10 da. $150. June 25 Cash $300. Mav 1 15 da. 540. Julv 31 Cash 360. June 3 10 da. 450. Aug. 10 Cash 250. July 2 10 da. 323. When is the above due by average ? What is the cash balance Aug. 29, 1916? 7. Dr. Charles Smith Cr. 1916 1916 Sept. 1 2 mo. $315.60 Oct. 5 Cash $200. Oct. 25 60 da. 419.10 Nov. 1 Cash 150. Nov. 16 10 da. 216.05 Dec. 2 Cash 375. When is the above due by average '^ What was the cash balance Mar. 5, 1917 ? 8. D r. Robert Brown Cr. 1915 1915 Jan. 2 1 mo. $1800. Feb. 18 Cash $300. 30 10 da. 600. Feb. 27 Cash 300. Mar. 5 Cash 300. When is the above due by average ? What was the cash balance Mar. 10, 1915 ? 98 RATIONAL ARITHMETIC 9. Dr. E. BOWDOIN Cr. 1916 1916 Oct. 1 30 da. $350. Oct. 21 Cash $300. Nov. 8 10 da. 340. Nov. 24 Cash 300. Dec. 9 15 da. 210. 1917 Jan. 20 10 da. 116. When is the above due by average ? What was the cash balance Feb. 1, 1917? 10. Dr. Sidney Berry Cr. 1915 1915 June 1 60 da. $410. Aug. 1 Cash $300. July 5 30 da. 135. Aug. 31 Cash 200. Aug. 1 10 da. 216.39 Sept. 4 Cash 100. Aug. 31 15 da. 162.54 When is the above due by average ? What was the cash balance Oct. 6, 1915 ? TAXES The general principles of percentage are used in figuring taxes. Study 401-404 inclusive. References 242-*263 inclusive. 71. 1. What is the tax on property assessed for $17,400, the rate of taxation being |% ? 2. What is the tax on property assessed for $8500, rate of taxation being 16f mills on the dollar? 3. What is the tax on property assessed for $23,500, the rate of taxation being $19.20 on the thousand .^^ 4. What is the tax on property assessed for $7588, the rate of taxation being $1.20 on the hundred.^ 5. I own real estate worth $19,500 upon which I pay a tax at the rate of $21.40 a thousand. I also pay an income tax of 6% on a net taxable income of $1400 and^ a poll tax of $2. What is nay entire tax.^ 6. My real estate is assessed at $6500, my personal property at $1570 ; my net taxable income is $2400. Tax on the tangible property is levied by the city at the rate of $19.40 a thousand ; an income tax is levied by the state at the rate of 3% ; my poll tax is $2. What is my entire tax ? 7. The assessed value of real estate in a town is $1,869,000; personal property is $2,450,000. It is 99 100 RATIONAL ARITHMETIC necessary to raise by taxation $412,560. What would be the rate a thousand if there are 1742 polls at $2 each ? 8. In a town whose valuation is $25,000,000, there is an increase in the budget to cover additional expenses of the public schools amounting to $40,000. How many cents a thousand is the tax increased thereby? How much will the improvement cost a citizen who is worth $30,000 ? CUSTOMS AND DUTIES Ad valorem duties are estimated according to the value of the goods in conformity to the principles of percentage. Study 405-422 inclusive. References 242-263 inclusive. Values of units of foreign currency expressed in United States money will be found in 468. 72. 1. What is the ad valorem duty upon an im- portation valued at £430, Ss, 9d, allowing 10% for breakage, duty being at 25% ? 2. Find the ad valorem duty on an invoice of 15,834 marks at 23%,. 3. Find the ad valorem duty on an invoice of 3446.18 francs, duty being 4^3%. 4. Find the ad valorem duty on a bill of 1475 pesos if the duty is 24%. 6. What is the specific duty on 13 tons of tan bark on which there is a duty of 3 cents a hundred pounds ? 6. What is the specific duty on an invoice amount- ing to $760, allowing 10% for breakage, duty being at 35% ? RATIONAL ARITHMETIC 101 7. Find the duty at 10 cents a square yard and 40% ad valorem on a rug 12'X18', imported from Eng- land and invoiced at £14. 8. What is the total duty on 140 cases of plate glass, each containing 25 plates, 20''x48'' at 8^ a square foot ? 9. What is the duty on an invoice of 2300 yards of 27-inch goods, invoiced at 8^ 9^ a yard, subject to an ad valorem duty of 40% and a specific duty of 6^ a square yard ? 10. What is the duty at 60% on a bill amounting to £736 9s Sd ? 11. W^hat is the duty at 30% ad valorem on U\o bales of burlap, each bale containing 40 webs, each web being 48 yd. long and 30 in. wide, invoiced at 30^ per square yard ? 12. What is the duty at 20^ per square yard and 35% ad valorem on 1750 yards of cloth invoiced at 7 francs per yard ? 13. A merchant imported a lot of steel knives from England as follows: 75 doz. at 12^ Qd; 50 doz. at 18.? 6d\ 30 doz. at £l 5s Qd; 20 doz. at £l 8^ 6d; 12 doz. at £2 9s 6d; 10 doz. at £2 10^ 6d. The charges in England amovmt to £7 12^* Qd. The consul's fee was 125 6d. Marine insurance was 20 (^ per hundred on the value of the invoice. The cartage amounted to $2.50. The duty was 30% ad valorem and 30 p per dozen. Find the total cost of the invoice. INSURANCE Study 423-434 inclusive. References M^-263 inclusive. 73. 1. My house is insured for $4500 for a period of five years at 21%. What is the premium ? 2. A merchant insured his stock of goods for $5600 at the rate of li% per annum. What annual premium does he pay ? 3. A factory is insured for $1 '25, 000 in four com- panies. A carries i of the insurance, B carries i, C carries i, and D i. A fire occurs causing a damage of $50,000. For how much will each company be responsible ? • 4. A stock of goods is insured in four companies as follows : $1500 in A, $2400 in B, $3200 in C, and $2500 in D. The goods are damaged to the extent of $8000. How much should each company pay.^ 5. A building worth $85,000 was insured for $68,000, and afterwards damaged by fire to the extent of $4500. The policy contains the average clause. What amount of insurance can be collected from the company ? 6. A vessel worth $50,000 is insured for $20,000 in company A and $18,000 in company B. The vessel is damaged to the extent of $20,000. What amount is to be paid by each company ? 102 RATIONAL ARITHMETIC 103 7. I insured my building worth $80,000 for 80% of its value at f% premium with the iEtna Insurance Company. The iEtna Insurance Company later re- insured $20,000 in the Niagara Insurance Company and $18,000 in the Massachusetts Fire and Marine Insurance Company. The property is damaged to the extent of $30,000. What was the net loss to each of the companies ? 8. I have a policy, containing the average clause, for $7500 on merchandise in stock worth $9000, upon which I have paid a premium of |%. A fire occurs by which the goods are damaged to the ex- tent of $4000. What was my total loss and the net loss to the company ? LIFE INSURANCE Study 435-441 inclusive. Reference 442, 74. 1. What would be the annual premium on a policy for $2500, premiums payable annually during life, at the age of 21 years ? 26 years ? 32 years ? 38 vears ? 2. What would be the annual premium on a fifteen - year endowment policy for $5000, at the age of 23 years ? 27 years ? 32 years ? 37 years ? 3. What would be the annual premium on a policy for $4000, premiums due annually for a period of ten years, policy payable at death only, at the age of 20 vears ? 25 vears ? 35 vears ? 104 RATIONAL ARITHMETIC 4. What would be the annual premium on a twenty- year endowment policy for $6500, age of insured at nearest birthday 23 years ? 29 years ? 37 years ? 5. What would be the annual premium on an or- dinary life policy for $3500, premiums to be paid annually for twenty years, policy to mature at death, age of insured at nearest birthday 28 years ? 23 years ? 38 years ? 6. A man insured his life at the age of 23 years on a twenty-year endowment plan, payments to be paid annually, amount of policy $5000. He died at the age of 33 years. How much less would he have paid in premiums if he had been insured by the ordinary life plan? 7. A man at the age of 35 took out a fifteen -year endowment policy for $2000. What annual premium must he pay ? He lives 20 years and receives the face of the policy. How much less will this amount to than it would have if he had invested the premium at 4% compound interest "^ 8. A man at the age of 25 took out a $3000 twenty payment life policy. He died after paying ten pre- miums. What was the annual premium ? How much more did his family receive than the premiums amounted to, making no allowance for interest ? 9. Three men, aged 24, take a policy for $1000 each. One takes an ordinary life policy, one a twenty-year life policy, and one a twenty -year endowment policy. At the end of five years how much had each paid in premiums ? EXCHANGE DOMESTIC EXCHANGE TO FIND THE VALUE OF A SIGHT DRAFT Study 443-449 inclusive. References 255-258 inclusive. 75. Find the value of the following drafts : 1. $2300 bought at i% discount. 2. $1400 bought at lf% premium. 3. $1740 sold at 1% premium. 4. $3000 bought at li% discount. 5. $2450 sold at li% premium. 6. $4500 bought at $1.50 premium. 7. $1240 sold at $1.25 discount. 8. $1450 bought at $.50 premium. 9. $4300 sold at |% discount. 10. $9000 sold at i% discount. TO FIND THE VALUE OF A TIME DRAFT 76. To find the cost or selling price of a time draft : Find the net proceeds of the draft according to the principles of bank discount (381-382). From this deduct the exchange discount, or to it add the exchange premium, found as in 75. 105 106 RATIONAL ARITHMETIC 77. What is the cost of a 1. 60-day draft for $6000, |% premium, interest at 6% ? 2. 30-day draft for $2200, i% premium, interest at 7% ? 3. 15-day draft for $2500, f% discount, interest at 7% ? 4. 30-day draft for $750, |% premium, interest at 5% ? 5. 30-day draft for $5000 at |% discount, interest at 6% ? 6. 90-day draft for $2300 at ^% premium, interest at 4i% ? 7. 60-day draft for $2500 at |% discount, interest at 4% ? 8. 30-day draft for $2350 at f% premium, interest at 4i% ? 9. 60-day draft for $1240 at $1.25 premium, interest at 5% .^ 10. 30-day draft for $2350 at $1.50 discount, interest at 6% ? TO FIND THE FACE OF A DRAFT 78. Find the value" of a draft of $1 as explained in 75 and 76, and divide the given value by this. 79. What is the face of a sight draft which can be bought for 1. $1207.50 if exchange is at f% premium ? 2. $1091.75 if exchange is at f% discount .^^ RATIONAL ARITHMETIC 107 3. $2453.28 if exchange is at i% premium? 4. $1636.02 if exchange is at f% discount? 5. $4234.62 if exchange is at i% discount? 80. What is the face vakie of a 30-day draft which can be bought for 1. $1183.50 at i% discount, interest 6% ? 2. $1453.84 at i% premium, interest 5% ? 3. $2493.62 at $1.20 premium, interest 4i% ? 4. $3977.33 at $1.50 discount, interest 5%? 5. $2843.55 at f% premium, interest 6% ? FOREIGN EXCHANGE Study 450-452 inclusive. Reference 468. 81. Find the exchange value of a bill for ' 1. £540 at 4.83i. 2. £1476 at 4.85f. 3. £250 9s 8d at 4.85^. 4. £783 13s lid at 4.841. 5. 15,642 francs at 5.18|. 6. 8575.75 francs at 5.19. 7. 8462.73 francs at 5.20^. 8. 2648.55 francs at 5.19f. 9. 1284 marks at 94^. 10. 2556 marks at 95i. 11. 6742 marks at 94f. 12. 1287.5 marks at 94^. 108 RATIONAL ARITHMETIC 13. 789.7 guilders at 40i. 14. 2345 guilders at 40^. 15. 1286 guilders at 401. 16. 1286.5 guilders at 39J. 82. What is the face of an English bill of exchange that cost 1. $2213.88 at 4.85i.^ 3. $585.37 at 4.84f,P 2. $6060.95 at 4.84^.^ 4. $1209.38 at 4.83f.? 83. What is the face of a French bill of exchange that cost 5. $819.29 at 5\19f.? 7. $225.44 at 5.19 ? 6. $2316.04 at 5. 18i.? 8. $88.42 at 5.201.^ 84. What is the face of a German bill of exchange that cost 9. $348.60 at 95f? 11. $393.46 at 95i? 10. $2945.31 at 94i? 12. $13795.19 at 94|.^ 85. What is the face of a Dutch bill of exchange that cost 13. $226.10 at 40f? 15. $235.52 at 40i? 14. $458.97 at 39i ? 16. $6415.60 at 40i? STOCKS AND BONDS The general principles of percentage are involved in solving the following problems (242-263). Studv 453-464 inclusive. 86. 1. A railroad with a capital stock of $2,500,000 declared a dividend at the rate of 5%. What was the total amount of the dividend.^ How much did A, the owner of 350 shares, receive ? 2. What will be the total dividend at 5%, declared by a $3,000,000 corporation ? 3. A manufacturing corporation with a capital of $50,000 levies an assessment of 8% upon its stock- holders. What is the total assessment, and what will B be called upon to pay, if he holds 230 shares of $100 each ? 4. What dividend would I receive on 163 shares of $100 stock at the rate of 5% ? 5. A corporation with a capital stock of $475,500, divided $38,040 among its stockholders. What was the rate of this dividend ? 6. A corporation of which I am a stockholder declares a dividend of 4^%. My dividend check is $652.50. How many shares, par value of $100, do I own? 109 110 RATIONAL ARITHMETIC 7. A mining corporation of which I am a stock- holder declares a dividend of 10%. I receive $125 as my dividend. How many shares of $10 par value do I own ? 8. A corporation with $1,500,000 capital had a gross income of $975,000. Its total expenses were $785,000. Its directors set $100,000 aside as a reserve fund ; the rest was divided among the stockholders. What per cent dividend was declared ? 9. What is the market value of 130 shares, par value $100, of Q., O. & K. C. R. R. quoted at 113^.^ 10. What is the market value of 640 shares, par value of $10 each, of New England Manufacturing Company, quoted at 85^ ? 11. How much must I pay for 75 shares ($100 par value) B. & M. R. R. at 49i, brokerage i%? 12. What is the cost of 130 shares ($100 par value) Bell Telephone at 418i, brokerage i% ? 13. Find the total cost of $1000 L. & E. R. R. 2d 4's at 1021; $4000 C. & N. W. 5's at 102f ; 40 shares ($100 par value) of A. T. & S. F. R. R. at 43f ; 75 shares ($100 par value) B. & M. R. R. at 34f ; brokerage on all i% ? 14. What is the net cost of 150 shares ($100 par value) of B. & A. R. R. at 134f ; 75 shares ($100 par value) M. C. R. R. at 104f ; $5000 S. E. L. Co. 6's at 95i ; brokerage on all i% ? 15. What is the proceeds of 450 shares ($50 par value) sold at 102f , brokerage i% ? RATIONAL ARITHMETIC 111 16. How much must I invest in U. S. 4's of 1932 to secure a quarterly income of $450, bonds selling at 108i, brokerage i% ? 17. I invested $3376.25 through my broker at i% commission, in U. S. 4% bonds at 115f. What will be my annual income ? 18. How much must I invest in U. S. 4's of 1935 to secure a quarterly income of $600, bonds selling at 108f , brokerage i% ? 19. Sold 75 shares ($100 par value) railroad stock through a broker and received $7388 net proceeds. At what quotation did the broker sell the stock? 20. At what price may 6% stock be bought to re- ceive 5% on the investment, brokerage i%? 21. What price can I afford to pay for 7% bonds in order to realize 8% income on the investment, broker- age i% ? 22. What price will I pay for 5% bonds bought through a broker so as to bring in a net income of 4% on the investment ? 23. At what quotation could 8% preferred stock be bought through a broker to realize 5% income on the investment ? 24. What price can I afford to pay for 7% bonds bought through a broker so as to receive a net income of 6% on the investment .^^ 25. What per cent income on the investment will be realized if 4% stock is bought at 79|, brokerage i% ? 26. What per cent is realized on the investment if 6% stock is bought at 74 1, brokerage i% ? 112 RATIONAL ARITHMETIC 27. 5% bonds bought at 124 J would bring what per cent on the investment, brokerage i% ? 28. Stock bought at 79|, brokerage i%, yields 4% on the investment. What is the rate of dividend .^ 29. Which is the better investment and how much : stock paying 6% dividend, bought at 74|, or stock paying 9% dividend, bought at 119J, brokerage i%? 30. I bought, through a broker, 52 shares of stock at 84. I paid an assessment of 5% and then sold them at 99f . How much did I gain, brokerage i% ? 31. I have $8000 to invest. I am offered bank stock at 375 yielding 3^% each three months, or stock in a shoe manufacturing company at 150 paying 4% semi-annually. I have made up my mind to invest in the stock which will give me the greater dividend. Which shall I buy and what will be the total dividend each year ? 32. By investing $18,750 in 150 shares of stock I am able to realize 4% on the investment. What rate of dividend does the stock pay ? 33. What can I afford to pay for 8% stock to realize 5% on the investment ? 34. I have been offered a block of 4v Libertv Bonds at 98. What per cent would they yield on the invest- ment ? PART TWO RATIONAL ARITHMETIC PART TWO 87. Arithmetic is the measure of values or quantities expressed in figures. All arithmetic consists of increasing or decreasing values or quantities. 88. Addition is the simple or basic operation of increasing values or quantities. The sign of addition is +, read plus. 89. Subtraction is the simple or basic operation of decreasing values or quantities. The sign of subtraction is — , read minus. 90. Multiplication is a short method of addition by which quantities or values are increased at a fixed ratio — by a given number. The sign of multiplication is X, read times. 91. Division is a short method of subtraction by which a certain quantity or value is reduced at a fixed ratio — by a given number. The sign of division is -^ , read divided by. Inasmuch as these operations are quite different in their applications, they are treated separately, and are known as the four fundamental operations of arithmetic. 1 2 RATIONAL ARITHMETIC NOTATION 92. For a thorough understanding of arithmetic it is necessary to be famihar with the system of notation used in expressing values and quantities in figures. Ten characters (figures) are used, nine of which have a positive or integral value. These are repre- sented by the figures 12345678 9. The tenth figure, (read cipher^ zero, or naught), represents nothing and has no integral value. In other words, the figure 3 stands for three individual units ; 5, for five individual units ; 7, for seven ; 9, for nine ; while the cipher is used to visualize nothing. These nine integral units, with their accompanying cipher, are given a distinct value according to their position in relation to a fixed line represented by the decimal point. Thus, one in the units column — the first column to the left of the line — means one whole unit. Move this 1 to the next column, one place to the left ; fill in the space from which it has been taken with a cipher to show that nothing is there; it then represents the value of " ten " and is so read. Move it one more column to the left and it repre- sents ten times ten, or one hundred, and so on. Writing the 1 in the second, or tens column, and the figure 3 in the first, or units column, we show 10 units (in the tens column) and 3 units (in the units column), that is, 13, read thirteen. 93. It will readily be seen that the removal of a figure one column to the left multiplies its value by ten. It will RATIONAL ARITHMETIC also be apparent that bringing it back one place to the right divides its value by ten. This is the governing principle of notation and may be appHed on either side of the decimal hne. The figure 1 starting to the left of the line, in the units column, and moving one place to the right becomes yq^ of 1 ; moved another place to the right it becomes -^o oi moved another place to the right it becomes j^ 1 10' w hich IS 1 100 ' of or and so on without limit. 10 0' "^ 10 00' If a student has difficulty in learning the value of a figure to the right of a decimal point (which is merely the line of division between whole numbers and their fractional parts represented by tenths), it is suggested that he take an ordmary sheet of writing paper, turn it sidewise, write the names of the various places in the columns thus formed, draw a heavy line to represent the decimal line, write the decimal notation to the right of this line, and then place figures in such columns as may appeal to him, calling them by the names of the values written in the columns. m a .2 'C o -a •a 6 on a .2 « 3 en C 4 C i -a 3 8 tn i a; H 9 5 tn a CD o H T3 OJ f- '^ c 3 6 d !D O d a 3 «3 T3 d 03 3 O H 1 2 ■c d 3 K 1 1 4 d 1 1 1 8 c < 'd 1 3 3 3 D d H 1 c r i CO, .d -o 1 1 Ire nil lur tn d 03 tc 3 2 H 1 dt ioi idr 73 -d -tj -a a 03 CO 3 O -d d « H hir a, ed m d o3 CO 3 O -c o d 3 ty- six eij 73 d fo h 73 d d = ^^ The value of three units is twenty-four eighths. 24-(-5 = 29 Twenty-four eighths pUis five eighths equals j4_flS, ^, twenty-nine eighths. Therefore, 3f equals -^^. Note. For practice problems in changing mixed numbers to improper fractions see par. 15. CHANGING A DECIMAL FRACTION TO A COMMON FRACTION 193. To Change a Decimal Fraction to a Common Fraction: Write the decimal in its common fraction form and then reduce to lowest terms. ILLUSTRATED SOLUTIONS 194. Problem: Change .125 to a common fraction. 1 Write •I'^o in its fraction form. It is apparent ■^^<^ _ t that 1*25 is a common divisor of both the numerator iOOO" 8 and the denominator. 125 is contained once in 8 the numerator. U25 is contained eight times in the A.ns. ^. denominator. 195. Problem: Change .16f to a common fraction. Writing .16§ as a common fraction produces a complex fraction. Reduce this complex fraction to its lowest terms (par. IS'i). The numerator con- tains fifty thirds, the denominator contains three hundred thirds, making -^^-q or ^. Note. For practice problems in changing decimal fractions to common fractions see par. 16. 100 = = 300 1 m _ _1 %m 6 6 Ans. -5- 30 RATIONAL ARITHMETIC CHANGING A COMMON FRACTION TO A DECIMAL FRACTION 196. Every common fraction is the statement of an unperformed division (par. 175). The result of performing this division is a decimal. (a) The principle involved in changing a common fraction to a decimal is practically the same as that for changing an improper fraction to a mixed number. (6) In changing from an improper fraction to a mixed number all the work is on the integral, or left, side of the decimal line ; while in changing from a common fraction to a decimal fraction the work is all to the right of the decimal line. (c) In changing from mixed numbers to mixed decimals, the work is on both sides of the line. 197. To Change a Common Fraction to a Decimal Fraction : Divide the numerator by the denominator. Decimal values are seldom carried beyond the sixth decimal place. Any fraction remaining at this point is usually disregarded, although when absolute accuracy is desired the decimal should be carried out until exact, or the fraction should be retained, making a complex decimal. ILLUSTRATED SOLUTIONS 198. Problem : Change i to a decimal fraction. 1 125 Divide the numerator by the denominator, 8)1 1 000 fi^^* placing the decimal point in the quotient, Ans 125 ^^ shown in pars. 139, 140, 141. Three decimal places will be used, making the result .l!25. RATIONAL ARITHMETIC 31 199. Problem : What is the decimal value of {i ? lftRR2 Solve this problem in the same way as the previous one. After two decimal places have 1 fir \ 1 o f\f\f\ ID) lo uuu been used, we find that the remainder will i-^ ^ continue to repeat itself. This shows that 1 00 the division will never be exact. No matter 90 how far carried, the decimal figure will be 6. Tq We may, therefore, stop at any decimal place, J ftf»fi2 retain the fraction ^, and make the result a ^* complex decimal, .866|^, .8666|^, or .86§, etc. 200. Problem : Change 2^ to its decimal form. P Reducing the common fraction ^ as shown above, 5)4 10 the result is .8. The integer 2 remains the same Ans. 2.8. ^^^ t^^^ result is, therefore, 2.8. Or, ^5 — 5 2|8 Changing 2f to fifths, we have ^, which, when 5)1410 reduced according to rule, gives 2.8. Ans. 2.8. Note. For practice problems in changing common fractions to decimal fractions see par. 17. ADDITION OF FRACTIONS 201. Only numbers representing like values or like quantities or like parts of such values and quantities can be added (par. 96, a, b). (a) In adding decimals all that is necessary is to carefully arrange the numbers so that the decimal points form a decimal line. In 32 RATIONAL ARITHMETIC this way tenths come over tenths, hundredths over hundredths, and so on, thus making it possible to add Uke parts. (6) In adding common fractions it is necessary to change all frac- tions to equivalent fractions having the same denominator. 202. To Add Common Fractions : Find the L. C. M. of all the denominators (par. 150). Use this L. C. M. as a common denominator. Change each given frac- tion to a fraction having this denominator (par. 185). Add the numerators of the new fractions. The result is the numerator of the sum. The L. C. M. of the given denominators is the denominator of the sum. Reduce the sum-fraction to its lowest terms. ILLUSTRATED SOLUTIONS 203. Problem: f+f+TV+l= ? 2 )3-8-12-6 2 )3-4- 6-3 3)3_2— 3 — 3 We find the L. C. M. of the given de- 1 _Q \ —\ nominators, 3, 8, 1 "2, 6, to be 24 (pars. 150, /-. /-» r. ./-» r»j 5k 151). Change each of the given fractions ^X^X^X^ ^^ to twenty-fourths, f = if , f = M' T2 =M --16 andf = |f. 3 As the denominators are all the same 5 ^ and we are to add only the numerators, 8 it will save time and confusion if we simply 7 write the numerators 16, 15, 14, 20. ]^2 Added, we have fl^, which reduced to a K mixed number (pars. 188, 189) equals 2^x« 65 24 * This L. C. M. can be determined mentally. a,=^ii Ans. m 24 RATIONAL ARITHMETIC 33 204. Problem: 12t+9i+23f+19f = ? 2)5-2-8-6 2 )5-1-4-3 5 — 1 — 2 — 3 Arrange the mixed numbers in a 2X2X5X2X3 = 120 column. Add the fractions as ex- -JQ4. qrj plained in the previous solution. The ^^ result will be 3 0j. = £>_6_i_ qj_ r^r\ lesLUL will ue J 20 ^12 0' ^ The Y2V ^i^^ ^^ t^^ fraction of the 238 — 45 gj^j^l gyj^^ 19f — 100 Add the 2 with the given integers. 2 tI i = 2 1^ The total sum is Q5^\. ^^1 2 ./xTlS . \)D -^ 2 • 205. Problem: 12.8 + 19i+14.875+5.8i= ? 12.8 Mixed numbers, mixed decimals and com- jg ^ plex decimals can be added, but it is neces- sary first to change the mixed numbers to mixed decimals ; that is, 19| must be changed to 19.5. The complex decimals must be re- 14.875 5.833^ oJ.UUo3^ duced to the same order as the longest Ans. 53.008^. decimal; that is, 5.8^ must be changed to 5.833^. x\fter these reductions have been made, add as explained in the previous solutions. Note. For practice problems in addition of fractions see par. 18. SUBTRACTION OF FRACTIONS 206. Subtraction of fractional values is performed in the same way as subtraction of integral quantities. As in addition, it is necessary to change the given fractions to fractions having a common denominator. 34 RATIONAL ARITHMETIC ILLUSTRATED SOLUTION 207. Problem: 412i-204f = ? The given denominators are 9 and 2. The ^ ^' common denominator is 18. ^ = ^8^5 l^~T§- 41/2^ p Ten cannot be taken from 9. Take one from 204|- 10 the units cohimn of the integers. l=Tf- rtf\iy iT _9 -|-J_8___2 7 27_X0_17 ryyi rlifff^rpnp*^ 207 T¥ 18^ I r8~T8- T¥ T8^ — T¥- ^'^^ omercnce between the integers is 207. The total dif- Ans. 207ii. Note. For practice problems in subtraction of fractions see par. 19. ference is 207^1^. MULTIPLICATION OF FRACTIONS 208. When an integer is multiplied by an integer, the product shows a value greater than either. 209. When an integer is multiplied by a fraction, the product shows a value less than the integer. (a) 5X5 = ^^5. Both factors are integral numbers and the pro- duct, 25, shows an increase in valuCo (6) ^X2 = f. f is a fraction and the product shows a value less than 2. (c) The expression ^X2 may be read ^ of 2. |^X2, and ^ of 2 represent the same arithmetical operation. (d) The sign "X" and the word "of" are interchangeable in fractional computation. 210. To Multiply a Fraction by a Fraction : Multiply the numerators. The result will be the numerator of the product. Multiply the denominators. The result will be the denominator of the product. Reduce the new fraction to its lowest terms. The work may be simplified by first casting out factors that may be common to one of the numerators and one of the denominators. RATIONAL ARITHMETIC 35 ILLUSTRATED SOLUTIONS 211. Prohlem : Multiply f by |. Multiply the numerators, 2X4 = 8. This is %yi^ = ^j the numerator of the product. Multiply the de- Ans. #y. nominators, 3X9 = 27. This is the denominator of the product. 212. Problem: f of|ofT'o=? - of - of — = — Multiply the numerators, 5X2X7 = 70. o o lu /*-*u This is the numerator of the new fraction. 70 ^^ 8X3X10 = 240. This is the denominator 24^ '24 of the new fraction. 2^*0 reduced to the J 'no JL- lowest terms equals ^V- Or, 1 1 The solution may be simplified by cast- ^ f ^ £ 7 _ 7 ing out factors as follows : Five is con- 8 3 1(^ 24 tained in the numerator of the fraction |^ of and in the denominator of the fraction yo . \ Casting out makes the first fraction ^ and J >7 9 -J- the last ^. Two is a common factor of the ' numerator of f and the denominator of ^ and may be cast out of each. Then multiply 1X1X7 = 7 for the numerator of the product. 8 X3 = 24 for the denominator of the product. 213. To Multiply an Integer by a Fraction : Multiply the integer by the numerator and divide the product thus obtained by the denominator of the fraction. 36 RATIONAL ARITHMETIC ILLUSTRATED SOLUTION 214. Problem : Multiply 324 by -. 5 324 4 5 Applying the above rule, 4 X324 = 1296. 5 )1296 * 1296-^5 = 259i. 259i Ans. 259i. 215. To Multiply Mixed Numbers : Multiply the fractions ; multiply the integers ; combine the results. ILLUSTRATED SOLUTIONS 216. Problem : Multiply 246 by 23^. 246 1964 First multiply the integer by the fraction. 23- 5)984 "x.vfe^x ^j ...V. x.«.^..xwxx. ' The product of 246 by the numerator, 4, is 1^"5 984, divided by 5 gives 196f as the partial 738 product by \. 492 Multiply 246 by 3 and by 2 as explained in 5854^ P^^* ^^^' ^^^ the results. The answer is Ans. 5854t. ^^''^^• 217. Problem : Multiply 68f by 24. . Multiply as in the previous problem. Multi- plying the integer 24 by the numerator, 2, of the fraction f , gives 48. Dividing this by 3 gives 16. This is the partial product by f. Multiply 68 by 24 (par. 117). Add, and the 1648 result is 1648. Ans, 1648. 68f 16 24 3)48 16 272 136 RATIONAL ARITH^ylETIC 37 218. Problem : Multiply 243| by 42 2431 2k/ i — 8 3 A 5 "~ 15 42t 5)972 8 1 5 8 1941 1941 6 3)84 28 28 486 972 10428 1 4 1 5 Multiply the fraction of the multiplicand by the fraction of the multiplier. The product of the numerators, 4 and 2, equals 8. The product of the denominators, 5 and 3, equals 15. The first partial product is j-^. Multiply the integral part of the multiplicand by the fraction of the multiplier. Four times 243 equals 972 (par. 214). 972 Ans. 10428|i. divided by 5 equals 194f, which is the second partial product. Multiply the fractional part of the multiplicand by the integral part of the multiplier (par. 214). 42X2 equals 84. 84 divided by 3 equals 28, which is the third partial product. Multiply the integer of the multiplicand by the integer of the multiplier (par. 117). Add the partial products and the total is 10428if. Note. For practice problems in multiplication of fractions see par. 20. DIVISION OF FRACTIONS 219. Division of fractions, like division of integers, is the direct opposite of multiplication, 220. When an integer is divided by an integer, the result shows a decrease. When an integer is divided by a fraction the result shows an increase. Ten divided by 5 equals 2. That is, one integer divided by another gives a decrease. Ten divided by one-half equals 20, for in 10 units there are 20 halves. 38 RATIONAL ARITHMETIC 221. To Divide Fractions by Fractions : Multiply the numerator of the dividend by the denominator of the divisor. The result is the numerator of the quotient. Multiply the denominator of the dividend by the numerator of the divisor. The result is the denomina- tor of the quotient. Reduce the quotient to its lowest terms. Or : Invert the divisor and proceed as in multi- plication. ILLUSTRATED SOLUTIONS 222. Problem : Divide t by f. -|-i-f = ij=li The numerator of the dividend is 4 and the J -ii denominator of the divisor is 3. 4X3 = 12, ^' which is the numerator of the quotient. The denominator of the dividend is 5 and the numerator of the divisor is 2. 2X5 = 10. The denominator of the quotient is 10. f| is an improper fraction, reduced to a mixed number equals l^^. Or, t"^f = ? In the problem f ^f, the dividend is f and tX f = i^ = li the divisor f . Inverting f gives f . 4X3 = 12. 223. To Divide Mixed Numbers : Change both the divisor and the dividend to improper fractions having a common denominator. Divide the numerator of the dividend by the numerator of the divisor. The result will be the quotient required. RATIONAL ARITHMETIC 39 ILLUSTRATED SOLUTIONS 224. Problem : Divide $124.50 by 23f. 5 26 71)373 50 355 18 5 14 2 4 30 4 26 In tliis problem, both the dividend and the divisor must be changed to thirds. $124.50 should be changed to thirds by multiplying by three, which gives 373.50 thirds. 23f = ^ (pars. 191, 192). Divide 373.50 by 71 (par. 140). The result is $5.26y*Y- A 71 Ans. $5.26-^. Problem : Divide 248^ by 34. n'2 7 85 170)1244 1190 54 27 Changing the dividend, 2484, to fifths equals 1244 fifths. Changing the divisor, 34, to fifths equals 170 fifths. Divide the numerator of the dividend, 1244, by the numerator of the divisor, 170. The result is 7||. 170 85 225. Problem : Divide 433^ by 18f . 23 112)2601 224 361 336 25 Change the fraction in the dividend and the fraction in the divisor to fractions having a common denominator. 433^ = 433|^ and 18f =18|. Then 433i-M8f is the same as 433f divided by 18|^. Changing the divi- dend to sixths, we have 2601 sixths. Chang- ing the divisor to sixths, we have 112 sixths. 25 Ans. 23i^. 112 Dividing 2601 by 112, gives 23^ Note. For practice problems in division of fractions see par. 21. For practice problems involving the use of fractions and decimals see par. 22. DENOMINATE NUMBERS 226. A Denominate Number is a number expressed in units of weight, measure, or value. 227. A Simple Denominate Number is a quantity expressed in a single denomination. 4 pounds, 5 bushels, 2 quarts are simple denominate numbers. 228. A Compound Denominate Number (usually called a compound number) is a quantity expressed in two or more different denominations. (a) 1 year, 9 months, and 9 days ; 3 pounds, 9 shillings, and 4 pence are compound numbers. (6) Tables of weight, measure, distance, values, etc., will be found on pages 130 to 146. (c) In the follow^ing illustrated solutions, English money is used throughout for the purpose of uniformity. The solutions are equally applicable to any or all the tables. REDUCTION OF DENOMINATE NUMBERS ILLUSTRATED SOLUTION 229. To Reduce a Compound Denominate Number io a Simple Denominate Number of Equivalent Value: 40 RATIONAL ARITHMETIC 41 Problem : Reduce £5 Ss 9d to pence. 5 100 8 108 1296 9 1305 = £ = s = d J. ins. ISOBd. Since there are 205 in £l, in £5 there are 5 times 20 or IOO5. £0 Ss equals lOSs. Since there are 12(Z in Is, in IO85 there are 12 times 108, which is 1296J. If there are UQQd in £5 8s, in £5 8s 9d there are lS05d. Note. For practice problems in reducing compound denominate num- bers to simple denominate numbers of equivalent value see par. 23. ILLUSTRATED SOLUTION 230. To Change to Higher Denomination: Problem : Reduce 1305 pence to higher denomina- tion. Since there are 12cZ in \s, in ISOofZ there are as many shillings as 12 is contained in 1305 $128.25 is 37i% of $342. Or, Inasmuch as $128.25 is a certain or.2. per cent of $342, it ($128.25) is the QzLQ MQQ QK product obtained by multiplying one lOQ r ^^^*^^ *^^^^^ desired rate) by another I5?_^ factor ($342). Therefore, if we di- 25 65 vide $128.25 (the product) by 342 23 94 (one factor) , we shall obtain the other 171 1 factor (par. 123). Perform this 349 ~ Q operation as described in par. 140. The result is .37^, which equals Arts. .37ior37i%. 263. Problem : $9072 is what per cent of $7560 ? 1|20 $9072 is the product obtained by 7560)9072|00 multiplying one factor (7560) by another factor (the desired rate). If we divide 9072 (the product) by 7560 (one factor), we shall obtain the other factor (par. 123). The result, 7560 1512 1512 Arts. 1.20 or 120%. 1.20, equals 120%. Note. For practice problems in finding the rate see par. 35. For general problems in percentage see par. 36. PROFITS AND LOSSES 264. In measuring and comparing profits and losses, it has been found best to do so by means of percentage. 265. For this purpose it is necessary to understand the exact meaning of the special terms used in connection with this subject. 266. Cost is the value of the investment. 267. Prime Cost of an article is the amount actually paid for it. The prime cost is sometimes called the net cost and also the^r^^ cost. 268. The Gross Cost of an article is the total amount invested in it, and includes the prime cost and the incidental expenses, such as freight, cartage, insurance, etc. The simple term " cost " usually means the gross cost. 269. The Gross Selling Price is the total amount received for the goods sold. 270. The Net Selling Price is the amount of the gross selling price left after incidental expenses of the sale have been deducted, such as freight, commission, insurance, etc. 57 58 RATIONAL ARITHMETIC 271. Profit is the difference between the net seUing price and the gross cost, ivhen the selling price exceeds the cost. 272. Loss is the difference between the net selHng price and the gross cost, when the cost exceeds the selling price. 273. In the subject of profit and loss : Base = Gross Cost Rate = Rate Percentage = Profit or Loss Cost =100% In solving problems in this subject, always use the easiest method. 274. To Find Profit, Loss, or Selling Price: ILLUSTRATED SOLUTIONS Problem : A farm costing $4200 increased in value ^i% when it was sold. Find the profit and the selling price. $4200 Qgj, The cost is the standard of value, 100%. Then $4200 is 100%. Eight and one-third per cent may be found, as ex- $350.00 Profit plained in par. 255, to be $350. 4200.00 If the cost is $4200 and the profit $4550.00 Selling Price is $350, the selhng price must be the sum of the two, $4550. 1400 33600 Ans. \ $350. $4550. RATIONAL ARITHMETIC 59 Or, 12 )4200 350 4550 Arts. Profit Selling Price f $350. [$4550. Since 8^% is -^^, the profit is yV of the cost. One-twelfth of $4200 is $350. Added to the cost, the amount is $4550. 275. Problem: 1200 bushels of wheat were pur- chased at 75 cents per bushel and later sold at a loss of 17%. What was lost, and for what was the wheat sold? 1200 bu. @ $.75 = $900 $900 .17 $153.00 Loss $900 153 $747 Selling Price $153. 1200 bushels of wheat at $.75 cost $900 (found by the use of aliquot parts, par. 241). 17% of $900 (par. 255) equals $153. Since the goods cost $900 and are sold for $153 less than cost, they will be sold for $900 minus $153, which is $747. Ans. \ [ $747. Note. For practice problems in finding the profit or loss and selling price see par. 37. 276. To Find Cost: ILLUSTRATED SOLUTIONS Problem : My profit on a certain transaction, figured at 12^%, would be $720. What was the cost of the transaction ? 60 RATIONAL ARITHMETIC 5 760 .125)720.000 6^5 95 87 5 7 50 7 50 In this problem, the rate is 12|% and the percentage $720, The base equals the cost, which may be found as explained in par. 259 or 260. Ans. $5760. — i Or 12i $720 8 $5760 Cost Ans. $5760. Twelve and one-half per cent equals ^. Since ^ is gained and the gain is $720, then $720 is i of the cost. Therefore, the cost would be 8 times $720. 277. Problem : I bought goods and afterwards sold them at a loss of 23%, receiving $412.80 for them. What did they cost? 100% 23% 77% 5 361103 .77)412.80 000 385 27 8 23 1 4 70 4 62 80 77 300 231 The base, or cost, is 100%, from which Vq is lost, leaving 77%, the measure of the value for which the goods were sold. Then $412.80 equals 77% of the cost, which may be found as explained in pars. 259, 260, 261. 69 Ans. $536.10. RATIONAL ARITHMETIC 61 278. Problem : Goods are sold for $126, which shows a loss of lli%. What did they cost? 100 IH 88f 126 = .88f $141|75 8.00)1134|OO Or, 8 )126.00 15.75 9 Arts. $141.75. One hundred per cent is the standard of measure, or cost ; tl^% has been lost. The goods then sold for 88 1% of the cost. The cost may be found as explained in par. 261. Eleven and one-ninth per cent is equal to If ^ is lost, the goods Therefore, $126 is f of the cost. If $126 is f of the cost, ^ of the cost would be found by dividing $126 by 8, which is $15.57. If $15.75 is ^ of the cost, the whole cost the common fraction ^ are sold for |^. 141.75 Ans. $141.75. would be 9 times $15.75, which is $141.75. Note. For practice problems in finding the cost see par. 38. 279. To Find Rate of Profit or Loss: ILLUSTRATED SOLUTIONS Problem : Goods costing $723.45 are sold so as to gain $241.15. What is the gain per cent? 1 J7^37*5- 3 = - = 33^ 3/0 M^tr^ 1 3 Ans. 33i%. The cost, or base, is $723.45. The gain, or percentage, is $241.15. Find the rate as explained in par. 262. Or, I33i 723.45)241. 15|00 217 03 5 24 11 50 21 70 35 2 41 15 Ans, 33i%. Use the fractional method when its reduction may be determined at a glance ; use the decimal method when this is not the case. m RATIONAL ARITHMETIC 280. Problem : Goods that cost $414 are sold for $492.66. What per cent is gained ? $492.66 414. $ 78.66 If the goods cost $414 and sell for $492.66, 1^" the difference, $78.66, must be gain or profit. 414)78|66 The problem then is: $78.66 is what per cent 41 4 of $414, the cost.'* This can be ascertained as 37 26 explained in par. 262. 37 26 Ans. 19%. Note. For practice problems in finding the per cent of gain or loss see par. 39. For general problems in profit and loss see par. 40. DISCOUNT 281. A Discount is an amount deducted from a sum owed by one person to another. In measuring dis- counts the principles of percentage are used. There are two kinds of discount, trade discount and time discount. 282. Trade Discount is the discount allowed by a manufacturer or jobber to a retail dealer. 283. Time Discount is a discount allowed as a con- sideration for paying an amount during a certain time. TRADE DISCOUNT 284. Manufacturers, wholesalers, and others doing business of a similar nature, and handling goods the value of which is likely to fluctuate from time to time, have a fixed list price for their goods. These list prices remain fixed, and fluctuations in market rates are met by allowing different discounts from time to time ; that is, if the value drops, the discount is made larger, and if the price of the goods rises, the discount is made smaller. 285. The List Price of goods is the price at which they are listed in the catalogue and from which dis- counts are allowed. Goods are always billed at the list price and then the discount is deducted from the total. 63 64 RATIONAL ARITHMETIC 286. The Gross Amount of the bill is the total amount before any discounts have been deducted. 287. The Net Amount is the amount to be paid after all discounts have been deducted. 288. The Discount is the sum deducted from the gross amount. 289. A Discount Series is several discounts deducted one after another ; as, 25%, 10%, and 5%. The first discount is deducted from the gross amount ; the second from the remainder, and so on ; the final remainder being tlie net amount. 290. The principles of percentage are used in per- forming all operations in discount. Base = Gross Amount Rate = Rate Percentage = Discount Difference = Net Amount Gross Amount = 100% 291. To Find the Discount or Net Amount: Always use the easiest possible method. ILLUSTRATED SOLUTIONS Problem : Goods listed at $420.34 are sold at a discount of 28%. What is the discount and what is the net amount of the bill ? RATIONAL ARITHMETIC 65 $420.34 .28 3362 72 8406 8 $117.69 52 Discount $420.34 117.70 $302.64 Net Amount [$117.70. ^''^- ($302.64. . The gross amount of the bill, $420.34, is 100%. Find 28% of this bv the easiest method. This will give $117.70. The difference will equal the net amount, $302.64. 292. Problem : A certain line of goods is sold at a discount of 25%, 20%, 10%, and 5%. What would be the net amount of a bill of $1214.43, purchased under these terms ? What would the discount amount to ? $1214.43 rpj^jg problem involves a dis- 303.60/.^ 25% count series. The first discount is 910.82^^ deducted from the gross amount. 182.16^ 20% Twenty-five per cent of $1214.43 ifoQ gg is $303.61 ; deducted from the 79 87 100/ gross amount leaves $910.82. ^ J — Twenty per cent of this, found in ^^^•^^ the easiest way, is $182.16; de- 32.78 5% ducted, leaves $728.66. Proceed- $623.01 Net Amount ing in the same way, deducting $1214.43 Gross Amount ^^^^ ^^^ ^^^^ ^^^' ^^^ "^^ ^^^^^^ nr^c ^-. TVT . A . of the bill is $623.01. If the net -^?M1 Net Amount ^^^^^^^ ^^ ^^^ ^.^^ .^ ^^,3^^ ^^^ $591.42 Discount the gross amount is $1214.43, the [ $623 01. difference must be the discount, ^"*- i $591.42. «^91-*2- Note. For practice problems in finding the net amount see par. 41. 66 RATIONAL ARITHMETIC 293. To Find a Single Rate of Discount Equal to a Discount Series: ILLUSTRATED SOLUTION Problem : What single discount is equal to a dis- count series of 25%, 20%, 10%, and 5% ? 4)100% One hundred per cent represents the gross 25 amount of the bill. The first discount is 25% or \ of this. One-fourth of 100% equals 25%. Deduct the first discount (25%) from 100%. This leaves 75% from which to deduct the second discount. The second discount is 20% or one-fifth. One-fifth of 75% equals 15%. Deduct this from 75%. This leaves 60%. The next discount in the series is 10% or one- tenth. One-tenth of 60% equals 6%. 60% minus 6% equals 54%. The last discount must then be deducted from 54%. The last discount is 5% or one- twentieth. One- 48.7% twentieth of 54% equals 2.7%. 54% minus 4 dft 70/ ^•'^% equals 51.3%. This is the net amount * ^^* to be paid. Then the discount is the differ- ence between the gross amount (100%) and the net amount (51.3%). This is 48.7%. Note. For practice problems in finding a single rate of discount equal to a discount series see par. 42. 294. To Find the Gross Amount or the List Price: ILLUSTRATED SOLUTIONS Problem : On a bill of goods sold at a discount of 33 i% the discount equals $55.40. What is the gross amount of the bill ? 5) 75 15 10) 60 6 ^0) 54 2.7 51.3 100 51.3 RATIONAL ARITHMETIC 67 33i%of ? = $55.40 $55.40 33^% 3 $166.20 100% Ans. $166.20. The discount, 33^%, is $55.40. If $55.40 is 33^%, 100% found as ex- plained in par. 259, is $166.20. 295. Problem : A check for $953.80 was given in full payment for a bill of goods bought at 24% dis- count. What would the gross amount of the bill be ? $12 55 .76)953.80 76 193 152 418 38 3 80 3 80 The gross amount of the bill is 100% and the discount 24%. The net amount of the bill must be 76%. Therefore, $952.80 equals 76% of the gross amount. Find the gross amount as explained in par. 259. Ans. $1255. 296. Problem : The net amount of a bill of goods sold at a discount of 40%, 30%, and 20% was $114.24. Find the gross amount of the bill. 100% 40 60 42- 8.4 33.6% Goods sold at a discount of 40%, 30%, and 20% are sold at a net price which equals 33,6% of the original bill (par. 293). If $114.24 equals 33.6%, 100%, the gross amount, may be found, as explained in par. 259, to be $340. 68 RATIONAL ARITHMETIC $340 .336)114.240 100 8 13 44 13 44 Ans. $340. Note. For practice problems in finding the gross amount or the list price see par. 43. 297. To Find the Rate of Discount: ILLUSTRATED SOLUTIONS Problem : The gross amount of a bill of goods is $346.40. The net amount is $259.80. What is the rate of discount ? $346.40 259.80 $86.60 The difference between the gross amount 25 and the net amount equals the discount, 346 40)86.60 00 which is $86.60. The problem then is : 6Q^8 $86.60 is what per cent of $346.40.? This is iiWoo ^'''''''^ ^"^ ^^ ^^^"^ ^^^'' ^^^^* 17 32 00 or 1 B600r _l ^04^ 4 4 Ans. 25%. or 25%. Note. For practice problems in finding the rate of discount see par. 44. RATIONAL ARITHMETIC 69 298. To Find What Price to Mark Goods in Order to Allow a Certain Discount and Still Make a Certain Profit: ILLUSTRATED SOLUTION Problem : What price must we mark goods costing $*214 in order that we may allow a discount series of 20%, 10%, and 5% and still make a profit of 25% ? $214 53.50 $267.50 100% 20 80 g By the principles of profit and loss, we see ^ that to make a profit of 25%, goods costing o ^ $214 must be sold for $267.50 (par. 274). — - — ^ Then we must sell the goods for a net amount 68.4% Qf $267.50. If a discount series of 2q^c, $391 108 10%, and 5% is to be allowed from the gross 684^267 500 loo ^^ount of the bill, the net amount of the bill oQx Q will be 68.4% of the gross amount (par. 293). Therefore, $267.50 equals 68.4% of the gross amount, or asking price, which will be found, as explained in par. 259, to be $391.08. 62 30 61 56 740 684 56 00 54 72 128 Ans. $391.08. Note. For practice problems in finding what price to mark goods in order to allow a certain discount and still make a certain profit see par. 45. For general problems in trade discount see par. 46. COMMISSION AND BROKERAGE 299. A Commission Merchant is a person or firm who buys or sells merchandise for another person or firm. A commission merchant actually handles the goods and buys and sells them as if for himself, but in reality for another person or firm. 300. A Broker is a person or firm who arranges transactions between other persons. A broker does not handle the merchandise himself, but simply brings the buyer and seller together in the interest of one or the other. 301. The Principal is the person or firm for whom the business is transacted. 302. The Commission is the compensation allowed the commission merchant or the broker. 303. The Gross Proceeds of a sale or collection is the entire amount received from the purchaser or debtor by the commission merchant. 304. The Charges are the incidental expenses of the sale or purchase. 305. The Net Proceeds is the amount remaining after the charges have been deducted from the gross proceeds. It is the amount to be returned by the commission merchant to his principal. 70 RATIONAL ARITHMETIC 71 306. Prime Cost is the first cost of goods purchased by the commission merchant in the interest of his principal. 307. The Gross Cost is the prime cost plus the charges incidental to the purchase. 308. Account Sales is an itemized statement of sales of merchandise by a commission merchant. It shows the amount for which the goods were sold, the charges, and the net proceeds of the sale. 309. Account Purchase is an itemized statement covering the merchandise purchased by a commission merchant and shows the prime cost plus the charges. 310. In solving problems in commission, the general principles of percentage are used. Base = Gross Sales or Prime Cost Rate = Rate of Commission Percentage = Commission Net Proceeds = Difference Gross Cost = Amount Gross Sales or Prime Cost = 100% 311. To Find the Commission and Net Proceeds or Gross Amount of Purchase : ILLUSTRATED SOLUTIONS Problem : A commission merchant sold goods for $2346.40. His commission was 2^%. Charges for insurance, freight, etc., amounted to $98.40. Find the commission and net proceeds. • 72 RATIONAL ARITHMETIC $2346.40 58.66 98.40 157.06 $2189.34 Ans. 2i% Com- mission Charges Net Proceeds $58.66. $2189.34. Gross sales, 100%, is $2346.40. Two and one-half per cent of this, the commission, is $58.66. Since the commission merchant spent $98.40 for expenses and kept $58.66 for commission, the sum of these, $157.06, must be deducted from $2346.40, leaving a net proceeds of $2189.34. 312. Problem : A commission merchant bought 500 barrels of apples at $2.75 on a commission of 5% and paid $15 for cartage and $52.50 for cooperage. For what sum must his principal write a check to cover the entire transaction ? 500 bbl. @ $2.75 = $1375 $1375 68.75 15. 52.50 5% $1511.25 Ans. $1511.25. Five hundred barrels of apples at $2.75 cost $1375. The com- mission, 5%, equals $68.75. The principal will, therefore, have to send the commission merchant $1375 to pay for the goods, $68.75 for his commission, $15 to pay for cartage, and $52.50 for cooperage, or $1511.25 in all. Note. For practice problems in finding the commission and net pro- ceeds or gross amount of purchase, see pars. 47 and 48. 313. To Find the Gross Sales or Net Purchase Price: ILLUSTRATED SOLUTIONS Problem : A commission merchant working on 2^% commission received $134.50 for selling a consignment of flour. What did- the flour sell for.^^ RATIONAL ARITHMETIC 73 $134.50 = 2i% $5 380 .025)134.500 125 $134.50 is ^%. Find 100% as explained in par. 259. Ans. $5380. Or, QJ-Or — JL ^2/0 — 4 134.50 = A 40 95 75 2 00 2 00 $5380.00 Ans. $5380. 314. Problem : The net proceeds is $568.40. The charges for freight, insurance, etc., are $27.40. The commission is 3%. For what were the goods sold ? $568.40 27.40 $595.80 6 14 22 .97)595.80 00 582 13 8 97 4 10 3 88 22 19 4 2 60 194 The net proceeds is $568.40. Charges for freight, insurance, etc., are $27.40; added to $568.40 equals $595.80. This is the amount remaining from the sales after the commissioin alone has been deducted. If the commission is 3%, then $595.80 is 97%. Find the total amount of the sales as explained in par. 259, which is $614.22. Ans. $614.22. 74 RATIONAL ARITHMETIC 315. Problem : I sent a commission merchant $927 to invest in apples. How many barrels at $3.75 can be purchased after deducting a commission of 3%? 9 00 1.03)927.00 927 2 40 3.75)900.00 750 150 150 Ans. 240. $927 includes the amount of purchase and the commission of 3%. Therefore, $927 is 103%. 100% may be found, as explained in par. 261, to be $900. If one barrel of apples cost $3.75, for $900 we can buy as many barrels as $3.75 is contained in $900, which is 240. Or, 3% of $3.75 = .1125, commission on 1 bbl. $3.75 Cost of 1 bbl. .1125 Commission on 1 bbl. $3.8625 Gross cost of 1 bbl. 240 3.8625)927.0000 772 50 154 500 154 500 Ans. 240. Since the market price of 1 bbl. is $3.75, the commission on one barrel would be 3% of $3.75 which is .1125, and the gross cost of one barrel would be $3.8625. For $927 the commission merchant could buy as many barrels as $3.8625 is contained in $927, which is 240. Note. For practice problems in finding the gross sales or net purchase price see par. 49. RATIONAL ARITHMETIC 75 316. To Find the Rate of Commission: ILLUSTRATED SOLUTION Problem : A commission merchant charges $80.16 for selling a bill of goods for $1336. What is his rate of commission ? |06 lSS6')80ll6 Since the commission is figured as a certain en 1 « per cent of the amount of sales, this problem — really is: $80.16 is what per cent of $1336, which, solved as explained in par. 262, is .06 or 6%. Ans. 6%. Note. For practice problems in finding the rate of commission see par. 50. For general problems in commission see par. 51. INTEREST 317. Interest is the amount paid for the use of money. (a) When the use of real estate is allowed to someone other than the owner, the compensation is called rent ; the compensation for the use of personal property is called hire; the compensation for the use of manual labor is called wages; the compensation for the use of mental labor and time is called salary. (b) The amount of interest to be paid depends upon the time that the money is used, the sum that is used, and the way it is used (risk involved). (c) The first two are self-fixing. The third is subject to agree- ment between the parties. If there is danger of the original sum being lost, a larger rate should be paid for its use. Also when money is plentiful and easy to hire, the rate should be lower than if it were scarce and hard to hire. (d) It has been found that the best way of figuring interest is on a percentage basis. Therefore, the principles of percentage with another element, time, govern the subject of interest. 318. Principal is the sum for the use of which interest is charged. 319. Rate is the per cent of the principal charged for the use of the principal for one year. 320. The Legal Rate is the rate fixed by law to be understood when no rate is mentioned by the parties. It differs in different states. The legal rate in a majority of the states is 6%. 76 RATIONAL ARITHMETIC 77 Charging more than a reasonable compensation is called usury. Some of the states name a definite maximum rate ; in such states to charge more is usury. Where no maximum rate is fixed by law, it is a question for the courts to decide whether or not usury is charged in a given case. 321. Time is the period for which the principal is used. 322. The Amount is the sum of the principal and interest. 323. The interest on $1 for one year at 6% would be 6% of $1, which is $.06. The interest on $1240 for one year at 6% would be 6% of $1240, which is $74.40. The interest for one year, then, is always equal to the percentage of the principal represented by the rate. 324. The general principle on which all interest is based is : Principal X Rate X Time = Interest In figuring the interest for part of a year, two dif- ferent methods arise : Accurate Method and Ordinary Method. ACCURATE INTEREST 325. Accurate Interest gives a year, or any part of a year, its exact value. (a) In computing the accurate interest for any part of a year the time is counted in days, and each day given its actual value, 3^^ of a year. (6) This method is used only in figuring interest on United States bonds, on foreign moneys, and by special agreement. 326. Accurate Interest is found by applying the general principle of interest with absolute accuracy. 78 RATIONAL ARITHMETIC ILLUSTRATED SOLUTIONS 327. Problem : Find the accurate interest of $1140 for 93 days at 6%. (h-, 1 j^Q Six per cent of $1140 equals $68.40. The interest on $1140 for one year is $68.40. The mterest for 93 days ^ — equals -^^^ of $68.40, which, found as explained in par. $68.40 214, is $17.43. Find ^^5 of $68.40. $68.40 $17 427 93 365)6361 200 205 20 365 6156 2711 $6361.20 2555 156 2 146 10 20 7 30 2 900 2 555 345 Ans. $17.43. 328. Problem : Find the accurate interest of $1140 for 458 days. 458 days = 1 year 93 days. * This problem differs from the preceding ^ ' •'*'^ one only in the matter of time. The interest $85.83 for one year is $68.40. For 93 days the Ans. $85.83. interest, found as in the previous problem, is $17.43. The sum of the two will equal the interest for 1 year and 93 days. RATIONAL ARITHMETIC 79 ORDINARY INTEREST 329. Ordinary Interest Method is that used by busi- ness men under ordinary circumstances. It differs from exact interest simply in that one day is roughly considered one three-hundred-sixtieth of a year. This is arrived at in this way : one-twelfth of a year is called one month, and one-thirtieth of a month is called a day. So that one day is one-thirtieth of one-twelfth, or one three-hundred-sixtieth of a year. 330. To figure the ordinary interest for parts of a year, business men sometimes count the time in months and days, considering each month to have thirty days ; sometimes the time is figured in exact days. The latter plan is always followed by banks. Some authorities divide ordinary interest into two classes. Common and Bankers' . 331. The best method of calculating ordinary in- terest is by what is known as the Sixty-day Method. By this method the interest is always found at 6% first and then changed to the rate desired. EXPLANATION At 6%, the interest on any sum for one year would be six one-hundredths of the principal. One one- hundredth of the principal, then, would be the interest at 6% for one-sixth of a year. One-sixth of a year is 2 months, or 60. days. Then the interest on any principal for 60 days, or 2 months, at 6%, would be one one-hundredth of itself. Thus the interest at 6% on $2420 for 60 days is $24.20. For 12 days it would be one-fifth of the interest for 60 days, or $4.84, and 80 RATIONAL ARITHMETIC so on. The interest at 1% would be one-sixth of the interest at 6%, or $4.0333 for 60 days. The interest at 3i% would be 3i times $4.0333, which is $14.11 for 60 days. SIXTY-DAY METHOD — ORDINARY INTEREST RULE 332, Write the principal. Set off the interest for 2 months, or 60 days, at 6%. Using the interest for 60 days as a- basis, the interest at 6% for any other period may be easily ascertained. When the interest for the desired period has been found at 6%, change to the rate desired. Interest in partial results should always be carried to the fourth decimal place. ILLUSTRATED SOLUTIONS 333. Problem : Find the interest on $1246.40 for 1 year 9 months 16 days at 6% ; at 4%. $1246.40 1 yr. 9 mo. 16 da. at 6% ; at 4% 12.4640 Int. for 60 da. at 6% 74.7840 Int. for 1 yr. at 6% 49.8560 Int. for 8 mo. at 6% 6.2320 Int. for 1 mo. at 6% 2.0773 Int. for 10 da. at 6% 1.2464 Int. for 6 da. at 6% 6 )134.1957 Int. for 1 yr. 9 mo. 16 da. at 6% 22.3659 Int. for 1 yr. 9 mo. 16 da. at 1% 4 $89.4636 Int. for 1 yr. 9 mo. 16 da. at 4% RATIONAL ARITHMETIC 81 Set off the interest for 2 months or 60 days at 6%, $12.4040. We now find the interest for 1 year, 8 months and 1 month, 10 days and 6 days. One year is 6 times 2 months. Eight months is 4 times 2 months. One month is one-half of 60 days. Ten days is one-sixth of 60 days. Six days is one-tenth of 60 days. 12.4640X 6 = $74.7840, ^Yhich is interest for 1 yr. at 6% 12.4640 X 4 = $49.8o60, which is Interest for 8 mo. at 6% 12.4640^ 2 = $6.2320, which is interest for 1 mo. at 6% 12.4640^ 6 = $2.0773, which is interest for 10 da. at 6% 12.4640 ^10 = $1.2464, which is interest for 6 da. at 6% Adding these partial results, we have a total of $134.1957, which is the interest for 1 year 9 months 16 days at 6%. Dividing this by 6 gives us the interest at 1%, which is $22.3659. Multiplying this by 4 gives $89.4636, which is the interest for 1 year 9 mouths 16 days at 4%. Or, $134 1957 at 69^ Divide the interest at 6% by 3, which ^ gives $44.7319, which is the interest at 2%. Subtract this from the interest at 6%. The 44.7319 at 2% $89.4638 at 4% difference is $89.4638, the interest at 4%. 334. Problem : What is the interest on $428.75 for 214 days at 6% ? $428.75 214 da.. at 6% 4.2875 Int. for 60 da. at 6% 12.8625 Int. for 180 da. at 6% 2.1437 Int. for 30 da. at 6% .2143 Int. for 3 da. at 6% .0714 Int. for 1 da. at 6% $15.2919 Int. for 214 da. at 6% ns. $15.29. 82 RATIONAL ARITHMETIC In this problem, the most convenient periods of time would be 180-30-3-1 days. Set off the interest for 60 days, which is $4.2875. One hundred eighty days is 3 times 60. Thirty days is one-half of 60. Three days is one-tenth of 30. One day is one-third of 3 days. Adding, gives us $15.2919, the interest for 214 days at 6%. Note. For practice problems in finding the interest see par. 54. 335. Problem : Find the Interest on $1234.28 from January 3, 1915, to August 1, 1916, at 6%. 1916 8 1 $1234.28 1 yr. 6 mo. 28 da. at 6% 1915 1 3 12.3428 Int. for 60 da. at 6% 1 6 28 74.0568 37.0284 4.9368 .8228 Int. for 1 yr. at 6% Int. for 6 mo. at 6% Int. for 24 da. at 6% Int. for 4 da. at 6% $116.8448 Int. for 1 yr. 6 mo. 28 da. at 6% Ans. $116.8448. In this problem it is necessary to find the time between Janu- ary 3, 1915, and August 1, 1916. This will be found (par. 238) to be 1 year 6 months and 28 daj's. One year is 6 times 2 months, 6 months is 3 times 2 months, 24 days is 4 times 6 days (6 days being one- tenth of 60 days). 4 days is one-sixth of 24. The total is $116.8448, which is the interest at 6% for 1 year 6 months and 28 days. 336. Problem : Find the interest on $514.95 from January 18, 1915, to June 7, 1915, at 4i%. Time computed in exact days. RATIONAL ARITHMETIC 83 Jan. 13 $514.95 140 da. at 4>i% Int. for 60 da. at Feb. 28 Mar. 31 Apr. 30 5.1495 10.2990 1.7165 Mav 31 June 7 4)12.0155 3.0038 140 days Ans. $9.0117. $9.0117 Int. for 120 da. at Int. for 20 da. at Int. for 140 da. at 6% Int. for 140 da. at li% Int. for 140 da. at 4^% Find the exact number of days between January 18, 1915, and June 7, 1915 (par. 239), which is 140 days. This consists of 120 days and 20 days. One hundred twenty days is twice 60 days. Twentv davs is one-third of 60 davs, making the total interest $12.0155 for 140 days at 6%. The difference between 6% and 4^% is 1^%. One and one-half per cent is one-fourth of 6%. Divide $12.0155, the interest at 6%, by 4, which gives us $3.0038, the interest at 1^%. Subtract this from the interest at 6%, which leaves $9.0117, the interest at 4^%. Note. For practice problems in finding the interest when the time has to be found either in exact days or by compound subtraction see par. 55. SIXTY-DAY METHOD — ACCURATE INTEREST 337. The difference between ordinary interest and accurate interest for any part of a year is one seventy- third of the ordinary interest. 338. The interest for one year or any number of years is the percentage yalue of the rate and is the same for ordinary and accurate because it is the standard for both methods. 339. Ordinary interest for any part of a year is greater than accurate interest for the same time. • 84 RATIONAL ARITHMETIC 340. To change the ordinary interest for any part of a year to the accurate interest for the same time, divide the ordinary interest by 73. Subtract this from the ordinary interest. 341. Problem : Find the accurate interest on $246 for 73 days at 6%. $240. 2.40 .48 .04 73)2.92 .04 73 da. at yjyo Int. for 60 da. at Int. for 12 da. at Int. for 1 da. at Ordinary interest c 2.88 Accurate interest Ans. $2.88. First find the ordinary interest, as explained in par. 334, on $240 for 73 days at 6%, which is $'2.92. One seventy-third of $2.92 is $.04. The ordi- nary interest, therefore, is $.04 larger than the ac- curate interest, which is found by subtracting $.04 from $2.92. 342. Prohlem : Find the accurate interest for 41^ days at 4i% on $425. 419 days = l year and 54 days $425. 4>.^5 2.125 1.700 73)3.825 .0523 3.7727 25.50 4)29.2727 7.3181 $21.9546 Ans. $21.95. 1 yr. 54 da. at 4^' Int. for 60 da. at Int. for 30 da. at Int. for 24 da. at uyo Ordinary interest for 54 da. Accurate interest for 54 da. at Accurate interest for 1 yr. at o RATIONAL ARITHMETIC 85 This problem differs from tlie previous one in that the time is more than a year. Three hundred sixty-five days should be sub- tracted from 419, leaving 54 days more than a 3''ear, so that the full time is 1 year and 54 days. Find the interest by the ordinary interest method for 54 days (par. 334). This is $3,825 at 6%. Change this to accurate interest as ex- plained above. The accurate interest for 54 days at 6% is $3.7727. Six per cent of $425 gives the interest for one year (both accurate and ordinary). Six per cent of $425 is $25.50. Add this to the accurate interest for 54 days and we have the accurate interest for 1 vear and 54 davs, which is $29.2727. Dividing this by 4 gives the interest at 1^%, which is $7.3181. Deduct this from the accurate interest at 6% and we have $21.9546, the accurate interest for 1 year and 54 days at 4^%. 343. Prohlem : Find the accurate interest on £214 \\s 9d for 115 days at 5%. 12)9 11 20)11 d 75s 75s 5875£ £214.5875 115 da. at 5% 2.1458 Int. for 60 da. at 2.1458 Int. for 60 da. at 1.0729 Int. for 30 da. at 6% .7152 Int. for 20 da. at 6% .1788 Int. for 5 da. at 6% 73) £4.1127 Ordinary interest at 6% .0563 4.0564 Accurate interest at 6% .6760 Accurate interest at 1% £3.3804 Accurate interest at 5% 86 RATIONAL ARITHMETIC Reduce £'^214 lis 9d to pounds (par. £3804 .608 ^32), which is £214.5875. Find the QQ TQ ordinary interest (par. 334). Change to ^ „„„^ ^ ^^^ 7 accurate interest by subtracting one 7.60805 7.29Da + +i • i ^ •. u / onx a.,. seventy-third oi itseli (par. 341). Ihis Ans. i/O tS la. equals £4.0564, which is the accurate interest at 6%. Subtract one-sixth (par. 333). This equals accurate interest at 5%. Note. For practice problems in accurate interest see par. 56. 344. The best combinations for finding the interest by the sixty-day method for from one to thirty days : 1 day = eV of 60, or ^ of 6 2 days = 3V of 60, or i of 6 3 days = 2V of 60, or i of 6 4 days = 3 days + 1 day 5 days = iV of 60 days 6 days = ro of 60 days 7 days = 6 days + 1 day 8 days = 6 days + 2 days 9 days = 6 days + 3 days 10 days = i of 60 days 11 days = 10 days + 1 day 12 days = i of 60 days 13 days = 10 days + 3 days 14 days = 12 days + 2 days 15 days = i of 60 days or i of 30 9 RATIONAL ARITHMETIC 87 16 days = 10 days + 6 days 17 days = 15 days + 2 days 18 days = 12 days + 6 days or 3X6 19 days = 15 days + 3 days + 1 day 20 days = |^ of 60 days 21 days = 20 days + 1 day 22 days = 20 + 2 23 days = 20+3 24 days = 20 + 4 or 4X6 25 days = 20 + 5 26 days = 20 + 6 27 days = 24 + 3 (i of 24) 28 days = 24 + 4 (i of 24) 29 davs = 24+5 30 days = i of 60 345. To Find the Interest at Common Rates Other Than 0' \% = A of 6% 2% = i of 6% 5% = 6% - 1% i% = \ of i% 2i% = 2% + ¥7o 7% = 6% + 1% 1% = 4+1% 3% = i of 6% 8% = 6%+2% 1% = i of 6% 4% = 6% - 2% 9% = 6% + 3% H% = 1 of 6% 4i% = 6% - U% 10% = lOX 1% 88 RATIONAL ARITHMETIC 346. To Find the Time : Divide the given interest by the interest on the principal for one year, at the given rate. (a) The divisor should be carried to the fourth decimal place if necessary. (6) The quotient should be carried to the fourth decimal place. (c) In the final result, more than half should be considered another day. ILLUSTRATED SOLUTIONS 347. Problem : In what time wih $532.56 produce $48, interest at 5% ? $532.56 .05 $26.628C 1 1 8026 26.628)48.000 0000 26 628 21 372 21 302 4 69 600 53 256 16 3440 15 9768 3672 1.8026 yr. 12 9.6312 mo. 30 18.9360 da. Five per cent of $532.56 is the interest on $532.56 for one year. It is $26.6280. If $532.56 earns $26.6280 in one year, it will take as manv years to earn $48 as $26.6280 is contained in $48, which is 1.8026 years. Reducing 1.8026 j^ears to years, months, and days (par. 231), we have 1 year 9 months and 19 days. Ans. 1 yr. 9 mo. 19 da. KATIONAL ARITHMETIC 89 348. Problem : In what time will $560 amount to $625.71 at 6% interest? $625.71 $560 560. .06 $ 65.71 $33.60 1 9556 33.60)65.71 10000 33 60 32 11 30 24 187 00 1 68 00 19 000 16 800 2 2000 2 0160 First find the interest by deducting the principal, $560, from the amount, $625.71. This shows that the interest is $65.71. The interest at 6% on $560 for one year is $33.60. Proceeding as in the previous illustrated problem, we find the time required to be 1.9556 years. Reduced to years, months, and days, this equals 1840 1 year 11 months and 14 days 1.9556 yr. (par. 231). I 12 11.4672 mo. I 30 14.0160 da. Ans. 1 yr. 11 mo. 14 da. Note. For practice problems in finding the time see par. 57. 349. To Find the Rate: Divide the interest by the interest on the principal for the given time at 1%. The divisor should be carried to the fourth decimal place if necessarv. ^0 RATIONAL ARITHMETIC ILLUSTRATED SOLUTIONS 350. Problem : At what rate will $475 earn $8.84 in 134 days ? $475. 4.75 9.50 .95 ^ .1583 6)10.6083 $1.7680 134 da. Int. for 60 da. at u/o Int. for 120 da. at 6% Int. for 12 da. at 6% Int. for 2 da. at 6% Int. for 134 da. at 6% Int. for 134 da. at 1% 1.768)8.840 8.840 j^lXS , O /q. The interest on $475 for 134 days at 1% is $1.7680. The interest at 1% is contained in the given interest 5 times ; therefore the inter- est must be reckoned at 5% in order to produce $8.84 in 134 days. Note. For practice problems in finding the rate see par. 58. 351. To Find the Princijpal: Divide the Given Interest by the hiterest on one dollar for the given time at the given rate. 352. Divide the Given Amount bv the Amount of one dollar for the given time at the given rate. The divisor must be absolutely correct. The interest on $1 should be carried to three places and all fractions must he retained. RATIONAL ARITHMETIC 91 ILLUSTRATED SOLUTIONS 353. Problem : What principal will be required to earn $152.50 in 1 year 7 months 20 days at 8% ? 1. .01 Int. for 2 mo. at 6% .06 Int. for 1 yr. at 6% .03 Int. for 6 mo. at 6% .005 Int. for 1 mo. at 6% .003i Int. for 20 da. at 6% 3).098i Int. for 1 yr. 7 mo. 20 da. at K^yo .032J - Int. for 1 yr. 7 mo. 20 da. at 2% $.131i Int. for 1 yr. 7 mo. 20 da. at 8% .13H)152.50 $11 63|135 1.180)1372.5O|OOO 118 74 5 70 8 IQQ First find the interest on $1 for 1 1 1 o year 7 months and 20 days at 8% by the sixty-day method, retaining all fractions (par. 333). This is $.131^. If $1 produces $.131^ in 1 year ^'^^ 7 months and 20 days at 8%, it will •^ ^^ take as many dollars to produce $152.50 16 as $.131i is contained in $152.50. 11 8 Performing this division (par. 224), we 4 20 find that $1163.14 is the principal 3 54 required. 660 590 70 Arts. $1163.14. 92 RATIONAL ARITHMETIC 354. Problem : What principal will amount to $1250 in 287 days at 6% ? 1. .01 .04 .005 .002i .000^ 1.047 Int. for 60 da. at \t-/o Int. for 240 da. at 6% Int. for 30 da. at 6% Int. for 15 da. at 6% Int. for 2 da. at 6% Amount on $1 for 287 da. at 1.0471)1250. $1 192 937 6.287)7500.000 000 6287 1213 628 7 584 30 565 83 18 470 12 574 5 896 5 658 3 237 70 188 61 49 090 44 009 First find the interest on $1 for 287 days at 6%. This is $.047f. Add this to $1 and we find that $1 will amount to $1.047f. It will take as many dollars to amount to $1250 as $1.047f is con- tained in $1250, which is $1192.94. 5 081 Ans. $1192.94. RATIONAL ARITHMETIC 93 355. The work of finding the interest on $1 at 6% may be simpHfied by using the following rule : Multiply the number of years by 6. Call the result cents. Divide the number of months by 2. Call the result cents. Divide the number of days by 6. Call the result mills, or tenths of a cent. 356. Applying this rule in the above illustrated solutions we would have : In the first : Find the interest on $1 for 1 year 7 months 20 days at 6%, thus : 6X1 = 6 written .06 7 -f- Q = 3i written .035 20 -^ 6 - 3i written .0331 .128^ = Int. on $1 at 6% In the second : To find the interest on $1 for 287 days at 6%, thus : 287-^6 = 471 written .0471 = Int. on $1 at 6% Note. For practice problems in finding the principal see par. 59. For general problems in interest see par. 60. COMMERCIAL PAPERS 357. Commercial Papers comprise notes, drafts, and checks. 358. A Note is a written promise of one party to pay a second party a certain sum at a certain time. There are two parties to a note. 359. The Maker is the party who promises to pay. 360. The Payee is the party to whom, or to whose order, payment is to be made. 361. A Draft is a written order of one party telling a second party to pay a third party a certain sum at a certain time. There are three parties to a draft. 362. The Drawer is the party requesting payment. 363. The Drawee is the party to whom this request is addressed : the party told to pay. 364. The Payee is the party to whom, or to whose order, payment is to be made. 365. A Sight Draft is a draft which by its terms is to be paid by the drawee immediately upon its presen- tation to him. (a) In some states, three days, called days of grace, are allowed on sight drafts. 94 RATIONAL ARITHMETIC 95 366. A Time Draft is a draft payable at a certain time after presentation to the drawee, or after date. 367. The Date of Maturity of either a note or a draft is the date upon which the payment is due. (a) To fix the maturity of a time draft due "after sight," it is necessary to present the draft to the drawee to see if he is wiUing to pay it. If he is wiUing to pay it, he so indicates by writing the word "accepted," together with the date, over his signature, across the face. (b) The maturity of a note is ascertained by reckoning the specified time from date of the note. (c) The maturity of a draft drawn "after sight" is ascertained by reckoning the specified time from the date of acceptance. (d) The maturity of a draft drawn "afterdate" is ascertained by reckoning the specified time from the date of the draft. 368. The Face of a note or draft is the amount of money mentioned in it. 369. The Amount due at maturity is the sum that is to be paid. This may be the face, or it may be the face plus interest. 370. A Check is an order by one who has funds on deposit in a bank, telling the bank to pay a certain sum from this deposit to a certain party. Checks are treated as cash. (a) The drawer of a check is the depositor. (6) The drawee of a check is the bank where the funds are on deposit. (c) The payee of a check is the party in whose favor the check is made : The party to whom funds from the deposit are to be paid by the bank. PARTIAL PAYMENTS 371. It is sometimes necessary to make a part pay- ment on a promissory note or other obligation. It is usually customary in such cases to cancel the original note and issue another for the reduced amount. When it is not feasible to do this, the amount of the part payment, together with the date, is indorsed on the back of the original instrument. Various rules are in use for finding the balance due on obligations upon which part payments have been indorsed. The more important of these are the United States Rule and the Merchants' Rule. The United States Rule has been sanctioned by the Supreme Court of the United States. The Merchants' Rule is used by most bankers and business men because of its sim- plicity. THE UNITED STATES RULE FOR PARTIAL PAYMENTS 372. To Find the Balance Due at a Given Time: Find the interest on the principal from the date of the instrument to the date of the first payment. If this interest is less than the first payment, add the interest to the original principal and subtract the payment from this amount. Treat the remainder as 96 RATIONAL ARITHMETIC 97 a new principal and proceed as before, so continuing until the date of settlement is reached. If at any time the interest is greater than the pay- ment, the interest should be disregarded and the interest found to such date as the sum of the payments exceeds the interest. ILLUSTRATED SOLUTION 373. Problem : What is the balance due July 1, 1916, on a note of $1200, dated January 1, 1914, upon which the following payments have been made : June 24, 1914, $250 ; August 16, 1914, $100 ; July 8, 1915, $40 ; January 1, 1916, $300.^ $1200 Face Jan. 1, 1914 34.60 Int. to June 24, 1914, 5 mo. 23 da. $1234.60 250. 1st payment 984.60 New principal June 24, 1914 8.53 Int. to Aug. 16, 1914, 1 mo. 22 da. 993.13 100. 2d payment 893.13 New principal Aug. 16, 1914 73.68 Int. to Jan. 1, 1916, 1 yr. 4 mo. 15 da. (Third payment not equal to interest) 966.81 340. 3d and 4th payments 626.81 New principal Jan. 1, 1916 18.8 Int. to July 1, 1916, 6 mo. $645.61 Ans, $645.61. 98 RATIONAL ARITHMETIC $1200 draws interest from January 1, 1914, to June 24, 1914. We find this time (par. 238) to be 5 months and 23 days. The interest on $1200 for 5 months and 23 days is $34.60 (par. 335), making the amount due on June 24, 1914, $1234.60, upon which $250 is paid, leaving the balance of $984.60. This is on interest from June 24, 1914, to August 16, 1914, a period of 1 month and 22 days. The interest on $984.60 for 1 month and 22 days is $8.53, making $993.13 due August 16, 1914. On this $100 is paid, leaving a balance of $893.13 to draw interest from August 16, 1914. The next payment is made on July 8, 1915. The interest on $893.13 from August 16, 1914, to July 8, 1915, is greater than the payment ; therefore we disregard the interest at this time and find the interest to the date of the next payment, January 1, 1916. The time from August 16, 1914, to January 1, 1916, is 1 year 4 months and 15 days. The interest on $893.13 for 1 year 4 months and 15 days is $73.68, making $966.81 due. On this $40 was paid on July 6, 1915, and $300 on January 1, 1916, which is $340 in all, leaving $626.81 to draw interest from January 1, 1916, to the date of settlement, July 1, 1916, 6 months. The interest on $626.81 for 6 months is $18.80, making $645.61 due on July 1, 1916. Note. For practice problems in partial payments (United States Rule) see par. 61. MERCHANTS' RULE 374. To Find the Balance Due at a Given Time: Find the interest on the face of the obligation from the date at which it begins to draw interest to the date of final settlement. Add this interest to the face of the debt. Find the interest on each payment from the date of the payment to the date of final settlement. Add the payments and the interest on the payments. Subtract this sum from the amount of the principal and interest. The difference will be the balance due. RATIONAL ARITHMETIC 99 ILLUSTRATED SOLUTION 375. Problem : What is the balance due Julv 1, 1916, on a note of $1200, dated January 1, 1914, upon which the following payments have been made : June 24, 1914, $250; August 16, 1914, $100; July 8, 1915, $40; January 1, 1916, $300? $1200 Face Jan. 1, 1914 180 Int. to July 1, 1916, 2 yr. 6 mo. $1380 $250. Paid June 24, 1914 30.29 Int. to July 1, 1916, 2 yr. 7 da. 100. Paid Aug. 16, 1914 11.25 Int. to July 1, 1916, 1 yr. 10 mo. 15 da. 40. Paid Julv 8, 1915 2.35 Int. to July 1, 1916, 11 mo. 23 da. 300. Paid Jan. 1, 1916 9. Int. to Julv 1, 1916, 6 mo. $742.89 $1380 742.89 $637.11 Balance due July 1, 1919. A $1200 note given January 1, 1914, would earn $180 interest to July 1, 1916, making its value at maturitj^ $1380. $250 paid June 24, 1914, would earn $30.29, interest to July 1, 1916, a period of 2 years 7 days. $100 paid August 16, 1914, would earn $11.25 to July 1, 1916. $40 paid on July 8, 1915, would accrue $2.35 in- terest to July 16, 1916. $300 would accrue $9 interest to July 1, 1916. The payments and accrued interest amount to $742.89. The value of the note at maturity, $1380, minus the payments and accrued interest $742.89, leaves $637.11 due on July 1, 1916. Note. For practice problems in partial payments (Merchants' Rule) see par. 62. BANK DISCOUNT 376. Bank Discount is the charge made by a bank for cashing an obHgation before it is legally due. It is the interest on the amount due at maturity for the unexpired time. 377. The Maturity of a debt is the date upon which it becomes legally due. {a) A few states allow three days in addition to the time men- tioned in a note or draft. Tliese are called days of grace. {b) Most states allow days of grace on sight drafts only. 378. The Term of Discount is the number of days between the date of discount and the date of maturity. 379. The Bank Discount is the interest on the amount due at maturity for the term of discount. 380. The Proceeds is the difference between the amount due at maturity and the bank discount. It is the cash value of the debt on the date of discount. 381. To Find the Proceeds : Find the date of maturity. Ascertain the amount due at maturity. Find the time in exact days from the date of discount to the date of maturity. Compute the interest on the amount due at maturity for this time. The result will be the bank discount. Deduct the bank discount from the amount due at maturity. The result will be the proceeds. 100 RATIONAL ARITHMETIC ' ^ ^ X5l - ,' :> ILLUSTRATED SOLUTIONS , >'^ '■> '-> >'\}l^} i\5 > .V 382. Problem : Find the bank discount and the proceeds of a note for 60 days for $5000, dated March 3, 1916, discounted April 1, 1916, at 5%. | March 3, 1916 + 60 days = May 2, 1916 = date of i maturity. i April 1, 1916, to May 2, 1916 = 31 days = term of discount ' A note given March 3, $5000. 1916, for 60 days would 50. fall due on May 2, 1916, 25 30 days which would be the date ~ '8333 Idav f '"T'f, ^,t/™' - — — " from April 1, 1916, to 6)25.8333 6^ jy^^^ 2^ l9jg^ j^ 3j ^^^.^ 4.30oo 1% This is the term of dis- 21.5278 5% Bank Discount count. The interest on $5000 for 31 days at 5% $5000. is $21.53. This is the 21 53 bank discount. The face $4978.47 Proceeds °f the note was $5000, the bank discount $21.53 ; Ans. $4978.47. the net proceeds would be i the difference, $4978.47. Note. For practice problems in finding the bank discount and proceeds of non-interest-bearing notes see par. 63. I 383. Problem : Find the bank discount and proceeds of a sixty-day note for $5000, dated January 1, 1915, . bearing interest at 6%, discounted February 6, 1915, at- o /q. January 1, 1915 + 60 days = March 2, 1915 j February 6, 1915, to March 2, 1915 = 24 days $5000 Face 50 Interest $5050 Due at Maturity 50.50 60 days 6)20.20 24 days at 6% 3.3666 $16.8334 Bank Discoun $5050 16.83 ■> 102 RATIONAL ARITHMETIC A note, dated January 1, 1915, to run 60 days would fall due on March 2, 1915. If it were discounted on February 6, 1915, it would then have 24 days to run. The term of discount, there- fore, is 24 days. xA.s this note is given with interest, its face value, plus 60 days' inter- est on $5000, is $5050. As the amount due at maturity ^5033 17 Proceeds ^^ $5050, the bank discount would be figured on this Ans. $5033.17. amount for 24 days at 5%, which is $16.83. The amount due at maturity being $5050 and the bank discount $16.83, the net proceeds would be the difference, or $5033.17. Note. For practice problems in ifinding the bank discount and proceeds of interest-bearing notes see par. 64. 384. In making a loan at a bank when a definite amount is desired, the note must be made for a sum that, when discounted, will leave as the net proceeds the amount of loan desired. To Find the Sum for Which a Note Must Be Drawn So That, if Discounted at Date, the Proceeds Will Be a Given Sum : Find the proceeds of a note for $1 for the given time at the given rate. Divide the given proceeds by this. The quotient will be the face required. The divisor must be absolutely correct. Carry the discount on $1 to the third place and retain all fractions. RATIONAL ARITHMETIC 103 ILLUSTRATED SOLUTION 385. Problem : For what sum must a ninety-day note be drawn so that if discounted on its date at 4^% the proceeds may be $1875 ? $1.00 .01 60 days .005 30 days 4).015 6% .0031 li% .01 li Bank Discount on $1 .9881 Proceeds on $1 $1875 --.9881 1 896 333 3.955)7500.000 000 3955 3545 3164 38100 355 95 25 050 23 730 1 320 1 186 5 133 50 118 65 14 850 11865 The bank discount on a note of $1 for 90 days at ^% would be $.01 1^. The net proceeds of $1 would be $.988f. If $1 yields proceeds of $.988f, it would take as many dollars to yield $1875 as $.988f is contained in $1875, which is 1896.33, the face of the note required. 2 985 Ans, $1896.33. Note. For practice problems in finding the sum for which a note must be drawn so that if discounted at date the proceeds will be a given sum, see par. 65. COMPOUND INTEREST 386. Compound Interest is interest on the principal and interest combined, as fast as the interest falls due. (a) Compound interest can only be charged by special agreement, and then care must be exercised that the laws of usury are not violated. {b) Interest is usually compounded annually, semi-annually, or quarterly. (c) Compound interest is little used except in savings banks. 387. To Find Compound Interest : Find the amount of the principal and interest at the end of the first interest period. Use this amount as a new principal for the next period, and so on. Deduct the original principal from the final amount. The difference will be the compound interest. ILLUSTRATED SOLUTION 388. Problem : Find the compound interest on $1400 from March 8, 1912, to June 15, 1916, at 5%, interest compounded annually. 1916 6 15 1912 3 8 4 3 7 = Four full periods and 3 mo. 7 da. extra 104 RATIONAL ARITHMETIC 105 $1400 70 $1470 73.50 $1543.50 77.18 $1620.68 81.03 $1701.71 17.0171 17.0171 8.5085 1.7017 .2836 6)27.5109 4.5851 22.9258 $1724.6358 1400 Original Principal Int. at 5% for 1st period New Principal Int. at 5% for 2d period New Principal Int. at 5% for 3d period New Principal Int. at 5% for 4th period New Principal Int. for 2 mo. at Int. for 2 mo. at Int. for 1 mo. at Int. for 6 da. at Int. for 1 da. at u/o Int. for 3 mo. 7 da. at u/o Int. for 3 mo. 7 da. at 1% Int. for 3 mo. 7 da. at 5% Final Amount $324.6358 Compound Int. Ans. $324.64. Note. For practice problems in compound interest see par. 66. PERIODIC OR ANNUAL INTEREST 389. Periodic Interest, often called annual interest, is interest on the principal and interest on each overdue payment of interest. (a) It is the result of a business custom in certain lines, and is merely an application of the general principles of simple interest. (6) It is not entitled to be considered as a separate arithmetical subdivision, and is only so treated in this book because many authors have seen fit to introduce it as a separate kind of interest. (c) It has no legal status. 106 RATIONAL ARITHMETIC ILLUSTRATED SOLUTIONS 390. Problem : $1400 was loaned on January 8, 1915, for 2 years at 6%, interest payable semi-annually ; each installment of interest to draw interest at 6% from its due date until paid. What sum would be required to cancel the debt and all interest on January 8, 1917, nothing having been paid previously ? $1400. 14. Int. for 2 mo. at 6% $42. Int. for 6 mo. at 6% 4. $168. Int. for 2 yr. at 6% 1 yr. 6 mo. — 1st installment overdue 1 yr. — 2d 6 mo. — 3d 2 yr. 12 mo. = 3 yr. $42. '0 .42 Int. on interest for 2 mo. at 6% 2.52 Int. on interest for 1 yr. at 5.04 Int. on interest for 2 yr. at 7.56 Int. on interest for 3 yr. at 168. Int. on the Principal 1400. Principal $1575.56 Amount due Jan. 8, 1917 Ans. $1575.56. The interest on $1400 for 6 months at 6% is $42. Then $42 should be paid every 6 months. There would be four such pay- ments due in 2 years. 4 X42 = $168, which must be paid as interest RATIONAL ARITHMETIC 107 on the principal. The first installment of this, amounting to $42, was due on July 8, 1915. Tliis is overdue 1 year 6 months. The next installment of $42 is overdue 1 year ; the next, 6 months ; and the last is just due. Besides the interest on the principal, then, there is interest on the interest due. This last is the interest on $42 for 1 year 6 months, and for 1 year, and for 6 months. In other words, the interest on the interest equals the interest on $42 for 3 years, w^iich is $7.56. Adding interest on interest, interest on principal, and principal gives $1575.56, the amount due at maturity. Note. For practice problems in periodic interest see par. 67. AVERAGING ACCOUNTS 391. Averaging of Accounts is the process of ascer- taining the date on which an account may be paid without loss of interest to either the debtor or the creditor. ^ Averaging of accounts has been abandoned by most lines of busi- ness. It is now the general custom to settle each item, separately, at maturity. The subject, however, is not entirely obsolete. 392. Cash Balance is the amount of cash required to settle an account without loss of interest to either party on any date other than the average due date. GENERAL PRINCIPLES OF AVERAGE 393. If an account is paid before it is due, the payer loses the interest on the sum paid, and the receiver gains it. 394. If an account is paid after it is due, the payer gains the interest on the money paid and the receiver loses it. 395. The average due date of several items, due at different dates, is the date when the payer's losses of interest and gains of interest would be equal, or within, one half day's interest of being equal. 108 RATIONAL ARITHMETIC 109 396. To Average an Account: Assume a date of settlement. Find the net gain or loss of interest by paying on that date. Find how long it will take the amount to be paid to earn this interest. Count that time forward or backward from the assumed date according to whether the payer would lose or gain. (a) The assumed date Is called the focal date. {h) The focal date may be any date. (c) The easiest focal date to use is the zero date of the earliest month in which any one of the items is due ; thus, if items are due June 8, July 15, and August 9, the easiest focal date to use would be June 0, which is, in reality. May 31. ILLUSTRATED SOLUTIONS 397. Problem : Average the follow^ing : Charles S. Chase 1916 Jan. 5 $434. 27 123.50 Feb. 8 215. Apr. 9 310.65 Focal date Jan. 0. Jan. 5 434. 5 da. .3616 27 123.50 27 da .5557 Feb. 8 215. 39 da. 1.3975 Apr. 9 310.65 99 da. 5.1256 1083.15 7.4404 110 RATIONAL ARITHMETIC 41 6 )1.0831 Int. for 6 da. .1805)7.4404 .1805 Int. for 1 da. 7 220 2204 1805 399 Ans. Jan. 0+41= Feb. 10. Assuming Januarj^ as the focal date, on the first item, $434, Chase would lose 5 days' interest, because he would pay it 5 days before it was due. The interest on $434 for 5 days is $.3616. By paying $123.50 on January 0, the interest for 27 days would be lost, which would be $.5557. By paying $215, due February 8, on January 0, the interest for 39 days would be lost, which would equal $1.3975. If $310.65 is paid 99 days before it is due, the interest lost would be $5.1256. Then by paying $1083.15 on January 0, $7.4404 interest would be lost. The interest on $1083.15 for one day is $.1805. It will take as many days to earn $7.4404 as $.1805 is contained in $7.4401, which is 41 days. Then Chase should pay the money 41 days later than January 0, which is February 10. Note. For practice problems in averaging accounts see par. 68. 398. Problem : Average the following : Find cash balance on Jan. 1, 1916. 1915 Sept. 4 $625 for 2 mo. Credit Oct. 23 350 for 30 da. Credit Nov. 18 215 for 10 da. Credit Dec. 8 643 for 60 da. Credit RATIONAL ARITHMETIC 111 Sept. 4 2 mo. Nov. 4 $ 625 4 da. $ .4166 Oct. 23 30 da. Nov. 22 350 22 da. 1.2832 Nov. 18 10 da. Nov. 28 215 28 da. 1.0032 Dec. 8 60 da. Feb. 6 643 98 da. 10.5023 Focal date Nov. 0. $1833 $13.2053 43 )1.833 Int. for 6 da. .3055)13.2053 .3055 Int. for 1 da. 12 220 9853 9165 688 Nov. + 43 da. = Dec. 13. Dec. 18 1833. 19 days at 6% Jan. J. 18.33 Int. for 60 da. at 6% 19 days 4.5825 Int. for 15 da. at 6% .9165 Int. for 3 da. at 6% .3055 Int. for 1 da. at 6% 5.8045 Int. for 19 da. at 6% 1833 $1838.80 Cash Bal. Jan. 1, 1916 . f Dec. 13, 1915. ^^^' I $1838.80. Goods billed on September 4 for 2 months' credit would be due on November 4. A bill bought October 23 on 30 days' credit would be due November 22. A bill bought November 18 on 10 days' credit would be due November 28. A bill bought December 8 on 60 days' credit would be due on February 6, Assuming Novem- ber as the focal date and averaging as in par. 397, we find the average due date to be December 13, 1915. If the debtor should pay $1833 on December 13, but did not do 112 RATIONAL ARITHMETIC so until January 1, he would owe $1833 plus 19 days' interest. The interest on $1833 for 19 days is $5.8045, making the amount due January 1, $1838.80. Note. For practice problems in averaging accounts and finding the cash balance see par. 69. 399. Problem : Dr. C. E. Batchelor Cr. 1915 1915 May 8 30 days $525.30 May 15 Cash $300. June 7 60 days 415.40 June 5 Cash 425.50 When is the above due by average ? What was the cash balance December 15, 1915 .^^ May 8+30 da. = June 7 $525.30 38 $3.3269 June 7 + 60 da. = Aug. 6 415.40 98 6.7848 940.70 10.1117 725.50 3.303 $215.20 $6.8087 May 15 $300 15 $.75 June 5 425.50 36 2.553 $725.50 $3,303 Focal date May 0. 190 6 ).2152 Int. for 6 da. at 6% .0358)6.8087 .0358 Int. for 1 da. at 6% 3 58 3 228 3 222 67 RATIONAL ARITHMETIC 113 May + 190 days = Nov. 6, 1915. Nov. 6 to Dec. 15 is 39 days. $^15.20 2.15 Int. for 60 da. at 6% 1.076 Int. for 30 da. at 6% .2152 Int. for 6 da. at 6% .1076 Int. for 3 da. at 6% 1.3988 Int. for 39 da. at 6% 215.20 $216.60 Cash Bal. Dec. 15, 1915 Ans. Nov. 6, 1915. $216.60. In the above account C. E. Batchelor owes the items on the debit side of the account and we owe him, theoretically, the items on the credit side. The debit side being the larger, he owes us the balance of the account. First find the due dates of all the items. On the debit side $5'25. 30 is due June 7. $415.40 is due August 6. On the credit side $300 is due May 15. $4*25.50 was due June 5. Assuming the zero date of the earliest month as the focal date, this w^ould be May 0. Batchelor, by paying $525.30 on May 0, would lose 38 days' interest, which is $3.3269, and by paying $415,40 on May he would lose 98 daj's' interest, which is $6.7848. If the account were settled on May 0, Batclielor would gain the interest on $300 for 15 days, which is $.75 and he would gain the interest on $425.50 for 36 days, which is $2,553. This would amount to $3,303. The balance of the account is $215.20. If Batchelor paid this on May 0, he would lose $10.1117 and gain $3,303, or he would make a net loss of $6.8087. The interest on $215.20 for one day is $.0358. To make up $6.8087, it would take as many days as $.0358 is contained in $6.8087, which is 190 times, or 190 days. Since by settling the account on May the interest for 190 days is lost, it should be settled 190 days later, which would be Novem- ber 6. 114 RATIONAL ARITHMETIC Whenever the balance of interest and balance of account fall on the same side, the payer will lose the interest and the time should then be counted forward. 400. Problem Dr. F. H. Bray Cr. 1916 1916 June 6 2 mo. $2200 Aug. 12 Cash $ 108 Aug. 12 12 da. 1400 Sept. o Cash 2892 Find the cash balance for Sept. 12, 1916, at ytyc. Focal date August 0. Aug. 6 24 $2200 1400 $3600 3000 $ 600 6 24 2.20 5.60 7.80 Aug. 12 $ 108 12 .216 Sept. 5 2892 36 17.352 $3000 17.568 7.80 9.768 6 ). 600 Int. for 6 da. at u/o .10 Int. for 1 da. at 6% 97 .10)9.76 90 76 70 6 8 68 RATIONAL ARITHMETIC 115 97.6 days = 98 days. 98 days counted backward from Aug. 0= April 24, Average Date. April 24 to Sept. 12 = 141 days. Balance of account 600 $600. 6. 14.10 12. -- = 120 $614.10 2. = = 20 .10 = = 1 14.10 = = Int. for 141 da • Ans. $614.10. The above problem is similar to 399, except that the balance of account and the balance of interest fall on opposite sides. This amount shows that F. H. Bray owes the balance of $600. By pay- ing this balance on August he would lose the interest on $2''200 for 6 days and the interest on $1400 for 24 days, or $7.80 on both items, because he would be paying before due. However, his loss of interest on the items on the debit side of the account is more than offset by his gain of interest on the items on the credit side (the interest on $108 for 12 days and the interest on $2892 for 26 days) which is $17,568. The difference between $17,568 and $7.80 is $9,768, which is the amount of interest that Bray would gain by paying the balance on August 0. The balance, $600, requires 98 days to earn $9,768. Then Bray, in order to neither gain nor lose, should settle the account 98 days before iVugust 0, which is April 24. When the balance of account and the balance of interest fall on opposite sides, count the time backward from the focal date. Note. For practice problems in finding the amount due by average and the cash balance of two-sided accounts see par. 70. TAXES 401. A Tax is a sum of money levied upon a citizen or his property to meet the expenses of maintaining the Government. (a) Taxes are levied to pay the expenses of the city, county, state, and United States. (h) The first three are levied directly upon the person, property, and, in some cases, the income of the individual. (c) In case of the United States, the tax is levied through duties and customs and by a tax on incomes. 402. A Poll Tax is a tax levied on the person, and in most states is assessed upon all male citizens of 20 years of age or more. 403. A Property Tax is a tax assessed on property, either real or personal, and is levied upon all persons owning taxable property, irrespective of age or sex. 404. An Income Tax is a tax upon incomes, and is levied alike on all citizens receiving certain incomes, regardless of age or sex. Income taxes are levied by the United States Government and by some states, in accordance with laws passed by Congress or State Legislatures. Note. For practice problems in taxes see par. 71. 116 RATIONAL ARITHMETIC 117 DUTIES AND CUSTOMS 405. Duties and Customs are taxes assessed by the United States Government on imported merchandise. 406. An Ad Valorem Duty is a certain percentage of the net cost (value) of the importation. 407. A Specific Duty is a specified sum levied on each article, or on each unit of measure, regardless of the value. (a) Ad valorem duties are not computed on fractions of a dollar. Cents are disregarded for less than 50 and are considered another dollar for more than 50. (b) Some articles are subject to both ad valorem and specific duties. (c) Specific duties are not computed on fractions of a unit. The long ton, or 2240 pounds, is used in computing specific duties. 408. A Tariff is a schedule showing the different rates of duties imposed by Congress on different articles. 409. A Free List is a schedule of articles upon which no duties are to be levied. 410. A Customhouse is a branch office of the Treasury Department of the United States Govern- ment. Customhouses are established at various ports ; each custom- house has jurisdiction over certain territory. 411. A Port of Entry is a port where a customhouse is established. 412. All ports, whether of entry or otherwise, are called ports of delivery. 118 RATIONAL ARITHMETIC 413. The Customhouse Business is distributed among three departments. 414. The Collector's Office takes charge of entries and papers, issues permits, and collects the duties. 415. The Surveyor's Office takes charge of the vessels and cargoes, receives the permits, ascertains the quantities, and delivers the merchandise to the importer. 416. The Appraiser's Office examines the mer- chandise and determines the value and rate of duty on the goods. 417. Internal Revenue is a revenue raised by the Government by placing duties on such articles of luxury as may be determined by Congress. These duties are collected by the Treasury Department and vary from time to time according to the needs of the country. 418. A Manifest is a memorandum, signed by the master of the vessel, showing the name of the vessel, its cargo, and the names and addresses of the con- signors and consignees. 419. An Invoice is a detailed statement showing the items and value of the goods imported and is made out in the weights and measures of the country of export. The values of foreign moneys are periodically proclaimed by the Secretary of the Treasury, and these values must be taken in estimating duties. See par. 468. RATIONAL ARITHMETIC 119 420. A Bonded Warehouse is a warehouse provided for the storage of goods upon which duties have not yet been paid. (a) Any importer may deposit goods in the warehouse by giving bond for the payment of duties on the goods thus stored. (6) On goods remaining in bond more than a year, 10% addi- tional duty is charged. (c) Goods left in a government warehouse for three years are forfeited to the Government and sold at auction. (d) Goods may be withdrawn from a warehouse for export without payment of duty. 421. An Excise Duty is a tax levied upon goods pro- duced and consumed in the United States. In this class come taxes upon tobacco and such articles of luxury as Congress may from time to time prescribe. 422. If goods on which either excise or import duties have been paid are exported, the amount of duty is refunded. This is called a drawback. Note. For practice problems in duties and customs see par. 72. INSURA.NCE • 423. Insurance is a contract bj^ which one party (the insurer) agrees to reimburse another party (the insured) in case of damage or loss to the latter's prop- erty or person. 424. The insurance business is usually conducted by corporations called Insurance Companies, which limit their operations to certain classes of risks. Some companies handle fire insurance, others marine in- surance, others accident insurance, others life insurance, etc. 425. There are two kinds of insurance companies, stock companies and mutual companies. 426. A Stock Company is one whose capital is owned by stockholders who share the profits and who are liable for the losses. 427. A Mutual Insurance Company is one in which there are no stockholders, but in which the parties insured share the profits and losses. 428. A Policy is the contract of insurance. 429. A Premium is the sum paid for the insurance. It is the consideration. 120 RATIONAL ARITHMETIC 121 (a) Sometimes the rate is expressed as a certain percentage of the value insured and sometimes at so much per hundred dollars of insurance. (b) The rate of premium depends upon the amount and the nature of the risk and the length of time for which the risk is taken. 430. Fire Insurance is insurance against loss or damage by fire, or from the means employed for ex- tinguishing it, or to prevent its spread. 431. Owners of property may insure in one or more companies. When the risk is placed in several com- panies, care should be taken to have the policies uniform in every particular. Each company will then pay such part of the total loss as its risk is of the total risk, 432. The Average Clause, contained in manj^ policies, is to the effect that the liability of the company in case of a partial loss shall be such part of the loss as the insured value is of the actual value of the property. Thus, a building worth $20,000 is insured for $18,000, which is nine-tenths or 90% of the value. In case of loss the company would pay, under the average clause, 90% of the loss. 433. Short Rate is the rate for less than a year. 434. Marine Insurance is insurance against loss or damage to a vessel or her cargo by storm or other dangers of the sea. Marine policies always contain the average clause. Note. For practice problems in insurance see par. 73. 122 RATIONAL ARITHMETIC LIFE INSURANCE 435. Under this head are considered life insurance^ accident insurance, and health insurance. 436. Life Insurance is indemnity for loss of life. 437. Accident Insurance is indemnity for loss or disability caused by accident. 438. Health Insurance is indemnity for loss oc- casioned by sickness. 439. There are two kinds of life insurance policies, life policies and endowment policies. (a) Under the ordinary life policy, premiums are paid annually during the life of the insured. {b) There is another kind of life policy known as the Limited Payment Life Policy. Under this policy the premiums are paid during a certain number of years only. 440. A Life Policy is a contract on the part of the insurance company to pay the beneficiary a designated sum upon the death of the insured. 441. An Endowment Policy is a contract on the part of the insurance company to pay the beneficiary at the death of the insured, or after the lapse of an agreed period of time, if the insured is then alive. 442. The following table shows the rates charged by an insurance company : RATIONAL ARITHMETIC ns LIFE PREMIUMS ENDOWMENT PREMIUMS Insurance of $1000, payable at death only Insurance of $1000, payable as specified or on prior decease Annual Premiums DURING Annual Payments Age Age Life 10 Years 20 Years In 15 Years In 20 Years 20 $18.95 $43.85 $27.65 20 $68.10 $49.45 21 19.35 44.55 28.10 21 68.20 49.55 22 19.75 45.25 28.55 22 68.25 49.65 23 20.20 46.00 29.00 23 68.35 49.75 24 20.65 46.75 29.55 24 68.45 49.85 25 21.15 47.55 30.05 25 68.55 50.00 26 21.65 48.40 30.60 26 68.70 50.10 27 22.20 49.25 31.15 27 68.80 50.25 28 22.75 50.15 31.75 28 68.95 50.40 29 23.35 51.10 32.35 29 69.10 50.55 30 23.95 52.05 33.00 30 69.25 50.75 31 24.60 53.05 33.65 31 69.40 50.95 32 25.30 54.10 34.35 32 69.55 51.15 33 26.05 55.20 35.05 33 69.75 51.35 34 26.80 56.30 35.80 34 69.95 51.60 35 27.65 57.45 36.60 35 70.20 51.90 36 28.50 58.65 37.45 36 70.40 52.15 37 29.40 59.95 38.30 37 70.70 52.50 38 30.35 61.25 39.20 38 71.00 52.85 39 31.40 62.60 40.15 39 71.30 53.25 40 32.50 64.00 41.20 40 71.65 53.70 By this table the premium on an ordinary life policy of $ 1000 at the age of 20 would be $18.95, for a $5000 policy the premium would be 5 X $18.95, or $94.75. At the same age, a $1000 ten-payment life policy would cost $43.85 per year, and a $5000 policy 5 X $43.85, or $2 19. 25 per year. A $1000 fifteen-year endowment policy would cost $68.10 a year, and a $5000 policy would cost $340.50. Note. For practice problems in life insurance see par. 74. EXCHANGE 443. Exchange is a system of paying debts in distant places by means of drafts. 444. A Draft, or Bill of Exchange, is an order of one party directing a second party to pay a third party a certain sum of money at a certain time. 445. There are two kinds of bills of exchange, do- mestic and foreign. 446. A Domestic Bill of Exchange, sometimes called an inland hill of exchange, is one drawn and payable in the same state or country. 447. Foreign Bills of Exchange are those drawn in one state or country and payable in another state or country. 448. The Face or Par Value of a bill of exchange is the sum of money for which it is written. Bills of exchange are always written in the coinage of the country in which they are to be paid. DOMESTIC EXCHANGE 449. Domestic Exchange quoted at a premium is worth the given percentage of the face more than the face. 124 RATIONAL ARITHMETIC 125 Domestic Exchange quoted at a discount is worth the quoted percentage of the face less than the face. (a) Thus, ^% premium means that $1 would cost $1,005. Ex- change quoted at |% discount would mean that $1 would cost $.995. (b) When Boston owes New York the same that New York owes Boston, exchange will be at par in botli places. AMien Boston owes New York more than New York owes Boston, exchange on New York will be at a premium in Boston, since there will be more buyers of New York exchange than sellers ; and when New York owes Boston more money than Boston owes New York, exchange on New York will sell at a discount in Boston and exchange on Boston will sell at a premium in New York. The general principles of percentage (pars. 242-254) are used in solving problems in exchange. Note. For practice problems in domestic exchange see pars. 75 to 80 inclusive. FOREIGN EXCHANGE 450. Foreign Exchange is exchange drawn in one country and payable in another country. Foreign bills are always made in the coinage of the country where they are to be paid. 451. The Intrinsic Par of Exchange is the actual value of the money of one country expressed in the monev of another. Intrinsic value of foreign money expressed in the money of the United States will be found in the table, par. 468. 452. Commercial Rate of Exchange is the market value of the money of one country expressed in the money of another. 126 RATIONAL ARITHMETIC (a) This value changes from time to time, according to the demand that may exist and according to the different conditions of commerce that may arise. (&) The rate of exchange on Great Britain is expressed by giv- ing the market value of a pound in United States money. (c) On France, Belgium, and Switzerland the rate of exchange is expressed by giving the number of francs that may be secured for $1. (d) On Germany the rate of exchange is expressed by giving the market value of 4 reichsmarks in United States money. (e) On Holland the rate of exchange is expressed by giving the value of 1 guilder in United States money. (/) Gold is exported at a profit when the cost of foreign ex- change is enough greater than the intrinsic value of the bill to pay the cost of safe shipment and yet leave a margin ; and gold is im- ported at a profit when the cost of exchange is enough less than the intrinsic value of the bill to pay the same expenses and leave a margin. Thus under normal conditions, the commercial rate is not allowed to vary from the intrinsic par by more than enough to pay the expense of shipping gold. Note. For practice problems in foreign exchange see pars. 81 to 85 in- clusive. STOCKS AND BONDS 453. A Corporation is an association of persons authorized by law to act as one person. 454. The Capital Stock of a corporation is the value of its investment. This is divided into equal parts called shares. 455. The Par Value of a share of stock is the value placed upon each share at the time of the original division of its capital stock. The usual par value of one share of stock is $100. Stock is frequently issued in other sized shares, however, usually $50, $25, $10, $5, or $1. 456. A Stock Certificate is a document issued by the company to the shareholder specifying the number and par value of the shares to which he is entitled. 457. The Market Value of a share is the sum for which it will sell in the open market. Sometimes stock is worth more than par and sometimes less. This depends upon the condition of the business. 458. A Dividend is that part of the net earnings of a corporation that is divided among its stockholders. 127 128 RATIONAL ARITHMETIC 459. An Assessment is a sum levied upon the stock- holders to make up losses. There are two kinds of stock, common and preferred. 460. Common Stock participates in the net earnings of the company, after all other expenses have been met, in such proportion as the directors of the cor- poration may determine. 461. Preferred Stock participates in the net earnings of the corporation at a fixed rate before any dividend may be declared on the common stock. 462. A Bond is an obligation of a corporation to pay money on a long term of credit. (a) Bonds are usually secured by deeds of trust and mortgages. They are generally issued as securities for loans. They are similar to promissory notes, but are more formal and are also made under seal. (6) Bonds are usually issued in $500, $1000, or multiples thereof. (c) Quotations on bonds are given on $100 par value. {d) Bonds are issued in two classes, registered and coupon. 463. Registered Bonds are those payable to the order of the owner and can be transferred only by acknowledged assignment. Interest on registered bonds is paid by check from the corpo- ration made to the holder of record. 464. A Coupon Bond is one made payable to the bearer, and has interest certificates attached. These certificates, called coupons, are to be cut off as they RATIONAL ARITHMETIC 129 become due and presented at the designated place for payment. (a) Bonds are named from the nature of the security ; the name of the corporation issuing tliem ; the date on which they are payable ; the rate of interest they bear ; or the purpose for which they are issued. (h) Both stocks and bonds are quoted at some per cent of par value. (c) The regular commission allowed to brokers for buying or selling either stocks or bonds is |% of the par value. There is a minimum charge for small transactions, however. (d) Dividends and assessments are always figured on the par value of the stock. Note. For practice problems in stocks and bonds see par. 86. TABLES UNITED STATES MONEYS 465. United States Money consists of gold coins, silver coins, United States Treasury notes and certifi- cates, and national bank notes. The unit of measure is the gold dollar of 25.8 grains. 10 mills =lcent(^) $ .01 10 cents =1 dime (d.) $ .10 10 dimes =1 dollar ($) $ 1. 10 dollars = 1 eagle (e.) $10. 20 dollars = 1 double eagle (d. e.) $20. ENGLISH MONEYS 466. English or Sterling Money is the legal currency of Great Britain. The unit of measure is the pound, worth $4.8665 in United States money. 4 farthings = 1 penny (d) 12 pence = 1 shilling (s) 20 shillings = 1 pound (£) Note. 21 shillings = 1 guinea (used in the retail trade). 130 RATIONAL ARITHMETIC 131 FOREIGN MONEYS 467. Once each year the Director of the United States Mint is required to compare the values of foreign coins with the United States Gold Dollar and certify the result of his comparison to the Secretary of the Treasury, who then proclaims the value of foreign money thus found to be the value to be used in esti- mating the worth of all foreign merchandise imported. Values thus found are called intrinsic or real values and should be distinguished from commercial or ex- change values. 468. The table of values on pages 132 and 133 was proclaimed Oct. 1, 1918. WEIGHT Troy Weight 469. Troy Weight is used in weighing precious metals. TABLE 24 grains (gr.) = 1 pennyweight (dwt.) 20 penny weights = 1 ounce (oz.) 12 ounces = 1 pound (lb.) Diamond Weight 470. Diamond Weight is used in weighing precious stones. The unit is 3^ Troy grains and is called a carat. This carat is not the same as that used in estimating the rela- tive purity of gold in coins and jewelry. Pure gold is 24 carats fine ; 18 carats fine means ^f pure gold and ^ alloy. 132 RATIONAL ARITHMETIC to •a M o O 'A o •H « O b O OQ P < > CO -a c3 s Pi u C o ^ o O 1^ =^ a; a^a c3 o a X T3 •- ^^ •r; e3 . % >y tH fl c3 ..*^ > >.$ t4 c o bl) S 03 bi O a 03 o o ^ -" a o 1-1 o o 03 a a u X o S c OS B 13 fl O 0) 03 to G C3 o u o a 03 ft CO EC (X) . "^ c 2 1* ^ g ^ "^ o >>o ^ +-> 03 4^ +i Ol i-i ri OJ o ITt H > 3 -f^ 0) ^.2 II .So-- '=' c=? - o ^^ >, ^ c^ c €©£11 £ ni - "^ '^ ft 3 O 3 I3O £o "^ .. Q .S ^ OT3io O 03^ C !«! 3 oi 09% HO ^ ii :S-'-5 . > o,j. o fl • o o <*- 03 CO o 00 CD CD O o 03 CO bO fl o t-l ^ w h_, ?^or CO o3 o — ^ o o o o o CO V ■3 § 43 »-iOt^ t^OiO a C co.i; -2 03 > c3 00 •-^ ■+^ V t-> m & a> +^ 03 ca ■^ a> o ii e8 O a> 03 o o fl 10 b< C3 bS fl c3 03 fl e8 'S 03 1-1 > • fl c4 2 O 0000 22 00 222 o'o'o 000 (-1 (a > 2 O fl fl fl a: 03 (1 43 bO fl 03 rfl o - C fl flajOut--*^Ti) t-.fl > •-; ^. 03 2 =s fc fl t- o ^-C c3 C3 fl > Office -2 fl -fl o RATIONAL ARITHMETIC 133 bo's o 12 o « • — a; ' C3 o o o a a c3 c O •So 53 - rt cl a; 5 rt „ t5 a J Co 2 III o tx, o "^ Si c3 ;:oo C3 3 o ^ > a 0-2 t, M -^ Sao 03 g C c3 O) O 13 c3 c3 o - Coo s 73 n o3 o 03 <» bCcc c *^^ 13 .lO OlCCv) .2 lO TJH c « (u Q =3 "3 . ^ > C " c G =3 c3 s- o o 3 >^ >< OHW o 00 o CO M C c3 O c3 a o3 o X a c3 a a> -*^ _c3 'o o a M C 03 o IX >^ " o ^ tc C O 4) :3 H O C3 Q (4 a 03 a M O (N a> (A a> bfl C =3 o « O c o c QJ (1 • jhCO o • C CO Is u o3 00 (a a o 13 ^" O 03 MJ II ^ CO O.S^ «*^ J o ? ^^ ^ HO • - .~oo o(N_a II <^ nO .S S 03' .2 '^ "Sc 3 S 03^ ■^ "^ a 5 cooot^ t-H 00 coooo o cococo t~- c o CO "# o; CTi c; o (N •* 03 -H i-H (M iC CO 00 CO COO CO CD 03 10 (MOO --KM TfiOLOOiOOOOOX TtiCOXOXC^OXOTH iM cr- CO>-i -t OC: C CO O :_ C Tf TTI O C^l C t> O (M i-O O lO o o c o c c Lococoocco-^ococco C CO O c X ; Cl— iXiCO>-HiOO»-iCO'-H r^03 CO X CO o OiMC 'J'^t CO ;o3 OC-"-! CO IH OJ +i CO "a o o (3 ^ So ^ " =^ ;:; S?L,QO o . . t; . fi b (3 ^ c oj .2 c3 > g ac3 == o-g= §£ Ph £ C 'S ^IJ^ O CO ^ - 13 -^ g ^_. _^~ 2 03 o -< X'^:- o p 03 Ofo 03 CMfi^eQ o ^ '» 13 c c3 13 "^ 13 13 "O "0 13 ©"o o o o"© o 0000000 > 13 • c o o3 o 0000 I- o > 13 • C! 02 13 13 13 13 T! 13 73 13 13 P "o o o'o'o'o o o"© — OOOOOOOOOm (1 13 • G c3 ^ C^ 'C "C "^ X5 '^ 13 1j '^ "o— o o'o'o'o o c"o o OmOOOOOOOOO 22 o o 00 'o'o3 000 03 s o So -o „ 03 ~ c3 aeS 0305 M.b £; a; • "So S'ol «-^ os*^ a; 03 QJ ti o a; O fc- (-1 „ „ 53 OOOB J3 t4 m c3 r^ 03 >>5 c3 g & c3 c u c -22 1^ ^2 o 5.0 2 >>5 S"x-^"? ■fij §'^»S'S ocas c3 3 c3 03 CO 13 e 03 Q^ 03 .3-3"S o tlO a •3 -"^ 5 'Q o ^ p o 3 3 ^3 .2 D 02 02 o3 >.2 3 a) -^ K! i; =^ f^ 3 »-i 4) 134 RATIONAL ARITHMETIC Apothecaries' Weight 471. Apothecaries' Weight is used by physicians and apothecaries in writing and preparing prescriptions for dry medicines. TABLE 20 grains (gr.) = 1 scruple (sc. or 3) 3 scruples = 1 dram (dr. or 3) 8 drams = 1 ounce (oz. or §) 12 ounces =1 pound (lb.) Avoirdupois Weight 472. Avoir d 'pois Weight is used in commerce in all cases excepting those requiring Troy or Apothecaries' weight. TABLE 16 ounces =1 pound (lb.) 25 pounds = 1 quarter (qr.) 4 quarters = 1 hundredweight (cwt.) 20 hundredweights = 1 ton (T.) 2240 pounds = 1 long ton 473. Comparison of Troy and Avoirdupois Weights. 1 pound Troy = 5760 grains 1 pound Avoirdupois = 7000 grains 1 ounce Troy = 427 grains 1 ounce Avoirdupois = 480 grains 474. The following table shows the weight of a bushel used commercially in measuring grain and other farm products : RATIONAL ARITHMETIC 135 Barley 48 1b. Oats 321b Beans 60 " Onions 57 " Buckwheat 48 " Peas 60 " Clover Seed 60 " Potatoes 60 " Corn, shelled 56 " 'I'imothy Seed 45 " Corn, in the ear 70 " Rye 56 " Corn Meal 50 " Rye Meal 50 " Flaxseed 56 " Wheat 60 " Hemp Seed 44 " Wheat Bran 20 " Malt 34 " Liquid Measure 475. Liquid Measure is used in measuring liquids. TABLE 4 gills (gi.) = 1 pint (pt.) 2 pints = 1 quart (qt.) 4 quarts = 1 gallon (gal.) 476. Standard liquid gallon contains 231 cubic inches. There are various kinds of casks for containing liquids. In commerce each is gauged and its capacity marked upon it. The various kinds of casks are : 1 lerce about 42 gal. Puncheon 84 " Pipe 126 " Butt 126 " Tun 252 " Hogshead (hhd.) 63 " 136 RATIONAL ARITHMETIC Apothecaries' Liquid Measure 477. Apothecaries'' Liquid Measure is used in pre- scribing and compounding liquid medicines. TABLE 60 minims ("l) =1 fluid drachm (f3) 8 fluid drachms = 1 fluid ounce (f i) 16 fluid ounces = 1 pint (O) 8 pints = 1 gallon (Cong.) The gallon of this measure is the same as the gallon of the liquid measure. Dry Measure 478. Dry Measure is used in measuring grain, fruits, vegetables, etc., which are not sold by weight. TABLE 2 pints (pt.) = 1 quart (qt.) 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bu.) Long Measure 479. Long Measure is used in measuring lengths, or distances. TABLE 12 inches (in.) = 1 foot (ft.) 3 feet = 1 yard (yd.) 5i yards (16i ft.) = 1 rod (rd.) 40 rods = 1 furlong (fur.) 8 furlongs (320 rods) = 1 mile (mi.) RATIONAL ARITHMETIC 137 Surveyors' Long Measure 480. Surveyors' Long Measure is used by surveyors in measuring distances. TABLE 7.92 inches = 1 link (1.) 100 links = 1 chain (ch.) 80 chains = 1 mile (mi.) Square Measure 481. Square Measure is used in measuring extent of surfaces. TABLE 144 square inches (sq. in.) = 1 square foot (sq. ft.) 9 square feet = 1 square yard (sq. yd.) 30i square yards = 1 square rod (sq. rd.) 40 square rods = 1 rood (R.) 4 roods = 1 acre (A.) 640 acres = 1 square mile (sq. mi.) Surveyors' Square Measure 482. Surveyors' Square Measure is used by surveyors in finding the area of land. TABLE 625 square links (sq. 1.) = 1 square rod or pole (sq. rd. or p.) 16 poles =1 square chain (sq. ch.) 10 square chains = 1 acre (A.) 640 acres = 1 square mile (sq. mi.) 138 RATIONAL ARITHMETIC Cubic Measure 483. Cubic Measure is used in measuring the con- tents of anything which has length, breadth, and thickness. TABLE 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) Wood Measure 484. Wood Measure is used in measuring wood. TABLE 16 cubic feet = 1 cord foot (cd. ft.) 8 cord feet (128 cu. ft.) = 1 cord (cd.) A cord of wood is a pile 8 feet long, 4 feet wide, and 4 feet high, or its equivalent. 485. TIME 60 seconds (sec.) = 1 minute (min.) 60 minutes = 1 hour (hr.) 24 hours = 1 day (da.) 7 days = 1 week (wk.) 30 days = 1 month (mo.) 52 weeks = 1 year (yr.) 12 months = 1 year (yr.) 365 days = 1 common year 366 days = 1 leap year 100 years = 1 century RATIONAL ARITHMETIC 139 486. The day is the time during which the earth makes one revolution on its own axis. 487. The Solar Year is the time the earth requires to make one complete revolution around the sun. It actually takes the earth 365^ days to make this revolu- tion. Therefore, every fourth year is given 366 days. This extra day is added to the month of February, the shortest month, and the year is called Leap Year. These figures are not absolutely accurate but are practically so. 488. Any year whose number can be divided by 4 is a leap year, except that a century year must be divisible by 400. (a) The year 1916 could be divided by 4 and was a leap year, while 1915 could not be divided bv 4 and was an ordinarv year. (b) The year 1900 was divisible by 4 but not by 400 and was not, therefore, a leap year. The year 2000 being divisible by 4 and 400 will be a leap year. 489. Months of the Year, and Days in Each : 1. Januarv 31 7. July 31 2. February 28 or 29 8. August 31 3. March 31 9. September 30 4. April 30 10. October 31 5. Mav 31 11. November 30 6. June 30 12. December 31 140 RATIONAL ARITHMETIC MISCELLANEOUS 490. Some articles are sold by quantity according to the following table : TABLE 12 units = 1 dozen (doz.) 12 dozen = 1 gross (gr.) 12 gross = 1 great gross (g. gr.) 20 units = 1 score Paper Measure 491. Paper is measured according to the following table : TABLE 24 sheets = 1 quire (qu.) 20 quires = 1 ream (rm.) 2 reams = 1 bundle (bdl.) 5 bundles = 1 bale (bl.) THE METRIC SYSTEM 492. The Metric System is a decimal system of weights and measures, similar to the decimal system used in measuring United States money. It was originated in France early in the nineteenth century, and has been adopted by nearly all the commercial nations except United States and England. The Metric System was made legal in the United States in 1866, but is not generally used except in scientific work. RATIONAL ARITHMETIC 141 493. The Meter. The basic unit is the meter. The other units, those of weight and of capacity, are based on it. 494. The length of the meter was originally deter- mined by taking one ten-millionth of the distance from the equator to the pole. This length is 39.37 inches. 495. The primary units are : For length — meter For capacity — liter For weight — gram 496. The desired integral multiples of these are formed by using the following Greek prefixes : Deca =10 (decameter = 10 meters) Hecto = 100 (hectometer = 100 meters) Kilo = 1000 (kilometer = 1000 meters) Myria= 10,000 (myriameter= 10,000 meters) 497. To designate decimals of a meter, the following Latin prefixes are used : Deci = To (decimeter = ro meter) Centi = -TWO (centimeter = two meter) Milli = ToVo (millimeter = x^oo meter) The most commonly used denominations in the following tables are indicated by heavy-faced type. 142 RATIONAL ARITHMETIC Linear Measure 498. The unit of Linear Measure is the meter. TABLE 10 millimeters (mm.) = 1 centimeter (cm.) 10 centimeters = 1 decimeter (dm.) 10 decimeters = 1 meter (m.) 10 meters = 1 decameter (dm.) 10 decameters = 1 hectometer (hm.) 10 hectometers = 1 kilometer (km.) 10 kilometers = 1 myriameter (mm.) Square Measure 499. The unit of Square Measure is the square meter. TABLE 100 square milHmeters = 1 square centimeter (cmq.) 100 square centimeters = 1 square decimeter (dmq.) 100 square decimeters = 1 square meter (mq.) 100 square meters = 1 square decameter (dcmq.) 100 square decameters = 1 square hectometer (sq. hm.) 100 square hectometers = 1 square kilometer (sq. km.) Land Measure 500. The unit of Land Measure is the are. TABLE 100 centiares (ca.) = 1 are (a.) = 100 sq. m. 100 ares = 1 hectare (ha.) = 10,000 mq. RATIONAL ARITHMETIC 143 Cubic Measure 501. The unit of volume is the cubic meter. TABLE 100 cubic millimeters (cmm.) = 1 cubic centimeter (cmc.) 100 cubic centimeters = 1 cubic decimeter (dmc.) 100 cubic decimeters = 1 cubic meter (mc.) Wood Measure 502. The unit of wood measure is the stere. TABLE 10 decisteres (dst.) = 1 stere (st.) = 1 cu. m. 10 steres = 1 decastere (dast.) = 10 cu. m. Measure of Capacity 503. The unit of capacity for either solids or liquids is the liter, which is equal in volume to 1 cu. dm. TABLE 10 milliliters (ml.) = 1 centiliter (cl.) 10 centiliters = 1 deciliter (dl.) 10 deciliters = 1 liter (1.) 10 liters = 1 decaliter (dl.) 10 decaliters = 1 hectoliter (hi.) 10 hectoliters = 1 kiloKter (kl.) Measure of Weight 504. The unit of weight is the gram, which is the weight of 1 dmc. of distilled water in a vacuum, at its greatest density. It weighs 15.4324 gr. 144 RATIONAL ARITHMETIC TABLE 10 milligrams (mg.) = 10 centigrams = 10 decigrams = 10 grams = 10 decagrams = 10 hectograms = 10 kilograms = 10 myriagrams = 10 quintals = centigram (eg.) decigram (dg.) gram (g.) decagram (dg.) hectogram (hg.^ kilogram (kg.) myriagram (mg.) quintal (q.) tonneau (t.) 505. TABLES OF EQUIVALENTS Convenient Equivalent Values 1 cu. cm. of water = 1 ml. of water, and weighs 1 gram = 15.432 gr. 1 cu. dm. of water = 1 1. of water, and weighs 1 kg. = 2.2046 lb. 1 cu. m. of water =1 kl. of water, and weighs 1 tonneau = 2204.6 lb. 506. Measures of Weight 1 grain, Troy = .0648 of a gram 1 ounce, Troy =31.104 grams 1 ounce, Avoir. =28.35 grams 1 lb. Troy = .3732 of a kilogram 1 lb. Avoir. = .4536 of a kilogram 1 ton (short) = .9072 of a tonneau or ton RATIONAL ARITHMETIC 145 1 gram 1 gram 1 gram 1 kilogram 1 kilogram 1 tonneau = 15.432 grains, Troy = .03215 of an oz. Troy = .03527 of an oz. Avoir. = 2.679 lb. Troy = 2.2046 lb. Avoir. = 1.1023 tons (short) 507. Measures of Capacity 1 dry quart =1.101 liters 1 liquid quart 1 liquid gallon 1 peck 1 bushel = .9463 of a liter = .3785 of a decaliter = .881 of a decaliter = .3524 of a hectoliter 1 liter 1 liter 1 decaliter 1 decaliter 1 hectoliter = .908 of a dry quart = 1.0567 liquid quarts = 2.6417 liquid gal. = 1.135 pecks = 2.8377 bushels 508. 1 inch 1 foot 1 yard 1 rod 1 mile Linear Measure = 2.54 centimeters = .3048 of a meter = .9144 of a meter = 5.029 meters = 1.6093 kilometers 146 RATIONAL ARITHMETIC 1 centimeter = .3937 of an inch 1 decimeter = .328 of a foot 1 meter =1.0936 yards 1 dekameter =1.9884 rods 1 kilometer = .62137 of a mile 509. Surface Measure sq. inch =6.452 sq. centimeters sq. foot = .0929 of a sq. meter sq. yard = .8361 of a sq. meter sq. rod =25.293 sq. meters acre =40.47 ares sq. mile =259 hectares sq. centimeter = .155 of a sq. inch sq. decimeter =.1076 of a sq. foot sq. meter =1.196 sq. yards are =3.954 sq. rods hectare =2.471 acres sq. kilometer = .3861 of a sq. mile 510. Cubic Measure 1 cu. inch = 16.387 cu. centimeters 1 cu. foot =28.317 cu. decimeters 1 cu. yard = .7646 of a cu. meter 1 cord =3.624 steres 1 cu. centimeter = .061 of a cu. inch 1 cu. decimeter = .0353 of a cu. foot 1 cu. meter =1.308 cu. vards 1 stere = 275) of a cord INDEX (Figures refer to paragraph numbers.) Accident insurance, 437. Account purchase, definition, 309. Account purchases, problems in making, 48. Account sales, definition, 308 ; prob- lems in making, 48. Accurate interest, definition, 325 ; illustrated solutions, 327, 328, 341, 343; problems in finding, 56. Addend, definition, 97. Addition, compound numbers, 233; decimals, 120, 131 ; definition, 88; fractions, 18, 201, 205; in- tegers, 1, 92, 99; proof of, 100. Ad valorem duty, 406. Aliquot parts, definitions, 240; illus- trated solution, 241 ; problems, 27, 30; table of, 241. Amount, 250, 322, 369. Amount of purchase, to find, illus- trated solution, 311. Annual interest, 389; illustrated solution, 390. Apothecaries' liquid measure, 477. Apothecaries' weight, 471. Appraiser, 410. Arithmetic, definition, 87. Asking price, problems in finding, 45. Assessment, 459. Average clause, 432. Averaging accounts, definition, 391 ; general principles of. 393, 395 ; illustrated solutions, 397, 400; problems, 68; rule, 396. Avoirdupois weight, 472. Balancing accounts, 109 ; problems, 9. Bank discount, definition, 376, 379; illustrated solution, 382, 385 ; to find face, 64 ; to find proceeds, 63. Base definition, 247; problems in finding, 34; to find, illustrated solution, 259, 261. Bill of exchange, 444. Billing, exercises in, 31 ; problems, 46. Bond, 462 ; coupon, 464 ; registered, 463. Bonded warehouse, 420. Broker, definition, 300. Brokerage, see Commission. Butt, 476. Capital stock, 454. Cash balance, 392 ; problems in find- ing, 68. Casks, 476. Charges, definition, 304. Check, definition, 370. Collector of customs, 414. Commercial paper, definition, 357. Commercial rate of exchange, 452. Commission, definitions, 302, 310; to find, illustrated solution, 310, 311. Commission and brokerage, illus- trated solutions, 311, 316. Commission merchant, definition, 299. Commission problems, to find com- 147 148 INDEX mission, 47 ; to find gross cost, 47 ; to find gross proceeds, 49; to find net proceeds, 47 ; to find rate, 50 ; general problems, 51. Common divisor, definition, 153. Common fraction, definition, 165 ; changing to, 193. Common multiple, definition, 148. Common stock, 460. Complex decimal, definition, 173. Complex fractions, definition, 174. Composite numbers, definition, 144. Compound interest, definition, 386; illustrated solution, 388 ; problems in finding, 66 ; rule, 387. Compound subtraction, problems in, 52. Corporation, 453. Cost, definition, 266, 268; problems in finding, 38 ; to find, illustrated solution, 276, 278. Coupon bond, 464. Cubic measure, 483. Customhouse business, 410, 416. Customs, 405. Customs and duties, problems, 72. Date of maturity, definition, 367. Decimal fraction, definition, 166. Decimal fractions, changing to, 196. Decimals, 10; division problems, 11 ; multiplication problems, 10; com- plex, 173; mixed, 171. Denominate numbers, addition, 233; change to higher denomination, 230; change to lower denomina- tion, 231 ; changing to simple, 232 ; definitions, 226, 228 ; divi- sion, 236 ; multiplication, 235 ; reduction, illustrated solution, 229 ; reduction problems, 23, 24, 25, 26 ; subtraction, 234. Denominator, definition, 192. Diamond weight, 470. Difference, definition, 251, 105. Discount, 288; definitions, 281, 283; series, 289; to find, illustrated solution, 291, 292. Dividend, 458. Dividend, definition, 120. Division, compound numbers, 236; decimals, 11, 136; decimals, illus- trated solutions, 139, 141 ; deci- mals, rule, 137 ; definition, 91 ; fractions, 219 ; fractions, problems, 21; integers, definition, 119; in- tegers, illustrated solutions, 124, 126; integers, proof, 125, 127. Divisor, definition, 121. Domestic bill of exchange, 446. Domestic exchange, 449. Draft, 444; definition, 361. Drawee, definition, 363. Drawer, definition, 362. Dry measure, 478. Duties, 405. Endowment policy, 441. English money, 466. Exact days, problems in finding, 53. Exchange, definition, 443 ; domestic, to find face value of a draft, 78, 79 to find value of sight draft, 75 to find value of time draft, 76 foreign, 450; to find value of draft, 81. Excise duty, 421. Face, 448. Factor, definition, 114. Factoring, 142. Fire insurance, 430 ; problems, 73. Foreign bill of exchange, 447. Foreign exchange, 450. Fraction, definition, 160. Fractions, addition, 201 ; addition, illustrated solution, 203, 205; addition, problems, 18; addition. INDEX 149 rule, 202; changing to common, illustrated solution, 195; chang- ing to common, rule, 190; illus- trated solution, 198, 200 ; changing to a decimal, rule, 197; division problems, 21; division, rule, 221, 223; changing to higher terms, illustrated solutions, 186 ; chang- ing to higher terms, rule, 185 ; changing to improper fractions, rule, 191 ; change to lower terms, illustrated solutions, 181, 183; change to lower terms, rule, 180; changing to mixed numbers, illus- trated solution, 189; changing to mixed numbers, rule, 188 ; com- plex, 174 ; general problems, 22 ; multiplication, illustrated solution, 211, 212, 214, 216, 218; multipli- cation problems, 20; multiplica- tion, rules, 210, 213, 215; prob- lems in reduction, 12, 22; sub- traction, 206 ; subtraction, illus- trated solution, 207. Free list, 409. Gain, see Profit. Gram, 495, 504. Greatest common divisor, definition, 154 ; illustrated solution, 156, 158, 159. Gross amount, definition, 286 ; prob- lems in finding, 43; to find, illus- trated solution, 294, 296. Gross cost, definition, 268, 307. Gross proceeds, definition, 303; problems in finding, 49. Gross sales, to find, illustrated solu- tion, 312, 314. Gross selling price, definition, 269. Health insurance, 438. Higher terms, changing to, 184. Hogshead, 476. Improper fraction, 190; definition, 169. Income tax, 404. Insurance, definition, 423 ; problems, 73; table, 442. Interest, combinations of time, 344 ; definition, 319; problems, com- pound interest, 66; general prob- lems, 60 ; periodic interest, 67 ; to find accurate interest, 5Q; to find interest, 54, 55, 56; to find principal, 59; to find rate, 58; to find time, 57 ; rates other than six, 345; to find interest on $1, rule, 355 ; illustrated solution, 356 ; to find principal, illustrated solu- tions, 352, 354 ; to find principal, rules, 351, 352; to find rate, illus- trated solutions, 350 ; to find rate, rule, 349; to find the time, illus- trated solutions, 347, 348 ; to find the time, rule, 346. Internal revenue, 417. Intrinsic par, 451. Invoice, 419. Least common multiple, definition, 149; illustrated solution, 151. Legal rate, definition, 320. Life insurance, 436; problems, 74. Life policy, 440. Liquid measure, 475. List price, definition, 285 ; to find, illustrated solution, 294, 296. Liter, 495, 503. Long measure, 479. Loss, definition, 272; to find, 274. Lower terms, changing to, 180. Maker, definition, 359. Manifest, 418. Marine insurance, 434. Market value, 457. 150 INDEX Marking price, to find, illustrated solution, 298. Maturity, date of, 367; definition, 377. Merchants' rule, partial payment, 374; problems, 62. Meter, 493. Metric equivalents, 505, 509. Metric system, 492, 510. Minuend, definition, 103. Miscellaneous measures, 490. Mixed decimal, definition, 171. Mixed numbers, changing to, 188, 189; definition, 170. Money, English, 466; foreign, 467; table of, 468 ; United States, 465. Multiplicand, definition. 111. Multiplication, compound numbers, 235 ; problems in decimals, 10 ; problems in fractions, 20; deci- mals, 133; decimals, illustrated solution, 135; decimals, rule, 134; definition, 90 ; fractions, 208 ; integers, definition, 110; integers, illustrated solutions, 115, 117; proof, 116, 118. Multiplier, definition, 112. Mutual Insurance company, 427. Net amount, definition, 287 ; to find, illustrated solution, 291, 292. Net proceeds, definition, 305 ; prob- lems in finding, 47, 48; to find, illustrated solution, 313. Net selling price, definition, 270. Notation, 92. Note, definition, 358. Numerator, definition, 162. Ordinary interest, definition, 329; rule, 332 ; sixty-day method, 331 ; sixty-day rule, illustrated solu- tions, 3.S2, 336. Paper measure, 491. Par value, 448, 455. Partial payments, merchants' rule, 374; problems in merchants' rule, 62, 375; problems in United States rule, 61, 373; problems, 371, 374; liuited States rule, 372. Payee, 364. Pavee, definition, 360. Percentage, definitions, 243, 254; general problems, 36; to find, 33, 255, 258; to find base, 34, 259, 261 ; to find rate, 35, 262, 263. Periodic interest, 389; illustrated solution, 390. Pipe, 476. Policy, 428. Poll tax, 402. Port of delivery, 412. Port of entry, 411. Preferred stock, 461. Premium, 429. Prime cost, definition, 267, 306. Prime factors, definition, 145 ; illus- trated solution, 146. Prime numbers, definition, 143. Principal, definition, 301, 318; prob- lems in finding, 59. Proceeds, 380; definition, 303, 305; problems in finding, 63, 64 ; to find, 381. Product, definition, 113. Profit, definition, 271; to find, illustrated solution, 274. Profit and loss, general problems, 37, 38, 39, 40; illustrated solu- tions, 274, 280; problems, 42; to find cost, 38; to find profit or loss, 37; to find rate of profit or loss, 39, 279, 280. Profits and losses, definitions, 264, 272. Proper fraction, definition, 168. INDEX 151 Property tax, 403. Puncheon, 476. Quotient, definition, 122. Rate, definitions, 248, 319, 320; problems in finding, 35, 39, 44, 50, 58; to find, illustrated solutions, 262, 263, 297. Rate of discount, to find single rate equal series, illustrated solution, 291. Reduction, denominate numbers, problems, 23, 24, 25, 26 ; fractions, 12, 13, 14, 15, 16, 17. Reduction of fractions, definition, 176. Registered bond, 463. Remainder, definition, 105. Selling price, definition, 269, 270; to find, illustrated solution, 274, 275. Short rate, 433. Sight draft, 365. Sixty-day method, ordmars'' interest, illustrated solution, 333, 336; ordinary interest, rule, 332. Specific duty, 407. Square measure, 481. Stock, 454; certificate, 456; com- mon, 460; company, 426; pre- ferred, 461. Stocks and bonds, 453, 464 ; general problems, 86. Subtraction, compound numbers, 234; decimals, 120; definition, 89; fractions, 207; fractions, problems, 19; illustrated solu- tion, 132; integers, 9, 102; illus- trated solution, 106, 109; proof, 107. Subtrahend, definition, 104. Sum, definition, 98. Surveyor, customs, 415. Surveyor's long measure, 480 ; square measure, 482. Tariff, 408. Taxes, 401 ; problems, 71. Term of discount, 378. Tierce, 476. Time, 485 ; definition, 321 ; method? of computing, 238, 239; problems in finding, 52, 53. Time discount, definition, 283. Time draft, 365. Trade discount, definition, 281, 284^ 289; general problems, 46; illus- trated solutions, 291, 298; tc find asking price, 45; to find gross amount, 43; to find net amount, 50 ; to find rate, 44 ; to find single discount equal to a series, 45. Troy weight, 469. Tun, 476. United States money, 465. United States rule, partial payment, illustrated solution, 373 ; problems in partial payments, 61 ; partial payments, rule, 372. Warehouse, bonded, 420. Weight, apothecaries', 471 ; avoir- dupois, 472 ; comparison, 473 : diamond, 470; troy, 469. Weights, miscellaneous, 490. Wood measure, 484. ye 35873 \ 4SU858 r A UNIVERSITY OF CALIFORNIA LIBRARY