GIFT OF Mrs. W. Barstow PREFACE. rpHE PACKARD COMMERCIAL ARITHMETIC, of which this is a **- revision, has been before the public for a space of five years, during which time it has attained a large sale and has given satis- faction to its patrons. No special effort has been made to increase the sale by advertising or making strong statements on its behalf. The authors and publisher have well understood from the begin- ning that the success of a text-book to be used by intelligent teachers was not in any sense dependent upon promises made in advance, or claims of particular merit which might otherwise escape attention. The book grew naturally out of the practice of the school-room, and the manuscript lessons which were after- wards utilized as copy were submitted to the most exacting tests, and their value in building up the student in a sound knowledge of principles and in the details of practice thoroughly established. And the satisfaction expressed by those who have used the book, has amply sustained the judgment of the authors. The present revision has not been undertaken on account of any dissatisfac- tion expressed with the old edition, nor because it had not proved in the broadest sense effective, but because the authors have felt that some additions, and particularly in the direction of preparatory work, would make it more acceptable to a large number who now use it, while it would not in any way detract from its utility or symmetry as a complete text-book. The omission of the funda- mental rules in the old edition emphasized the fact that it was an advanced commercial text-book, intended for schools not requir- ing primary instruction. It has come to our knowledge, however, IV PREFACE. that in many of such schools, the better methods of applying the fundamental rules would be acceptable, and that such an addition would make the book more useful in the class-room. In present- ing this introductory matter, great care has been taken to divest it of all unnecessary rules and exercises, and to bring it into harmony with the other portions of the book for directness and forcible application of principles. The book, as now presented, is deemed to be a complete Commercial Arithmetic, with little that is redundant, and with all that is requisite to establish the learner in a thorough knowledge of commercial usages. A large number of practical examples have been added in the various departments, not with a view of parading them, but in order to satisfy teachers who wish a wider field to cull from. In making this addition the same rule has been observed which did so much to commend the old edition to practical teachers, viz. : to admit no puzzles nor conundrums, and in every case to test the example by the busi- ness standard. In all matters as to business customs or local laws care has been taken to get information from the highest source and of the latest date. In short, in presenting this practically new book the authors have sought to meet the reasonable expectation of their friends and the public generally in keeping abreast with the times, and to show their faith in honest work. CONTENTS. PAGE NOTATION AND NUMERATION 7 Roman Notation . . . . 9 ADDITION 10 SUBTRACTION 18 Short method of finding the balance of an account 21 MULTIPLICATION 22 Short methods 20 DIVISION 36 Short methods. 40 UNITED STATES MONEY 41 PROPERTIES OF NUMBERS 49 Prime Factors 51 Common Multiples 52 Cancellation 54 FRACTIONS 56 Reduction 58 Addition 62 Subtraction 64 Multiplication 65 Division 70 DECIMALS 77 Reduction 81 Addition 82 Subtraction 84 Multiplication 85 Division 86 To find the value of goods sold by the hundred or thousand 88 DENOMINATE NUMBERS 91 Reduction of Denominate Integers 91 Reduction of Denominate Fractions 93 Addition of Denominate Numbers 97 Subtraction of Denominate Numbers 98 Multiplication of Denominate Numbers 98 Division of Denominate Numbers < . 99 Divisions of Time 100 To find the interval of time between two dates 102 Linear Measures. 104 Square Measures. 106 Cubic Measure 112 Broad Measure 114 Liquid Measures 116 Dry Measure. 118 Measures of Weight 119 English Money 122 Miscellaneous Tables 124 Circular Measure 125 Longitude and Time 125 VI CONTENTS. PAGE THE METRIC SYSTEM 128 Linear Measure 128 Square Measure 130 Cubic Measure 131 Dry and Liquid Measure 132 Weight 133 Table of Equivalents 134 Approximate Rules 135 ALIQUOT PARTS 139 PERCENTAGE 141 PROFIT AND Loss 150 DISCOUNTS 153 BILLS 157 COMMISSION AND BROKERAGE 166 INTEREST 171 Accurate Interest 184 PROBLEMS IN INTEREST 186 PRESENT WORTH AND TRUE DISCOUNT 190 COMPOUND INTEREST 193 COMMERCIAL PAPER 197 BANK DISCOUNT 203 PARTIAL PAYMENTS 207 United States Rule 207 Mercantile Rules 210 RATIO AND PROPORTION 213 Simple Proportion 214 Compound Proportion 217 INSURANCE 219 Fire Insurance 220 Marine Insurance 221 EXCHANGE 225 Domestic Exchange 226 Foreign Exchange 229 EQUATION OF ACCOUNTS 237 When the items are all debits or all credits 237 When the account contains both debit and credit items 246 Equation of Account Sales 251 ACCOUNTS CURRENT 255 STOCKS AND BONDS 263 New York Stock Exchange 266 TAXES 276 DUTIES 279 PARTNERSHIP 286 NATIONAL BANKS 300 SAVINGS BANKS 303 LIFE INSURANCE 306 APPENDIX 317 Greatest Common Divisor 317 Annual Interest 319 New Hampshire Rule 321 Vermont Rule 322 Storage 323 Alligation 327 Square Root 331 Cube Root 332 Mensuration 334 General Average 337 Foreign Weights and Measures 342 Detection of Errors in Trial Balances. . . 344 ARITHMETIC. NOTATION AND NUMERATION. 1. Arithmetic is the science of numbers and the art of com- putation by them. 2. A Unit, or Unity, is one, or a single thing ; as one, one foot, one dollar. 3. A Number is a unit v or a collection of units ; as one, four, three feet, five dollars. Numbers are expressed by words, by letters, and by figures. 4. Notation is a system of representing numbers by symbols. There are two methods of notation in use, the Roman and the Arabic. 5. Numeration is a system of naming or reading numbers. 6. The Arabic method of notation employs ten characters or figures, viz. : One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Zero, The first nine of the above are called significant figures, because each, standing by itself, represents a value, or denotes some number. They are also called digits, from the Latin word digitus, which means a finger. The last one is called zero, naught, or cipher, because when standing alone it has no value, or signifies nothing, NOTATION AXD JV UM ERATION. [Art. 6. FRENCH AND AMERICAN NUMERATION TABLE. s 3 - 1 .2 5 ' '-'S3 ac CH DO (S ulUil! H ^ W En ^ K H 4 S, 1 9 5, 7 3 2, 4 3 6, 8 7, 5 9 3. 7Yh period' eth period, Sth period, 4th period, 3d period, 2d period, 1st period. Quintillions. Quadrillions. Trillions. Billions. Millions. Thousands. Units. ENGLISH NUMERATION TABLE. H 508642195732436807593 Billions. Millions. Units. NOTE. It will be observed by a comparison of the French and English systems, that numbers consisting of nine figures or less are read the same. 7. Copy and read the following numbers : 73 102 616 1064 8174 12741 69 333 348 3604 8006 20809 48 570 222 4364 7070 47038 90 895 843 7208 3300 68605 8. Express by figures the following : 1. Nineteen. 2. Twenty-two. S. Forty-six. 4. Sixty-eight. 5. Ninety-two. 6. Eighty-seven. 7. One hundred forty-four. 8. Three thousand sixteen. 9. Four thousand forty-four. 10. Six million two thousand six. 11. Sixteen million eight hundred two. 12. Eighty-seven thousand sixty-two. Art. 9.] ROMAN NOTATION. ROMAN NOTATION. 9. In the Roman Notation, seven capital letters are used to express numbers, as follows : I V X L C D M One, Mve, Ten, fifty, One Hundred, Five Hundred, One Thousand. Other numbers are expressed by combining the letters according to the following principles : 1. If a letter is repeated, its value is repeated. Thus, III represents three ; XX, twenty ; CCC, three hundred. 2. If a letter of less value is placed before one of greater value, the less is taken from the greater. Thus, IV represents four ; IX, nine ; XL, forty. 3. If a letter of less value is placed after one of greater value, the less is added to the greater. Thus, VI represents six ; XI, eleven ; LX, sixty. 4. A bar (~~) placed over a letter increases its value a thousand times. Thus, X represents ten thousand ; M, one million. The Roman Notation is used for numbering dials, chapters, pages, etc. 10. TABLE OF ROMAN NOTATION. Roman. Arabic. Eoman. Arabic. Roman. Arabic. Roman. Arabic. I, I. IX, 9. XX, 20. xc, 90. II, 2. x, 10. XXI, 21. c, 100. III, 3. XIII, 13. XXX, 30. CCC, 300. IV, 4. XIV, 14. XL, 40. D, 500. v, 5. XV, 15. L, 50. DCC, 700. VI, 6. XVIII, 18. LX, 60. M, 1000. VIII, 8. XIX, 19. LXXX , 80. MD, 1500. 11. Express by Roman notation : 1. Eighteen. 2. Thirty-six. 3. Forty-eight. 4. Seventy-six. 5. Sixty-four. 6. Eighty-seven. 7. Three hundred sixty. 8. Six hundred forty-nine. 9. Five hundred eighty-eight. 10. Two thousand sixty-two. 12. Express by Arabic notation : 1. LXXVII. 6. DCCLXVI. 11. MMCC 2. CCXIX. 7. DCXLIV. 12. MMDC 3. XCVIII. 8. DCXLIV. IS. MMCC 4. CCCLIV. 5. DCXXVI. 9. MDCXLVI. 10. MCCLXXIX. 14. MMDC 15. MDCCC 11. 12. IS. 14. 15. 584. 777. 1638. 1886. 80000. ADDITION. 13. The Sum or Amount of two or more numbers is a number which contains as many units as all the numbers com- bined. 14. Addition is the process of finding the sum of two or more numbers. 15. The sign of addition is -f , and is read plus. 16. The sign of equality is =, and is read equals, or is equal to. Thus, 6 + 2 = 8 is read 6 plus 2 equals 8, or the sum of 6 and 2 is equal to 8. 17. The sign of dollars is $ ; of cents ^, ct., or cts. 18. To find the sum of two or more numbers. Ex. Find the sum of 416, 578, 695. OPERATION. ANALYSIS. Write the numbers so that like units stand 416 in the same column and begin to add at the right. The 578 sum of the units (6 + 8 + 5) is (14, 19) 19 units, equal to ggg 1 ten 9 units. Write the 9 units under the column of units, and add the 1 ten to the column of tens, obtaining for the 1689 Sum. sum (2, 9, 18) 18 tens, equal to 1 hundred 8 tens. Write the 8 tens under the column of tens, and add the 1 hundred to the column of hundreds, obtaining for the sum (5, 10, 16) 16 hundreds, equal to 1 thousand 6 hundreds, which write in the hundreds' and thousands' places. Hence, the entire sum is 1689. NOTES. 1. Write the numbers in vertical lines. Irregularity in the placing of figures is the cause of many errors. 2. Think of results and not of the numbers themselves. Thus in the above example, do not say 6 and 8 are 14 and 5 are 19, but 14, 19. 3. To avoid repeating the work, in case of interruption, write the figures to be carried in pencil underneath. Art. 19.] ADDITION. 11 19. RULE. Write the numbers to be added so that like units stand in the same column. Commencing at the right, add each column separately, and if the sum is less than 10, write it under the column added. If the sum of any column is 10 or more than 10, write the right-hand figure under the column added, and add the remaining figure or figures to the next column. PROOF. Find the sum by adding the columns in the opposite direction, thus forming new combinations of fig- ures. If the results agree, th e work is probably correct. EXAMPLES. 20. Copy or write from dictation and add the following : (1} (2} (3) (4) (6) (6) 789 682 1234 1357 7812 9876 123 109 5678 9135 3625 6789 456 375 9012 8642 .4875 9787 246 488 3456 4109 '9850 8678 (7) (8) (9) (10) (11) (12) 568 431 9672 7812 8796 808 134 866 8738 1357 809 7612 680 219 4126 404 4205 37 419 581 1886 9686 6666 4123 723 49 7143 8072 7777 2264 842 376 8275 9706 8088 7714 906 408 9325 5555 4144 9008 294 792 4444 2009 9995 3348 21. There is nothing of more importance to the student than the ability to add a column of figures easily, accurately, and rapidly. In order that his labor may be lightened and much valuable time saved, not only in after-life but in his school work, he should have various kinds of daily drill exercises in addition, especially in the earlier part of his course of study. The follow- ing suggestions will be found valuable in securing accuracy and rapidity : 12 ADDITION. [Art. 22. 22. The 45 simple combinations should be used as an exer- cise in addition. They may be copied on the blackboard in the following or in irregular order, and the sum should be announced by the student at sight : 123243543654765 121212123123123 48765987659876!* 412341234523453 8 4 M I 5 6 6 9 4 8 5 7 6 9 5 8 6 7 7 9 6 8 7 9 7 8 8 9 8 9 9 23. The above should be supplemented by exercises similar to the following : 74 64 44 94 34 24 14 54 84 45 75 35 15 55 65 95 25 85 _9_9_9_9_9_9_9 9 9 It is just as easy to add 74 and 8 as 4 and 8. It should be impressed on the mind of the student that 4 and 8 when added always produce 2 in units' place, whatever the number of tens, and the tens are increased by 1. If the student is thoroughly drilled, he will not hesitate when near the end of the column or when the sum is above 20 or 30. 24. Make combinations of 10, 20, 30, or other numbers as often as possible, and add them as single numbers. Thus, add 9 and 1, 8 and 2, 5 and 5, 4, 3, and 3, etc., as 10; 7 and 2, 6, 2, and 1, etc., as 9; 2 and 3, 4 and 1, 2, 2, and 1, as 5; 8, 7, and 5, 9, 7, and 4, etc., as 20; etc., etc. In Example 1, Art. 27, think only of the following results: 9, 19, 30, 50. Drill on the following and similar combinations (10, 20, 30, etc. ) until the student can announce the sums at sight : 321453822612213 434111176281433 355546112217464 Irt. 24.] ADDITION. K 7 6 6 8 4 8 9 4 3 2 5 8 9 7 9 6 8 5 4 7 3 2 i% 8 9 7 7 8 7 7 7 6 9 8 9 9 9 9 9 9 ' 8 5 3 6 4 1 1 1 3 3 6 8 4 7 7 7 6 9 8 9 2 4 3 3 3 6 4 4 7 4 7 M f 6 8 4 3 2 2 1 3 6 1 4 4 4 7 8 7 9 8 4 3 4 3 1 2 7 8 2 5 9 9 8 5 9 When a figure or number comes between two numbers that make 10, think of the total at once; as 7 + G + 3 = 16, 6 + 7 + 4 = 17. NOTES. 1. When three figures are in regular order, the sum may be found by multiplying the middle figure by 3. Thus, 9 + 8 + 7 - 24 (3 x 8) ; 5 + 6 + 7= 18; 6 + 7 + 8 = 21. 2. When five figures are in regular order, the sum may be found by multiplying the middle figure by 5. Thus, 1 + 2 + 3 + 4 + 5 = 15 (ox 3); 5 + 6 + 7 + 8 + 9 = 35. 3. When a figure is repeated several times, multiply it, instead of adding. 4. Do not think of numbers between 10 and 20 as a certain number of units and 1 ten as they are named, but as 1 ten and a certain number of units. Thus, think of 14 as 1 ten and 4 units (onety-four, or one-four), not 4 units and 1 ten (fourteen). 5. Add downwards, for then the sum is found just where it should be placed at the foot of the column. In proving results, add upwards. 25. Adding two columns at once. Drill on the follow- ing or similar combinations of numbers of two figures each, until the student can announce the sums at sight : 12 24 45 24 37 41 37 62 27 57 16 33 33 26 42 58 45 34 48 27 When the above have been mastered, give exercises containing three or more numbers, as : 12 16 22 24 19 42 56 37 51 27 31 29 33 36 31 24 21 33 25 34 42 38 56 22 44 17 24 28 38 45 26. The following " magic square" may be used as a drill exercise in addition. The sum downwards, from left to right, or diagonally is 54351. To vary the exercise, the teacher may die- 14 ADDITION. [Art, 26. tate, to be added, all the numbers but one in any line or column. The sum can be found by subtracting the number omitted from 54351. The sum of the digits of any result will be a multiple of 9. 4536 9477 3726 8667 2916 7857 2106 7047 1296 6237 486 567 4617 9558 3807 8748 2997 7938 2187 7128 1377 5427 5508 648 4698 9639 3888 8829 3078 8019 2268 6318 1458 1539 5589 729 4779 9720 3969 8910 3159 7209 2349 6399 6480 1620 5670 810 4860 9801 4050 8100 3240 7290 2430 2511 6561 1701 5751 891 4941 8991 4131 8181 3321 7371 7452 2592 6642 1782 5832 81 5022 9072 4212 8262 3402 3483 7533 2673 6723 972 5913 162 5103 9153 4293 8343 8424 3564 7614 1863 6804 1053 5994 243 5184 9234 4374 4455 8505 2754 7695 1944 6885 1134 6075 324 5265 9315 9396 3645 8586 2835 7776 2025 6966 1215 6156 405 5346 EXAMPLES. 27. Copy or write from dictation and add the following : w 4) 5 f 15 w 13 (3) 12 85] 21 96 1 (4) 123 456 382 648 3 ) ( 95 ) (44 21 789 584 7'f 1 30 (62) 20 3 M66 462 765 8 ) 3 f 40 36 j2 l 315 829 406 483 6 ) eUo 55. r 68) 54) {? 1- 918 234 163 852 1 j y 8) (46) 63 77 49 789 574 w (7) W m (j (") (12) (18) 48 71 39 12 77 312 514 376 13 43 34 34 88 123 627 499 82 36 46 56 66 456 842 678 67 94 25 78 99 789 462 437 54 V N dfe fc 83 f90 % , 41. 987 460 245 87 25 31 89 63 654 329 536 43 38 63 76 74 321 411 984 Art. 27.] ADDITION. 15 (14) (15) (16) (17) (18) (19) 1234 4121 1728 3416 17642 18114 5678 1865 5280 4725 176 285 9212 3760 2246 8850 20048 28510 3456 4825 4153 4975 248 30048 9753 7145 4839 2137 24800 400 8642 3333 2437 8910 1149 17512 7531 7163 4627 2048 1216 8 1594 4943 7342 175 385 14150 7777 7289 8916 1075 19175 30032 20. Find the sum of the following numbers : Forty-five thousand forty-five ; sixteen thousand three hundred sixty ; one hundred sixty-seven thousand ; eight hundred fifty thousand ninety-two ; nine million twenty-four. 21. 46 + 72 + 89 + 93 + 75 + 31 + 58 -f 45 + 52 = ? 82. 376 + 416 -f 287 + 123 + 456 + 789 + 916 + 328 = ? 23. 42 + 175 + 287 + 56 + 63 + 324 + 189 + 172 + 96 = ? 24. 365 + 1728 -f 64 + 172 + 89 + 38 + 9 + 5280 +176 = ? 25. A bushel of corn weighs 56 pounds, a bushel of rye 56 pounds, a bushel of wheat 60 pounds, a bushel of barley 45 pounds, a bushel of oats 32 pounds, and a bushel of buckwheat 48 pounds. What would be the total weight of one bushel of each of the above grains ? 26. A farmer raises 375 bushels corn, 419 bushels barley, 849 bushels wheat, 668 bushels oats, 957 bushels barley, and 389 bushels rye. Find how many bushels in all. 27. Find the total distance around a rectangular field 1728 feet long and 1683 feet wide. 28. An exporter of provisions buys 187 barrels hams, 428 bar- rels shoulders, 475 barrels pork, 229 barrels beef, and 392 barrels bacon. How many barrels in all ? 29. In an orchard there are 375 apple trees, 416 pear trees, 37 quince trees, 98 cherry trees, 238 peach trees, and 276 plum trees. How many trees in all ? SO. A man pays for a house and lot $6375. For repairs as follows : mason-work, $68 ; plumbing, $78 ; carpenter-work, $164 ; painting and decorating, $277. For how much must he sell it to gain $567 on the total cost ? 16 ADDITION. [Art. 27. 31. A manufacturer sells on Monday 2387 barrels flour, on Tuesday 2618 bbls., on Wednesday 2178 bbls., on Thursday 2125 bbls., on Friday 2348 bbls., and on Saturday 2496 bbls. How many does he sell during the week ? 32. Find the total number of pounds of tobacco produced in the following states in 1879 : Kentucky, 171,121,134; Virginia, 80,099,838; Pennsylvania, 36,957,772; Ohio, 34,725,405; Ten- nessee, 29,365,052; North Carolina, 26,986,448; Maryland, 26,082,147; Connecticut, 14,044,652; Missouri, 11,994,077; Wisconsin, 10,878,463. Add the following numbers as they stand, from left to right, and from right to left. [In making out bills and in other com- mercial operations, a great deal of time can be saved by adding in this manner, without re-arranging the numbers.] S3. 17, 27, 36, 14, 43, 42, 65, 73, 81, 35. 84. 176, 340, 203, 62, 177, 96, 398, 75, 148, 96. 35. 137, 414, 528, 345, 678, 975, 864, 357, 121, 234. S6. 6716, 512, 375, 475, 3842, 5927, 3875, 17525. 37. 2345, 16, 375, 4218, 376, 7, 8475, 247, 39. 38. 123427, 34825, 775, 716, 8976, 37412, 567356, 39723. NOTE. In tally-sheets of pounds, gallons, yards, feet, etc., for conveni- ence in adding, place 10 (20 or 30) numbers in each column as in the following example. (See Note 3, Art. .24.) Add the totals from left to right. 39. Find the total weight of the following 100 boxes of cheese: 67 64 62 61 06 68 64 62 61 60 61 67 60 64 //*j o< 60 63 61 68 64 60 63 63 64 66 65 67 61 60 63 63 60 62 61 68 64 65 66 63 67 62 62 61 65 61 66 63 67 62 65 64 61 60 68 66 64 63 67 69 66 66 65 62 61 68 67 64 63 66 66 65 66 65 64 67 67 68 66 63 61 61 68 64 62 65 60 61 60 62 65 62 64 66 _66 63 _61 62 __67 _68 _69 *** *** *** *** *** *** #** *** #** *** **** Art. 27.] ADDITION. 17 40. Find the total estimated value of the following crops for the year 1884: Corn, $640,735,589; wheat, $330,861,254; rye, $14,855,255; oats, $161,528,470; barley, $29,781,155. 41. Find the total number of bales of cotton produced in the following states in 1879 : Mississippi, 955,808 ; Georgia, 814,441 ; Texas, 803,642; Alabama, 699,654; Arkansas, 608,256 ; South Carolina, 522,548; Louisiana, 508,569; North Carolina, 389,598. J$. Find the total number of bushels of wheat produced in the following states in 1879 : Illinois, 51,136,455 ; Indiana, 47,288,989; Ohio, 46,014,869; Michigan, 35,537,097; Minne- sota, 34,625,657; Iowa, 31,177,225; California, 28,787,132; Missouri, 24,971,727; Wisconsin, 24,884,689. Complete the following statements by adding downwards and from left to right. The sums of the totals should be equal. 43. SALES FOR THE WEEK EN-DING JAN. 21, 1887. Days. Goods. Shoes. Furniture. Books. Crockery. Millinery. Totals. Monday 64718 17642 37640 117 13 92 17 Ill 40 **** ** Tuesday 35625 97.20 184.00 6472 3815 5765 *** ** Wednesday Thursday Friday 716.40 828.30 316 17 200.48 227.59 8714 417.65 476.00 148 12 156.25 171.14 10080 127.16 287.80 7744 328.40 116.38 7940 ****^** ****^** *** ** Saturday 929.12 249.67 417.13 426.00 225.98 429.75 **** ** Totals **** ** **** ** **** ** **** ** *** ** **** ** **** ** ' 44. SALES FOR 1887. Months. Domestics. White Goods. Notions. Woolens. Dress Goods. Totals. January 12248 10 6478 10 3429.12 811740 5276 40 ***** ** February 13375 16 7149 37 4176 19 9190 56 6189 24 ***** ** March 17177 48 8214 92 4375 94 8237 41 3416 48 ***** ** April 15119 43 7175 12 4040 40 7116 40 5255 90 ***** *# May 16284 19 813437 4287 38 3697 82 1716 32 ***** ** June 13484 25 6375 28 3343 72 2419 38 2100 58 ***** ** July 9119 47 4569 34 2417 75 3129 16 3269 43 ***** ** August 10219 36 5162 12 3412 96 4289 50 3716 96 ***** ** September . . 14196 42 70QQ 4ft 4.198 4Q 5391 42 4128 31 ***** ** October November 13184.16 14174 12 6517.58 7288 49 3692,29 4140 50 7214.97 8345 68 6247.39 711640 ***** ** ***** ** December 13184.9G 6697.13 3990.82 8175.75 7348.19 *****^** Totals ****** ** ***** ** ***** ** ***** ** ***** ** ****** ** SUBTRACTION. 2S. The difference between two numbers is a number which,, added to the smaller, will produce a result equal to the greater. 29. Subtraction is the process of finding the difference between two numbers. The greater of two numbers whose difference is required is called the minuend, and the smaller the subtrahend. The result is called the re- mainder. 30. The sign of subtraction is , and is read minus or less. Thus, 8 5 is read 8 minus 5, or 8 less 5, and means that 5 is to be taken from 8. 31. To find the difference between two numbers. Ex. Find the difference between 967 and 384. OPERATION. ANALYSIS. Write the smaller number under the 967 Minuend. greater so that units are under units, tens under tens, 384 Subtrahend. etc ' Commence to subtract at the right. 4 units from ~T7 7 units are 3 units, which write below the line under the 58d Kemamder. co i umn o f un i ts gi nce 3 tens cannot be taken from 6 tens, take 1 hundred from 9 hundreds, leaving 8 hun- dreds, and add it (1 hundred = 10 tens) to the 6 tens, making 16 tens. 8 tens from 16 tens are 8 tens, which write under the column of tens. 3 hundreds from 8 (9 1) hundreds arc 5 hundreds. Hence the result is 583. Instead of subtracting 1 from the figure of the upper number of the next higher order when it has been necessary to add 10 to the figure of the minu- end, some persons add 1 to the figure of the lower number of the next higher order. This method depends on the principle that adding equivalent num- bers to both minuend and subtrahend does not affect the remainder. In practice, do not think of explanations, nor say 4 from 7 is 3, etc., but think only of results and write them at once. Thus, in the above example, say or think only 3, 8, 5. Art. 32.] SUBTRACTION. 19 32. RULE. Write the subtrahend under the minuend so that units of the same order stand in the same cuiumn. Commencing at the right, subtract each figure in the lower number from the one above it, and write the differ- ence in the line below. If any figure is greater than the one above it, add 10 to the latter, perform the subtraction, and then consider the next figure in the upper number decreased by 1 (or, con- sider the next figure in the lower number increased by 1). EXAMPLES. 33. Find the difference between 1. 8716 and 4379. 11. 80706040 and 23456789. 2. 917642 and 9819. 12. 76483672 and 87132191. 3. 64321 and 23456. 18. 123456789 and 9897960. 4. 428165 and 317618. 14. 72081099 and 87643229. 5. 9371641 and 876543. 15. 16417528 and 90716801. 6. 7642878 and 6789119. 16. 43184296 and 37529510. 7. 8090403 and 7090508. 17. 100010001 and 9890978. 8. 6380912 and 5270937. 18. 30040050 and 29917168. 9. 7654321 and 1234567. 19. 20103040 and 19181746. 10. 7060509 and 6987969. 20. 40020003 and 20807064. Find the difference between the numbers in each of the fol- lowing groups. [In all of these cases the subtrahend is placed above the minuend, the purpose being to give the student practice in subtracting doivnward rather than upward, as the general cus- tom is. It is often requisite in business to perform the work in this way, and the accountant should practice both methods.] (21) (22) (23) (24) (25) 76534 19827 26347 72016 12345 81907 81279 84362 71356 99385 54321 94371 (27) (28) (29) (30) (31) (32) 12467 31617 46789 24681 46897 36478 75112 42131 50000 30502 50901 41516 20 SUBTRACTION. [Art. 33. 83. There were 50017 post-offices in the United States in 1884 and 51252 in 1885. What was the increase during the year ? 34. In 1880, the population of the United States was 50,152,866, and in 1870, 38,558,371. What was the increase during the decade ? 35. The area of Alaska is 369,529,600 acres. How much greater is it than Texas, whose area is 175,587,840 acres ? 86. The public debt of the United States Nov. 1, 1885, was $1,447,657,568, and Nov. 1, 1886, $1,354,347,947. What was the reduction of the debt during the year ? 37. The gross weights (weights of barrels and sugar) and tares (weights of barrels) of ten barrels of sugar are as follows: 326-19, 332-19, 307-18, 321-18, 324-19, 330-19, 313-18, 313-19, 317-17, 327-19. Find the total net weight. NOTE. Find the total gross weight and total tare, and then the difference, or the total net weight. Population of the following cities of the United States in 1880: New York, Philadelphia, - Brooklyn, Chicago, Boston, - St. Louis, Baltimore, Cincinnati, San Francisco, New Orleans, - Cleveland, Pittsburg, 1,206,590 846,984 580,370 503,304 362,535 350,522 332,190 255,708 233,956 216,140 160,142 156,381 Buffalo, - Washington, - Newark, Louisville, Jersey City, - Detroit, Milwaukee, - Providence, - Albany, Rochester, Allegheny, Indianapolis, - 155,137 147,307 136,400 123,645 120,728 116,342 115,578 104,850 90,903 89,363 78,681 75,074 88. What is the total population of the first column of the above cities ? Of the second column ? What is the total popula- tion of all ? 39. What is the difference between the sums of the first and second columns ? 40. How much does the total population of New York, Brook- lyn, Newark, Jersey City, Hoboken (30,999), Yonkers (18,892), and Long Island City (17,117) lack of being 2,500,000 ? 4J. How much does the total population of Pittsburg and Allegheny exceed that of San Francisco ? Art. 34.J S UB TR A C TIO N . 21 34. Short method of finding the balance of an account. Ex. Find the balance of the following ledger account : Dr. C. E. & W. F. PECK. Or. 1882. 1882. Mar. 16 Merchandise. 1192 97 Apr. 22 Cash. 800 M 30 Sundries. 567 40 " 22 Bills receivable. 1000 31 Merchandise. 384 30 May 1 Merchandise. 317 Apr. 22 Interest. 16 48 " 17 Cash. 424 i 24 Merchandise. 846 51 July 1 Balance. 852 May 17 " 387 25 3394 91 3394 July 1 Balance. 852 84 91 ANALYSIS. It can readily be seen that the debit side is greater; therefore add that side first and write the sum as the total or footing of each side. Then pass to the other side of the account. The sum of the first column is 17, which subtracted from the next higher number, 21, ending with 1, the corresponding figure of the total, leaves 4, which write as the first figure of the balance, carrying the 2 to the next column. (If the right-hand figure of the sum of any column is the same as the corresponding figure of the total, subtract it from itself, and not from the next higher number ending with the same figure ; or write in the balance and carry the left-hand figure of the sum.) The sum of the figures in second column plus 2 carried is 11, which subtracted from 19 leaves 8, the second figure of the balance. Proceed in like manner until all the figures of the balance are obtained. Prove by adding all the numbers, including the balance. EXAMPLES. 85. Find the balances of the following accounts : (*) (*) (*o Dr. Or. Dr. Or, Dr. Or, 817.20 812.20 237.25 112.27 1075. 375.60 222.22 214.13 900. 218.36 2318.42 218.24 427.30 375. 800. 717.49 812.10 717.37 810.75 412. 718.24 648. 938.40 244.45 416.30 717. 218.75 118.75 4312. 946.33 225. 538.98 222.48 719.46 203.13 108.75 MULTIPLICATION 36. Multiplication is the operation of taking one number as many times as there are units in another. The number taken or multiplied is called the Multiplicand. The number which indicates how many times the multiplicand is taken or multiplied, is called the Multiplier. The result obtained is called the Product. 37. The sign of multiplication is x , and is read times, or multiplied by. Thus, 5 x 4=20, is read 5 times 4 equals 20, or 5 multiplied by 4 equals 20. 38. Multiplication Table. 21 9 lO:il! 18 20 22 24 27j 30 33 36 33 40 44 48 35J 40; 45! 50 1 55 60 42j 48j 54| 60 66 72 49 56 63 1 70: 77| 84 56 64 1 72 ! 80; 88; 96 63| 72J 8l! 90j 99108 70' 80 a 0^1 84! 96108:120132144 40 72 90.100110190 13 14 15 10 17 18J19 20 31 22 23 34 35 40 26; 28 30 1 32 39| 42, 45| 48J 51 1 54J 57 52 56 60j 64, 68, 72, 76 8 65! 70! 75! 80| 85 90! 95100 78 91 104 112J120J128jl36!l44'152J160 117 126 135 144 153162 17l|l80 130:i40'l50 160 170 180190200 84| 90 96102108114120 98 105112;119 126133140 99!llO;i21;132|l43 154 165 176 187 198,209 220 156168 180:198 204 216 228:240 78j 911041171130,143156169182 195^208 221 234247,260 42 .0611 26132138144150 47 154' 161 168 175 .68 176 184 192 200 189198207216225 21012202301240250 231242'253264275 2264276288300 273286,299312325 294'303 322,336 350 84 98112126|l40 i 154168l82196210224238252'266j280 90 105 120 135J150165 180195 210^225!240!255|270|285;300 315|33o|345,360 375 64' 80 96112128144J160176 l 192 l 208 l 224 i 240'256272288304320336352368384400 68' 85 102 119 136 153 170;187'204 I 221;238 255 2721289J306 323 340 357J374 391 408 425 72: 90108'l26144162180198216'234|252270288|306|324342360 378|396 414 432 450 4 oon or if i ' 761 95,114 133 152 171J190 209 228 247|266|285,304|323|342 361 380 80!lOO 120 140 160180 200 220240 260:280 300'320 340 860J880I40C 456475 84105126147 88110:i32154176|l{ 92115 ! 138161 420:4401460480500 1(58 189 210 231 252 273 294 315 336 357 378 399 420j44l|462 | 483 | 504 525 1220 242 264 286 374|396i415 440462484 ( 506|L !550 264 288,312 336 360,384 408 482 456,480 504|528;552 576 600 75 100125150175 200J225|250 275 300 325 350J375 400 ! 425 450J475 500 525 550 575 600 625 1| 3 3 4| 5| 6| 7 8 9110111313114115161718,1913013133333435 Art. 39.] MULTIPLICA TION. 23 39. To find the product of two numbers when the multiplier does not exceed 12. Ex. Multiply 456 by 7. OPERATION. ANALYSIS. 7 times 6 units are 42 units=4 tens and 2 units. 456 Write the 2 units under the figure of the multiplier (the column 7 of units) and add the 4 tens to the next product (the column of tens). 7 times 5 tens are 35 tens, plus 4 tens from the preced- ing product, are 39 tens = 3 hundreds and 9 tens. Write the 9 tens under the column of tens, and add the 3 hundreds to the next product (the column of hundreds). 7 times 4 hundreds are 28 hundreds, plus 3 hun- dreds from the preceding product are 31 hundreds = 3 thousands and 1 hun- dred. Write the 1 hundred under the column of hundreds and the 3 thousands in the column of thousands. 40. RULE. Commencing at the right, multiply each figure of the multiplicand by the multiplier, writing the result and carrying as in addition. EXAMPLES. 41 . Multiply Multiply 1. 23456 by ?; by 8. 8. 789123 by 2; by 3. 2. 37804 by 9; by 6. 9. 123567 by 4; by 5. 3. 24687 by 2; by 4. 10. 781693 by 6; by 9. 4- 36925 by 3; by 8. 11. 417009 by 8; by 7. 5. 48716 by 5; by 9. 12. 509048 by 8; by 7. 6. 90809 by 9; by 8. 13. 637485 by 6; by 9. 7. 26048 by 5; by 7. U. 748596 by ?; by 5. 15. There are 5280 feet in one mile. How many feet in 11 miles ? 16. There are 4 gills in one pint, 2 pints in one quart, and 4 quarts in one gallon. How many gills in 63 gallons ? 17. There are 2 pints in one quart, 8 quarts in one peck, and 4 pecks in one bushel. How many pints in 379 bushels ? 18. There are 12 inches in one foot, and 3 feet in one yard. How many inches in 1760 yards ? 19. In one gross there are 12 dozen, and in one dozen 12 units. Find the value of 45 gross lead-pencils at 3 cents each. 20. How many buttons on 6 dozen pair of shoes, if there are 9 buttons on each shoe ? 24 MULTIPLICATION. [Art. 42. 42. To find the product of two numbers, when the multiplier is more than 12. Ex. Multiply 456 by 237. OPERATION. ANALYSIS. Write the multiplier under the multipli- 456 cand so that their right-hand figures are in the same ver- 237 tical line. Since the multiplier consists of 7 units, 3 tens, and 2 hundreds, the multiplicand is repeated or multiplied by 7, by 30, and by 200. 7 times 456 is 3192, the first par- 1368 tial product ; 30 times 456 is 13680, the second partial 912 product; 200 times 456 is 91200, the third partial product. The sum of these partial products is 108072. the entire product. In practice, the ciphers are omitted. In the operation, observe that the first or right-hand figure of each partial product is directly under the figure of the multiplier used. 43. RULE. Write the multiplier under the multipli- cand so that their right-hand figures are in the same vertical line. Multiply the multiplicand by each significant figure of the multiplier, writing the first or right-hand figure of each partial product under the figure of the multiplier used. Add the partial products. The sum will be the desired product. EXAM PLES. 44. Multiply Multiply 1. 1728 by 37 ; by 481. 9. 23456 by 294 2. 2893 by 26 ; by 506. 10. 40607 by 144 3. 3904 by 18 ; by 624. 11. 32738 by 176 4. 5107 by 41 ; by 375. 12. 91609 by 201 6. 6079 by 59 ; by 208. 13. 24135 by 345 6. 8125 by 67 ; by 567. 14. 38246 by 678 7. 9236 by 78 ; by 781. 16. 94538 by 987 8. 7438 by 89 ; by 936. 16. 10908 by 406 by 3742. by 4803. by 5964. by 6075. by 7186. by 8297. by 9410. by 2465. 17. How many hours in the month of January ? 18. How many minutes in the month of April ? 19. How many seconds in the month of February, 1899 ? 20. Find the cost of 375 barrels pork at $14 per barrel. 21. There are 5280 feet in one mile. How many feet in 96 miles ? In 208 miles ? Art. 44.] MULTIPLICATION. 25 22. How many pounds in 471 bushels corn,, if there are 5G pounds in one bushel ? 23. In a bushel of timothy seed, there are 45 pounds. How many pounds in 2367 bushels ? In 3416 bushels ? 24. How many shoes in 24 boxes, if each box contains 12 pair ? 25. A certain building has 192 windows, and each window contains 24 panes of glass. How many panes in all ? 26. How many feet of wire will be required to fence a field 209 feet square, the fence being 6 wires high and on all sides of the field ? 45. To find the product of two numbers when there are ciphers at the right of the significant figures (6) of one or both. Ex. Multiply 37600 by 47000. OPERATION. ANALYSIS. Write the numbers so that the right-hand 37600 significant figures are in the same vertical line. 37600 47000 = 376 x 100, and 47000 = 47 x 1000. Since the product of two or more numbers is the same in whatever order they are multiplied, multiply 376 by 47, and their product 1504 by 100000 (100 x 1000), by annexing 5 (3 + 2) ciphers to 1767200000 the right 46. RULE. Write the numbers so that their right-hand significant figures are in the same vertical line. Multiply the significant figures together as if there were no ciphers, and to their product annex as many ciphers as are found on the right of both numbers. EXAM PLES. 47. Multiply Multiply 1. 3600 by 40 ; by 300. 9. 48400 by 200 ; by 1400. 2. 1728 by 80 ; by 500. 10. 37000 by 500 ; by 2500. 3. 3456 by 70 ; by 420. 11. 12345 by 600 ; by 3600. 4. 3710 by 50 ; by 360. 12. 28000 by 420 ; by 4700. 5. 4000 by 30 ; by 800. 13. 19700 by 340 ; by 5800. 6. 2800 by 90 ; by 370. 14. 14320 by 560 ; by 6900. 7. 1360 by 60 ; by 200. 15. 84000 by 800 ; by 7320. 8. 4200 by 20 ; by 500. 16. 96000 by 900 ; by 4800. 26 MULTIPLICATION. [.\rt.48- SHORT METHODS.* 48. To multiply any number of two figures by n. 49. RULE. Place the sum of its digits between them when the sum is less than 10. When the sum is 10 or more than 10, write Us right-hand figure in the second place and carry one to the left-hand figure of the multi- plicand. EXAMPLES. 50. 1. Multiply 34 by 11. ANALYSIS. 3 + 4 = 7, which placed between 3 and 4 produces the product 374. 2. Multiply 68 by 11. ANALYSIS. 6 + 8 = 14. Write 4 in the second place and carry 1 to the 6, the left-hand figure of the multiplicand producing the product 748. 3. Multiply the following numbers by 11 : 24, 16, 18, 32, 43, 33, 72, 81, 37, 44, 92, 87, 93, 64, 35, 36, 47, 17, 19, 48, and 57. 51. To multiply any number by n. 52. EULE. Write the 1st j*ight-hand figure, add the 1st and 2nd, the 2nd and 3rd, and so on ; finally write the left-hand figure, carrying as usual. EXAMPLES. 53. 1. Multiply 783742 by 11. Ans. 8621162, ANALYSIS. Write the right-hand figure 2 ; for the remaining figures of the product, add 2 to 4, 4 to 7, 7 to 3, 3 to 8, 8 to 7, and write the left-hand figure, carrying when necessary. 2. Multiply the following numbers by 11 : 245, 346, 325, 416, 784, 517, 875, 918, 4218, 7324, 7218, 1728, 4375, and 8376. 54. To multiply by any number of two figures ending with I. 55. RULE. Multiply by the tens of the multiplier, writing the product under the multiplicand one place to the left, and add. Or, * It is suggested that these short methods be studied in connection with the more advanced work one method with each lesson; or they may be presented to the student, one at a time, with the daily drill exercises on the fundamental rules. Art. 55.] SHORT METHODS. 27 Write as the first figure of the product the unit figure of the multiplicand; multiply each figure of the multipli- cand by the tens of the multiplier, and at the same time, add mentally to each product the figure to the left of the one multiplied, carrying as usual. EXAMPLES. 56. 1. Multiply 456 by 61. 1ST OPERATION. 2ND OPERATION. ANALYSIS, SND METHOD. Write 6 in the product, 6 x 6 + 5 = 41. Write 1 and carry 2736 61 4. 6 x 5 + 4 (carried) +4 = 38. Write 8 and 27816 27816 carry 3. 6x4 + 3 (carried) = 27. Multiply Multiply 2. 864 by 61 ; by 41. 5. 2345 by 121 ; by 111. 3. 717 by 31 ; by 71. 6. 7416 by 51 ; by 81. 4. 447 by 21 ; by 81. 7. 8324 by 41 ; by 21. NOTE. The first method may be used with the following multipliers, by placing the products two places to the left. Multiply Multiply 8. 375 by 301 ; by 401. 11. 483 by 701 ; by 801. 9. 425 by 201 ; by 101. 12. 376 by 201 ; by 901. 10. 469 by 601 ; by 501. 18. 875 by 301 ; by 401. 57. To multiply by any number between 12 and 20. 58. EULE. Multiply ~by the units of the multiplier, writing the product under the multiplicand one place to the right, and add. Or, Multiply the units of the multiplicand by the units of the multiplier, write the units of the product, and carry the tens, if any, to the next product ; multiply the remain- ing figures of the multiplicand by the units of the multi- plier, and at the same time add mentally to each product the figure to the right of tfie one multiplied, carrying as usual ; finally, to the left-hand figure of the multiplicand, add the number to be carried, if any, and write the result. 28 UL TIP LIC A TION. [Art. 59. EXAM PLES. 59. 1. Multiply 456 by 18. 1ST OPERATION. 2ND OPERATION. 456 456 3648 18 8208 8208 ANALYSIS, SND METHOD. 8 x 6 = 48. Write 8 and carry 4. 8x5 + 4 (carried) + 6 = 50. Write and carry 5. 8x4 + 5 (carried) + 5 = 42. Write 2 and carry 4. 4 + 4 = 8. Multiply 2. 785 by 13 3. 378 by 14 4. 522 by 15 by 17. by 16. by 19. Multiply 6. 1234 by 14 7. 2345 by 16 8. 3456 by 19 by 16. by 18. by 13. 5. 376 by 18 ; by 16. 9. 7891 by 17 ; by 15. NOTE. The first method may be used with the following multipliers by placing the products two places to the right. Multiply 10. 875 by 101 ; by 108. 11. 936 by 102 ; by 103. 12. 877 by 104 ; by 106. 18. 736 by 105 ; by 109. Multiply 14. 147 by 108 ; by 101. 15. 385 by 104 ; by 107. 16. 783 by 105 ; by 103. 17. 546 by 107 ; by 106. 60. To multiply by any number ending with 9. 61. RULE. Multiply by 1 more than the given multi- plier, and from the result subtract the multiplicand. EXAMPLES. 62. 1. Multiply 387 by 49. OPERATION. 387 product by 1 19350 " " 50 49 (Subtracted downwards.) 18963 Multiply 2. 76 by 49 ; by 39. 3. 87 by 29 ; by 99. 4. 45 by 59 ; by 69. Multiply 5. 312 by 19 ; by 89. fc. 427 by 39 ; by 79. 7. 825 by 29 ; by 69. Art. 63.] SHORT METHODS. 29 63. To multiply by any multiple of 9 less than 90. 64. RULE. Multiply by the multiple of ten next higher than the given multiplier, and from the result subtract one- tenth of itself. EXAM PLES. 65. 1. Multiply 785 by 63. OPERATION. 785 70 ANALYSIS. 63 = 70-7. 785 x 70 = 54950. 54950 product bv 70 Divide 5495 by 10 by placing its digits one place to the right. 54950-5495 = 48455. 7 49455 63 Multiply Multiply 2. 67 by 18 ; by 27. 6. 345 by 36 ; by 45. 8. 34 by 36 ; by 45. 7. 567 by 18 ; by 72. 4. 77 by 54 ; by 63. 8. 518 by 27 ; by 63. 5. 84 by 72 ; by 81. 9. 724 by 54 ; by 81. 66. To multiply by 25. 67. RULE. Add two ciphers and divide the result by 4- Or, Divide the number by 4 / if there is no remainder, add two ciphers ; if there is a remainder of 1, add 25 ; of 2, add 50; of 3, add 7 5. EXAMPLES. 68. 1. Multiply 446 by 25. ANALYSIS. Since 25 is equal to 100 divided by 4, multi- plying by 100 and dividing the result by 4, is the same as 11150 multiplying by 25. 2. Multiply the following numbers by 25 : 24, 36, 37, 49, 62, 387, 448, 512, 746, 424, 817, 9S7, 544, 717, 318, 324, 256, 556, 9224, 8378, 5280, 1728, 5648. 30 MULTIPLICATION. [Art, 69. 69. To multiply by any number one part of which is a factor of another part. EXAMPLES. 70. 1. Multiply 576 by 287. 2. Multiply 567 by 936. OPERATION. OPERATION. 576 567 287 936 4032 product by 7. 5103 product by 9. 16128 " " 28(4x7). 20412 " " 36(4x9). 530712 "' " 287. 530712 " " 936. Multiply Multiply 3. 227 by 369 ; by 427. 8. 932 by 183 ; by 927. 4. 516 by 246 ; by 568. 9. 718 by 284 ; by 832. 5. 344 by 126 ; by 124. 10. 529 by 546 ; by 756. 6. 728 by 426 ; by 189. 11. 638 by 217 ; by 618. 7. 325 by 147 ; by 273. 12. 435 by 248 ; by 428. 71. To multiply by any number near and less than 100, 1000, etc. 72. The Complement of a number is the difference between the number and the unit of the next higher order. 73. RULE. Add to the multiplicand as many ciphers as there are ciphers in the unit next higher than the mul- tiplier, and from the result subtract the product obtained by multiplying the multiplicand by the complement of the multiplier. EXAMPLES. 74. 1. Multiply 456 by 98. OPERATION. 45600 product by 100.' 912 " " _2. 44688 " " ~98. Multiply Multiply 2. 77 by 99 ; by 93. 6. 387 by 93 ; by 999. 8. 84 by 98 ; by 95. 6. 416 by 95 ; by 994. 4. 72 by 94 ; by 96. 7. 528 by 93 ; by 992. Art. 75.] SHORT METHODS. 31 CROSS MULTIPLICATION. 75. Cross Multiplication depends upon the following principles : Units multiplied by units Tens units Units tens Hundreds " units Tens tens Units hundreds Thousands " " units Hundreds tens Tens hundreds Units thousands Ten-thousands " " units Thousands " " tens Hundreds " " hundreds Tens thousands Units ten-thousands Etc., etc. Ex. Multiply 68 by 74. produce units " tens. hundreds. " thousands. " ten-thousands. Am. 5032. OPERATION. 68 74 4x6 + 3 (carried) + 7 8 = 3 8 = 8 5032 7x6 + 8 (carried) = 50 Ex. Multiply 579 by 42. Am. 24318. OPERATION. 579 42 ANALYSIS. 24318 2x9 = 1 2x7 + 1 (carried) +4x9 = 5 2x5 + 5 (carried) +4x7 = 4 Ex. 4x5+4 (carried) = 24 Ans. 197316. Multiply 567 by 348. OPERATION. ANALYSIS. 567 8 x 5 = 40 8 x 6 = 48 8 x 7 = 56 348 4x5 = 20 4x6 = 24 4x7 = 28 197316 3x5 = 15 3x6 = 18 3x7 = 21 . 7 3 1 ~ 6 19 32 MULTIPLICATION. [Art. 7G. 76. To multiply together numbers of two figures each, whose units are alike. Ex. Multiply 76 by 46. Ans. 3496. OPERATION. ANALYSIS. 6 6x6 = 3 46 * 4 j- 6x11 + 3 (carried) = 6 6 9 3496 4x 7 + 6 (carried) = 34 Ex. Multiply 135 by 65. Ans. 8775. ANALYSIS. 5x5 = 2 1-5x19 + 2 (carried) = 9 OPERATION. ANALYSIS. 135 5x5 = 65 5 x 13 8775 5x 6 6x13 + 9 (carried) = 77. RULE. Multiply units by units for the first figure of the product, the sum of the tens by units for the second figure, and tens by tens for the third figure, carrying when necessary. EXAMPLES. 78. Multiply 1. 56 by 56 ; 72 by 32 ; 94 by 44. 2. 65 by 75 ; 87 by 37 ; 46 by 36. 3. 99 by 49 ; 85 by 75 ; 34 by 24. 4. 47 by 37 ; 67 by 57 ; 85 by 45. 5. 125 by 65 ; 126 by 36 ; 154 by 84. 6. 76 by 76 ; 36 by 36 ; 114 by 114. 79. To multiply together numbers of two figures each, whose tens are alike. Ex. Multiply 87 by 85. Ans. 7395. OPEBATION. ANALYSIS. 87 5x7 = 3 85 8 x 7395 X 8 x 8 + 9 = 7 9 Art. 79.] SHORT METHODS. 33 Ex. Multiply 127 by 122. Ans. 15494. OPERATION. ANALYSIS. 127 2x7=1 122 12 x 2 ) 1^494 12x7J 12 >< 9+ 1 = 10 12 x 12 + 10 = 15 4 80. RULE. Multiply units by units for the first figure of the product, the sum of the units by tens for the second figure, and tens by tens for the remaining figures, carrying when necessary. EXAMPLES. 81. Multiply 1. 87 by 82 ; 81 by 87 ; 65 by 63. 2. 47 by 44 ; 56 by 52 ; 58 by 57. 8. 73 by 76 ; 79 by 75 ; 68 by 63. 4. 44 by 43 ; 52 by 55 ; 67 by 63. 5. 116 by 117 ; 107 by 105 ; 125 by 122. 82. To multiply together two numbers whose tens are alike, and the sum of whose units is ten. 83. RULE. Multiply the units together for the two right-hand figures of the product, one of the tens by 1 more than itself for the remaining figures. EXAMPLES. 84. 1. Multiply 76 by 74. Ans. 5624. ANALYSIS. 6 x 4 = 24, the two right-hand figures of the product. G x 7 (6 + 1) = 42, the remaining figures. Multiply mentally 2. 24 by 26 ; 85 by 85 ; 128 by 122. 3. 17 by 13 ; 94 by 96 ; 112 by 118. 4. 34 by 36 ; 37 by 33 ; 104 by 106. 5. 25 by 25 ; 43 by 47 ; 143 by 147. 6. 35 by 35 ; 56 by 54 ; 152 by 158. 34 MULTIPLICATION. [Art. 85. 85. To multiply by means of complements (72). Ex. Multiply 991 by 996. OPERATION. ALGEBRAIC MULTIPLICATION. 991 .. 9 991 = 1000 9 ) 996.. 4 996 = = - 13 987036 1000 x 1000 9 x 1000 ___ 4 x 1000 + 36 (1000 13) x 1000 + 36 ANALYSIS. From the above algebraic multiplication, it is observed : 1st, that as many of the right-hand figures as there are ciphers in the unit of comparison may be obtained by multiplying the complements together ; 2nd, that the second part of the result is equivalent to the sum of the numbers less the unit of comparison multiplied by that unit. The sum of the numbers less the unit of comparison may be obtained by adding the numbers and omitting the 1 at the left-hand, or by subtracting either complement from the opposite number. Thus, 991 4 = 987. 86. RULE. From either number subtract the comple- ment of the other, and to the right of the remainder write the product of the complements. NOTES. 1. When there are less figures in the product of the comple- ments than ciphers in the unit of comparison, write ciphers in the result to supply the deficiency. 2. When there are more figures in the product of the complements than ciphers in the unit of comparison, add the excess on the left-hand to the second part of the result. 3. After practice, the complements may be omitted in the operation. EXAMPLES. 87. 1. Multiply 88 by 95 ; 975 by 993 ; 9999 by 9999. (-) (*.) ('-) 88.. 12 775.. 225 9999... 1 95. ..5 993 .... 7 9999... 1 8360 769575 99980001 Multiply Multiply 2. 97 by 99 ; by 94. 8. 993 by 992 ; by 994 3. 88 by 91 ; by 95. 9. 990 by 991 ; by 988. 4. 89 by 93 ; by 96. 10. 982 by 994 ; by 995. 5. 75 by 97 ; by 98. 11. 925 by 996 ; by 994. 6. 92 by 98 ; by 93. 12. 875 by 992 ; by 993. 7. 86 by 94 ; by 95. 13. 847 by 990 ; by 988. Art, 88.] SHORT METHODS. 35 88. To multiply together two numbers of the same number of figures over and near 100, 1000, etc. Ex. Multiply 116 by 103. OPERATION. ALGEBRAIC MULTIPLICATION. 116 116 = 100 + 16 I -900+19 103 = 100+ af 811111 " 200 + 9 100 x 100 + 16 x 100 H948 + 3 x 100 + 48 (100 + 19) x 100 + 48 89. RULE. From the sum of the numbers subtract the unit of comparison, and to the right of the result write the product of the excesses. (See Notes to Art. 86.) EXAMPLES. 90. Multiply Multiply 1. 112 by 106 ; by 111. 5. 145 by 107 ; by 112. 2. 102 by 103 ; by 104. 6. 176 by 111 ; by 108. 3. 122 by 108 ; by 105. 7. 1004 by 1006 ; by 1007. 4. 116 by 107 ; by 112. 8. 1125 by 1008 ; by 1012. 91. To multiply together two numbers, one of which is more and the other less than 100, 1000, etc., Ex. Multiply 109 by 97. OPERATION. ALGEBRAIC MULTIPLICATION. 109 9 excess. 109 = 97 3 complement. 97 = 10600 , 100 x 100 + 9 x 100 f Product of excess _ 3 x 100 - 27 - ( and complement. 10573 (100 + 6) x 100 - 27 92. RULE. Multiply the sum of the nuinbers less the unit of comparison ~by that unit, and from the product subtract the product of the excess and complement. EXAMPLES. 93. Multiply Multiply 1. 107 by 97 ; by 95. 5. 1005 by 91 ; by 93. 2. 112 by 96 ; by 92. 6. 1007 by 95 ; by 97. 3. 116 by 94 ; by 98. 7. 1012 by 99 ; by 92. 4. 108 by 91 ; by 99. 8. 1018 by 94 ; by 96. DIVISION. 94. Division is the operation of finding how many times one number is contained in another. The number divided is called the dividend. The number by which it is divided is called the divisor. The result obtained is called the quotient. The part of the dividend which remains after the operation is completed is called the remainder. 95. The sign of division is -- and is read divided by. Thus, 16 -f- 2 = 8 is read, sixteen divided by two equals eight. 96. To divide when the divisor does not exceed 12. NOTE. When the work is performed mentally, as in the following opera- tion, the process is called Short Division. Ex. Divide 1859 by 4. OPEBATION. ANALYSIS. Write the divisor at the left of the dividend, 4 ) 1859 as in the operation, and begin to divide at the left. 4 is 4645. n k contained in 1 thousand, the highest order of the dividend, therefore, divide 18 hundreds by 4. 4 is con- tained in 18 (hundreds), 4 (hundred) times, and 2 hundreds remain. Write the 9 hundred under the line in hundreds' place, and reduce the 2 hundreds remaining to tens, making 20 tens, which added to the 5 tens of the dividend, make 25 tens. 4 is contained in 25 (tens), 6 (tens) times and 1 ten remains. Write the 6 tens under the line in tens' place, and reduce the 1 ten remaining to units, making 10 units, which added to the 9 units of the dividend, make 19 units. 4 is contained in 19 (units), 4 (units) times, and 3 units remain. Write the 4 units in units' place, and write the remainder over the divisor, with a line between them in the form of a fraction, thus, f (three-fourths). The complete result is 464f . Observe that each quotient figure is placed directly under the last figure of the dividend used. In practice, do not think of explanations, etc. ; but, think only of the par- tial dividends and quotient figures. Thus, in the above example, say or think, 4 into 18 4 times, into 25 6 times, into 19 4 times, etc. Art. 97.] DIVISION. 37 97. RULE. Write the divisor at the left of the dividend with a curved line between them. Beginning at the left, divide each figure of the dividend by the divisor, and place the quotient beneath the figure divided. Whenever a remainder occurs, prefix it to the following figure of the dividend, and divide as before. Continue the operation until all the figures of the dividend have been divided, and place the remainder, if any, over the divisor at the right of the quotient. 98. PROOF. Multiply the quotient by the divisor, and to the product add the remainder. If the result equals the dividend, the work is probably correct. 99. 1. 2. 3. 4. 5. 6. 7. 8. 9. 21. In one square yard there are 9 square feet. How many square yards in 41652 square feet ? 22. There are 12 pence in one shilling. How many shillings in 124656 pence ? 23. In a barrel containing 1068 eggs, how many dozen ? What is their value at 23 cents per dozen ? 24. In one foot there are 12 inches. How many feet in 63360 inches ? 25. There are 2 pints in one quart, and 4 quarts in one gallon. How many gallons in 160048 pints ? 26. There are 8 quarts in one peck, and 4 pecks in one bushel. How many bushels in 349056 quarts ? EXAM PLES Divide Divide 78912348 by 2 ; by 3. 11. 103050709 by 2; by 5. 97652464 by 4 ; by 6. ]2. 214161810 by 3; by 6. 16327620 by 5 i by 6 . 13. 425262728 by 4; by 7. 78070804 by 4 ; by 7 . 14. 123456789 by 3; by 8. 12345678 by 6 j by 9 . 15. 246801234 by 6 ; by 11. 988654320 by 5 ;by 8. 16. 789123650 by 7; by 10. 234568836 by 4 ;by 9. 17. 287236450 by 5; by 12. 357212254 by 2 ;by 7. 18. 176111888 by 6; by 11. 246886425 by 5 ;by 9. 19. 1010101010 by 7 ; by 9. 217181916 by 7 ;by 9. 20. 200200200 by 8; by 12. 38 DIVISION. [Art. 100. 100. To divide by any divisor greater than 12. NOTE. When the work is all written, as in the following operation, the process is called Long Division. Ex. Divide 13218 by 43. OPERATION. ANALYSIS. Since 43 is not contained in 13 Divisor. Dividend. Quotient. (thousands), we take 132 (hundreds) for the 43 ) 13218 ( 307i-J first partial dividend. 43 is contained in 132 129 (hundreds), 3 (hundred) times. 43 x 3 (hun- TT7 dreds) = 129 (hundreds), which write under the 132 (hundreds), and subtract. The re- mainder is 3 (hundreds), to which annex the 1 17 Bemainder. (ten) of the dividend, and the second partial dividend is 31 (tens). 43 is not contained in 31 (tens), therefore write as the next figure of the quotient. Annex to the partial dividend, 31 (tens), the 8 (units) of the dividend, and the next partial dividend is 318 (units). 43 is contained in 318 (units), 7 (units) times. 43 x 7 (units) = 301 (units), which write under the 318 (units) and subtract. The remainder is 17 (units). Indicate the division of this remainder in the form of a fraction, thus : $, and annex it to the quotient, producing 307|| for the complete quotient. 101. RULE. Write the divisor at the left of the dividend, with a curved line between them. Take for the first partial dividend the least number of figures on the left that will contain the divisor, and write the quotient figures at the i^ight. Multiply the divisor by the quotient, ivrite the product under the partial dividend, and subtract. To the remain- der, annex the next figure of the dividend, for the second partial dividend. Divide as before, and thus continue until all the figures of the dividend have been used. Write the remainder, if any, over the divisor in the form of a fraction, and annex it to the quotient. The result will be the complete quotient. 102. PROOF. Multiply the divisor by the quotient, and to the product add the remainder. If the sum equals the dividend, the work is probably correct. Art. 103.] DIVISION. 39 EXAM PLES. 1O3. Divide Divide 1. 307845 by 26 ; by 143. 11. 8712460 by 73 ; by 817. 2. 248916 by 19 ; by 249. 12. 1428716 by 84 ; by 365. 8. 375428 by 38 ; by 375. 13. 2893429 by 69 ; by 144. 4. 481369 by 48 ; by 116. 14. 7364128 by 14 ; by 128. 5. 423706 by 25 ; by 208. 15. 2125639 by 70 ; by 320. 6. 3064028 by 18 ; by 429. 76?. 3756425 by 64 ; by 231. 7. 1289434 by 64 ; by 567. 17. 4183691 by 36 ; by 365. 8. 7090805 by 73 ; by 432. 18. 3804072 by 96 ; by 729. 9. 6321457 by 87 ; by 618. 19. 1653891 by 33 ; by 640. 10. 2304802 by 92 ; by 729. 20. 2763940 by 95 ; by 160. 21. How many days in 8766 hours ? 22. In 20000 pens, how many gross ? (1 gross = 144.) 23. How many bushels in 21674 pounds of oats, if there arc 32 pounds in one bushel ? 24- There are 56 pounds in a bushel of rye. How many bushels in 19958 pounds ? 25. There were 31392893 gallons of molasses imported into the United States in 1885. How manys hogsheads of 63 gallons each ? 26. How many cords in 47164 cubic feet, if there are 128 cubic- feet in one cord ? 27. How many miles in 49164 rods, if there are 320 rods in one mile ? 28. The expenditures of the United States for the year 1880 were $287,034,182. How much was that per day (365 days in the year) ? 29. During the year 1882, 788992 immigrants arrived in the United States. What was the average number per day ? 30. The population of the 38 States was 49,371,340 in 1880. and there are 325 members in the House of Representatives. What is the average population to each member ? 31. The exports of cotton during the year 1885 were 1,889,514,368 pounds. How many bales averaging 476 pounds each ? 82. How many rails 18 feet in length would be required for a railroad 51 miles long ? (1 mile = 5280 feet.) 40 DIVISION. [Art. 104. SHORT METHODS OF DIVISION. 104. Leaving out the Products. In long division the process may be shortened by the following rule : 105. RULE. Subtract the several products from the next number greater ending with the corresponding figure in the dividend, and carry each time the left-hand figure of the minuend to the next product. NOTE. If the right-hand figure of any product is the same as the corres- ponding figure of the dividend, subtract it from itself, and not from the next higher number ending with the same figure ; or, write in the remainder, carrying the left-hand figure of the product. Ex. Divide 42343014 by 973. 973 ANALYSIS. The first quotient figure is 4, by which we multiply. 4 times 3 are 12, which sub- 42343014 3423 tracted from 14 (the next number greater ending with 4) leaves 2. Write 2 in the remainder and carry 1. 4 times 7 are 28, 1 carried makes 29, 7784 which subtracted from 33 (the next number 000 greater ending with 3) leaves 4. Write 4 in the remainder and carry 3. 4 times 9 are 36, 3 carried makes 39, which subtracted from 42 leaves 3. Write 3 in the remain- der and carry 4. 4 subtracted from 4 .leaves 0. Bring down 3, the next figure of the dividend. So proceed until the division is finished. NOTE. Perform any of the examples in Art. 1O3 by this method. 106. To divide by 25. 107. RULE. Multiply the dividend by 4> ana divide the product by 100 by cutting off two figures from the right. NOTE. To divide by 125, multiply by 8 and divide the product by 1000 by cutting off three figures from the right. Ex. Divide* 11175 by 25. OPERATION. ANALYSIS. Since 25 is one-fourth of 100, multiplying ___ 4 by 4 an( j dividing by 100, is the same as dividing by 25. 447.00 EXAMPLES. 108. 1. Divide the following numbers by 25 : 1175, 1650, 1700, 2875, 3825, 4950, 3800, 1725, 1775, 1825, and 2000. UNITED STATES MONEY. 109. United States Money is the legal currency of the United States. It consists of gold, silver, nickel, and copper coins, treasury and national bank notes, gold and silver certificates. 110. Legal Tender. The term legal tender is applied to money which may be legally oifered in the payment of debts. 111. The unit of value is the gold dollar of 25.8 grains. TABLE. 10 Mills = 1 Cent c., ct. 10 Cents = 1 Dime d. 10 Dimes = 1 Dollar $. 10 Dollars = 1 Eagle E. NOTES. 1. In business operations, dollars and cents are principally used. Eagles and dimes are used only as the names of coins. 2. The currency of the Canadian Provinces is nominally the same as that of the United States. 112. The legal coins GOLD. Weight in grains. 1 dollar piece, 25.8 2 dollar piece, or ) Quarter-eagle, J 3 dollar piece, 77.4 5 dollar piece, or ) Ig9 Half-eagle, \ 10 dollar piece, or ) Eagle, t* 20 dollar piece, or ) Double-eagle, ( of the United States are as follows : SILVER. Weight. Standard dollar, 412 grains. Half dollar, or , or ) > 12* grams, or 192.9 grains. 50 cent piece, ) Quarter dollar, or ) _. > 61 grams, or 96.45 grams. 25 cent piece, ) 24 grams, or 38.58 grams. piece Dime, or 10 cent piece, COPPER AND NICKEL. 5 cent piece, 5 grams, or 77.16 grains. 3 cent piece, 30 grains. 1 cent piece, 48 grains. 113. The gold and silver coins of the United States contain 9 parts by weight of pure metal and 1 part alloy. The alloy of silver coins is copper ; and of gold coins, copper, or copper and silver. (The silver in no case exceeds ^ of the whole alloy.) 42 UNITED STATES MONEY. [Art. 114. 114. Gold Coins are a legal tender in all payments at their nominal value when not below the standard weight * provided by law ; and, when reduced in weight, below said standard, are a legal tender at valuation in proportion to their actual weight. 115. Standard Silver Dollars are a legal tender at their nominal value for all debts except where otherwise expressly stipulated in the contract. 116. Silver Certificates. Any holder of standard silver dollars may deposit the same with the Treasurer, or any Assistant Treasurer of the United States, in sums not less than $10, and receive therefor certificates, corresponding with the denominations of United States notes (119). These certificates are receivable for customs, taxes, and all public dues. 117. Subsidiary Coins. The present silver coins of the United States of smaller denominations than $1 are a legal tender in all sums not exceeding $10. The holder of any of the silver coins of the United States of smaller denominations than $1 may, on presentation of the same in sums of $20, or any multiple thereof, at the office of the Treasurer or any Assistant Treasurer of the United States, receive therefor lawful money of the United States. 118. Minor Coins. The 5 and 3 cent pieces contain } copper and nickel. The 1 cent piece contains 95 per cent, copper arid 5 per cent, tin and zinc. These coins are a legal tender for any amount not exceeding twenty-five cents. 119. United States Notes ("Greenbacks") are a legal tender for all debts except duties on imports and interest on the public debt. Since Jan. 1, 1879, they have been redeemable in coin at the office of the Assistant Treasurers of the United States in the Cities of New York and San Francisco, in sums of not less than $50. They represent the values of $1, $2, $5, $10, $20, $50, $100, $500, $1000, $5000, $10,000. The Act of May 31, 1878, fixed their value at $346,681,016, and forbade their further contraction. * " Any gold coin of the United States, if reduced in weight by natural abrasion not more than one-half of one per centum below the standard weight prescribed by law, after a circulation of twenty years, as shown by its date of coinage, and at a ratable proportion for any period less than twenty years, is received at its nominal vaiue by the United States treasury and its offices." The " Coinage Act of 1873 " allows a deviation from the standard weight of { of a grain, or less, in the manufacture of the dollar piece. Art. 120.] UNITED STATES MONET. 43 120. National Bank Notes (64O) are not a legal tender ; but, since they are secured by bonds of the United States deposited with the U. S. Treasurer at Washington, and are redeemed in lawful money by the national banks and the Treasurer of the United States, they are usually accepted in the payment of debts in any part of the United States. They are receivable in all parts of the United States in payment of taxes and other dues to the United States except duties on imports, and for debts owing by the United States to individuals and corporations, within the United States except interest on the public debt. They represent the values of $1, $2, $5, $10, $20, $50, $100, $500, and $1000. Since Jan. 1, 1879, no notes of the denomination of $1 and $2 have been issued to national banks. Nov. 1, 1886, their total circulation was $301,529,889. 121. To write United States money. 122. In writing U. S. money, the decimal notation is used. Dollars are written at the left of the decimal point and form the integral part. Cents are written as hundredths of a dollar, and occupy the first two places at the right of the decimal point. Mills are written as thousandths of a dollar, and occupy the third decimal place. Thus, twelve dollars, forty-eight cents, and six mills, is written $12.486. When the number of cents is less than ten, a cipher must be written in the first place at the right of the point. Thus, eight dollars and six cents is written, $8.06. In the final results of business operations, if the mills are more than five, they are regarded as an additional cent ; if less than five, they are rejected. In checks, notes, drafts, etc., cents are usually written as hundredths of a dollar in the form of a fraction. Thus, twenty-five cents may be written,$ T 2 ^. 123. Express the following amounts by figures : 1. Eighty-six dollars, nineteen cents, five mills. 2. Fourteen dollars, seventy-five cents, three mills. 3. Five hundred twenty-six dollars, seventy cents. 4. Two thousand dollars, thirty cents, two mills. 5. Seven hundred dollars, nine cents. 6. Fifty thousand dollars, seven mills. 7. Four hundred eight dollars, two cents, five mills. 8. Two hundred fifty dollars, sixty cents, three mills. 44 UNITED STATES MONEY. [Art. 124. 124. To reduce dollars to cents and mills, or to reduce cents and mills to dollars. 125. Dollars may be reduced to cents by multiplying by 100 or by annexing two ciphers. Dollars may be reduced to mills by multiplying by 1000 or by annexing three ciphers. Thus, $64 = 6400 cents, or 64000 mills. If the amount consists of dollars and cents, reduce to cents by removing the decimal point 2 places to the right ; to mills, three places to the right. Thus, $17.28 = 1728 cents, or 17280 mills ; $34.658 = 34658 mills. 126. Cents may be reduced to dollars by dividing by 100, or by pointing off two decimal places. Mills may be reduced to dollars by dividing by 1000, or by pointing off three decimal places. Thus, 12345 cents = $123.45 ; 37560 mills = $37.56. 127. Reduce the following to cents : 1. $345. 4. $17.04. 7. $148.19 2. $2376. 5. $28.37. 8. $204.40. 8. $2004. 6. $49.75. 9. $317.04. 128. Change the following to mills : 1. 75 cents. 4. $38. ' 7. $14.172. 2. 19 cents. 5. $376. 8. $4.866. 8. 47 cents. 6. $408. 9. $10.012. 129. Reduce the following to dollars : 1. 148 cents. 4. 705 cents. 7. 18000 mills. 2. 2300 cents. 5. 4212 cents. 8. 9370 mills. 8. 4617 cents. 6. 13409 cents. 9. 12375 mills. 130. To add or subtract in United States money. 131. RULE. Write dollars under dollars, and cents under cents. Add or subtract as in simple numbers, and place the point in the result directly under the points in the numbers added or subtracted. NOTE. In subtraction of U. S. money, if there are cents in the subtrahend and none in the minuend, suppose ciphers to be added to the subtrahend in cents' place. Art. 132.] UNITED STATES MONEY. 45 EXAM PLES. 132. 1. Add 5 dollars, 16 cents ; 18 dollars, 5 cents ; 404 dollars, 75 cents ; 25 dollars, 8 cents ; 2376 dollars, 40 cents ; 8 dollars, 2 cents. 2. Add $170, $106.40, $240, $200.40, $70, $.70, $234.75. ' 3. Add $108.25, $2345, $6.04, $7.10, $192.43, $117.05. 4. Add $.06, $6, $108.16, $500.64, $564, $5.64, $117.10, $2081.48. 5. From $124.16 subtract $109.25. 6. From $117 subtract $98.49. 7. From $575 subtract 575 cents. 8. A merchant makes the following deposits in a bank i $1875.24, $416, $234.70, $558.96, and $437.10. He draws the following checks: $442.37, $120.92, $316.75, $242.71, $195, $716.32, $100.48, and $76.19. What is the balance of his bank account ? 133. To multiply United States money. Ex. Find the cost of 9 desks at $2.45 each. OPERATION. 2 45 ANALYSIS. Disregarding the decimal point, multiply as in q ordinary multiplication. 9 times 245 cents is 2205 cents = $22.05. 22.05 134. RULE. Multiply as in simple numbers, and from the right of the product point off as many figures as there are figures to the right of the decimal points in both numbers. NOTE. If, as in Examples 6 and 7, the number expressing cents would make a convenient multiplier, use it as such, considered as an abstract number, and point off the result according to the rule. 135. Multiply Multiply 1. 12 dollars and 18 cents by 8. 7. $.07 by 1239 ; by 13416. 2. 4 dollars and 25 cents by 12. 8. $20.04 by 20 ; by 108. 3. 16 dollars and 9 cents by 17. 9. $176 by 18 ; by 144. 4. 27 dollars and 8 cents by 25. 10. $36.25 by 36 ; by 117. 5. 43 dollars and 50 cents by 76. 11. $48.19 by 48 ; by 288. 6. 8 cents by 2345 ; by 3456. 12. $50.08 by 75 5 by 192. 40 UNITED STATES MONEY. [Art. 136. 136. To divide in United States money. Ex. If 9 desks are worth $22. 05, what is one worth ? OPERATION. ANALYSIS. If 9 desks are worth 2205 cents ($22.05), 1 9 ) 22.05 desk is worth one-ninth of 2205 cents or 245 cents. 245 cents $2.45 - $2.45. Ex. If 8 chairs are worth $18, what is one worth ? OPERATION. ANALYSIS. If the dividend consists of dollars only, and 8 ) 18.00 does not contain the divisor an exact number of times, reduce *2 ~T it to cents by annexing two ciphers. Ex. At $6.25 each, how many sheep can be bought for $50 ? OPERATION. ANALYSIS. If 1 sheep costs $6.25, as many $6.25 ) $50.00 ( 8 sheep can be bought for $50 as $6.25 is contained times in $50. $50 = 5000 cents. $6.25 = 625 cents. O r > 5000 cents -f- 625 cents = 8 times. Hence the result 625. ) 5000. (8 is 8 sheep. 137. RULE. Divide as in simple, numbers, and point off from the right of the quotient as many decimal places as those in the dividend exceed those in the divisor. NOTE. If the divisor alone contains cents, make the dividend cents by annexing two ciphers ; or, reduce both divisor and dividend to cents by annexing ciphers, omit the decimal points, and divide as in simple numbers. EXAM PLES. 138. 1. If 12 books are sold for $41.40, what is the price of one book ? 2. How many pounds of tea at 65 cents per pound can be bought for $9.75? NOTE. In the following examples, if the quotient is in U. S. money and the result is not an exact number of dollars, continue the division to cents. Divide Divide 3. $25.44 by 48 ; by 106. 9. $130.38 by $2.46 ; by $1.06. 4. $476 by 25 ; by 35. 10. $149.04 by $0.36 ; by $2.07. 5. $1728 by 36 ; by 48. 11. $156.24 by $0.72 ; by $4.34. 6. $73.08 by 84 ; by 87. 12. $1728 by $0.75 ; by $6.75. 7. $106.56 by 72 ; by 576. 13. $3456 by $2.25 ; by $13.50. 8. $1884 by 75 ; by 1535. 14. $7154 by $1.75 ; by $25.55. Art. 139.] REVIEW EXAMPLES. 47 REVIEW EXAMPLES. 139. 1. Find the sum of the following numbers : Twenty- six thousand forty-eight ; twelve thousand four hundred eighty ; one hundred thirty-six thousand ; seven hundred ninety thousand forty-three ; four million fifty-eight. 2. Subtract eight hundred fourteen thousand nine hundred sixteen from four million nineteen thousand. 3. Multiply five hundred sixty thousand seven hundred eight by eighteen hundred sixty. 4. A quantity of merchandise was bought for $27618.75, and sold for $32418.25. What was the gain ? 5. Find the total length of the Brooklyn bridge, its meas- urements being as follows : Length of river span, 1596 feet ; of' each (2) land span, 930 feet ; of New York approach, 1562 feet ; of Brooklyn approach, 971 feet. 6. If I sell goods for $23876, and gain $5389, what did the goods cost me ? 7. The estimated production of gold and silver of the world for 1884 was as follows : Gold, $98,990,772 ; silver, $116,525,949. For 1885, gold, $101,562,748; silver, $124,968,784. What was the total increase ? 8. If the quotient is 375 and the divisor 246, what is the dividend ? 9. If the product of two numbers is 450072, and one of the numbers is 987, what is the other number ? 10. Divide 76432801 by 783. Prove that your solution is correct. 11. A clerk receiving a salary of $1256, pays $468 a year for board, $180 for clothing, and $150 for other expenses. What amount has he left ? 12. If I take 24889 from the sum of 9872 and 24967, divide the remainder by 50, and multiply the quotient by 18, what is the product ? IS. If 160 acres of land cost $10720, how many acres can be bought for $8844 ? 14. If 75 head of cattle cost $2550, what will 59 head cost ? 15. Cash on hand at beginning of the day, $6492.75; cash received, $11456.75; cash paid out, $13285.26. Kequired the cash balance at the end of the day. 48 UNITED STATES MONEY. [Art. 139. 16. A merchant sold 426 barrels of flour for $2556, which was $639 more than he gave for it. What did it cost him a barrel ? 17. Mr. A has three farms, the first of which contains 158 acres, the second 32 acres less than the first, and the third as many as the other two. What is the value per acre, if all are worth $26128 ? 18. A merchant bought 387 yards of cloth at 79 cts. per yard ; he sold 298 yards at $1.16 per yard, and the remainder at 97 cts. per yard ; how much did he gain ? 19. The United States nickel and copper coinage for the year 1886 was 5,519 five-cent pieces, 4,519 three-cent pieces, and 1,696,613 one-cent pieces. Find total value of minor coinage. 20. The silver coinage for 1886 was as follows: 29,838,905 dollars, 6,105 half-dollars, 14,505 quarter-dollars, 1,767, 642 dimes. What was the total value of the silver coinage ? 21. The gold coinage for 1886 was as follows : 243,584 double- eagles, 1,042,847 eagles, 3,751,629 half-eagles, 101 three-dollar pieces, 4,086 quarter-eagles, 8,567 one-dollar pieces. What was the value of the gold coinage ? 22. There are four bidders to supply the government with 800 tons Lehigh, 500 tons Cumberland, and 700 tons Baltimore coal. A offers Lehigh at $6.29, Cumberland at $4.38, and Baltimore at $7.23. B offers Lehigh at $6.80, Cumberland at $4.12, and Balti- more at $7.24. C offers Lehigh at $6.40, Cumberland at $4.45, and Baltimore at $7.18. D offers Lehigh at $6.17, Cumberland at $4.19, Baltimore at $7.20. Who is the lowest bidder for the whole amount, and how much does each bid amount to ? 23. A drover bought a number of cattle for $12204, and sold the same for $13560, by which he gained $4 per head. How many cattle were purchased ? 24 A farmer raised in one year 512 bushels of wheat, the next year twice as much as he raised the first year, and the third year four times as much as he did the second year. What was the value of the three crops at $1.65 per bushel ? 25. Bought 75 tons of hay at $16 per ton ; gave in payment 56 sheep at $3.75 each, and the . remainder I paid in butter at 33 cts. per pound. How many pounds of butter were required ? 26. Bought 225 acres of land for $12,600, and sold 116 acres at $65 per acre, and the remainder at cost ; how much did I gain ? Art. 139.] PROPERTIES OF NUMBERS. 49 27. A sold to B 175 acres of land at $135 an acre, and by so doing gained $1925 ; B sold the land at a loss of $1750. What did A pay per acre, and what was B's selling-price per acre ? 28. A merchant sold 800 barrels of flour for $5867, 144 barrels of which he sold at $7 per barrel, and 225 barrels at $6.75. At how much per barrel did he sell the remainder ? 29. According to the following table, what was the average immigration per year ? What per month ? Years. Number. Years. Number. Years. Number. 1875 227 498 1879 177 826 1883 603 322 1876 169,986 1880 457 257 1884 518 592 1877 141,857 1881 . . . 669 431 1885 395 346 1878 138,469 1882 . 788 992 1886 334 203 PROPERTIES OF NUMBERS. 140. A Number is a unit, or a collection of units ; as one, four, three feet, five dollars. 141. All numbers are either integral or fractional, abstract or concrete. 142. An Integral Number, or Integer is a number which expresses whole things ; as two, four gallons, seven dollars. 143. A Fractional Number, or Fraction is a number which expresses one or more equal parts of a unit ; as one-half, three-fourths. 144. An Abstract Number is a number which does not refer to any particular object ; as one, six, ten. 145. A Concrete Number is a number applied to an object, or quantity ; as three apples, five pounds, ten dollars. 146. Integral numbers are either odd or even, prime or composite. 147. An Odd Number is a number whose unit figure is 1, 3, 5, 7, or 9 ; as 7, 21, 39. 50 PROPERTIES OF NUMBERS. [Art. 148. 14:8. An Even Number is a number whose unit figure is 0, 2, 4, 6, or 8 ; as 6, 40, 74. 149. A Prime Number is a number which can be exactly divided only by itself and unity ; as 1, 1, 13, 29. 150. Numbers are prime to each other when no integral number greater than 1 will divide each without a remainder. Numbers that are prime to each other are not necessarily prime numbers. Thus, 25 and 28 are prime to each other, but they are not prime numbers. 151. A Composite Number is a number which can l;e exactly divided by other integers besides itself and unity. Thus 28, the product of 4 and 7, is a composite number. It is exactly divisible by 4 and 7. DIVISIBILITY OF NUMBERS. 152. An Exact Divisor of a number is any number that will divide it without a remainder. Thus 2, 3, 4, 6, 8, and 12 are exact divisors of 24. 153. A number is said to be divisible by another when the latter will divide the former without a remainder. Any number is divisible 1. By 2, if it is an even number ; as 6, 28, and 32. 2. By 3, if the sum of its digits is divisible by 3 ; as 841) (8 + 4 + 9 = 21, 21 is divisible by 3), 7323, and 47892. 3. By 4, if the two right-hand figures are ciphers, or express a number divisible by 4 ; as 1100, 216, and 7328. 4. By 5, if the right-hand figure is or 5 ; as 40 and 135. 5. By 6, if it is an even number and the sum of its digits is divisible by 3 ; as 216, 840, and 732. 6. By 8, if the three right-hand figures are ciphers, or express a number divisible by 8 ; as 3000 and 7168. 7. By 9, if the sum of its digits is divisible by 9 ; as 216, 783, and 12348. Art. 154.] PRIME FACTORS. 51 PRIME FACTORS. 154. The Factors of a number are those numbers which when multiplied together will produce the number. Thus 4 and 7; 2 and 14; 2, 2, and 7 are factors of 28. The number itself and unity are not regarded as factors. The factors of a number are also the exact divisors of it. 155. A Prime Factor is a prime number used as a factor. Thus, 2, 2, and 7 are the prime factors of 28. 4 is a factor of 28, but not & prime factor. 156. To find all the prime factors of a composite number. Ex. What are the prime factors of 6930. OPERATION. ANALYSIS. Any prime number that is an exact divi- 2 ) 6930 sor of the given number is a prime factor of it. Divide o \ QAAP; ^ e gi ven number by 2 (153, 1), the least prime divisor of it, obtaining the quotient 3465. Next, divide this quo- 3 )_1155 tient successively by 3 (153, 2), 3, 5 (153, 4), and 7. 5 } 385 ^k e * ast q uot i ent; 11 i g a P r irae number and therefore a prime factor. The several divisors 2, 3, 3, 5, 7 and the 7 ) 7? last quotient 11 are the prime factors required. 11 2x3x3x5x7x11 = 157. RULE. Divide by the least prime number which will divide the given number without a remainder. In like manner divide the resulting quotient, and continue the division until the quotient is a prime number. Tl^e several divisors and the last quotient are the prime factors. EXAMPLES. 158. Resolve the following numbers into their prime factors : 1. 3465. r. 6552. 13. 8192. 19. 6660. 2. 3003. 3. 4158. 4. 3150. 5. 3675. 6. 2310. 8. 7826. 14. 6561. 20. 2448. 9. 6006. 15. 3125. 21. 8525. 10. 5368. 16. 1800. 22. 9926. 11. 3825. 17. 1935. 23. 9576. 12. 5324. 18. 2475. 24. 5075. 52 PROPERTIES OF NUMBERS. [Art. 159. COMMON MULTIPLES.* 159. A Multiple of a number is a number that is exactly divisible by it ; or, it is any product of which the given number is a factor. Thus, 12 is a multiple of 6; 15 of 5; etc. 160. A Common Multiple of two or more numbers is a number that is exactly divisible by each of them. Thus, 12, 24, 36, and 48 are multiples of 4 and 6. 161. The Least Common Multiple of two or more num- bers is the least number that is exactly divisible by each of them. Thus, 12 is the least common multiple of 4 and 6. 2. PRINCIPLES. 1. A multiple of a number contains all the prime factors of that number. 2. A common multiple of two or more numbers contains all the prime factors of each of those numbers. 3. The least common multiple of two or more numbers contains all the prime factors of each of the numbers, and no other factors. 163. To find the least common multiple of two or more numbers. Ex. What is the least common multiple of 12, 18, 20, and 40 ? FIRST OPERATION. ANALYSIS. Since 40, a multiple 12 = 2x2x3 of 20, contains all the prime factors of 18 = 2x3x3. 20, the number 20 may be omitted in 40 = 2x2x2x5 ^ e P erat i n - Resolve the numbers into their prime factors. The least 2x2x2x3x3x5 = 360 common multiple must contain 2 as a factor 3 times in order to be divisible by 40 ; it must contain 8 as a factor twice in order to be divisible by 18 ; and it must contain 5 as a factor, in order to be divisible by 40. 360, the product of the factors, 2, 2, 2, 3, 3, and 5, is the least common multiple of the given numbers, since it contains the different factors the greatest number of times that they occur in the given numbers, and no other factors (Prin. 3). * For Greatest Common Divisor, see Appendix, page 317. Art. 1 63.J C MM ON MUL TIP LES. 53 SECOND OPERATION. ANALYSIS. The factors of the F6- 2 ) 12, 18, 40 quired multiple may be found by the % \ Q g on following process. Divide the given num- bers by any prime number that will divide 3)3, 9, 10 two or more of them, writing the quo- -i o 10 tients and the undivided numbers be- neath. Treat the resulting numbers in 2x2x3x3x10 = 360 like manner, and continue the process until no two of the numbers have a com- mon factor or divisor. The product of the several divisors and the remaining quotients and undivided numbers will be the least common multiple. 164. BULE. Resolve the given numbers into their prime factors. The product of the different prime factors, taking each factor the greatest number of tunes it appears in any of the numbers, will be the least common multiple. Or, Divide the given numbers by any prime number (see Note 2) that will exactly divide two or more of them, writing the quotients and undivided numbers beneath. Repeat the operation with the resulting numbers until there is no exact divisor of any two of them. Tlie product of the several divisors and the last quotients and undivided numbers will be the least common multiple. NOTES. 1. In the operation, reject such of the smaller numbers as are divisors of the larger ; also reject such of the quotients and undivided num- bers as are divisors of the others. 2. Divide by composite numbers when they are exact divisors of all the numbers. EXAMPLES. 165. Find the least common multiple of the following numbers: 1. 6, 10, 15, and 30. 12. 24, 36, and 40. 2. 16, 24, and 48. 13. 32, 48, and 72. 8. 30, 40, and 60. U. 16, 22, 24, and 30. 4. 2, 4, 8, and 16. 15. 18, 28, 30, and 36. 6. 14, 21, and 28. 16. 12, 16, 20, and 24. 6. 5, 8, 15, and 18. 17. 33, 44, 55, and 66. 7. 6, 9, 21, and 24. 18. 27, 36, 42, and 48. 8. 12, 20, and 30. 19. 36, 45, 60, and 72. 9. 6, 10, 30, and 40. 20. 28, 35, 42, and 56. 10. 32, 48, and 60. 21. 45, 55, 60, and 75. 11. 24, 32, and 40. 22. 60, 72, 84, and 90, 54 PROPERTIES OF NUMBERS. [Art. 166. CANCELLATION. 166* Cancellation is a method of shortening an operation by rejecting equal factors from both dividend and divisor. 167. PRINCIPLE. Dividing both dividend and divisor by the same number does not affect the value of the quotient. Ex. Divide 84 x 36 by 27 x 14. OPERATIONS. ANALYSIS. Indicate the oper- 2 Or, ations to be performed as in the * 2 margin. Since 36 and 27 contain = 3 * /l ** the common factor 9, cancel or re- 10 4 ject it from both, retaining the factors 4 and 3 respectively. 14 and 84 contain the common factor 14 ; therefore reject it, retaining the factor 6 in the dividend. [Since cancellation is a process of division, the rejecting of 14 does not destroy it, but divides it, leaving 1 as a quotient. It is unnecessary to write 1 as a quotient, except when there are no other factors in the dividend.] 3 is a common factor of 6 and 3 ; therefore reject it from both, retaining the factor 2 in the dividend. The product of the remaining factors, 2 and 4, is the required quotient. 168. RULE. Write the numbers denoting multiplication above a horizontal Hue, and the numbers denoting division below. The numbers above the line will form a dividend, and the numbers below, a divisor. Cancel the factors common to both dividend and divisor. Tlie product of the remaining factors of the dividend divided by the product of the re- maining factors of the divisor will be the required quotient. EXAMPLES. 169. Find the value of the following expressions : 27 x 48 x 60 1760x175x6 360 x 28 x 27 x 5 * 54x36^40* *" 36 x 100 x 10 ' * 25 x 42 x 18x~12' 1500x144x5 40 x 36 x 42 x 18 17x36x25x144 ' 365x100 ' 9x35x30x8 * ' 48x60x106x51' 1760 x 6 x 145 24 x 30 x 54 x 35 144 x 625 x 37 x 12 100x365 ' 14x15x21x64* 288x^75x185 10. Multiply 72 by 3 x 18, divide the product by 8 times 9, multiply the quotient by 7 x 20, divide the product by 360, mul- tiply the quotient by 6 times 8. Art. 169.] CANCELLATION. 55 11. If 42 tons of coal cost $147, what will 16 tons cost ? 12. A man gave 9 pounds of butter at 17 cents a pound for 3 gallons of molasses ; how much was the molasses worth a gallon ? IS. If 20 pounds of beef cost 250 cents, what cost 75 pounds ? 14. How many potatoes at 65 cents per bushel will pay for 13 weeks' board at $7.50 per week ? 15. A merchant bought 375 barrels of flour at $5.50 per barrel, and paid in cloth at $2.75 per yard ; how many yards did it require ? 16. How many pounds of coffee at 27 cents per pound should be given for 57 bushels of corn at 63 cents per bushel ? 17. Sold 28 bushels of apples for $21 ; what should I receive for 42 bushels ? 18. How many cows worth $35 each must be given in exchange for 84 tons of hay at $15 per ton ? 19. How many bushels of corn at 52 cents a bushel must be exchanged for 324 bushels of oats at 39 cents per bushel ? 20. If 430 bushels of wheat are obtained from sowing 7 bush- els, how much would be obtained from sowing 21 bushels ? 21. What should be paid for the transportation of 3600 pounds of cheese at the rate of 47 cents per 100 pounds ? 22. What must be paid for transporting 31600 pounds of iron at $5 per ton of 2000 pounds ? 23. What will 7840 pounds of coal cost, at $6 per ton of 2240 pounds ? 24. If 3 men eat 7 pounds of meat in one week, how much would 6 men eat in 4 weeks ? 25. How many canisters, each holding 40 ounces, can be filled from 3 chests of tea, each containing 55 pounds of 16 ounces ? 26. How many times can 16 bottles, each holding 3 pints, be filled from 6 demijohns, each containing 10 gallons of 8 pints each ? 27. A man exchanged 275 barrels of potatoes, each containing 3 bushels, at 54 cents per bushel, for a certain number of pieces of muslin each containing 45 yards, at 11 cents per yard. How many yards did he receive ? 28. If a person travel 24 hours each day at the rate of 45 miles an hour, how many days would it require to pass around the globe, a distance of 25000 miles ? FRACTIONS. 170. A Fraction is one or more of the equal parts of a unit ; as one-half (J), two-thirds (f ), one-fourth (), seven-eights (J). If a unit be divided into four equal parts, each part is called a fourth. If one of these parts be taken, the expression will be one-fourth (\) ; if three parts, three-fourths (f), etc. 171. The greater the number of equal parts into which a unit is divided, the less will be each part ; the less the number of parts, the greater will be each part. One-half (|) is greater than one-third () ; one-fourth (J) is less than one- third i). A fraction is usually expressed by two numbers, one written above the other, with a line between. Fractions written in this form are called Common Fractions. 173. The number below the line is called the Denominator, because while indicating the number of equal parts into which the unit is divided, it denominates or names those parts. 174. The number above the line is called the Numerator, because it shows how many of the parts are taken to form the fraction. 175. The numerator and denominator, taken together, are called the Terms of the fraction. In the fraction f , 3 and 4 are the terms ; 4 is the denominator, and shows that the unit is divided into four equal parts, called fourths ; 3 is the numera- tor, and shows that three of these parts are taken to constitute the fraction. 176. A fraction is an expression of unperformed division The numerator is the dividend, the denominator is the divisor, and the value of the fraction is the quotient. 177. A Simple Fraction is a single fraction, both of whose terms are integers. 178. Simple fractions are proper or improper. Art. 179.] FRACTIONS. 57 179. A Proper Fraction is one that is less than a unit ; the numerator being less than the denominator. Thus, J-, J, and | are proper fractions. ISO. An Improper Fraction is one that is equal to, or greater than a unit ; hence the numerator must be equal to, or greater than the denominator. Thus, f, f, f, and ty- are im- proper fractions. 181. A Mixed Number is an integer and a fraction united ; as 2J, 4f, 18}. 182. A Complex Fraction is one whose numerator is a f- 105f 75} 3| 12 16| fraction or a mixed number; as |-, -^, , --, , g. Q3 The expression -j indicates division, and is not properly a fraction. ^ A unit cannot be divided into 5^ equal parts. 183. PRINCIPLES. 1. Multiplying the numerator or dividing the denominator by a number multiplies the fraction by that number. 2. Dividing the numerator or multiplying the denominator by a number divides the fraction by that number. 3. Multiplying or dividing both numerator and denominator by the same number does not change the value of the fraction. EXERCISES. 184. 1. Read the following fractions, and copy separately : 1, the simple fractions ; 2, the proper fractions ; 3, the improper fractions ; 4, the mixed numbers ; 5, the complex fractions : if; H;V; A-; -J; ; tt; ^l; y; *; 3 i; 13 S; \> ft; 7*; 8*5 46|; ifi; || ; ^ ; f; f; f. #. Write the following fractions : three fourths ; seven eighths ; nineteen sixteenths ; five, and one half ; one hundred three thirty- seconds ; one hundred, and three thirty-seconds ; forty-eight, and five twelfths ; eleven tenths ; nine forty-fifths. 3. Write the following fractions : eight ninths ; thirteen, and two-thirds ; sixteen twenty-fourths ; ten tenths ; fourteen, and forty-six hundredths ; sixty-five forty-eighths ; nineteen one hun- dred nineteenths ; thirty-six four hundred thirty-seconds. 58 FRACTIONS. [Art. 185. REDUCTION OF FRACTIONS. 185. Reduction of Fractions is the changing their form without changing their value. 186. A fraction is reduced to lower terms when the numerator and denominator are expressed in smaller numbers. 187. A fraction is in its lowest terms when its numerator and denominator have no common divisor. 188. A fraction is reduced to higher terms when the numerator and denominator are expressed in larger numbers. 189. To reduce a fraction to its lowest terms. Ex. Reduce -ffa to its lowest terms. OPERATION. ANALYSIS. Dividing both terms of the fraction, T/ir iH" f T 8 A> by the common divisor, 6, the result is ff ; dividing both terms of f by the common divisor, 7, the result is f. Since 2 and 3 have no common divisor, the fraction is reduced to its lowest terms (187). The value of the fraction has not been changed, because both terms have been divided by the same number (183, 3). 190. EULE. Divide the terms of the fraction by any number that ivill divide both without a remainder, and continue the operation with the resulting fractions until they have no common divisor. EXAMPLES. 191. Reduce to their lowest terms, i- if- * i%. 17. m- & iff. a. H- 10. in- 18 - tti ** HI*. & ** 11- * 4- If. 12. Hi 5. 3%. 18. iftt- 21. Jfr. 29. 6. -^ 14. f}f. **. H*. 80. 7- 1%. 15. fWV- 28. -MV. 81. 8. Ait. 192.] REDUCTION OF FRACTIONS. 59 192. To reduce a fraction to higher terms. Ex. Reduce f to a fraction whose denominator is 32. OPERATION. ANALYSIS. The fraction f is reduced to thirty- 32 -r- 4 = 8 seconds, without changing its value, by multiplying | '= JJ- the terms by the number that will cause its denomina- tor 4 to become 32 (183, 3). By dividing the required denominator 32 by the given denominator 4, this number is found to be 8. Multiplying both terms of f by 8, the result is ff . In practice, say or think, 4 into 32 8 times. 8 times 3 are 24. 193. RULE. Divide the required denominator ~by the denominator of the given fraction, and multiply the numerator of the given fraction by the quotient. EXAMPLES. 194. - 1. Reduce f to 48ths. 2. Change T 7 ^ to an equivalent fraction having 60 for its denominator. 3. Reduce -|, f, , | each to 24lhs. 4. Reduce , f, T \, J each to 36ths. 5. Reduce |, , T V each to 48ths. 6. Reduce \-, f, f each to 105ths. 7. Reduce ffc, f> i each to 56ths - 8. Reduce T V, H> if eacn to 96the. ^. Reduce }, f, -^ each to 360ths. 10. Reduce j|, f, ffc. each to 72ds. ^. Reduce f, f^, |J each to 108ths. 12. Reduce |, |, ^ each to 360ths. 195. To reduce two or more fractions to equivalent fractions having their least common denominator. 196. A Common Denominator of two or more fractions is a denominator to which they can all be reduced, and is the com- mon multiple of their denominators. 197. The Least Common Denominator of two or more fractions is the least denominator to which they can be reduced, and is the least common multiple of their denominators. 60 FRACTIONS. [Art. 197. Ex. Eeduce f, f , $, -fy to equivalent fractions having their least common denominator. -| 44}- 3 __ 4 5. S To -- "So OPERATION. 2 ) $, 4, G, 10 2 x 2 X 3 X 5 = 60 ANALYSIS. The least common multiple of the denominators is found to be 60 (163), which we take as the least common denominator. By Art. 193, | is reduced to . We proceed in the same manner with each of the other fractions. The value of each fraction remains unchanged, since both terms have been multiplied by the same number. In many cases, the least common denomina- tor can be readily found by inspection. 198. RULE. Find the least common multiple of the given denominators for the least common denominator, and reduce the given fractions to this denominator. EXAMPLES. 199. Reduce the following fractions to equivalent fractions having their least common denominator : i- iiVfV 5. , , H- 9 - . 3, V, A> tt- , t, I- if > i- 2OO. To reduce an integer or a mixed number to an improper fraction. Ex. In 18 units, how many fourths ? ANALYSIS. In 1 there are 4 fourths (f), and in 18, eighteen times 4 fourths, or 72 fourths (-\ 2 -). Hence, 18 = *. Ex. Reduce 16 J to an improper fraction. OPEBATION. 16} g - 128 eighths. 7 eighths. 135 eighths. ANALYSIS. In 1 there are 8 eighths (f ), and in 16, sixteen times 8 eighths, or 128 eighths (if 4). 128 eighths and 7 eighths are 135 eighths. Hence, 16| = if*. Art.2Ol.J REDUCTION OF FRACTIONS. 61 201. RULE. Multiply the integer by the required denom- inator, and to the product add the numerator of the frac- tion, and under the result write the denominator. NOTE. When the numerator of the fraction is a small number, add it mentally to the product of the integer and the denominator. EXAM PLES. 202. 1. In 27, how many ninths ? 2. Reduce 46 \ to halves. 3. How many eighths of a peck in 37-J pecks ? Reduce the following to improper fractions : 4. 37f ; 19J ; 208 T V 9. 81| ; 196J ; 375f. 5. 56} ; 49 ; 182f 10. 116ft ; 456^ ; 87}}. 6. 375*; 94^; 46}. 11. 24& ; 179^} ; 1767}. 7. 44} ; 37^ 5 19. ^. 87} ; 490^ ; 168ft. J^. 384f ; 161} ; 175f|. 203. To reduce an improper fraction to an integer or a mixed number. Ex. Reduce *- to a mixed number. ANALYSIS. 1 = | ; hence in -^-, there are as many units as 4 fourths aro contained times in 27 fourths, or 6|. 204. RULE. Divide the numerator ~by the denominator. EXAM PLES. 2O5. 1. Change -^ to a mixed number. 2. Reduce & of a dollar to dollars. Reduce to integers or mixed numbers : S. AJJL ; A^JL. 8. Afjp- ; J-ff-*. 75. ^ ; 4. l$a ; 1|1. P. A}A ; ^L. ^. ^ ; A}f i ^. H* ; A * A - ^- W ; W- ^ W ; W- ft W;W- ^ W;W- 16 - W; 1 ^ 7. ^ ; J1JJ1. ^. ^L ; 1||A. 17. *ff. ; FRACTIONS. ADDITION OF FRACTIONS. 206. Addition of Fractions is the process of finding the sum of two or more fractions. 207. PRINCIPLE. In order that fractions may be added, they must have like denominators and be parts of like units. Ex. What is the sum of -^ &, and T ^ ? OPERATTON. ANALYSIS. As these frac- & ~f~ A + iV If f li ^ ons nave a common denomina- tor, we add their numerators, and write their sum, 15, over the common denominator, 12. |f = 1|, the required result. Ex. Add f, f, and |. OPERATION. ANALYSIS. Reduce the given fractions to equivalent fractions having the least common denominator, 12 (198). Then proceed as in previous example. Ex. Find the sum of 29J-, 38}, 17f, and 42f OPERATION. 24ths. 4 -i o ANALYSIS. The sum of the fractions is |f = If, which added to the sum of the inte- gers, gives 127|, the required result. Ex. How many yards in 12 pieces of prints containing 46 1 , 48 2 , 51 2 , 49 3 , 44 1 , 48 2 , 47 1 , 49, 47 3 , 50 3 , 48 1 , 48 2 yards respect- ively ? OPERATION. 46 1 47 1 48 2 49 ANALYSIS. The small figures represent 512 4.73 fourths (quarters). The sum of the fourths is 49 3 50 3 "^ = ^^' wn ^ c ^ ft dded to the sum of the in- A . l .oi tegers gives 580, the total number of yards. 48 2 48 2 580 1 . Art. 208.] ADDITION OF FRACTIONS. 63 208. RULE. Reduce the given fractions to equivalent fractions having the least common denominator. Write the sum of the numerators over the common denominator, and reduce the resulting fraction to its simplest form. When there are mixed numbers or integers, add the integers and fractions separately, and then add the results. XOTE. Before adding, reduce all fractions to their lowest terms, and all improper fractions to mixed numbers. i EXAMPLES. 209. Add the following : -Z- A, H, A, and if 6. 127&, ft, 175}, and f. 2. f, }, }, and }. 6. 141 A, 197}, and 43ft. S. 12}, 7f, 16ft, and 38}. 7. 75}, |, 1028}, and }}. 48|, 46ft, 31f, and 17}. S. }, 119}, 240}, and 17ft. 5. 46 1 , 48 3 , 40 2 , 49, 47 3 , and 46 2 . (See Analysis opposite.) 10. 40 3 , 41 1 , 48 2 , 44 1 , 49 3 , 48 2 , 49 3 , 49^ 47 3 , 48 3 , 48 3 , and 49 1 . 11. 18|, 27}, 42}, 51|, and 14J-. J^. 146}, 1^, 53ft, and 68J. J5. 1172|, 19}, 440J, 6}, and 10ft. 14. ft, 106ft, 37f, 7|, and 176}}. 15. 175, 116ft, 143}, and 7f 76. 20}, 164, tt> and 43 f 77. 44}, 16}, 29ft, 13}, and 44}}. jtf. 31 1 , 48 3 , 62 1 , 19 3 , 27 2 , 48 l , and 37 3 . 19. 61 3 , 48 1 , 47 3 , 48, 48 2 , 49 1 , and 45 3 . 20. 19}, 444ft, 737}, and 385}. 21. A farmer sold 317}} bushels wheat, 176}} bushels timothy seed, 202}} bushels buckwheat, 526ff bushels corn, 1 75 1| bushels oats, and 276|f bushels clover seed. How many bushels did he sell altogether ? (See Note, Art. 208.) 22. A jeweler has nine diamonds whose respective weights expressed in carats are }}, |, ft, }, }}, }, -}, #, }}. Find their total weight. #3. How many inches of moulding would be required for three frames whose dimensions are as follows : first, 11} inches wide, 17} inches long ; second, 18} inches wide, 26} inches long ; third, 17} inches wide, 24} inches long ? 64 FRACTIONS. [Ait. 210 SUBTRACTION OF FRACTIONS. 210. Subtraction of Fractions is the process of finding the difference between two fractions. 211. PRINCIPLE. In order that fractions may be subtracted, they must have like denominators and be parts of like units. Ex. From f take |. OPERATION. ANALYSIS. As these fractions have a common -J f = | = i denominator, we take the difference between the numerators, and place it over the common de- nominator. | = | is the result required. Ex. What is the difference between f and f ? OPERATION. ANALYSIS. Reduce the given fractions 9 8 to equivalent fractions having the least 4 f -jlj ** common denominator (1O7). Then proceed as in the previous example. Ex. From 176f subtract 89f . OPERATION. 176$ -f ANALYSIS. | from f we cannot take ; we therefore 93. &. take 1 = f from 176, leaving 175. f + f- = -y-. V~ ~ f = f. 175 - 89 = 86. 86 + f = 86f . 5. RULE. Reduce the given fractions to equivalent fractions having the least common denominator. Write the difference between the numerators over the common denominator, and reduce the resulting fraction to its simplest form. When there are mixed numbers, subtract the integers and fractions separately, and add the results. EXAM PLES. 213. Find the difference between 1. | and f . 4. 24 and 1^. 7. 1J and f. 2. | and &. 5. -fa and ^. 8. f and ^-. 3. | and U. 6. 4 and 4. 9. 1 and U. Art. 213.] MULTIPLICATION OF FRACTIONS. 65 X Find the difference between 10. 17* and 9J. 17. 116$ and 48|. . 764J and 375^. 11. 175^ and 86J. 7<9. 381f and 17}. 25. 827-J- and 737f. 18. 138| and 17f 19. 157f and 19}. 26. 919} and 447 T V IS. 149 and 18f ^^118 3 and 48 2 . 27. 376 1 and 287 3 . 14. 416| and 49J.XJW. 387f and 116}. 28. 445 2 and 318 3 . 15. 512 and 53f 22. 248^ and 129. #0. 737 3 and 438 2 . 16. 100 and 13J. 23. 764^ and 375|. 80. 648 1 and 5263. MULTIPLICATION OF FRACTIONS. To multiply a fraction by an integer. PRINCIPLE. Multiplying the numerator or dividing the denominator by a number multiplies the value of the fraction by that number (183, 1). Ex. What will 4 pounds of tea cost @ $J a pound ? OPERATIONS. ANALYSIS. If 1 pound costs $, 4x7 4 pounds will cost 4 times $, or g ' a : ~ *\ $- 8 /-, equal to $3. Hence, 4 pounds of tea @ $| will cost $3. Or, To multiply by 4, multiply the 7 numerator 7 by 4, or divide the de- * * g -JL_ 4 ~ " 2 - * y nominator 8 by 4 ; either operation will give 3^, the required product Or > (Prm.). -^X^ = ] = 3^ By cancellation (166), the 2 operation is shortened, and the re- sult is obtained in its lowest terms. Multiplying the numerator, as in the first operation, increases the num- ber of parts, their size remaining the same ; dividing the denominator multi- plies the fraction by increasing the size of the parts, their number remaining the same. Ex. Multiply 123f by 9. OPERATION. ANALYSIS. Multiply the fraction f and the integer 123 separately, and add the products. In practice, when possible, 53 add the products mentally ; e. g., 9 times fare - 2 ^, equal to 6|. 1107 Write the f. 9 times 3 are 27, and 6 are 33. Write the 3, carry, and proceed as in simple numbers. 6G FRA CTIONS. [Art. 215, Ex. Multiply 227} by 175. ANALYSIS. As in preceding ex- ample. Or, by aliquot parts, when the fractions are fourths, eighths, etc., the fractions generally used in com- mercial operations. * = * + *(* of *). of 175 = 87^ \ of 175, or | of 87| = 43f . 216. RULE. Multiply the numerator or divide the de- nominator of the fraction by the integer. When the multiplicand is a mixed number, multiply the fraction and integer separately, and add the results. OPERATIONS. 227} Or, 227} 175 175 4 ) 525 131J 1135 1589 227 39856J 87|- 44f 1135 1589 227 39856i- EXAMPLES. 1. Find the cost of 20 yards of silk at $J a yard. 2. How much grain in 12 bins, each containing 76-J bushels ? 3. If 1 man earns $J in 1 day, how much will 16 men earn in 26 days ? 4. If a ton of hay cost $16}, how much will 22 tons cost ? 5. Required the cost of 60 yards of muslin at 35| cents a yard. Multiply S. ^ by 7. 17. 412f by 47. 7. -B-byS. 18. 148|by40. 8. 1% by 3. 19. 412f by 89. 9. 110 J by 12. 20. 775| by 65. 10. 117} by 16. 21. 119 T \ by 20. 11. 248| by 3. 22. 772} by 17. 12. 146f by 3. 23. 338| by 30. IS. 1971- by 7. 24. 550| by 27. j 420^ by 8. 5. 643} by 121. 15. 384f by 12. 26. 875 } by 234. Jf0. 375 by 48. 07. 916 by 275. 28. 234|by318. 29. 678f by 427. 50. 625} by 516. 51. 7181 by 542. 32. 275| by 287. 33. 813| by 319. 84. 444Jby412. 35. 555f by 875. 36. 81 7} by 416. 37. 913 by 375. 38. 787| by 525. Art. 218.] MULTIPLICATION OF FRACTIONS, 6? 218. To multiply an integer by a fraction, or to frnd a fractional part of an integer. 219. PRINCIPLE. Multiplying by a fraction is talcing such part of the multiplicand as the fraction is of a unit. Ex. If 1 ton of hay cost $18, what will } of a ton cost ? Or, OPERATIONS. 4 )18 Or, 18 3 of ^ = V = ANALYSIS. If 1 ton cost $18, f of a ton will cost f of $18. f of $18 is 3 times of $18. of $18 is $4 (taking is the same as dividing by 4), and 3 times $4| is $13. Or, f of $18 is | of 3 times $18. 3 times $18 is $54. of $54 is Ex. Find the product of 175 and 8|. OPERATIONS. 175 Or, 175 4) 525 43* 3 131} 1400 1400 1531} Ex. Multiply 275 by 47|. ANALYSIS. Multiply by the frac- tion f and by the integer 8 separately, and add the products. The first method is preferable, when the denominator of the fraction is not an exact divisor of the multipli- cand. FIRST OPERATION. SECOND THIRD OPERATION. OPERATION. 275 _47f 8) 825 103J 1925 1100 13028J 275 47| 275 _47| 68J 84| 1925 1100 13028| 34| 3 103J 1925 1100 ANALYSIS. For the first and second operations, as in the pre- ceding examples. When the fractions are fourths, eighths, etc., multiply by means of aliquot parts. | of 275 - 68f. | of 275, or i of 68| = 34|. 68 FRACTIONS. [Art. 220. 220. KULE. Multiply by the numerator of the fraction and divide the product by the denominator. Or, Divide by the denominator of the fraction and multi- ply the quotient by the numerator. When the multiplier is a mixed number, multiply by the fraction and integer separately, and add the results. EXAM PLES. 221. 1. Find the cost of 8| yds. of ribbon at 25 cts. a yard. 2. What is the cost of 42-J pounds of butter at 26 cts. a pound ? 3. Required the value of 48^ yards of flannel at 75 cts. a yard. Multiply 84byf. 10. 216byl4|. 16. 780 by 64f . 5. 126 by 4- . 11. 375 by 24f . 17. 512 by 37f 6. 49 by . 12. 375 by 22f . 18. 611 by 87f 7. 128 by 9J. 13. 146 by 28}. 19. 625 by 92|. 8. 156 by 8J. ^. 184 by 16f. 20. 937 by 75}. 9. 187 by 10}. 75. 110 by 41}. &?. 575 by 81}. 222. To multiply a fraction by a fraction. Ex. At I-J a pound, what will f of a pound of tea cost ? OPERATION. 37 i 7 ANALYSIS. If 1 pound cost $f f of a pound will cost f of |f | of $| is 3 times of |f Or, } X J = A i of $ is |^, and 3 times $^ is |ft, or $^. 3 Ex. What is the value of 8 x 8| x T "% X ^ ? 2 OPERATION. ANALYSIS. Reduce the inte^ 4 5 ^ ger 8 and the mixed number 8^ - -3 x ^ x TS - : V : = 3 i to improper fractions, and mul- tiply as in the preceding example. 223. KULE. Reduce integers and mixed numbers to im- proper fractions. Cancel oil factors common to the numerators and de- nominators. Multiply the remaining numerators together for the numerator, and the remaining denominators for the de- nominator. Art. 224.] MULTIPLICATION OF FRACTIONS. 69 EXAMPLES. 224:. Find the product of 1. I and f . 5. I and Y- 9. *, 13*, and *. & | and f 5. 6, 3*, and f JO. 26}, 4, and f . 5. f and T 5 f . 7. of, 4, and f}. JJ. }, $, and 164. 4- f and ^. 8. 12J, lOf, and A ^. 13*, f, and -J. Reduce the following compound fractions to simple ones. A Compound Fraction is a fraction of a fraction. The word " of " is equivalent to the sign x . 18. i of f of f . 17. | of | of 18. 21. I of ff of }J. 14. | of 34 of f 18. | of 11| of 4. 22. f of } of | of -f. 15. | of | of 7*. 19. *-} of ff. #5. } of 12J O f 6|. 16. } of } of 5J. ^. T 7 of | of T 5 . ^. | ofjfaei: 4}. Find the value of the following expressions : 25. | of 1728. 50. ($ -f ^) x (f + A)- ^. | x 375. 81. (} - f ) X (1 + }). 27. I times 864. 82. (A + I) x (y 5 |). ^<9. f of 75 x f of 16|. 88. 37-J times | of T V 29. I x } of A x |. 54. | of | x | of f. To multiply mixed numbers together. Ex. Multiply 147} by 41f OPERATION. ANALYSIS. Commencing at the 147f right as in multiplication of integers, 41 5 first multiply the fraction and inte- ger of the multiplicand by the frac- tf : "s x T tion of the multiplier ; and then 8 ) 735 91J = | X 147 multiply the fraction and integer of 4 ) 123 30f = 41 X f the multiplicand by the integer of j^ -v the multiplier. The separate steps j- = 41 X 147 are indicated at the right of the operation. The sum of the several 6150^- partial products will be the required product. 70 FRA CTIONS. [Art. 221 226. BULK Multiply the fractions tog^her, each integer by the fraction of the other number, and the integers to- gether. The sum of the partial products will be the product required. NOTE. When both mixed numbers are small, reducfe them to improper fractions and multiply as in multiplication of fractions (2 EXAM PLES. 221. Multiply /Multiply 1. 875 by 8J ; by 37 *. / 6. H6f by 27J- ; by 581. 2. 737i by 10 ; bv 12J. i f 7. 447Jby45i; by 641. S. 512} by 7J ; by 27f 8. 459} by 37} ; by 39} k 449f by 16i ; by 36f 1 9. 378jby43i; by4aj( 5. 612J by 13J ; by 42|. \ 10. 479 by 56J ; by 2/f . DIVISION OF FRACTIONS. 22S. To divide a fraction by an integer. PRINCIPLE. Dividing the numerator or multiplying the denominator by a number, divides the value of the fraction by that number (183, 2). Ex. "What cost 1 pound of tea, if 5 pounds cost OPERATIONS. ANALYSIS. If 5 pounds cost $3, 1 pound will cost of $&*, or$|. To divide ^ (3$) by 5, divide the numerator 10 by 5, or multiply the denominator 3 by 5 ; either operation will give f, the required quotient (Prin.). Dividing the numerator, as in the first operation, decreases the number of parts, their size remaining the same ; multiplying the denominator divides the fraction by decreasing the size of the parts, their number remaining the same. Or, Ex. Divide 867f by 4. OPERATION. 4 ) 867| 3f = V- 216|| if. .=. 4 = ANALYSIS. Dividing as in simple numbers, 4 is contained in 867f, 2 HI times and a remainder of 3f . 3f equals -\ 5 -, which divided by 4 is ||. Art. 330.] DIVISION OF FRACTIONS. 71 23O. RULE. Divide the numerator or multiply the denominator of the fraction by the integer. When the dividend is a mixed number, divide the integer and the fraction separately, and add the results. 231. x~*^S Divide /" 1. fby 6. fn. 2. fby 3. / 12. 3. fby 6. 1 18. 4. A by 4. / 14. 5. A by 4. 15. 6. 16J- by 5. 16. 7. 172J by 3. 17. 8. 875f by 6. 18. 9. 0. 935f 729J by 8. by 9. \ 19. \ 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 5316J 7144J 1729J 1749J 8763J 7385J 4255| 7134 9727| 6345| by by by by by by by by by by 4, 5. 3. 9. 6. 8. 9. 7. 12. 16. EXAMPLES 637 by 9. 875 T V by 12. 1716f by 8. 1729 by 3. 241 8 J by 5. 351 6f by 5. 2428J by 3. 6375| by 4. 4287| by 2. 3281^ by 8. 232. To divid\by a fraction. 233. The Reciprocal of a number is 1 divided by that number. Thus, the reciprocal of 4 is 1 divided by 4, or J. The Reciprocal of a Fraction is 1 divided by that fraction, 234. PRINCIPLE. 1 divided by a fraction is the fraction in- verted. Thus, 1 divided by f is |. This principle may be demonstrated as fol- lows : In 1 there are 4 fourths. 1 fourth is contained in 4 fourths 4 times. Since f is 3 times \, f is contained in 1 ^ as many times as ^. Hence, is contained in 1 ^ of 4 times, or f times. The reciprocal of a fraction is the fraction inverted. Ex. At $f a yard, how many yards of cloth can be bought for $5 ? OPERATIONS. ANALYSIS. Since 1 yard cost 5 _._ 3. 2_o _._ i _ 6 2 $f, as many yards can be bought for $5 as $f is contained times in $5. Or, 5 -f- f = = -f- X f = : -V- = = 6| 5 ig equal to ^ and 3 f ourt hs is con- Or, $f is contained in $1 equal to 6f times. tained in 20 fourths 6| times. times (Prin.\ and in $5, 5 times | or 72 FRACTIONS. [Art. 234. Ex. At If a yard, how many yards of cloth can be bought for If- ? OPERATIONS. ANALYSIS. Since 1 yard I _:- J = i-| -:- T 9 ^ 1 cost $|, as many yards can be Or JL ^ - 5 x 4 _ A o = ii or as s con - tained times in $f . f is equal Or 5 3 - 8 v 4- -_ 10 - 11 to A' and * is e( l ual to " ^ IT 4 - -f ' ^ is contained in }$ 1 times. Or, $| is contained in $1 | times (Prin.\ and in $f, f times or f, equal to 1 times. Ex. If 6f yards of cloth cost $5, what will 1 yard cost ? OPERATIONS. ANALYSIS. 6f yards are 5 ^_ zg. (5 H- 20) X 3 = I equal to * yards. Since - 3 ^ Or, 5 AA = 4-xA = U= : '|- yards cost $5 ' * of a yard will cost ^ of $5 or $i, and Or, 5 -T- Y = f X tfc == f 3 or j yard will cost 3 times 4 $J or $f. Or, the price per yard equals the cost, divided by the quantity as an ab- stract number. 5 divided by %- equals 5 times 1 divided by - 8 5 -, or 5 times % (Prin.), equal to f. Ex. Divide 7552 by 78f. 7-K9 ANALYSIS. Reduce both divisor and dividend to thirds as in the operation, omitting the 3 ___ 3 common denominators. *ifi. -s- |. j s ^ e same 236) 22656 (96 as 22656 ^236. 21 y. Or, multiplying both divisor and dividend by the same number does not affect the quotient. 1416 Multiply both divisor and dividend by 3, and then divide as in simple numbers. Ex. Divide 2195$ by 175|. OPERATION. 1751 ) 2195| f. ANALYSIS. Reduce both divisor and divi- dend to sixths, their least common denomina- 1054 ) 13175 ( 12 tor, reject the common denominator, and 1054. divide the numerators as in simple numbers. Or, multiply both divisor and dividend by 6, the least common denominator, and divide as in simple numbers (see preceding analysis). j>21 175| = 175f. 1054 ~~ * Art. 235.] DIVISION OF FRACTIONS. 235. RULE. Reduce the divisor and dividend to equiv- alent fractions having a common denominator, and divide the numerator of the dividend by the numerator of the divisor. Or, Invert the terms of the divisor and proceed as in multi- plication. In dividing mixed numbers, multiply both divisor and dividend by the least common denominator, and divide as in simple numbers. NOTE. If both mixed numbers are small, reduce them to improper frac- tions, and apply the rule for division of fractions. EXAMPLES. 236. Divide 1. Ibyf 16 by f . 28 by f . 49 by J. 88 by f . fbyf 2. 3. 4- 5. 6. 7. 8. 9. 10. 11. 12. 13. A by i. i a, Jo fbyf. 28 by 4. 33 by 3f . 64 by 5f . 27. 875 625 by 83 \. 516 by 34f . 917 by 43f . 864 by 86f . 702 by 920 by 73f. 720 by 43. 700 by 37. 560 by 26J. 682J by 45J. 847! by 89. 984 3 by 75^. 86&? by 18 3 . 731 1 by 56i. 431J by 18}. 983i by 29 f 504| by 36f . 5831 by 43f . Find the value of the following complex fractions (182) and expressions of division : 40. |A . 4| . 24f 9 ' 35 ' 36 /./ _1 _1 _ 1, 40 ' 13 ' 20 H . f . A 74 FRA CTIONS. [Art. 237. 11} cts. per REVIEW EXAMPLES. 237. 1- Reduce f f f to its lowest terms. Reduce -J to forty-eighths. j 3. Reduce 727| to an improper fraction. \ 4 Reduce ^jp- 1 to a mixed number. \J^&dd 17i, 37}, 18f , 49*, 13|, and 56^. 1 6. From 1728| take 865}. 7. Multiply ixSJx^x^x 16|. 8. Multiply 1727} by 175. f 9. Multiply 1727 by 175}. 10. Divide 1J by -%. 11. Divide 1736 by 144f. 12. Divide 5779| by 275f. , IS. Divide 12346J by 7 ; by 35. \ 14. -What is the cost of 1583 pounds sugar p 3Lyards broadcloth at $1.75 per yard ? V. 5$\ What is the value of 45 3 yards damask at 77 2 cts. per yr44^ 56. The salary of the President of the United States is $50000 per year ; how much is that per day ? 57. l-/g- pounds of beef and 1 T 6 ^ pounds of flour are allowed to a ration ; how much will 617 rations cost, if the price of beef is llf cts. per pound, and of flour 3J cts. per pound ? 58. What is the value of 36385 pounds of corn at 48f cents per bushel, each bushel containing 56 pounds ? 59. What is the least common multiple of the nine digits ? 60. The 'total production of gold and silver in the United States from 1792 to 1886 was $2,403,986,769. What was the average production per year ? DECIMALS. 238. A Decimal (from the Latin decem, ten) Fraction is a fraction whose denominator is 1 followed by one or more ciphers; as ilFJ TolT? Tooo? 10060' 239. Decimal fractions arise from dividing a unit into 10 equal parts, and then dividing these parts into 10 other equal parts, and so on. Thus, if a unit be divided into 10 equal parts, each part is called a tenth. If a unit be divided into 100 equal parts, or 1 tenth into 10 equal parts, the parts are called hundredths. If a unit be divided into 1000 equal parts, or 1 hundredth into 10 equal parts, the parts are called thousandths. 240. All the rules, principles, operations, etc., of common fractions may be applied to decimal fractions. Since decimal fractions increase and decrease uniformly according to the scale of ten, a more simple notation, similar to that of integers, has been devised for them. A hundred is written 100 ; a tenth part of a hundred (ten) is written 10, the 1 being written one place to the right ; a tenth part of one ten (one unit) is written 1, the 1 being written one place to the right ; in like manner, a tenth part of one unit (one-tenth) is written .1, the 1 being written one place to the right ; the tenth part of one-tenth (one hundredth) is writen .01, the 1 being written one place to the right, etc. , etc. Decimal fractions, like integers, decrease from left to right in a tenfold ratio, and increase from right to left in the same ratio. 241. In the decimal notation, the numerator only is written, the denominator being indicated by the position of a point ( . ) called the decimal point. The decimal point separates the inte- gral from the fractional part. 78 D E CIMA LS. [Art. 242. 24:2. The denominator of a decimal fraction is understood, and is 1 with as many ciphers annexed as there are figures in the decimal ; thus, Form of Form of common fraction. decimal fraction. T V is written .7 and is read seven tenths. T g-o " " .08 " " eight hundredths. yUcr " " -016 " " sixteen thousandths. Hereafter, the first form, that of the common fraction, will be called a fraction, and the second, that of the decimal notation, a decimal. 24:3. The first place to the right of the point is called tenths, the second place hundredths, the third place thousandths, and BO on. 244. The relation between integers and decimals is shown in the following NUMERATION TABLE. 2436807593. 689460582 S I 3o I I I! s s 3 I s 3 5 I I i Orders of Integers. Orders of Decimals. 245. In the above table, observe that the first place to the left of units is called tens, and the first place to the right, tenths ; the second place to the left of units is called hundreds, and the second place to the right, hundredths, etc. Hence the number of any order or place of the decimal, counting from the point, or from units' place, is the same as the number of ciphers in the denominator of the decimal. 246. A Complex Decimal has a fraction in its right-hand place. Thus, .16f r-^j is a complex decimal, and is read 16f hundredths, the fraction not being counted as a decimal place. Art. 247.] NUMERATION OF DECIMALS. 79 247. PRINCIPLES. 1. Annexing ciphers to a decimal does not alter its value. Annexing a cipher multiplies both the numerator and the denominator by 10, and hence does not alter the value of the decimal (183, 3). Thus, .7 ( T V) 2. Each removal of the decimal point one place to the right multiplies the value of the decimal by 10. Removing the point one place to the right does not change the numerator, but divides the denominator by 10, and hence multiplies the value of the decimal (183, 1). Thus, .072 (rffa) becomes .72 (flfc) ; ^ = T ^ x 10. 3. Each removal of the decimal point one place to the left divides the value of the decimal by 10. Removing the point one place to the left does not change the numerator, but multiplies the denominator by 10, and hence divides the value of the frac- tion by 10 (183, 2). Thus, .72 (^fr) becomes .072 (yflfo); T f = ft, + 10. NUMERATION OF DECIMALS. 248. KULE. Read the decimal as if it mere an integer, and give it the name of its right-hand order. EXERCISES. 249. Write in words, or read orally the following numbers : 1. .6. 8. 17.6. 15. 375. 18|. 2. .008. 9. 8.029. 16. 19.0033J. 3. .27. 10. 24.000488. 17. 6.148|. 4. .0375. 11. 400.00008$. 18. 648. 6|. 5. .0108. 12. 76.7071. 19. 347.18005. 6. ."75. 13. 3000.0045. 20. 808.008. 7. .1007. 14. .3045. 21. 600.06. NOTATION OF DECIMALS. 250. Write sixty-four thousandths in the form of a decimal. ANALYSIS. Since there are only two figures in the numerator 64, and the right-hand figure of the decimal must occupy the third decimal place to ex- press thousandths, it is necessaiy to prefix a cipher to bring the right-hand figure into its proper place. Therefore write point, naught, six, four (.064) in the order named. 80 DECIMALS. [Art. 251. RULE. Prefix the decimal point, and decimal ciphers if necessary, to the numerator written as an integer, so that the right-hand figure will occupy the order named. NOTE. Before writing, determine mentally the place of the right-hand figure and the number of ciphers required. Write in all cases from left to right. EXERCISES. 252. 1. What is the name of the third decimal order ? The .sixth ? The first ? The fourth ? The second ? The seventh ? 2. How many decimal places are required to express hun- dredths? Millionths ? Ten-thousandths? Tenths? Hundred- millionths ? Hundred-thousandths ? 8. How many ciphers must be written after the decimal point in writing 375 millionths ? 27 hundredths ? 875 thousandths ? 446 ten-millionths ? 37 ten-thousandths ? 4. Write the following as decimals, so that the decimal-points stand in the same vertical line : 8 tenths ; 16 hundredths ; 175 thousandths ; 1804 millionths ; 56 ten-thousandths ; 3004 ten- millionths ; 1728 ten-thousandths. 5. Seventeen, and seventy-five hundredths. 6. Twenty-six, and twenty-six thousandths. 7. Two hundred forty-six ten-millionths. 8. Two hundred, and forty-six ten-millionths. 9. Three hundred seventy-five, and eighteen hundred-thou- sandths. 10. Eight thousand, and sixty-five ten-thousandths. 11. Eight thousand sixty-five ten-thousandths. Art.253.] REDUCTION OF DECIMALS. 81 REDUCTION OF DECIMALS. 253. To reduce a fraction to a decimal. Ex. Reduce f- to a decimal. OPERATION. 4. \ 3 00 ANALYSIS. f equals of 3 units. 3 units equal 300 hundredths. of 300 hundredths equal 75 hundredths. .75 254. RULE. Annex decimal ciphers to the numerator, and divide by the denominator, pointing off as many deci- mal places in the quotient as there are ciphers annexed. 255. A fraction in its lowest terms can be reduced to a pure decimal only when its denominator contains no prime factors but 2 and 5. If the denominator or divisor contain any prime factor other than 2 and 5, the divisor will not end. The decimals thus produced are called Interminate or Repeating Decimals, and the figures repeated, Repetends. When a fraction is in its lowest terms, its numerator and denominator have no common factors (187). Annexing ciphers to the numerator introduces the factors 2 and 5 only ; hence, if the denominator is an exact divisor of the numerator with the ciphers annexed, it must contain these prime factors and no others. EXAMPLES. 256. Reduce to equivalent decimals : 1. i- 4. f. 7. H. 10. T V . 13. 16f. - 5. T V 8. f. 11. f 14. 27|f. 6. ff. 9. f - 12. f . 15. 36|f. 257. To reduce a decimal to a fraction. Ex. Reduce .075 to an equivalent fraction. ANALYSIS. A decimal is changed to a OPERATION. fraction by writing its denominator, and omit- .075 = yj^ -f- -fe ting the decimal point and prefixed ciphers. = A 82 DECIMALS. [Art. 257. Ex. Change .83J to a simple fraction. OPERATION. ANALYSIS. Reduce the 83 * = = = * = * complex fraction to ' simple fraction by multi- plying both terms by the denominator 3. (183, 3.) 258. RULE. Qmitthe decimal point, supply the proper denominator, and reduce the fraction to its lowest terms. EXAM PLES. 259. Reduce to equivalent fractions : 1. .25. 8. .128. 15. .33^. ff)0) %rC. .44*. 2. .75. 9. .00144. 16. .41|. 23. .142857f 3. .375. 10. .512. 17. .066f. 24. .0833^. 4- .625. 11. .5625. 18. .37f 25. 28.0375. 6. .875. 12. .1875. 19. .104. 26. 107.166|. 6. .125. 13. .12|, 20. .097f. 27. 175.096. 7. .016. 14. .16|, 21. .0053f 28. .6.0175. ADDITION OF DECIMALS. 26O. Since decimals, like integers, increase and decrease uni- formly according to a scale of ten, with the exception of placing the decimal point in the result (usually called pointing off), they may be added, subtracted, multiplied, and divided in the same manner as integers. Ex. What is the sum of 28.7, 175.28, .037, 25.0045, and 4.08 ? OPEBATION. 28.7 ANALYSIS. Write the numbers so that units of the 175.28 same order stand in the same column. ^037 If the decimal points are in the same vertical l\ne. 25 0045 tenths will necessarily be under tenths, hundredths under hundredths, etc. Add as in integers, and place the point ^0 in the result directly under the points of the numbers. 233.1015 Art. 260.] ADDITION OF DECIMALS. 83 Ex. Add .6, .37|, 16.0484, 8.12344, and 24.125. OPERATION. ANALYSIS. Reduce the complex deci- .6 = .6 mals as far as the decimal places extend .37} .3775 in the other numbers. Since the fractios__ 16.048 1 = 16.0483 1 now express parts of the same fractional ft 1 9*A2 ft 1 93A2 unit ' the ^ ma ^ be adcle(i - O. 1/&O4:*- := O. JL/wOrHf- T . , . . . 7 * In practice, the fractions may be re- : ^4:- 1%5 jected if the decimals are carried one 49.2742}f place, at least, farther than accuracy is required. RULE. Write the numbers so that their decimal points are in the same vertical line. .Add as in integers, and place the decimal point in the result directly under the points in the numbers added. EXAMPLES. 262. 1. Add ninety-seven hundredths ; three hundred forty-seven thousandths ; sixteen, and seventy-five hundred-thou- sandths ; four hundred seventy-five, and two thousand thirty- seven millionths. 2. Add four, and eighty-one thousandths ; thirty-seven, and two hundred one ten-thousandths ; seven thousand eight hundred- thousandths ; seven thousand, and eight hundred-thousandths ; nineteen hundredths ; three hundred sixty-four, and nine tenths ; and fifty-six, and fifty-four thousandths. 3. Add three hundred seventy-five, and eight hundredths ; eighteen thousandths ; ninety-six, and eighty-four hundredths ; four, and four tenths ; and eight hundred seven ten-millionths. 4. What is the sum of 18 hundredths ; 716 hundred-thou- sandths ; 6342 millionths ; 11567 ten-millionths ; 625 ten-thou- sandths ; 9 tenths ; 99 hundredths ; and 512 thousandths ? ^_^Add 81.86; 12.593; 4.004; 18.00129; .443; 400.043; .12875; 175.00175; 17.3008; 9000.0016; and .9016. 6. Find the sum of 99 ten-thousandths ; 157} thousandths ; 789J millionths ; 6 tenths ; 18J hundredths ; 1728 ten-millionths ; and 88 hundredths. 7. Add $1728.64; $0.37}; $18.44}; $10.18}; $6.25; and 84 DECIMALS. [\rt.262. 8. What is the sum of $12.37 J ; $144.18}; $6.62; $175.06^; $40.17|; and $398? P. Add .1263$; 12.875; 187.25; 9.1414J; .12; 5.7604 T V; and .0008|. 10. Add .264; 4.18}; .0017f; .00864$; .04|; 17.387$; and .0102075. SUBTRACTION OF DECIMALS. 263. Ex. From 12.75 subtract 8.125. OPEBATION. ANALYSIS. Write the subtrahend under the minuend so 12. 75 that units of the same order stand in the same column. Sub- 8.125 tract as in integers, and place the point in the result directly ~~"~ under the points of the numbers. If, as in this example, the minuend has not as many deci- mal places as the subtrahend, suppose decimal ciphers to be annexed until the right-hand figures are of the same order. (247, 1.) Reduce complex decimals as in addition (26O). 264. RVLE. Write the numbers so that their decimal points are in the same vertical line. Subtract as in inte- gers, and place the point in the remainder directly under the points in the minuend and subtrahend. EXAMPLES. 265. 1. From four, and sixty-five thousandths, subtract eight hundred forty-seven ten-thousandths. 2. From twenty-seven hundredths take twenty-nine hundred- thousandths. 3. From nine thousand, and thirty-four ten-thousandths, sub- tract nine thousand thirty-four ten-thousandths. Find the difference between 4. 8.3644 and 7.8996. 12. 17.864| and 16.94. 5. 17.4586 and .785. 13. 144.43$ and 113.3875. 6. 1.010101 and .999999. 14. 54.37| and .98f. 7. $173.46 and $87.29. 15. 117.48} and 49.434. 8. 3 and .873845. 16. 448.987$ and 38^28$. 9. 17.24$ and 18.973}. 17. 5556.8} and 44.48. 10. $510. 60 and $389. 45$. 18. 968.44$ and 37.386f. 11. $1728 and $.06}. 19. 49.45$ and 48.9876$. Art, 266.] MULTIPLICATION OF DECIMALS. 85 MULTIPLICATION OF DECIMALS. 266. Ex. Multiply '.144 by .12. OPERATION. ANALYSIS. .144 x .12 = ^% x jfo = tfffo. Multiply .144 the numerators of the two factors for the numerator of the .12 product, as in multiplication of fractions. In the above multiplication of fractions, it will be observed that the m *79ft number of ciphers in the denominator of the product equals the sum of the ciphers in the denominators of the two factors. Since, each cipher represents a decimal place, the product should have as many decimar places as both factors. If the number of figures in the product is less than the number of decimal places in the two factors, supply the deficiency by prefixing ciphers. RULE. Multiply as in integers, and from the right point off as many decimal places in the product as there are decimal places in the two factors. NOTE. To multiply a decimal by 10, 100, 1000, etc., remove the decimal point as many places to the right as there are ciphers in the multiplier, annexing ciphers to the multiplicand, 11 necessary. EXAM PLES. / 268. 1. Multiply three hundred forty-four ten-thousandths / by twelve thousandths. 2. Multiply one hundred ninety-two thousandths by four, and \ nineteen hundredths. V ^What is six hundredths of six hundred five millionths ? 4. What is five hundredths of $864.32 ? Of $3645.75 ? \5. What is .058 of 784.65 ? Of 943.25 ? 6. What is .99 x 1.106 x .25 ? 4.105 x .625 x .512 ? Multiply fr; Multiply f 7. 8.716 by .39 ; by .047. jl/& 17.28 by ^16f ; by 2.55f 8. .00865 by .625 ; by 97.75. J13. 64.325 by 1.44f ; by .06J. 9. .00128 by 8756.8; by 7.865. f-^. 86.75 by 1.33$; by 5.76. 10. 387.25 by .0147$; by .087-|.,J 15. 5.78 by .0885; by .66f. \1. 58^625 by .488} ; by .375. \ 16. 237.5 by .345$ ; by 4.468$. 17. Multiply 1728 by "v33 ; by T25 ; by .125 ; by .20. 18. Multiply .01837 by 1000 ; .00145 by 100000 ; .6874 by 100 ; 5.375 by 10 ; 17.056 by 10000. Find the sum of the products. 86 DECIMALS. [Art. 269. DIVISION OF DECIMALS. 269. Ex. Divide .01728 by 1.44. OPERATION. ANALYSIS. Dividing as in integers, with- 1.44 ) .01728 ( .012 out reference to the decimal points and pre- \^ fixed ciphers, the quotient is 12. Since the dividend is the product of the divisor and quo- tient, it must contain as many decimal places 288 . as both of them. Hence the number of decimal places in the quotient must equal the number in the dividend less the number in the divisor. SECOND OPERATION. ANALYSIS. Since multiplying both divisor and .012 dividend by the same number does not affect the .. . . -> O1 '79S quotient, make the divisor a whole number by plac- ing the point two places to the right (or imagine the point to be omitted), and place the point of the divi- 288 dend the same number of places to the right. (Com- 2gg pare 3rd analysis, page 72). Indicate the new posi- tion of the point of the dividend by placing an index ( ' ) or .pointer between the figures as in tjje operation. 1728 thousandths divided by 144 is 12 thousandths. Observe that the number of decimal places in the quotient is equal to the number in the dividend at the right of the pointer. In practice, place the point in the quotient when all the figures in the dividend at the left of the pointer have been divided. Notice in the operation that the point of the quotient is directly above the pointer of the dividend, and that each figure of the quotient is directly above the figure of the dividend which produced it. RULE. Divide as in integers, and point off from the right of the quotient as many decimal places as the num* ~ber in the dividend exceeds the number in the divisor. Or, Make the divisor a whole number, by placing the point to the right, and place the point of the dividend the same number of decimal places to the right. Divide as in inte- gers, and place a decimal point in the quotient when the figures of the dividend have been used as far as the new position of the point in the dividend. NOTES. _ 1. If the number of figures in the quotient is less than the number of decimal places to be pointed off, supply the deficiency by prefixing ciphers. 2. If the divisor contains more decimal places than the dividend, before dividing make them equal by annexing ciphers to the dividend. If necessary to continue the division, more ciphers may be added. Art. 270.] DIVISION OF DECIMALS. 87 3. If, after dividing all the figures of the dividend, there is a remainder, the division may be continued by annexing ciphers (247, 1). The ciphers thus annexed must be regarded as decimal places of the dividend. 4. To divide a decimal by 10, 100, 1000, etc., remove the decimal point as many places to the left as there are ciphers in the divisor, prefixing ciphers to the dividend, if necessary. EXAMPLES. 271. 1. Divide three thousand four hundred fifty-six hun- dred-thousandths by seventy-two hundredths. 2. Divide six, and twenty-five hundredths by twenty-five thou- sandths. Divide Divide 3. 35.88 by .345 ; by 4.16. 8. .0648 by .00425 ; by .0288. 4. .89958 by .47 ; by .319. 9. .31752 by .648 ; by .00384. 5. 12.6 by 14.4; by-. 125. 10. .1898 by .33^; by .0048|. 6. 96.3 by .20 ; by .25. 11. 85.2451 by 4.56; by 8.27J. 7. 5.27 by 1.24 ; by .85. 12. 45.367 by .016| ; by l.OSOf 13. Divide 17.28 by .20 ; by .25 ; by .33^ ; by .125 ; by .66f . 14. 321 is .178J of what number ? 15. 186 is five hundredths of what number ? 16. What must 37.375 be multiplied by to produce 448.5 ? 17. What must 631.25 be divided by to produce 250 ? 18. Divide 176.824 by 100; 876.35 by 1000; 17380. 5 by 10000; 2886.57 by 10 ; 375 by 1000000. Find the sum of the quotients. NOTE. To produce a result in hundredths or cents, the dividend must contain two decimal places more than the divisor ; to produce thousandths, three places, etc. Find the results of the following examples in hundredths, and reduce the fractional remainders, if any, to their lowest terms. Divide 19. $12. 52 by $375.60; by $100.16. 20. $288 by $1728 ; by $720. 21. $232.50 by $3720 ; by $3875. 22. $60.40 by $2416 ; by $1812. 23. $72 by $3456 ; by $9000. 84. $212 by $1484 ; by $508.80. 88 DECIMALS. [Art. 211. NOTE. When the quotient is dollars and cents, it is customary in busi- ness operations to omit the fraction if less than cent, and add 1 to the cents if the fraction is more than ^ cent. In the following examples, the results are carried to cents only and the factions are omitted. 3 Divide Divide 26. $18.08 by .05 ; by .04. 28. $720 by .03^ ; by .07. 26. $648 by .06 ; by 13. ' 29. $12.25 by .06| ; by .08. 27. $17.28 by .48 ; by 21. 80. $960 by .OOf ; by .27. Divide approximately to thousandths f 81. 176.4 by 13 ; by .17. f 34. 120.96 by 70 ; by 64. 82. 229.48 by 50.72 ; by 57, 35. 348.50 by 36 ; by .84. 88. 91.20 by 65 ; by 14.60. f* 86. 1728 by 12.16 ; by 17.5. / 212. To find the value of goods sold by the hun- dred or thousand. \E&. Find the cost of 864 pounds of meal at $1.15 per dreq^ppunds (civt.). OPERA1TON. 8.64 ^ ^5 ANALYSIS. 864 pounds equal 8 hundred weight and .64 of a hundred weight. Hence the cost of 864 pounds equals 8.64 times $1.15, or $9.94. _9504_ C. is the sign for hundred, and M. for thousand (1C). $9.9360 273. RULE. Reduce the quantity to hundreds by point- ing two places at the right, or to thousands by pointing off three places. Multiply the price by this result and point off the product as in multiplication of decimals. NOTE. If preferable, multiply the price by the quantity as given, and point off two additional places if the price is per hundred, and three additional places if the price is per thousand. EXAM PLES. 274. 1. Find the cost of 500 pounds of feed at $1.20 per cwt. 2. Find the cost of 4000 feet of boards at $8.50 per thou- sand feet. 3. Find the cost of 4375 feet of lumber at $11 per thousand. 4. What is the cost of 13280 bricks at $7 per thousand ? Art. 274.] REVIEW EXAMPLES. 89 5. Find the cost of 6500 cigars at $48 per M. 6. What is the value of 640 pounds of hay at 85c. per cwt. ? 7. Find the freight on 18480 pounds of merchandise at 62 cents, per civt. /- What is the cost of 5967 Ibs. meal at $1.10 per cwt., and 4880 Ibs. bran at 75c. per cwt.? 9. Find the cost of 26728 Ibs. of feed at $1.05 per cwt. 10. Find the cost of 11760 feet joists at $14 per thousand. 11. Find the cost of 2 civt. of oatmeal at $2.40 per cwt., and 3 cwt. of cracked wheat at $3.84 per cwt. 12. What is the cost of 4 C. bolts at $2.70, and J 0. bolts at $3.20 ? 13. Find the cost of insuring a house for $4500 at 35c. per $100. 14. Find the cost of 25 M. needles at $1.55 per M. 15. Find the value of 13450 feet of scantling at $18 per thousand. f 16. What is the cost of 7J M. envelopes at $2.20 per M. ? r V 17. Find the cost of 12400 sjiingles at $16 per thousand V*^ REVIEW EXAMPLES. 275. 1. Add 16 hundredths, 137 millionths, 48 ten-thou- sandths, and 2016 ten-millionths. 2. Add 16.07, 240.127|, 6.044, 27.1234. 3. Eeduce ff- to a decimal. 9. Change .8375 to a fraction. 4. From 175 take 16.083J. 10. Multiply 117. 084- by 7.37|. 6. What is f of $175.75 ? 11. Eeduce .083$ to a fraction. 6. What is .33 of 187.5 ? 12. From 375. If take 198.88$. 7. Divide 43.75 by .0125. 18. 1.75 is of what number ? 8. Divide .06| by 1.66|. U- What is .33$ times 1728 ? 15. $3.75 is how many hundredths times $75 ? 16. $86.40 is how many hundredths of $2592 ? 17. 16.56 is .05 of what number ? 18. What will 17280 bricks cost at $3.25 per M. ? 19. If 278 barrels of pork cost $4378.50, what is the cost of 100 barrels ? 20. Find the cost of 12456 feet of plank at $8.75 per M. 21. What is the value of 5 bbls. sugar, containing 312, 304, 301, 305, 304 pounds respectively, at 9| cents per pound ? 90 DECIMALS. [Art. 275. 22. Find the cost of 13f pounds of crackers at 15 J cents per pound. ANALYSIS. Instead of multiplying by the methofl given in Art. 226, change one of the fractions to a decimal and then multiply. Thus, $.15 = $.155. 13f x |.155 = $3.131}, or $2.13. Multiply according to the above method 33. $3.17J by 13} ; by 19|, 28. $4.12} by 26} ; by llf 24. $.68fby 24J; by 16}. 27. $1.79 by 37f; by 44J. 25. $.88 by 32 ; by 36J. 28. $1.37 by 18} ; by 45 J. 29. A merchant paid for merchandise during the year $137618.75, and sold merchandise to the amount of $146347.87. What was the gain, if the net market value of the merchandise remaining unsold was $24378 ? 80. A quartermaster has $8345 on hand, and receives $4379.62 from each of six sales of property ; he turns over to. quarter- master A $2875.28, and pays $120 for corn. Upon being relieved from duty, he turns over to quartermaster B one-third of the residue, and divides the remainder equally among three others, C, D, and E. What was paid over to each ? 31. Merchandise on hand, Jan. 1, 1879, $46312.85 ; merchan- dise sold during the year, $317829.32 ; merchandise purchased in the same time, $301449.72 ; merchandise on hand, Dec. 31, 1879, $61378.12. What was the net gain or loss ? 82. A farmer sold land for $22.50 an acre, as follows : to A, 98| acres ; to B, f of the number sold to A ; and to 0, J the number sold to A and B both. How much land was sold, how much did B and each receive, and what was the amount realized ? 33. At $28. 75 per thousand, how many feet of lumber should be given for 2816 pounds of sugar at 7y\ cts. per pound ? 84. A man bequeaths J of his property to his wife, } to his son, -j- to his daughter, and the remainder, which is $36375, to charitable institutions. What is the, amount bequeathed to each, and the total amount ? 85. A gentleman after spending -J- of all his money, and } of the remainder,, had $177.50 remaining ; how much had he at first? 36. A merchant bought 100 yards of cloth at $3. 62 J per yard, and 87 yards at $4.12J per yard. At what average price per yard should he sell the whole, to realize a profit equal to -J- of the cost ? 37. If 31} bushels of corn cost $17.50, how many bushels can be bought for $616 ? DENOMINATE NUMBERS. 276. A Denominate Number is a concrete number (145), and may be either simple or compound. Denominate numbers are used to express divisions of time, weights, meas- ures, and moneys of different countries. The scale of integers and decimals is uniform ; that of most denominate numbers is varying. The moneys of nearly all countries excepting Great Britain, and the metric system of weights and measures have a uniform decimal scale. 277. A Simple Denominate Number refers to units of the same name and value ; as 7 inches, 4 pounds. 278. A Compound Denominate Number refers to units of different names, but of the same nature ; as 3 feet 6 inches, 4 pounds 8 ounces. REDUCTION OF DENOMINATE NUMBERS. 279. Reduction of Denominate Numbers is the process of changing their denomination without changing their value. 28O. To reduce denominate numbers from higher to lower denominations. Ex. How many pence in 8 16s. Id. ? OPERATION. s. d. ANALYSIS. Since there are twenty shillings in 1 pound, 8 16 ? in 8 pounds there are 8 times 20 shillings, or 160 shillings. 20 (For convenience multiply by 20 as an abstract number.) TTT 160 shillings plus 16 shillings equal 176 shillings. Since there are 12 pence in 1 shilling, in 176 shillings there are 176 times 12 pence, or 2112 pence. 2112 pence plus 7 176s. pence equal 2119 pence. When possible, add mentally .. - the number of the lower denomination to the product, as in the last step of this operation. 2119^. 92 DENOMINATE NUMBERS. [Art. 881. 281. EULE. Multiply the number of the highest denom- ination given by the nuiriber of the next lower denomina- tion required to make 1 of this higher, and to the product add the given number, if any, of such lower denomination. Treat this result, and the successive results obtained, in like manner until the number is reduced to the required denomination. EXAMPLES. 282. Reduce Reduce 1. 9 13s. Wd. to pence. 13. 8 bu. 3 pk. 6 qt. to pints. 2. 6 gal. 3 qt. 1 pt. to gills. 14. 13 gal. 3 qt. 1 pt. to pints. 3. 112 18s. 5d. to farthings. 15. 1 mi. 32 rd. 10 ft. to inches. 4. 6 T. 12 cwt. 65 Ib. to pounds. 16. 29 sq. rd. to square feet. 5. 8 mo. 16 da. to days. 17. 97 s<^. rrf. to square yards. 6. 75 17s. 7d. to pence. 7. 5 sq. mi. to acres. 7. 24515s. 3>r. to farthings. 75. 11 mo. 24 nomination to the required denomination. Ex. Reduce ^ of a to shillings and pence. ATI 6 N ' ANALYSIS. Multiplying by 20, ^ = 8f ^ X ^ 3 / = 8J5. shillings. Reserve the integral part of the 3 result, and reduce the fractional part to pence. I X \* = 9d. Multiplying by 12, f shilling = 9 pence. Hence ' * A - ^ M - 94 DENOMINATE NUMBERS. [Art. 288. EXAMPLES. 288. 1. Beduce -^ of a to pence. 2. Keduce ^ of a to integers- of lower denominations. 8. Keduce f of a mile to feet. 4. Change -f^ of a ton to pounds. 5. Reduce f J of a to shillings and pence. 6. Change -J-f of a bushel to pints. 7. Eeduce ff of a mile to feet. 8. Reduce -J of a hundred- weight to integers of lower denom- inations. 9. Reduce ^-J- of a gallon to pints. 10. Change |f of a cord to cubic feet. 289. To reduce denominate decimals from higher to lower denominations. Ex. Reduce .4375 of a to pence. ANALYSIS. Since there are 20 shillings in 1, in .4375, there are .4375 x 20 shillings, or 8.75 shillings. 8.7500.5. Since there are 12 pence in 1 shilling, in 8.75 shillings, 12 there are 8.75 x 12 pence, or 105 pence. OPERATION. .4375 20 105.0000^. Ex. Reduce .4375 of a to integers of lower denominations. OPERATION. ANALYSIS. Multiplying by 20, .4375 = 8.75 shil- lings. Reserve the integral part of the result, and reduce S. 8J.7500 the decimal part to pence. Multiplying by 12, .75 shilling 12 =9 pence. Hence, .4375 = Ss. Qd. d. 9|.0000 EXAMPLES. 29O. 1. Reduce .625 of a to pence. 2. Reduce .875 of a to shillings and pence. 8. Reduce .6375 of a to pence. 4. Reduce .6825 of a to shillings and pence. 5. Change 2.333J yrs. to integers of lower denominations. 6. Change 16.467 to integers of lower denominations. Avt. 290.] REDUCTION OF DENOMINATE FRACTIONS. 95 7. If 1 pound sterling can be bought for $4.87, how many pounds can be bought for $1000 ? 8. Reduce 2.417 yr. to integers of lower denominations. 9. Reduce 15.3375 to integers of lower denominations. 10. A certain sum at a certain rate will in 1 yr. produce $60 interest ; in what time will the same sum at the same rate produce $15.50 interest ? 291. To reduce denominate numbers to fractions of higher denominations. Ex. Reduce f of a penny to the fraction of a . OPERATIONS. ANALYSIS. Divide the given fraction A __:_ 12 = Jn- S. ^J the numbers of the scale required to re- i . o A i f duce pence to pounds. If the answer is required in the form of /-\ ;a i v i i-fi a decimal, reduce the resulting fraction to a < Y* X inr - decimal by Art. 254. ^ = .0025. Ex. Change 9 pence to a fraction of a . OPERATIONS. ANALYSIS. For first operation, as in pre- I X ^ X ,V = iro^- Vious exam P le - 4 Or, since there are 240 pence in 1, 1 Or, r^ = g V penny equals ^ of a , and 9 pence equal T&S, or & of a . Ex. Reduce 125. 9d. to the fraction of a . OPERATION. 12s. 9d. = 153^. ANALYSIS. 12 shillings 9 pence = 153 pence. ^ __ 240d Since 1 = 240 pence, 1 penny equals ^ of a , and 153 pence equal ff, or f of a . iff ft*' EXAMPLES. 292. 1. Reduce f of a penny to the fraction of a pound. 2. Reduce 420 grains to the fraction of an ounce Troy. 3. Change 275 feet to the fraction of a mile. 4. Reduce 49 feet to the fraction of a mile. 5. Reduce 165. lOd. to the fraction of a pound. 6. Reduce 3s. 6d. to the fraction of a pound. 7. Reduce 6 mo. 20 da. to the fraction of a year. 8. Reduce 84 pounds to the fraction of long ton (2240 pounds). 9. Reduce 3 qt. 1 pt. to the fraction of a bushel. 10. Reduce 16s. 8d. to the fraction of a pound. 96 DENOMINATE NUMBERS. [Art. 293. 293. To reduce denominate numbers to decimals of higher denominations. Ex. Reduce . 6 of a penny to the decimal of a . OPERATION. ANALYSIS. Divide the given decimal by the num- 12 ) .6 d. bers of the scale required to reduce pence to pounds. 20 ^ 05 = 1 Common Year .... yr. 12 Calendar Months J 366 Days = 1 Leap Year yr. 100 Years 1 Century ' C. NOTE. In many business transactions the year is regarded as 360 days, or 12 months of 30 days each. 304. The Solar Day is the interval between two consecutive returns of the sun to the meridian. On account of the varying motion of the earth around the sun, the solar days are of unequal length. For civil purposes in measuring time, the average of all the days in the year is taken as the unit. 305. The Solar Year is the time between two consecutive returns of the sun to the vernal equinox. Its exact length is 365 da. 5 lir. 48 min. 50 sec. in mean solar time. For civil pur- poses, the year consists of 365 or 366 days. In the calendar established by Julius Csesar, B.C. 46, and thence called the Julian calendar, three successive years were made to consist of 365 days each ; and the fourth, of 366 days. According to the Julian calendar, the average length of the year was 365|- days, thus making an error of 11 min. 10 sec. each year ; which in 400 years would amount to 73 hours, or about 3 days. In the sixteenth century, in consequence of the excess of the Julian year above the .true solar year, the error in the calendar was 10 days. To correct the calen- dar, and to prevent any error in the future, Pope Gregory XIII. decreed that 10 days should be omitted in the month of October, 1582, and that all centen- nial years not divisible by 400 should be common years. This calendar is sometimes called the Gregorian calendar. It is now used in all civilized countries except Russia. Art. 305.] DIVISIONS OF 101 The Julian and Gregorian calendars are also Style and New Style. In consequence of theaters i^CO and 1$$ being j .)op- mon years by the Gregorian calendar, the difference between the two styles is now 12 days. Thus, when it is July 4 in Russia, it is July 16 in other countries. 306. RULE FOR LEAP YEARS. All years divisible by 4> except centennial years, are leap years. All centennial years divisible by 400 are leap years. 307. The Calendar Months, with the number of days they contain, are as follows : Season. WINTER. SPRING f 3. Marc] NG. ! 4. April 1 5. Mav Days. January (Jan.) 31. February (Feb.) 28. in leap year 29. March (Mar.) 31. (Apr.) 30. May 31. SUMMER. AUTUMN WINTER. C 9. S JMN. -! 10. C 111. J Days. 6. June 30. 7. July 31. 8. August (Aug.) 31. 9. September (Sep.) 30. October (Oct.) 31. November (Nov.) 30. 12. December (Dec.) 31. The number of days in each month may be easily remembered from the following lines : " Thirty days hath September, April, June, and November; February twenty-eight alone, All the rest have thirty-one ; Except in Leap year, then is the time When February has twenty-nine." EXAMPLES. 3O8. 1. Reduce 2 ivk. 4 da. 16 lir. 40 min. to minutes. 2. How many days in 7 mo. 22 da. ? 3. Reduce 2.375 years to years, months,, and days. 4' If a person's income is $1000 per day, how much is that per minute ? 5. From 88 yr. 8 mo. 10 da. subtract 86 yr. 5 mo. 24 da. 6. Multiply 3 hr. 24 mm. 32 sec. by 15. 7. How many leap years from 1886 to 1897 ? From 1795 to 1827 ? From 1887 to 1903 ? 8. How long from 14 min. 40 sec. past 9 A. M. to 37 min. 30 sec. past 5 P. M. ? 9. Reduce 100000 sec. to higher denominations. 10. Find the value of 76 hours of labor at $3.50 per day of 8 hours. 102 DXXbMINATE NUMBERS. [Art. 3O9. 3O!>. To fip.d the interval of time between two dates. 31O. There are two raethods in common use for finding the time between two dates : 1, by compound subtraction, in which the result is given in years, months, and days, and in which 12 months are considered a year, and 30 days a month ; 2, the result is given in days, or in years and days, and the true number of days is taken for each month. Ex. Find the time in months and days from Apr. 24 to Nov. 10. OPERATION. ANALYSIS. Represent the months and days by their num- mo. da. bers and find their difference by compound subtraction (297), 11 10 writing the later date as the minuend and the earlier as the 4 24 subtrahend. In many examples the interval may be found mentally as follows : From Apr. 24 to Oct. 24 are 6 mo. ; in Oct. there are 6 more days after the 24th (regarding each month as 30 days), and in Novem- ber to Nov. 10th inclusive, there are 10 days. Hence the total time between the given dates is 6 mo. 16 da. The above methods may be used for finding the exact interval in days by making the necessary corrections, 6 mo. 16 da. = 196 da. From Apr. 24 to Nov. 10, there are 4 months containing 31 da. each ; hence the true answer is 196 da. + 4 da., or 200 da. NOTE. When the month of February is included, subtract 2 days in a common year, and 1 day in a leap year. Ex. Find the time from May 18, 1884, to Mar. 2, 189G. OFBRATION. yr. mo. da. 90 3 2 ANALYSIS. As in preceding example. 84 5 lo 5 9 14 Ex. What is the exact number of days from July 20. 1888, to Nov. 10, 1889 ? OPERATION. ANALYSIS. In finding 365 from July 20, 1888, to July 20, 1889. the interval between two 11 remaining in July. dates the last da r is count - 31 in August. * a nd not the fil ^ Since the time is more than one 30 in September. year> write down 365 dayg 31 in October. as the number of days from 10 in November. the first date to the same m from July *,, 1888, to Nov. 10, 1889. days in the month of July after the 20th, then the number of days in each Art. 31O.] DIVISIONS OF TIME. 100 of the full calendar months, and finally the number of days in November to Nov. 10 inclusive. The sum of these numbers will be the required time. In solving examples by this method, the student should remember that the first quarter contains 90 days, the second quarter, 91 days, the third quarter, 92 days, and the last quarter, 92 days; the first six months, 181 days, and the last six months, 184 days. EXAMPLES. 311. Find the time by compound subtraction from 1. Jan. 10 to Aug. 28. 2. Mar. 16 to Dec. 4. 3. Feb. 5, 1886, to Oct. 16, 1887. 4. Jan. 27, 1885, to July 4, 1887. 5. May 16, 1886, to Mar. 24, 1887. 6. June 28, 1885, to Apr. 10, 1886. 7. July 30, 1886, to May 12, 1887. 8. Aug. 16, 1887, to Jan. 1, 1888. Find also the exact number of days between the above dates. 9. How many days from Jan. 1, 1888, to Jan. 1, 1906 ? 10. Ninety days after June 21 is what date ? OPERATIONS. , ANALYSIS. Subtract from the given 90 Or, 9 June. number of days, the number of days re- 9 June. 31 July. maining in June, and from this remainder, 31 Auff subtract successively the number of days in the following months until the remain- 31 July. 71 der is equal to or less than the number of 5Q 90 days in the next following month. The last remainder represents the required 31 Aug. 19 Sept. date _ 19 Sept. Or, write the remaining number of days in June, and the number of days in a sufficient number of months to produce about the given number of days. Take their sum and subtract it (if possible) from the given number of days. The remainder will be the day of the following month representing the required date. If the sum is greater than the given number, subtract the excess from the number of days in the last month written. The remainder will be the required date. If the time be 30, 60, or 90 days, regard each 30 days as a calendar month, and correct by subtracting 1 day for each intervening month containing 31 days, and adding 2 days for February (in leap year 1 day). Thus 3 months after June 21 is Sept. 21, and by subtracting 2 days for July and August, the correct result is Sept. 19. 11. 63 days after Oct. 4 is what date ? 12. 90 days after Mar. 24 is what date ? 104 DENOMINATE NUMBERS. [Art. 312. LINEAR MEASURES. Linear or Long Measure is used in measuring dis- tances ; also the length, breadth, and height of bodies, or their linear dimensions. In measuring length, the yard derived from the standard yard of England is the standard unit, the yards of the United States and England being iden- tical. Theoretically, the yard is equal to ff iff of the length of a pendulum that vibrates seconds in a vacuum, at the level of the sea, in the latitude of London ; that is, a pendulum that vibrates seconds under the above conditions is 39.1393 inches in length. The standard yard is, in fact, the distance be- tween two points on a brass bar, preserved at Washington, the distance to be taken when the bar is at a temperature of 62 Fahrenheit. TABLE. 12 Inches (in.,")= 1 Foot . 3 Feet = 1 Yard . 5| Yards = 1 Rod . 320 Rods = 1 Mile . yd. rd. mi. ml. rd. yd. ft. I = 320 = 1760 = 5280 1 = 3 = 1 = in. = 63360 = 198 36 12 NOTES. 1. The inch is usually divided into halves, quarters, eighths, and sixteenths. 2. The foot and inch are divided by civil engineers and others into tenths, hundredths, thousandths, etc. 3. In measuring cloth, ribbon, and other goods sold by the yard, the yard is divided into halves, quarters, eighths, and sixteenths. . 4. At the U. S. Custom Houses, the yard is divided into tenths and hundredths. 5. The mile (5280 ft.) of the above table is the legal mile of the United States and England, and hence it is sometimes called the statute mile. . 6. 1 furlong = mile = 40 rods. (Rarely used.) 313. The following denominations are also used : 1 Size ^ Inch. Used by shoemakers. 1 Hand = 4 Inches. Used in measuring the height of horses. 1 Fathom 6 Feet. Used in measuring depths a , sea. 1 Cable-length = 120 Fathoms, or 240 yards. 1 Geographic Mile = 1.15+ Statute Miles. Used in measuring distances at sea. 1 Knot = 1 Geo. Mile. Used in determining the speed of vessels. 60 Geo. Miles, or \ _ J of latitude on a meridian, or of longitude 69.16 Stat. Miles J = * I on the equator. 360 Degrees = the Circumference of the Earth. Art. 314.] LINEAR MEASURES. 105 314. Surveyors' Linear Measure is used in measuring land, roads, etc. The unit of this measure is a chain, 4 rods or 66 feet in length, called Gunter's Chain. It is divided into 100 parts called links, each link being 7.92 inches in length. TABLE. 100 Links (1.) = 1 Chain . 80 Chains = 1 Mile mi. ch. ft. I. in. 1 = 80 = 5280 = 8000 = 63360 1 = 66 = 100 = 792 .66 = 1 = 7.92 NOTES. 1. Links are written decimally as hundredths of a chain. 2. For railroad and other purposes, engineers use a chain or tape 100 feet long, the feet being divided into tenths. 3. 1 rod = 25 links. EXAM PLES. 315. 1. Add 9/tf. 8 in., 12ft. 6 in., 16ft. 5 in., loft. 11 in., 21 ft. 4 in., lift. 3 in. 2. In | of a yard, how many inches ? S. How many feet in 17 miles ? In 35 rods ? 4. Find the difference between 5ft. 4^ in. and 16J- hands. 5. Reduce 49175 ft. to higher denominations. 6. Reduce 32 rd. 4 yd. 2ft. 10 in. to inches. 7. How many feet in -J of a mile ? S. In 4376 feet, how many chains ? How many inches ? 9. In 396 rods, how many chains ? How many feet ? 10. In 37.56 chains, how many feet ? How many rods ? 11. Children's size 1 of shoemakers' measure is 4J- inches long ; what is the length of boys' size 8, youths' size 1, and men's size 10 ? (Size 1 of the second series is one size longer than size 13 of the first series.) 12. How many fathoms in 1722 ft.? In 3136 ft.? 13. Reduce 48276 ft. to higher denominations. 14. Add 4.16 ch., 3.75 ch., 8.08 ch., 17.28 ch., 46.10 ch., and 38.09 ch. 15. How much will it cost at $3.25 per rod to fence a field whose sides are SOS ft., 975/tf., 822ft., and" 992 ft. respectively ? 16. How many posts placed 8 ft. apart will be required to fence a railroad 14 miles in length ? How many feet of wire will be required, the fence being 5 wires high ? 106 DENOMINATE NUMBERS. [Art. 316. SQUARE MEASURES. 316. Square Measure is used in measuring surfaces, as land, paving, painting, plastering, roofing, etc. The unit of square measure is a square bounded by lines of some known length. Thus, a square inch is a square whose sides are one inch long ; a square foot, a square whose sides are one foot long ; etc. TABLE. 144 Square Inches (sq. in. ) = 1 Square Foot . sq. ft. 9 Square Feet = 1 Square Yard . . . sq. yd. 30J Square Yards = 1 Square Eod . . . sq. rd. 160 Square Rods = 1 Acre A. 640 Acres = 1 Square Mile . . . sq. mi. NOTES. 1. 1 Rood = 40 sq. rds. = \ A. The rood has practically gone out of use. 2. All of the above, excepting the acre, are derived from the corresponding units of Linear Measure. Thus, 1 sq. ft. = 144 (12 x 12) sq. in. ; 1 sq. yd. = (3 x 3) sq. ft. ; 1 sq. yd. = 30^ (5 x 5*) sq. rd. 3. The acre is the common unit of land measure, and is equivalent to a square whose side is 208.71 feet, or a rectangle 10 rods by 16 rods (165 ft. by 264 ft.). 4. Roofers, plasterers, and carpenters sometimes call 100 square feet a square. 317. Surveyors' Square Measure is used in measuring land. TABLE. 10000 Square Links (sq. 1.) = 1 Square Chain . . . sq. cli. 10 Square Chains = 1 Acre A. NOTE. In the vicinity of St. Louis, and in other parts of the Mississippi valley that were settled by the French, the old French arpent is still used as the unit of land measure. It contains about f of an English acre. 318. U. S. Public Lands are divided by north and south lines run according to the true meridian, and by others crossing them at right angles, so as to form townships of six miles square. Townships are subdivided into sections, containing, as nearly as may be, 640 acres each, or 1 square mile. Sections are subdivided into half -sections, quarter-sections, half -quarter-sections, and quarter-quarter-sections. Art. 3 IS.] SQUARE MEASURES, 107 1 Township 1 Section 1 Half-Section 1 Quarter-Section 1 Half-Quarter-Section TABLE. = 6 mi. x 6 mi. = 36 sq. mi. 23040 A. = 1 " xl " = 1 " = 640 " = I " x-|- " = I " 320" = " xj- " = J " = 160" J- " x-J " = i " 80" 1 Quarter-Quarter-Section = J " xi " = T V = 40" The following diagrams show the method of numbering the sections of a township, and that of naming the subdivisions of sections. A TOWNSHIP. N A SECTION. N 6 5 8 4 9 16 3 10 2 11 1 7 12 13 24 25 18 17 20 29 15 14 19 30 81 21 22 23 28 27 34 26 32 33 35 3d X.i 320 A. N.W. i of S.W. i 40 yl. E J of S.W. i 80-4. S.J -w j_ ! S.W. 1 of S.W. i 40 A. 319. A Rectangle is a plane (flat) surface having four straight sides and four square corners (right angles). A rectangle whose sides are equal is called a square. 02 O. The Area of a surface is an expression for that surface in terms of square units. In the diagram each small square represents a square foot. Since there are 8 rows, and 4 square feet in each row, there are 3 times 4 square feet, or 12 square feet in the rectangle. Hence, the area of any rectangle may be found by multiplying together the numbers denot- ing its length and breadth, in the same denomination ; or, more briefly, 4 feet. To find the area of a rectangle, multiply its length by Us breadth. 108 DENOMINATE NUMBERS. [Art. 391. EXAMPLES. 321. 1. Reduce 28140 square rods to acres. 2. How many square feet in 3 acres ? 3. Reduce 4 A. 100 sq. rd. 20 sq. yd. to square yards. 4. Reduce 46.3125 A. to integers of lower denominations. 5. How many acres in the State of Wisconsin, whose area is 53924 square miles ? 6. How many square feet in a lot 25 feet front and 100 feet deep ? (32O.) 7. How many square feet in a roof 20 ft. wide and 45 ft. long? 8. How many square feet in a floor 42 feet long and 33 feet wide ? How many square yards ? 9. How many square feet in a tight board fence 8ft. high and 120 ft. long ? 10. How many building lots, each 36 ft. by 110 ft., can be made from a lot containing 5 acres ? 11. How many acres in a farm 384 rods long and 245 rods wide ? 12. How many acres in a rectangular field 28.50 chains by 46.38 chains ? 13. How many acres in a rectangular piece of land 224 links by 448 links ? 14. How many square yards in a floor 16 ft. 6 in. by 12 ft. 9 in. ? 15. How many square feet of floor in a 3-story building QQft. by 98/1? 16. A ceiling, whose area is 720 sq. ft., is 30 ft. long. What is its width ? 17. What is the value of a field 320 rd. long and 160 rd. wide, at $22.50 an acre ? 18. A rectangular lot contains 24 acres ; what is its width, its length being 1056 feet ? 19. What part of a square foot is a surface 3 in. by 8 in. ? 4 in. by 9 in. ? 8 in. by 12 in. ? 20. How many square yards of oil-cloth will cover a floor 15 ft. long, 13$ ft. wide ? 21. A railroad passes through 5808 feet of a farm. If the area occupied is 50 ft. wide, what is the cost of the right of way at $66 per acre ? Art. 321.] SQUARE MEASURES. 109 22. How many square feet in 60 panes of glass each 24 in. by 30 in. ? 23. How many square yards of paving in a street 1200 ft. long and 60/2. wide? 24. How many square feet in a sidewalk 6 ft. wide and | of a mile long ? 25. What part of a square yard is a surface 6 in. x 8 in. ? 10 in. x 14 in. ? 14 in. x 20 w. ? #0. How many shingles, 3 in. wide and 4 in. of the length exposed to the weather, would be required for a square yard of roofing ? 27. How many shingles, 5 in. wide and 4 w. of the length exposed to the weather, would be required for a roof 60 ft. long and 24 ft. wide ? #. How many square feet in a piece of tin 20 in. x 14 in. ? 29. How many pieces of tin, 14 in. x20 in., will be required for a roof 60 ft. long and 49 ft. wide, making no allowance for seams and waste ? 30. How many brick, upper surface 4 tw. x 8 in., will be required for a walk, 6/. wide and 660 ft. long ? 31. How many paving stones, 6 in. by 8 m., will be required for a street 50 ft. wide and 1248 ft. long ? ##. Find the value of a quarter-section (318) of land at $6.50 per acre. 33. | of the land in a western township (318) is assessed at $6 per acre and the remainder at $8. What is the total assessment ? 34. How much will it cost to build a road from the common corner of sections 6, 5, 7, and 8 (see diagram, Art. 318) to the common corner of sections 2, 1, 12, and 11 at $1.25 per linear rod ? How many acres of land will be occupied if the road is 4 rods wide ? 35. How many square feet in the walls of a room, 16/V. wide, IS ft. long, and 9ft. high ? NOTE. There are 2 ends each 16 ft. x 9 ft., and 2 sides each 18 ft. by 9 ft. The following method is frequently used by mechanics : Multiply the pe- rimeter (the distance around the room) by the height. To find the perimeter, add twice the length to twice the width. 36. How many square inches of gold leaf would be required to cover a box 7 in. x 4 in. X 3 in..? (Use crayon box as an illus- tration.) 110 DENOMINATE NUMBERS. [Art. 321. 37. How many square yards of plastering surface in the sides and ends of a room, 9 ft. high, 16 ft. long, 15 ft. wide ? How many square yards in the ceiling ? 38. How many square yards of plastering surface in a room 12 ft. high, 20 ft. long, and 18 ft. wide, deducting 120 sq. ft. for doors and windows ? 39. How many square feet of painting surface, excepting the bottom, 011 the outside of a car, '30ft. long, 8ft. wide, 7 ft. high ? 40. How many square feet of tin plate would be required for making 1000 rectangular cans 8 in. x 6 in. x 15 in., adding 7 sq. in. for seams and waste in making each can ? 41. How many yards of paper border, 5 strips in a piece, would be required for a room 16 ft. 3 in. long and 12 ft. 6 in. wide, adding 2ft. 6 in. for chimney, jambs, etc.? 42. How many squares (100 square feet) in a roof 40 ft. by 60 ft.? 45 ft. by 64/f.f NOTE. Divide by 100 by pointing off 2 figures at the right. 43 How many shingles, exposed portion 5 in. by 5 in., in a square cf 100 square feet ? If exposed portion is 4 in. by 4 in., how many ? 4 in. by 5 m., how many ? ,4. How many sheets of tin 20 in. by 14 in. in a square of 100 square feet ? 10 in. by 14 in. ? 8 in. by 10 in. ? ^5. How many shingles would be required for a roof 60 ft. by 80 //., if 500 shingles will cover a square ? How many, TOO shingles to a square ? How many, 800 shingles to a square ? 46. How many sheets of tin would be required for a roof 30ft. by 50ft,, if 50 sheets will cover a square ? How many, 62 sheets to a square ? 47. What part of a square yard in a piece of carpet 27 in. wide and 1 yd. long ? 48- How much will it cost to carpet a floor 15ft. by 18ft. with carpeting j- yd. wide, at $1.60 per yard, making no allowance for waste in matching figures, etc. ? NOTE. If no allowance is made for waste in cutting, divide the number of square yards in the floor by the number of square yards in one linear yard of carpet. Carpet dealers in estimating the number of yards of carpet required for a room, multiply the length of the room (plus a proper allowance for matching the design) by the number of full widths. Ingrain carpet is usually 1 yd. wide, Brussels, Moquette, Wilton, Velvet, and Axminster, f yd. wide. Art. 321.] SQUARE MEASURES. Ill 9. How many whole widths of carpet 1 yd. wide would be required for a room 14//. 8 in. wide, and how many inches would be folded under ? If the room is 19 ft. 6 in. long, how many yards of carpet would be required, supposing the excess at the sides to be folded under, and 2 yds. to be wasted in cutting and matching the figures ? 50. How many yards of carpet border would be required for a room 21 ft. by lG%ft.? 51. The height of a flight of stairs is 12 ft. How many steps, if they are each 8 in. high ? How many yards of carpet would it be necessary to purchase if the tread of each step is 10 in., and allowing one yard for moving ? (Find what part of a yard is required for one step.) 52. How many square yards in a roll of paper 8 yd. long and 18 in. wide ? How many sq. ft. ? 53. How many rolls of paper, 8 yd. long, 18 in. wide, would be required for the sides and ends of room 22 ft. 6 in. long, 13 ft. 6 in. wide, and 9 ft. high, deducting 12 sq. yd. for doors and windows, and making no allowance for waste in cutting ? NOTE. If no allowance is made for waste in cutting, divide the surface to be papered by the number of square feet (or square yards) in one roll of paper. In practice, there is a great deal of waste in cutting and matching wall paper. If the room is 9 ft. high, but two whole strips could be cut from a roll 8 yds. long. If double rolls (16 yds. each) are used, 5 whole strips could be cut from each roll. It is therefore more economical to use double rolls. Paper-hangers in estimating the number of rolls required for a room, cal- culate the number of full strips that will be necessary for the regular surface of the walls, and divide this number by the number of whole strips that can be cut from one roll. The ends of rolls are used for the surface above the doors, and above and below the windows, and other irregular places. 54. How many strips of paper, 18 in. wide, would be required for a surface %^ft. wide ? 55. How many whole strips of paper, 8 ft. 9 in. long, could be cut from a roll of paper, 8 yd. long ? How many from a double roll, 16 yd. long ? 56. How many rolls of paper (16 yd. long, 18 in. wide) would be required for the sides and ends of a room, 20 ft. long, 16/7. wide, and Sft. 6 in. high, deducting 31 ft. for the width of doors, windows, mantels, etc.? (Paper-hangers' method.) 112 DENOMINATE NUMBERS. [Art. 322. SOLID OR CUBIC MEASURE. 322. Solid or Cubic Measure is used in measuring solids, or bodies which have length, breadth, and thickness or depth. The unit of cubic measure is a cube, each of whose edges is a unit of some known length. Thus, a cubic inch is a cube, each of whose edges is one inch ; a cubic foot is a cube, each of whose edges is one foot ; etc. TABLE. 1728 Cubic Inches (cu.in.) = 1 Cubic Foot . 27 Cubic Feet = 1 Cubic Yard . 128 Cubic Feet = 1 Cord . . cu.ft. . cu. yd. . cd. NOTES. 1. The above units, excepting the cord, are derived from the corresponding units of linear measure. Thus, 1 cubic foot contains 1728 (12 x 12 x 12) cubic inches ; 1 cubic yard, 27 (3 x 3 x 3) cubic feet. 2. The U. S. measurement ton for freight contains 40 cubic feet. 3. The U. S. register tonnage (entire internal cubical capacity) of vessels is expressed in tons of 100 cubic feet each. 4. A perch of masonry is 1 rod long, 1| feet thick, and 1 foot high, and is equal to 24 (16 x 1| x 1) or about 25 cubic feet. 323. The Volume or Solid Contents of a solid is an expression for that solid in terms of cubic or solid units. 4 Feet \\ The diagram represents a solid 4 feet long, 3 feet broad, and 2 feet thick. Each small cube is a cubic foot. Since the end of the solid contains (3 x 2) 6 square feet of surface, it is evident, if a section 1 foot thick be cut off from this end, it can be divided into 6 cubes, with edges 1 foot in length, and therefore the section will contain 6 cubic feet ; and since the whole solid is 4 feet long, and contains 4 like sections, it must contain 4 times 6 cubic feet, or twenty-four cubic feet. Hence the volume of a rectangular solid may be found by multiplying together the numbers expressing its length, breadth, and thickness, in the same denomination ; or, more briefly, To find the volume of a rectangular solid, multiply together its length, breadth, and thickness. Art. 324.] CUBIC MEASURE. 113 324:. A Rectangular Solid is a solid having six rectangular sides or faces. A Cube is a rectangular solid whose sides are six equal squares. EXAM PLES. 325. 1. How many cords in 15744 cubic feet ? 2. How many cubic inches in 175 cubic feet ? 3. Eeduce 37368 cubic feet to cubic yards. 4. Add 7 cd. 49 cu. ft., 13 cd. 92 cu. ft., 12 cd. 28 cu. ft., 16 cd. 110 cu.ft., 3 cd. 16 cu. ft., 14 cd. 80 cu. ft. 5. How many cubic yards in an excavation, 42 ft. long, 40 ft. wide, and 9 ft. deep ? 6. How much will it cost to dig a cellar 36 ft. long, 30 /rf. wide, and 6 ft. deep, at 30 cents per cubic yard ? 7. What is a pile of wood, &0//. long, 4= ft. wide, and 7/. 6 in. high, worth at 15.75 per cord ? #. If a pile of bark is 40 ft. long and 4 /7. wide, how high must it be to contain 10 cords ? 9. How many cords in a pile of bark, 22.5ft. long, 4: ft. wide, and 4. 8 /. high? j?0. How many cubic feet in a box, 4/#. 6 in. high, 8/tf. long, and 3ft. 9 w. wide? 11. How many cubic inches in a rectangular cistern, 6 ft. x .?#. How many tons in a shipment which occupies a space, 16 ft. by Uft. by 28/f. ? (322, 2.) .?#. What is the freight of 350 cu. ft. of merchandise at $8 per ton ? At 50 shillings per ton ? 14- How many cubic feet in a vessel whose measurement is 2135 tons? (322, 3.) 15. How many perch of masonry in a wall, 40 ft. long, 9 ft. high, and Uft. thick ? (1 Perch = 25 cu. ft.) (322,4.) .#>. How many bricks 2 -iw. x 4 m. x 8 in. in one cubic foot ? 77. The space occupied by a bag containing 1000 standard silver dollars is 12 in. long, 9 in. wide, and 4 tw. deep. How many cubic feet would be occupied by 1,000,000 such dollars ? 18. How many cubic feet is a wall, 12 in. thick, 42 ft. long, and 30ft. high ? How many bricks would be required for the above wall allowing 21 to a cubic foot ? What would be their value at $9 per thousand ? 114 DENOMINATE NUMBERS. [Art. 325. Common North River brick are 8 in. x 4 in. x 2| in. Brick manufactured in other localities are of various sizes. Builders usually allow 7 common bricks for each square foot of the surface of the wall if the wall is one brick thick (4 in.), 14 to a square foot if 2 bricks thick (about 8 in.), 21 to a square foot if 3 bricks thick (about 12 in.), etc. 19. According to the above builder's rule, how many bricks would be required for a wall, 64 ft. long, 39 ft. high, and 2 bricks thick (about 8 in.)? 20. How many cubic feet of masonry in a cellar wall 2 ft. thick, 8 ft. high, outside measurement 25 ft. by 45 ft ? Builders sometimes multiply the total outside measurement (perimeter or girth) by the height and thickness to find the number of cubic feet. By this method, the corners are counted twice. To find the exact length of the wall, from the total outside measurement subtract four times the thickness of the wall. (See note, Ex. 35, Art. 321.) 21. How many bricks will be required to build a chimney 28ft. high, if there are 5 courses of brick to each foot, and 1C bricks in each course (flue 4 in. by 12 in., double wall) ? How many, 8 bricks in a course (flue 8 in. by 16 in., single wall) ? 22. How much stone, lime, and sand will be required for a wall 144 ft. by 10 ft. by 2 ft., if 128 cu. ft. of broken stone, 1J bbls. of lime, and a load (cubic yard) of sand will lay 100 cu. ft. of wall ? 23. How many bricks will be required for a house 24 ft. wide, 63 ft. long, and 30 ft. high, allowing 21 bricks for each square foot of surface (the walls being 3 bricks thick), if 225 square feet are deducted for doors and windows, and if the walls are con- sidered 1 ft. thick in making deductions for the corners ? BOARD MEASURE. 326. Lumber is measured by board measure. The board foot is 1 ft. long, 1 ft. wide, and 1 in. thick ; hence it is -^ of a cubic foot. In measuring boards one inch or less in thickness, the number of square feet of surface which they would cover is measured. Plank, joists, etc., more than one inch in thickness are reduced to inch boards and measured by board measure. Boards, plank, scantling, joists, beams, and sawed timber generally are measured by board measure ; hewn and round timber are sometimes measured by cubic measure. Art. 327.] BOARD MEASURE. 115 327. When lumber is not more than one inch thick, to find the number of feet board measure : Multiply the length in feet by the width in inches, and divide the product by 12. 16 pieces each containing a board foot could be cut from a board 16 ft. long, 12 in. wide. A board 11 in. wide is {% of a board, 12 in. wide and of the same length and thickness. When more than 1 inch thick : Multiply the length in feet by the width and thickness in inches, and divide the product by 12. Lumbermen use an automatic rule for measuring board, planks, etc. They also use table books for reducing logs or round timber to board measure. EX A M PLES. 328. 1. How many square feet would be covered by a board 16 ft. long and 9 in. wide ? 2. If a stick of timber 24 ft. long, 8 in. wide, 3 in. thick, is reduced to boards one inch in thickness, how many square feet would they cover ? 3. How many feet of boards would be required for a floor 20ft. wide and 24 ft. long ? What would be their value at $14 per thousand ? 4. How many board feet in 475 cubic feet ? Find the number of feet, board measure, in each of the follow- ing boards, joists, beams, etc. : Length. Width. Thickness. Length. Width. Thicknees. 5. 12 A 12 in. 1 in. 9. 16 ft. 8 in. 1J in. 6. 14 A 8 in. i in. 10. 20 ft. 10 in. 2 in. 7. 16 A 6 in. 1 in. 11. 12 ft. 6 in. 4 in. 8. 18 A 14 in. i in. 12. 24 ft. 9 in. 3 in. 13. How many square feet would be covered by 45 boards, 16 ft. long, 8 in. wide, if they are laid side by side ? (First find the total width in feet.) 14. How many board feet in a pile of lumber 8 ft. high, 7 ft. wide, and 18 ft. long ? Find its value at $15 per thousand feet. 15. How many feet, board measure, in 16 boards, each 18 ft. long, 10 in. wide, and 1 in. thick ? 16. How many board feet in 24 joists, 20 ft. x 8 in. x 4 in. ? 17. How many feet of boards, 4 in. wide and 1 in. thick, would be required for a fence 5 boards high and one mile long ? 18. How many posts 8ft. apart would be required for above fence ? 116 DENO MINA TE N UMB E R S . [Art. 328. 19. How many feet, board measure,, in 13 planks, each 10 ft. long, 12 in. wide, and 2 in. thick ? * 20. Making no allowance for the corners, how many feet of boards would be required to make a box 8ft. x 4= ft. x oft. ? 21. How many feet in a tapering board 18/2. long, 12 in. wide at the smaller end, and 16 in. at the other ? NOTE. To find the average width of a tapering board, measure it at the center, or take \ the sum of the widths at the ends. gal. qt. pt. gi. . pt. -j 4 Q 32 . qt. 1 = 2 = 8 . gal. 1 = 4 LIQUID MEASURES. 329. Liquid Measure is used for measuring liquids. The unit of this measure is the wine gallon, which contains 231 cubic inches. TABLE. 4 Gills (gi.) = I Pint . 2 Pints = 1 Quart . 4 Quarts = 1 Gallon NOTES. 1. In estimating the capacity of tanks, cisterns, reservoirs, etc., 1 barrel = 31^ gallons ; 1 hogshead = 2 barrels = 63 gallons. 2. In commerce, the barrel, tierce, and hogshead are not fixed measures, but their capacity is found by gauging, or actual measurement. 3. The imperial gallon of England contains 277.274 cubic inches, and is equivalent to 1.2 U. S. wine gallons. 4. 1 cubic foot = 7.48 or about 7 (1728 -5- 231 = 7.48) wine gallons. Hence to find the number of gallons in a rectangular cistern, multiply the number of cubic feet by 7.48 or 7^. 5. To find the number of gallons in a cylindrical vessel, multiply the square of the diameter by the height, and this product by 5| (.7854x1728-^-231 = 5.8752). 330. Apothecaries' Fluid Measure is used in prescribing and compounding liquid medicines. The gallon and pint of this measure are the wine gallon and pint. TABLE. 60 Minims (Tit) = 1 Fluidrachm . / 3 . 8 Fluidrachms = 1 Fluidounce . / . 16 Fluidounces = 1 Pint ... 0. 8 Pints = 1 Gallon . . . Cong. Cvng. 0. f I . f 3 . % 1 = 8 = 128 = 1024 = 61440 1 = 16 = 128 = 7680 1 = 8 = 480 1 = GO Art. 330.] LIQUID MEASURES. 117 NOTES. 1. Cong, is for the Latin congius, gallon ; 0., for the Latin octar vius, one-eighth. 2. The symbols precede the numbers to which they refer ; thus, 0. 6 /5 10, is 6 pints 10 fluidounces. EXAMPLES. 331. 1. Reduce 8 gal 3 qt. 1 pt. to pints. 2. Reduce 875 pints to gallons. 3. Add 4 gal. 2 qt. Ipt., 3 gal. 3 qt. I pt., 9 gal. 1 pt., 11 gal. 1 qt. 4. How many barrels in 100000 gallons ? 5. 480 English gallons (329, 3) equal how many U. S. gal- lons ? (Add|.) 6. How many cubic feet in 1000 gallons ? 7. How many gallons in 300300 cubic inches ? 8. How many gallons in a rectangular tank, 8ft. x 4//. x 6ft. ? OPERATION. 8x4x6x7^ = 1440, approximate result (see Art. 329, Note 4). 8x4x6x7. 48 = 1436.16, accurate result. 9. How many gallons in a rectangular cistern 6 ft. long, 4 ft. wide, 3ft. high? 10. How many gallons in a rectangular cistern 16 ft. long, 4 ft. high, and 6ft. wide ? 11. How many gallons in a rectangular reservoir, 40 ft. long, 16ft. wide, and 8ft. deep ? ./. How many gallons in a cylindrical vessel, 3ft. in diameter and 9 ft. high? OPERATION. 3 x 3 x 9 x 5| = 475f (see Art. 329, Note 4). 13. How many gallons in a cylindrical tank 4: ft. in diameter and 16ft. deep? ! How many gallons in a circular reservoir, 40 ft. in diameter and 6ft. deep ? .75. How many gallons in a circular reservoir 60 ft. in diameter and 8ft. deep ? ^<5. Reduce Cong. 2 0.6 / 3 10 /3 5 to fluidrachms. 17. From the sum of 51 ^aJ. 2 qt. 1 pt., and 45 gal. 1 gtf. 1 pt. 9 subtract 27 ##?. 1 qt., and divide the result by 9. 18. How many bottles, each holding 1 qt. 1 pt. 2 gi., can be filled from a barrel of cider ? 118 DENOMINATE NUMBERS. [Art. 332. DRY MEASURE. 332. Dry Measure is used in measuring dry articles; as salt, grain, fruits, etc. The unit of this measure is the Winchester bushel, which contains 2150.42 cubic inches. TABLE. 2 Pints (pt.} = 1 Quart . . . qt. 8 Quarts = 1 Peck . . . pk. 4 Pecks = 1 Bushel . . bu. bu. pk. qt. pt. 1 m 4 = 32 = 64 1 = 8 = 16 NOTES. 1. The half-peck or gallon of. this measure contains 268.8 cubic inches, and is 37.8 cubic inches larger than the liquid gallon (268.8 231 = 37.8). 2. The imperial bushel of England contains 2218.19 cubic inches, and is equal to 1.03 Winchester bushels. In certain localities in Great Britain, 8 bushels are called a quarter. 3. Grain, seeds, etc., are usually sold by weight. Portable of equivalents see Art. 338. 4. 36 bushels = 1 chaldron of coke or charcoal. 5. 1 bushel is equivalent to about 9.3 (2150.42 -4- 231) wine gallons. EXAMPLES. 333. 1. Reduce 2 bu. 3 pk. 5 qt. 1 pt. to pints. 2. Reduce 10000 pints to bushels. 3. Find the value of 5 bushels of nuts at Sc. per pint. 4. How many cubic inches in 75 bushels ? 5. How many bushels in 322563 cubic inches ? 6. How many bushels in 400 cubic feet ? NOTE. Since a bushel is about 1 cubic feet, the following approximate rules may be used for all practical purposes : To reduce cubic feet to bushels: Deduct one-fifth, or multiply by .8 The result will be too small by about 4 bushels for every 1000 bushels of the result. To reduce bushels to cubic feet: Add one- fourth, or divide by .8. The result will be too great by about 4 cubic feet for every 1000 cubic feet of the result. Solve the above example both exactly and approximately, and compare the results. 7. How many bushels will a box 10/tf. long, 5 ft. wide, and ft. high contain ? (Approximate method.) t ^r /\ L^ I- Art. 333.] MEASURES OF WEIGHT. 119 8. How many bushels of grain will a bin 14 ft. long, 3J ft. wide, and 6 ft. high contain ? 9. Find the capacity in bushels of a crib, 20 ft. long, 8 ft. high, 4/V. wide at the bottom and 6 ft. wide at the top. NOTE. To find the average width, take one-half the sum of the top arid bottom measurements. 10. A crib 24 ft. xSft. xQft. is filled with unshelled corn. How many bushels of shelled corn would this quantity produce, if two cubic feet of corn in the ear will make one bushel of shelled corn ? MEASURES OF WEIGHT. 334. Troy "Weight is used in weighing gold, silver, coins* and jewels ; also in philosophical experiments. The unit of weight is the Troy pound, which contains 5760 grains. A cubic inch of distilled water weighs 252.458 of these grains, when the height of the barometer is 30 inches, and the temperature of the air and water 62 Fahrenheit. TABLE. 24 Grains (gr.) =1 Pennyweight pwt. 20 Pennyweights = 1 Ounce . . . oz. 12 Ounces = 1 Pound . . Ib. lb. oz. pwt. gr. 1 = 12 = 240 = 5760 1 = 20 = 480 NOTE. The carat, used in weighing diamonds, equals 3.2 Troy grains. The term carat is also used to denote the fineness of gold, and means ^ part. Thus, gold 18 carats fine contains 18 parts pure gold and 6 parts alloy. 335. Apothecaries' "Weight is used in prescribing and compounding medicines not liquid. The pound, ounce, and grain of this weight are the same as those of Troy weight, the division of the ounce being different. TABLE. 20 Grains (gr.) = 1 Scruple . . sc. or 3 . 3 Scruples = 1 Dram . . dr. or 3 . 8 Drams = 1 Ounce . oz. or 5 . ft I 3 3 gr. 1 = 12 = 96 = 288 = 5760 1 = 8 = 24 = 480 1 = 3 = GO 12 Ounces = 1 Pound . . lb. or lb . NOTES. 1. The symbols precede the numbers to which they refer ; thus. 36 3 4, is 6 ounces 4 drams. 2. Drugs and medicines are sold in large quantities by Avoirdupois weight 120 DENOMINATE NUMBERS. [Art. 336. 336. Avoirdupois "Weight is used in weighing all articles, excepting gold, silver, precious stones, and medicines in small quantities. The Avoirdupois pound contains 7000 Troy grains. TABLE. 16 Ounces (oz.} 100 Pounds = 1 Pound . . . . Ib. _ ( \ Hundred-weight, or cwt. ~ \ 1 Cental .... 0. 20 Hundred-weight = 1 Ton T. T. cwt. Ib. oz. 1 = 20 = 2000 = 32000 1 = 100 = 1600 1 = 16 NOTES. 1. The ounce is divided into halves and quarters. 2. The dram, y 1 ^ of an ounce, is now little used, except by silk manu- facturers. 3. The Long or Gross ton, formerly used, contained 2240 pounds ; the hundred-weight, 112 pounds ; and the quarter, 28 pounds. These weights are still used in Great Britain, at the U. S. Custom Houses, in ocean freights, and by wholesale dealers in coal and iron. 337. Comparison of Troy and Avoirdupois weights. 5760 grains = 7000 grains = 1 Ib. Troy. 1 Ib. Avoirdupois. 480 grains = 1 oz. Troy. 437^ grains = 1 oz. Avoirdupois. 338. In buying and selling grain, seeds, and other produce, the bushel is regarded as a certain number of pounds. The Boards of Trade of the principal cities of the United States use the equivalents given in the following table : TABLE OF AVOIRDUPOIS POUNDS ix A BUSHEL. Commodities. Lbs. Commodities. Lbs. Commodities. Lbs. 48 Corn shelled 56 Peas 60 Beans 60 Corn in the ear. 70 Rve. . 56 Buckwheat 48 Malt 34 Timothy Seed . . 45 Clover Seed. . . , 60 Oats 32 Wheat 60 . In the Liverpool, San Francisco, and some other markets, produce is bought and sold by the cental of 100 pounds. Railway freight tariffs in the United States on grain, provisions, etc., are reckoned per cwt. or cental. 339. The following units are used in commerce : 1 Quintal of Fish = 100 Ibs. I Barrel of Flour = 196 Ibs. 1 Barrel of Pork or Beef = 200 Ibs. 1 Gallon Petroleum = 6J Ibs. 1 Keg of Nails = 100 Ibs. Art. 340.] MEASURES OF WEIGHT. 121 EXAM PLES. 34O. 1. Reduce 10000 grains to Troy pounds. 2. Reduce 2 Ib. 8 oz. 16 pwt. to grains. 3. What is the weight in Troy ounces of 1000 silver dollars Of 1280 silver dollars ? 4. Find the weight in Troy ounces of 1000 gold dollars 5. Find the weight in Troy ounces of 2000 half-dollars 6. What is the cost of a 14 Jc. watch chain weighing 37| pwt. at $1.15 per pennyweight ? 7. A watch case, 14 carats fine, and weighing 60 pwt., con- tains how many ounces of pure gold ? 8. Find the value of a diamond weighing ^-J- of a carat, at $100 per carat. 9. How many Troy ounces of pure silver would be required for the coinage of 2,000,000 standard silver dollars (113)? How much copper ? 10. Reduce fel 39 36 32 to grains. 11. How many powders, each containing 5 grains, can be made from 1 Ib. Apothecaries (Troy) of quinine ? How many from 1 Ib. Avoirdupois ? 12. Add 8 Ib. 9 oz., 10 Ib. 7 oz., 14 Ib. 15 oz., and 17 Ib. 13 oz. Avoirdupois. 18. How many grains in 16 Ib. Avoirdupois ? H. In 70 Ib. Avoirdupois, how many pounds Troy ? (337) 15. In 175 oz. Troy, how many ounces Avoirdupois ? 16. Find the cost of 58 Ib. 4 oz. of butter at 28c. per pound. 17. Find the cost of 875 pounds of feed at $1.15 per cwt. (273) 18. Find the cost of 17387 pounds of oats at $1.85 per cental. 19. Find the cost of 21370 pounds of hay at $8 per ton. OPERATION. 2 ) 21370 ANALYSIS. 21370 Ib. = 70.685 (21370 -*- 2000) tons. If 10. 685 1 ton costs $ 8 10.685 tons will cost 10.685 times $8, or g $85.48. To divide by 2000, divide by 2 and place the point in the quotient three places to the left. $85.480 20. What is the value of 28140 pounds of straw at $6.50 per ton? 21. Find the cost of 16480 pounds of hay at $11.50 per ton. 122 DENOMINATE NUMBERS. [Art. 340. 22. Find the value of 28160 pounds of coal at $5.25 per ton. 23. Find the value of 42250 pounds of coal at $3. 75 per ton of 2240 pounds. (336, 3) How many bushels in 24. 8375 pounds of wheat ? 29. 18174 pounds of rye ? 25. 9116 pounds of corn ? 30. 13275 pounds of wheat ? 26. 1128 pounds of beans ? SI. 20000 pounds of barley ? 27. 5172 pounds of peas ? 32. 11419 pounds of clover seed ? 28. 3375 pounds of oats ? 33. 12562 pounds of timothy seed? 84. What is the value of 49375 pounds of corn at 64c. per bushel ? NOTE. Usually in business computations, the number of bushels is also required. 49375 Ib. = 881|| bu. 881 ff times 64c. = $564.29. When the number of bushels is not required, to avoid fractions, multiply the number of pounds by the price per bushel and divide the product by the number of pounds in one bushel. 49375 x $.64 -*- 56 = $564.29. By both of the above methods, find the cost of 35. 8375 Ib. wheat at $1.10 per bushel. 36. 9416 Ib. corn at 85c. per bushel. 37. 7428 Ib. oats at 72c. per bushel. 38. 6224 Ib. beans at $2.25 per bushel. 39. 9118 Ib. barley at $1.25 per bushel. 40. 8128 Ib. rye at 82c. per bushel. 41. 4170 Ib. clover seed at $4.25 per bushel. 42. 5160 Ib. timothy seed at $1.75 per bushel. 43. What is the freight on 528^-- bushels corn at 32c. per cwt. ? 44- What is the freight of 16 T. 17 cwt. 20 Ib. at $5 per ton of 2240 Ib. ? ENGLISH MONEY. 341. English or Sterling Money is the legal currency of Great Britain. TABLE. value in U. S. Money. 4 Farthings = 1 Penny . . . d. ... $ .02 + 12 Pence = 1 Shilling . . . * 243 + Art. 341.] ENGLISH MONEY. 123 NOTES. 1. 1 Crown = 5 shillings, or of a pound ($1.216 + ). 2. 1 Guinea = 21 shillings ($5.11). It is not now coined. 3. The gold coins of Great Britain are 22 carats ({$), or .916f fine. (The old carat system (334, note) is generally abandoned except for jewelry,, 1 carat = ,041f .) The silver coins of Great Britain are .925 (f) fine. EXAMPLES. 342. 1. Add 27 16s. 10&, 6 10s. Sd., 47 15s. lid., 25 7s. 6d., 3 14s. Sd., and 23 16s. 3d. 2. From 17 8s. 4=d. subtract 10 12s. Sd. 3. Multiply 5 6s. 3d. by 8. 4. Eeduce 8375^. to shillings and pounds. 5. Reduce 12 16s. Sd. to pence. 6. What is the cost of 466 yards of cloth at 9J^. per yard ? 7. Find the value of 4120 bu. wheat at 4s. ^\d. per bushel. 8. In 47 guineas, how many pounds and shillings ? 9. Divide 16 5s. 6d. by 9 ; by 7 ; by 31. 10. How many yards of cloth at 3s. Id. per yard can be bought for 7 ? For 9 5s. Qd. ? 11. What is the cost of 20 yd. silk at 10s. Qd. per yard ? 12. Reduce 8 17s. Sd. to the decimal of a pound. (293) NOTE. The following method for reducing shillings and pence to the decimal of a pound is sufficiently accurate for most business purposes : Write one-half of the greatest even number of shillings as tenths, and if there be an odd shilling write 5 hundredths ; multiply the number of pence by ^, and write the product as thousandths. If the product is between 12 and 36, add 1 to the thousandths ; if between 36 and 48, add 2 to the thousandths. Thus, 8 17s. Sd. = 8 + .85 + .033 = 8.883. Reduce mentally the following to the decimal of a pound : 13. 16s. Zd. 15. 10s. Sd. 17. 7s. 3d. 14. 18s. 5d. 16. 17s. 4rf. 18. 13s. lid. 19. Reduce .821 of a pound to shillings and pence. (289) NOTE. This example can be performed mentally by reversing the opera- tion explained in note to Ex. 12. Multiply the number of tenths by 2, and write the product as shillings (2x8 = 16). Divide the number of thousandths expressed by the 2nd and 3rd figures by 4, and write the quotient as pence (21-*- 4 = 5). .821 = 16s. 5d. If the second figure to the right of the point is 5 or more than 5, it is evident that there is an odd number of shillings, and the decimal must be separated into two parts before applying the above rule. Thus, .875 = .85 + .025. .85 = 17s. (2 x 8). .025 = Qd. (25 -s- 4). 124 DENOMINATE NUMBERS. [Art. 342. Eeduce mentally the following to shillings and pence : 20. .425; .637. 22. .255; .183. 21. .817; .245. 23. .376; .496. . '"^.Divide 15 16*. Sd. by .10; by .20 ; by .25 ; by .40. 25. Multiply 16 12s. 9d. by .05 ; by .06 ; .04. 26. If 1 sterling is worth $4.87, what is the value of 225 18s. 6d. ? Of 140 8s. 8d. ? ^Jty. If 1 sterling is worth $4.88, how many pounds can be bought for $1000 ? How many for $1625 ? MISCELLANEOUS TABLES. 343. The following table is used in counting certain articles: 12 Units = 1 Dozen . . . doz. 12 Dozen = 1 Gross . . . gr. 12 Gross = 1 Great Gross . g.yr. g. gr. gr. doz. units. I = 12 = 144 = 1728 1 = 12 = 144 344. The following table is used in the paper trade : 24 Sheets = 1 Quire .... qr. 20 Quires = 1 Ream .... rm. rm. qr. sheets. 1 = 20 = 480 Manufacturers and wholesale dealers usually sell paper by the pound. EXAMPLES. 345. 1. Find the value of 5 gross pencils at 4c. each. 2. A merchant buys 6 gross pens at 95c. per gross, and sells them at Ic. each. What is his profit ? 3. How many sheets of paper in 12 quires ? In 2 reams ? 4* Find the difference between six dozen dozen and half a dozen dozen. 5. Combs are bought at $2. 70 per dozen. How much is that apiece ? 6. Find the value of 306 eggs at 22c. per dozen. 7. At 1 cent each, what is the value of 20 great gross pens ? 8. A grocer buys 81 dozen eggs* at 22c. per dozen, and sells them at the rate of 9 for 25 cents. What is his profit ? 9. If 32 pages of a book are printed on one sheet, how many reams of paper would be required for 2000 copies containing 384 pages each ? Art. 346.] CIRCULAR MEASURE. 125 CIRCULAR MEASURE. 346. Circular or Angular Measure is used in measuring angles and arcs of circles. It is employed principally by surveyors in determining directions, by navigators in determining latitude and longitude of places, and by astronomers in making observa- tions. The unit of this measure is the degree, which is ^ of the circumference of any circle. TABLE. 60 Seconds (") = 1 Minute '. 60 Minutes = 1 Degree . 360 Degrees = 1 Circle C. NOTES. 1. A quadrant is one-fourth of a circle, or 90. 2. A sextant is one-sixth of a circle, or 60. 3. 1 minute of the circumference of the earth is called a nautical, or geographic mile, and is about 1.15 statute or common miles. 4. An arc of 1 degree of the circumference of the earth measured at the equator equals 69.16 statute miles. EXAMPLES. 347. 1. Add 74 0' 3" and 77 49' 58". & Add 12 27' 14" and 122 26' 45". 8. From 84 29' 31" subtract 77 0' 45". 4. Multiply 13 11' 16" by 5 ; by 15. 5. Divide 76 11 ; 45" by 15 ; by 12. 6. Divide 179 42' 15" by 15 ; by 16. 7. Eeduce 1,000,000" to higher denominations. 8. Reduce 44 16' 40" to seconds. 9. The angles of a triangle are 67 18' 40", 72 39' 50", and 40 1' 30" respectively. What is their sum ? LONGITUDE AND TIME. 348. The whole circle of the earth, or 360, passes under the sun in 24 hours, and in 1 hour passes -fa of 360, or 15; in 1 minute, ^ of 15 (15 x 60'), or 15'; and in 1 second, -gV of 15' (15 x 60"), or 15". 126 DENOMINATE NUMBERS. [Art. 349. 349. Comparison of Longitude and Time. For a difference of There is a difference of 15 in Longitude 1 lir. in Time. 15' " " 1 min. " " 15" " 1 " 1' " 1 sec. " 4 min. " 4 sec. " " 35O. RULE. 1. Tlic difference in longitude of two places, expressed in ", divided by 15 ivill produce their differ- ence in time expressed in hours, minutes, and seconds. 2. The difference in solar time of two places, expressed in hr. min. sec., multiplied by 15 will produce their differ- ence in longitude expressed in ' ". 351. TABLE OF LONGITUDES. Albany 73 44' 50" W New York 74 0' 3" W Ann Arbor 80 43' W New Orleans 90 2' 30" W Boston 71 3' 30" W. Paris .... 2 20' 22" E Berlin 13 23' 45" E Philadelphia 75 10 7 W Calcutta 88 19' 2" E. Rome 12 27' 14" E 84 29' 31" W Richmond "Va 77 25' 45" W 87 37' 45" W San Francisco 122 26' 45" W Jefferson City, Mo. . . 92 8' W. St. Paul, Minn. . 95 4' 55" W. London 5' 38" W St Louis Mo 90 15' 15" W Mexico. . . 99 5' W. Washington. D. C.. .77 & 15" W. 352. Standard Time. In 1883, the principal railroads and cities of the United States and Canada adopted the time of four different meridians as the standard time of four belts or sections comprising the whole of the above countries. The most eastern of these sections embraces the Eastern and Middle States, Maryland and Virginia, and extends about 7 east and west of the meridian of 75 west of Greenwich (near Philadel- phia). The time of this meridian, called Eastern time, is used in this section, and is 5 hours slower than Greenwich time. The time of the meridian of 90 west of Greenwich (near St. Louis) called Central time, is used in the next section, and is 1 hour slower than Eastern time. The time of the meridian of 105 west of Greenwich (near Denver), called Mountain time, is used in the Rocky Mountain region, and is 1 hour slower than Central time and 2 hours slower than Eastern time. The time of the meridian of 120 west of Greenwich, called Pacific time, is used in the Pacific slope, and is 3 hours slower than Eastern time. Art. 353.] LONGITUDE AND TIME. 127 EXAMPLES. 353. Find the difference in longitude between 1. New York and London. 4. -St. Louis and Calcutta. 2. Boston and Paris. 5. Philadelphia and Berlin. 3. Chicago and San Francisco. 6. San Francisco and Calcutta. Find the difference in solar time between 7. New York and Greenwich. 10. Rome and London. 8. Chicago and New York. 11. Paris and Albany. 9. Richmond and Calcutta. 12. Calcutta and Jefferson City. Find mentally the difference in standard time between 18. Albany and Denver. 16. St. Louis and Richmond. 14. New York and Chicago. 17. St. Paul and Sacramento. 15. Boston and San Francisco. 18. Phila. and Portland, Me. Find the difference between the standard time and the solar time of the following cities: 19. Boston. 21. San Francisco. 28. Chicago. 20. Philadelphia. 22. St. Louis. 24. St. Paul. 25. A navigator finds that when it is noon at his place of observation, it is 16 min. 34 sec. past 10 P.M. by his chronometer, Greenwich time ; what is his longitude ? 26. When it is 6 o'clock P.M., standard time, at Richmond, Va., what is the time at St. Louis, Mo.? 27. If the difference of solar time between two places is 1 hr. 18 min. 4 sec., what is the difference of longitude ? 28. When it is 20 min. past 2 P.M., standard time, at Boston, Mass., what o'clock is it at San Francisco ? 29. When it is Monday, 10 P.M., standard time, in Chicago, what day and time is it in London (Greenwich time) ? 30. When it is 9 o'clock P.M., solar time, in San Francisco, it is 3 min. 3 T 2 ^ sec. past 11 A.M. in Calcutta ; what is the longitude of San Francisco, if the longitude of Calcutta is 88 19' 2" E. ? 81. When it is noon, solar time, in Chicago, it is 5 min. 29-J- sec. of 1 P.M., solar time, in New York ; what is the longitude of Chicago, the longitude of New York being 74 3" W. ? 128 DENOMINATE NUMBERS. [Art. 354. THE METRIC SYSTEM.* 354. In the Metric System of weights and measures, the Meter is the basis of all the units which it employs. 355. The Meter is the unit of length, and is. equal to one ten-millionth part of the distance measured on a meridian of the earth from the equator to the pole, and equals about 39.37 inches. The, standard meter is a bar of platinum carefully preserved at Paris. Exact copies of the meter and the other units have been procured by the several nations, including the United States, that have legalized the system. Comparisons with the standard units are made under certain conditions of temperature and atmospheric pressure. 356. The names of the higher denominations, or multiples, of the unit are formed by prefixing to the several units the Greek numerals, deka (10), hecto (100), Mo (1000), and myria (10000); as dekameter, 10 meters, hectometer, 100 meters, etc. To assist the memory, observe that the initial letters of the multiples are in alphabetical order ; thus, D, H, K, and M. 357. The names of the loiver denominations, or divisions, of the unit are formed by prefixing to the several units the Latin numerals, deci (-fa), centi (y^), milli ( i * ) ; as decimeter, fa meter, centimeter, y^ meter, etc. To assist the memory observe that the following words are derived from the same roots : dime, decade, decimal, decimate, decennial, etc. ; cent, cental, century, centennial, etc.; mill, millennium, etc. LINEAR MEASURE. 358. TABLE. 10mm. 10 cm. 10 dm = 1 1 1 1 Millimeter. . . Centimeter. . . Decimeter. . . . METER . ... (unnr of a meter) . . (x^ of a meter) ..(^ of a meter) (1 meter) = .03937 in. .3937 in. 3.937 in. 39 37 in. 10 m. _ 1 Dekameter. . . . .(10 meters) _ 32.8ft. 10 Dm. 10 Hm. = 1 1 Hectometer . . Kilometer ..(100 meters) ..(1000 meters) 328.09ft. .62137 mi. For other foreign weights and measures, see page 343. Art. 358.] THE METRIC SYSTEM. 129 NOTES. 1. The meter, like the yard, is used in measuring cloths, ribbons, laces, short distances, etc. 2. The kilometer is used in measuring long distances, and is about f of a mile. 3. The centimeter and millimeter are use by artisans and others in measuring minute lengths. The other denominations are rarely used. EXAMPLES. 359. 1. Eeduce 875275 meters to kilometers. ANALYSIS. Since 1 kilometer equals 1000 meters, in 875275 meters there are as many kilometers as 1000 is contained times in 875275, or 875.275. To divide by 1000, place the point three places to the left (27O, 4). 2. Eeduce 675.318 kilometers to meters. ANALYSIS. Since 1 kilometer equals 1000 meters, in 675.318 kilometers there are 675.318 times 1000, or 675318 meters. To multiply by 1000, place the point three places to the right (267, note). 3. Reduce 383.64 meters to centimeters ; to kilometers. 4. Reduce 175.16 centimeters to kilometers ; -to meters. 5. Reduce to meters and find the sum of 876.2 decimeters, 30347 centimeters, 176.48 meters, 8.175 kilometers. 6. A ship sails 5712 kilometers in 48 days ; how many kilo- meters does she sail per day ? 7. What is the value of 56.4 meters of silk at $1.75 per meter? 8. 16 pieces of cloth contain 38.5 meters each ; 18 pieces con- tain 39 meters each; and 24 pieces contain 41.2 meters each; how many meters in all? 9. How many meters of ribbon at 27 cents per meter can be purchased for $245. 70 ? 10. If the forward wheels of a carriage are 3.5 meters in cir- cumference, and the hind wheels 4.8 meters, how many more times will the forward wheels revolve than the hind wheels, in running a distance of 8.4 kilometers ? 11. How much will it cost to sewer a street .64 Km. long, at $3.75 per meter? 12. How many meters of wire will be required to fence a rectangular field, .72 Km. long and .56 Km. wide, if the fence is 4 wires high ? 13. How long will it take a railway train, running 60 Km. per hour, to go from New York to Chicago, the distance being 1440 Km.? 130 DENOMINATE NUMBERS. [Art. 36O SQUARE MEASURE. 360. The unit of square measure is the square meter. TABLE. 100 Square Centimeters, sq. cm. = 1 Square Decimeter = 15.5+ sq.in. 100 Square Decimeters, sq. dm. = 1 SQUARE METER, Sq. M. 1.196+ sq. yd. NOTES. 1. The square meter is used in measuring flooring, ceilings, etc. ; the square decimeter and the square centimeter are used for minute surfaces. 2. Since units of square measure form a scale of hundreds, each denomi- nation must have two places of figures. 361. The unit of Land Measure is the are, and is equal to a square dekameter (100 square meters), or 119.6 square yards. TABLE. 1 Centare .... (1 square meter) = 1550 sq. in. 100 Centares, ca. = 1 Are (100 square meters) = 119.6 sq. yd. 100 Ares, A. = 1 Hectare. . . .(10000 square meters) = 2.471 acres. NOTE. The hectare is the ordinary unit for land. EXAMPLES. 362. 1. Write 16 sq. m., 8 sq. dm., 24 sq. cm., having the square meter as the unit. (36O, 2.) Ans. 16.0824. 2. Write 83 sq. m., 9 sq. dm., having the sq. m. as the unit. 8. In 47 ares how many square meters ? 4. In 60.25 hectares how many centares ? 5. How many square meters in a building lot 8 m. by 32 m. ? 6. How many building lots, each containing 225 sq. m., can be formed from a field containing 9 hectares ? 7. How many hectares in a farm 1.024 Km. in width and 1.625 Km. in length? 8. What is the cost of a mirror 2.25 m. by 1.44 m., at $3.84 per sq. m. ? 9. How many lots 25 m. wide and 60 m. deep, or having an equivalent area, can be laid out from 6 hectares ? 10. A man bought a piece of land for $6950.50, and sold it for $7603.30, by which transaction he made $6.80 a hectare; how many hectares were there ? Art. 363.] THE METRIC SYSTEM. 131 CUBIC MEASURE. 363. The unit for measuring ordinary solids is the cubic meter. TABLE. 1000 Cu. Millimeters, cu. mm. = 1 Cu. Centimeter = .061 cu. in. 1000 Cu. Centimeters, cu. cm. = 1 Cu. Decimeter = 61.027 cu. in. 1000 Cu. Decimeters, cu. dm. = 1 Cu. METER = ( ? 5 '^ 7 CU ' &' \ 1.308 cu. yd. NOTES. 1. The cubic meter is used in measuring embankments, excava- tions, etc. ; cubic centimeters and cubic millimeters for minute bodies. 2. Since units of cubic measure form a scale of thousands, each denomi- nation must have three places of figures. 364. The unit of "Wood Measure is the ster, and is equal to a cubic meter, or 35.317 cubic feet. TABLE. 10 Decisters, ds. = I Ster. . . .(1 Cubic Meter) = ( l^fj 50 * 1 ' {. OO.ol/ CU. jt. 10 Sters, s. = 1 Dekaster, Ds. .(10 Cubic Meters) = 2.759 cords. EXAMPLES. 365. 1. Write 29 cu. m., 75 cu. dm., having the cubic meter as the unit. (363, 2) Ans. 29.075 cu. m. 2. Write 17 cu. m., 218 cu. dm., 27 cu. cm., having the cubic meter as the unit. 3. How many cubic meters in a box 3.5 m. by 3.2 m. by 2.5 m. ? 4. Bought 12 sters of wood ; having sold 8. 7 cubic meters, how much remained ? 5. There are 13 blocks of marble, each containing 370.16 cu. dm. how many cubic meters in all ? 6. How many cubic meters in an excavation 13.2 m. by 18.5 m. by 8.4 m.? 7. At $1,25 a cubic meter, what will it cost to dig a cellar 6.5 m. long, 5.4 m. wide, and 2.5 m. deep ? 8. How many sters of wood in a pile of wood 2.5 m. high, 2 m. wide, and 16.5 m. long ? What is the length of a pile of the same height and width containing 216 sters ? DENO Ml NA TE 3 r UKB ERS. [Ar . 3C6. DRY AND LIQUID MEASURE. 366. The unit of Dry and Liquid Measure is the liter, which is equal to a cubic decimeter, 1.0567 wine quarts, or .908 dry quart. TABLE. Dry Measure. Liquid Measure. 1 Milliliter (y^ of a liter) = .06103 cu. in., or, .0338 yJ. oz. 10ml. = 1 Centiliter (^ of a liter) .6103 cu. in., or, .338^.0,2. 10 el. 1 Deciliter ( T V of a liter) = 6.1027 cu. in., or, .845 gi. 10 dl. = 1 LITER (1 liter) = .908 qt., or, 1.0567 qt. 10 I. = 1 Dekaliter (10 liters) = 9.08 qt., or, 2.6418 gal. 10 Dl. = 1 Hectoliter . . . .(100 liters) = 2.8375 bu., or, 26.418 gal. 10 HI. = 1 Kiloliter (1000 liters) = 28.375 bu., or, 264.18^. NOTES. 1. The liter is commonly used in measuring wine, milk, etc., in moderate quantities. For minute quantities the centiliter and milliliter are employed ; and for large quantities the dekaliter. 2. For measuring grain, etc., the hectoliter (2.8375 bushels) is commonly used. 3. Instead of the kiloliter and milliliter, it is customary to use their equals, the cubic meter and cubic centimeter. EXAM PLES. 367. 1. How many liters in a vessel whose capacity is 1 cubic meter ? 2. What is the cost of sixteen liters of milk at 8 cents a liter ? 3. How many hectoliters of wheat can be bought for $396 at $5.50 per hectoliter ? 4. How many hectoliters of grain can be put in a rectangular bin, 4 m. long, 3.5 m. wide, and 1.2 m. high ? 5. How many liters in 63.5 dekaliters ? In 83.75 hectoliters ? 6. At $1.75 a liter, what is the cost of 85.6 dekaliters of wine? 7. How many hectoliters in 16 cubic meters ? 8. How many bags, each holding 1 hectoliter, can be filled from a bin, 1.5 m. high, 2.4 m. wide, and 5 m. long ? 9. A cistern 3.5 m. by 3.2m., and 9 m. deep, will hold how many dekaliters ? 10. A merchant bought 4 hectoliters of nuts at $8.50 per hectoliter, and retailed them at 12 cents a liter ; what was his profit ? Art. 368.] THE METRIC SYSTEM. 133 WEIGHT. 368. The unit of weight is the gram, which is equal to the weight of a cubic centimeter of distilled water in a vacuum; at its greatest density (39.2 F.), or 15.432 grains. TABLE. 1 Decigram (-^ of a gram) = 1.543 gr. Tr. 10 dg. = 1 GRAM (1 gram) = 15.432 gr. Tr. 10 g. = 1 Dekagram (10 grams) = .3527 oz. Av. 10 Dg. = 1 Hectogram (100 grams) = 3.5274 oz. Av. 10 Hg. = 1 Kilogram (1000 grams) = 2.2046 Ib. Av. 10 Kg. 1 Myriagram (10000 grams) = 22.040 Ib. Av. 100 Kilos = 1 Quintal. . (100000 grams) = 220.46 Ib. Av. 10 Q., or \ / 1 Tonneau, ) ( 2204.6 Ib. Av. 1000 Kilos / = ( or TON \-^ )0 grams) : = j 1.1023 T. NOTES. 1. The above table is used in computing the weights of all objects from the smallest atom to the largest known body. The gram, kilo- gram (or kilo), and ton are principally used. 2. The gram is used in weighing letters, gold, silver, and medicines. 3. The kilogram, or kilo, like the pound, is used in weighing groceries and coarse articles. It is approximately 2i pounds Av. 4. The ton is the weight of a cubic meter of water, and is used in weigh- ing very heavy articles, as coal, iron, etc. 5. The pound of Germany, Austria, and Denmark is equal to of a kilo- gram ; the centner, to 100 pounds, or | of a quintal. EXAMPLES. 369. 1. What is the weight in grams of a cubic meter* of water ? Of a cu. dm. of water ? 2. A farmer sells to A 3.716 T. of hay, to B 4.325 T., to C 8775 kilos ; how many tons does he sell ? 3. The U. S. 50-cent piece weighs 12.5 grams ; how many can be coined from a kilogram of standard silver ? 4. The U. S. 5-cent piece weighs 5 grams ; how many 5-cent pieces are equivalent in weight to 12 50-cent pieces ? 5. How much alloy must be used in making 1200 U. S. twenty- five-cent pieces ? (See Art. 113.) 6. What is the cost of 75.6 kilos of sugar at 18 cents a kilo ? 7. How many powders, each containing 6 grams, can be made from .372 kilogram ? 8. What is the weight of 10 cii. m. of ice, it being .93 as heavy as water ? 134 DENOMINATE NUMBERS. [Art. 37 O. 37O. TABLE OF EQUIVALENTS. 1 inch = 2.54 centimeters 1 centimeter = 0.3937 inch. 1 foot = 3.048 decimeters 1 decimeter = 0.328 foot. 1 yard = 0.9144 meter 1 meter = 1.0936 yards = 39.37 in. 1 rod = 0.5029 dekameter 1 dekameter = 1.9884 rods. 1 mile = 1.6093 kilometers 1 kilometer = 0.62137 mile. 1 sq. inch = 6.452 sq. centimeters 1 sq. centimeter = 0.155 sq. inch. 1 sq. foot = 9.2903 sq. decimeters 1 sq. decimeter = 0.1076 sq. foot. sq. yard = 0.8361 sq. meter 1 sq. meter = 1.196 sq. yards. sq. rod 25.293 sq. meters 1 are = 3.954 sq. rods = 119.6 sq. yards. acre = 0.4047 hectare 1 hectare = 2.471 acres. sq. mile = 2.59 sq. kilometers 1 sq. kilometer 0.3861 sq. mile. cu. inch = 16.387 cu. centimeters. .. 1 cu. centimeter = 0.061 cu. inch. cu. foot = 28.317 cu. decimeters 1 cu. decimeter = 0.0353 cu. foot, cu. yard = 0.7645 cu. meter 1 cu. meter = 1.308 cu. yards. 1 cord = 3.624 sters 1 ster = 0.2759 cord. 1 liquid quart = 0.9463 liter 1 liter = 1.0567 liquid quarts. 1 gallon = 0.3785 dekaliter 1 dekaliter = 2.6417 gallons. 1 dry quart = 1.101 liters 1 liter = 0.908 dry quart. 1 peck = 0.881 dekaliter 1 dekaliter = 1.135 pecks. 1 bushel = 3.524 dekaliters 1 hektoliter = 2.8375 bushels. 1 ounce av. = 28.35 grams 1 gram = 0.03527 ounce av. 1 pound av. = 0.4536 kilogram 1 kilogram = 2.2046 pounds av. 1 pound av. = 0.9072 German pounds. 1 German pound = 1.1023 pounds av. 1 ton (2000 Ibs.) = 0.9072 met. ton. ... 1 met. ton = 1.1023 tons = 2204.6 Ib. av. 1 grain Troy = 0.0648 gram 1 gram = 15.432 grains Troy. 1 ounce Troy = 31.1035 grams 1 gram = 0.03215 ounce Troy. 1 ppund Troy = 0.3732 kilogram 1 kilogram = 2.679 pounds Troy. EXAMPLES. 371. 1. In 226 meters how many yards ? How many inches ? 2. Reduce G miles to kilometers ; to meters. 8. Eeduce 640 acres to hectares ; to ares. 4. In J.O kilometers, how many feet ? How many miles ? 5. In 375.6 kilos, how many pounds ? 6. How many German pounds in 225 English or U. S. pounds? 7. What is the weight of the U. S. standard silver dollar in grams ? Of the trade dollar ? 8. In 5000 U. S. bushels, how many hectoliters ? How many dekaliters ? 9. In 875 cu. yd. how many cu. m. ? Art. 371.] APPROXIMATE RULES. 135 10. In 1000 cu. m. how many cu. yd. ? 11. Reduce 1728 gal. wine to liters ; to dekaliters. 12. In 244 sq. m. how many sq. yd. ? How many sq. ft. ? IS. Reduce 220 oz. Av. to grams ; to kilograms. 372. APPKOXIMATE VALUES. When no great accuracy is required, we may consider 1 decimeter = 4 inches. 1 meter = 39 inches. 5 meter's 1 rod. 1 kilometer = f mile. 1 square meter = lOf square feet. 1 hectare = 24 acres. 1 cu. met. or ster = 1 cu. yd. or \ cord. 1 liter 1 hectoliter 1 gram 1 kilogram 1 ton = 1 quart. = 2^ bushels. = 15^ grains. = 2 pounds. = 2200 pounds. APPROXIMATE RULES. 373. To reduce avoirdupois pounds to kilograms : Divide by 2, and then deduct one-tenth. NOTE. Answer too small by about 8 kilos for every 1000 kilos of the result. If -jJj-, instead of T V, be deducted, the answer will be too great by 2 kilos for every 1000 kilos of the result. 374. To reduce avoirdupois pounds to half-kilograms, or German pounds : Deduct one-tenth. NOTE. Answer too small by about 8 German pounds for every 1000 Ger- man pounds of the result. If ^ be deducted, the answer will be too great by 2 German pounds for every 1000 German pounds of the result. 375. To reduce yards to meters: Deduct one-twelfth. NOTE. Answer too great by 2 m. for every 1000 m. of the result. 376. To reduce square yards to square meters : Deduct one-sixth. NOTE. Answer too small by about 3 sq. m. for every 1000 sq. in. of the result. 377. To reduce cubic yards to cubic meters : Divide by 1.3. NOTE. Answer too great by about 6 cu. m. for every 1000 cu. m. of the result. 136 DENOMINATE NUMBERS. [Art. 378. 378. To reduce U. S. gallons to liters : Multiply by h and then subtract one-twentieth (5 per cent.). NOTE. Answer too great by about 4 /. for every 1000 I. of the result. 379. To reduce U. S. bushels to hectoliters : Divide by 8, and then add one-twentieth (5 per cent.). NOTE. Answer too small by about 7 hi. for every 1000 hi. of the result. 380. To reduce kilograms to avoirdupois pounds : Multiply by 2, and then add one-tenth. NOTE. Answer too small by about 2 Ib. for every 1000 Ib. of the result. 381. To reduce German pounds, or half-kilograms, to avoirdupois pounds : Add one-tenth. NOTE. Same error as in Art. 38O. 382. To reduce meters to yards : Add one-twelfth and 1% 'of the original number. NOTE. Answer too small by only \ yd. for every 1000 yd. of the result. Dealers in dry goods add only T ^ in reducing meters to yards, and thus make the result too small by about 9 yd. for every 1000 yd. of the result. If -fa be added, the answer will be too small by about 2 yd. for every 1000 yd. of the result. If y 1 ^ be added, the answer will be too great by about 6 yd. for every 1000 yd. of the result. 383. To reduce square meters to square yards : Add one-fifth. NOTE. Answer too great by about 3 sq. yd. for every 1000 sq. yd. of the result. 384. To reduce cubic meters to cubic yards : Multiply by 1.3. NOTE. Answer too small by about 6 cu. yd. for every 1000 cu. yd. of the result. 385. To reduce liters to U. S. gallons : Multiply by 2. 11, and then divide by 8. NOTE. Answer too small by about 1.7 gal. for every 1000 gal. of the result. 386. To reduce hectoliters to U. S. bushels : Multiply by 3, and then subtract one-twentieth (5 per cent.). . NOTE. Answer too great by about 4 bu. for every 1000 bu. of the result. Art. 387.] REVIEW EXAMPLES. 137 REVIEW EXAMPLES. 387. 1. Add 174, 26J, 35f, 4S, and 8 T V ; multiply the sum by 59 ; subtract 2309 T \ from the product ; and divide the remain- der by 162f . 2. Divide fourteen, and twenty-five hundredths by one hun- dred twenty-five thousandths ; add nineteen, and sixty-four hun- dredths to the quotient ; and multiply the sum by eight, and five tenths. 3. Find the time by Compound Subtraction and in exact days from March 24 to Sept. 18. 4. How many lengths of pipe, each 10 ft. (') 3 in. (") long, will be required for a well 130 ft. deep ? 5. A horse trots a mile in 2 min. 45 sec. How many feet is that per second ? 6. A grass plot 13 ft. by 54 ft. is surrounded by a stone walk \\ft. wide. The stone walk is surrounded by a gravel road 7J/rf. wide. How many square feet are covered by the grass, the stone, and the gravel respectively ? (Make diagram.) 7. What is the cost of 15669 pounds meal at $1.10 per cwt. ? 8. What is the cost of 16450 pounds of hay at $15.50 per ton? 9. How many square rods in a triangular piece of land, 360 rd. long and whose perpendicular width is 240 rd. ? NOTE. To find the area of a triangle, take one-half the product of the length (base) and the height or width (altitude). 10. How many square feet in the gable of a house, 40 ft. long and 24: ft. high ? 11. How many feet of siding would be required for a house 40 ft. long, 24 ft. wide, IS ft. high, with two gables each 24 ft. wide and 12 ft. high, adding one-fifth for the lap and waste in cutting ? 12. How much will it cost to make an excavation, 40 ft. long, 30 ft. wide, and 9ft. deep, at 32c. per cubic yard. 13. How many feet of 2-inch plank, making no deduction for the corners, would be needed to build a rectangular tank, without a cover, 10 ft. long, 6$ ft. deep, Sft. wide ? 14> The circumference of any circle is equal to the diameter multiplied by 3.1416 (about 3^). Find the circumference of a circle, whose diameter is 5 feet. 138 DENOMINATE NUMBERS. [Art. 387. "^15. The area of any circle equals the square of the radius multiplied by 3.141G (3|), or the square of the diameter multi- plied by . 7854. What is the area of a circle whose diameter is 6 ft. ? Whose radius is 5 ft. ? 16. How many feet of 2-inch plank would be required to make a cylindrical cistern without a cover, 7 ft. in diameter and 8 ft. high? 17. How many pounds in 16 T. 3 qr. 18 Ib. (Long Ton Table)? 18. How many quarts in 3 Vbl. 24 gal. cider ? 19. In 27318 pounds of corn, how many bushels ? What is the value of the same at 48f cents per bushel ? ^ W. What is the value of 27318 pounds of corn, at 87.1 cents per cental ? NOTE. Examples 19 and 20 illustrate the present and the centai systems of buying and selling produce, and show the calculations saved by using the latter. 21. Paid $222.75 for boards at $13.50 per M.; how many feet were purchased ? 22. What is the value of 27315 ft. of lumber at $12 per M. ? 23. A quartermaster purchased 75000 pounds of corn, at 31J- cents per bushel ; 32113 pounds of oats, at 32] cents per bushel ; and 79500 pounds of hay, at $22. 37^ per ton (2000 pounds). "What was the total cost of the purchase ? 24. A farmer sold 18360 pounds of corn, at 64 cents per cen- tal ; 22450 pounds of oats, at 94 cents per cental ; and 36650 pounds of hay, at $1.31 per cental. How much was realized from the sale ? 25. Reduce 19 16s. 9d. to the decimal of a pound. 26. If 1 sterling is worth $4.87, what is the value of 225 185. Qd. ? 27. From 16 12s. 9rf. deduct .05 of itself. 28. What is the value of 20 yd. silk at 10s. Gd. per yard ? 29. The difference in the local time of two places is 3 Ur. 43 min. 12 sec. ; what is the difference in longitude ? 30. What is the capacity in liters of a cistern 25 meters long, 2.2 meters wide, and 3 meters deep ? 81. The specific duty on Brussels carpet is 44 cents per square yard ; what is the duty per square meter ? 32. The duty on tallow candles is 2 cents per pound ; what | is the duty per kilogram ? ALIQUOT PARTS. 388. An Aliquot Part of a number or quantity is a number that will divide it without a remainder ; as 20 of 60, 12 of 100, 4 of 12, etc. Any fraction having 1 as its numerator is an aliquot part of a unit. Many of the ordinary business computations can be shortened by the use of aliquot parts. EXAMPLES. 389. 1. Find the cost of 217 pounds of sugar at 8'. ham at 10}c. per pound. 12. 644 Ibs. lard at lljc. per pound. 13. 2957 /6s. sugar at S-foc. (J-f J of J) per pound. 140 ALIQUOT PARTS. [\rt.390. 39O. Aliquot parts of 100. 3* = A- 16 f = * 62i = i + i (i of 1). 4 =A- 20 = t. 75 =J + i(*ofi). 5 = ,V- 25 - i- S?i = * + i + *- 6J - A, 33i = J. 18| = i + A (* of 1). 10 = . 50 = 31 = of . NOTE. In the following commercial problems, use as few figures as possible. EXAMPLES. 391. 1. Find the cost of 13756 pounds of meal at $1.05 per cwt. OPERATION. 137.56 ANALYSIS. At $1 per cwt. the cost would be $137.56. g gg 5c. = fa of $1. To divide by 20, divide by 2 and place the quotient figures one place to the right. 144.44 2. Find the value of 16345 Ibs. of feed at $1.10 per cwt. (We = A of *!) 3. What is the cost of 12 doz. hats at $4.12 () per doz. ? 4. Find the cost of 471f# ( + ) bushels of corn at 41c. 5. What is the cost of 96 doz. buttons at $1.75 (J i + J) per dozen ? 0. Find the cost of 711ff (J+i) bushels oats at 39c. per bushel. 7. Find the cost of 24 boxes note paper at 16f c. per box. 8. Find the cost of 24116 Ibs. bran at $1.20 per cwt. 9. Find the cost of 1750 Ibs. soap at 5c. per pound. 10. Find the cost of 131 Ibs. coffee at 16Jc. per pound. 11. Find the cost of 60 Ibs. crackers at 12Jc. per pound. 12. Find the cost of 4880 Ibs. feed @ 75c. per cwtf . JT5. Find the cost of 20 half -barrels fish at $4.25 per half- barrel. At $5.35 (J-f A) P er half -barrel. U. Find the cost of 75 books at 25c. each. 16. Find the cost of 36 pairs shoes at $2.25 per pair. At $2.50 per pair. 16. Find the cost of 3019 Ibs. bran at 62|c. per cwt., and 24375 Ibs. feed at $1.05 per cwt. PERCENTAGE. 392. Percentage is a term applied to all operations in which 100 is used as the basis of computation. It is also the name given to any number of hundredths of a number. 393. Per Cent. (%) is an abbreviation of the* Latin per centum, meaning on or by the hundred. Thus, 5% means 5 of every hundred, or 5 hundredths ( T 7 , or .05). 394. Any per cent, may be expressed in the form of a decimal or fraction. Thus 5 per cent. = 5^ = 5 hundredths = .05 T ^ = &. The first two forms are used in the statements of questions ; the others in the operations. 395. In percentage, three elements are considered, viz : the Base, the Rate, and the Percentage. Any two being given, the other can be found. 396. The Percentage is the result obtained by taking a certain number of hundredths of a number. 397. The Base is the number of which a certain number of hundredths are taken. 398. The Rate is the number of hundredths, or the num- ber per cent. Thus, in the statement, 6% of 300 is 18, the Percentage is 18, the Base 300, and 6 per cent. (.06) is the Rate. 399. Applications of Percentage. The principles of per- centage are applied to many of the most common business trans- actions. Among the most important of these are Trade Discounts, Commission, Insurance, Profit and Loss, Duties, Interest, and Exchange. 142 PERCENTAGE. [Art. 40O. 400. To find the percentage, the base and rate being given. Ex. What is 6 per cent, of 300 ? OPERATION. ANALYSIS. $% means 6 himdredths. 6% of 300 is 300 Base. equivalent to .06 of (or times) 300. .06 x 300 = 18. The .06 Rate. percentage is the. product of two factors, the base and - , , the rate. Or, \% of 300 is 3, and 6$ is 6 times 3, or 18. To find 1 % of any number, place the point two places to the left. 401. EULE. 1. To find the percentage, multiply the base by the rate expressed decimalli/. EXAM PLES. 402. What is What is What is 1. .03x1728? 6. 8% of $414? 11. 16$ of $375.60 ? 2. .16 times 375 ? 7. .08 of 8716 ? 12. 8% of $414.60 ? 3. 4% of 448 ? 8. 1.12 x $575 ? 18. 6% of $875.75 ? 4. 6 per cent, of 387? 9. 107% of $385 ? 14. 113% of $913.25 ? 5. .06 of 945 ? 10. 9% of $456 ? 15. 32% of $485.50 ? 16. What is the difference between 2$% of $16000 and 5% of $8475 ? 17. A merchant bought goods amounting to $375.60, and sold them so as to gain 30% of the cost ; how much did he gain ? 18. A lawyer collected $2875, and charged 5% for his services ; how much did he retain for his services, and how much did he pay over ? The amount paid over is what per cent, of the amount collected ? * 19. An agent sells a house and lot for $16450, and receives 2% for his services ; what does he pay to the owner of the property ? 20. What is the duty, at 25% of the value, on twelve watches worth $75 each ? 21. Jan. 10, a merchant buys a bill of goods amounting to $876.40 on the following terms : 4 months, or less 6% if paid in 10 days. How much would settle the bill Jan. 18 ? 22. A merchant, failing in business, pays 43% of his indebted- ness. He owes A $3750, and B $6280. How much does he pay each ? * In order to prepare the student for examples in which the conditions are the reverse of those in this example, the teacher should ask oral questions similar to the above in all examples in which an amount or difference is involved. Art. 402.] PERCENTAGE. 143 23. A commission merchant sold 450 barrels of flour at $5.30 per barrel. How much should he send to the miller,, if he charges 2 per cent, for making the sale ? 24. A manufacturer's list price of cans is $2.50 per dozen. He sells a dealer 48 dozen at a discount of 30%. How much does he receive for them ? 25. A broker sells merchandise amounting to $916.64, at a commission of 1-J per cent. What is his commission ? OPERATION. ANALYSIS. \% of $916.64 is $9.166, to which add \ 1.146 \%. of itself as in the operation. (See Art. 389.) 10.312 1#. According to the above method, find 26. 1% of $375.60. 29. If ( + )% of $287.96. 7- i (i + i)# of $ 875 - so - l ?% of *5275. 28. 1|% of $1176.40. 81. If % of $3075.75. 32. A contracts to make an excavation containing 3456 cubic yards, at 28^ per cubic yard for earth, and $1.20 per cubic yard for rock. When completed, it is found that 16% is rock, and the remainder earth. How much does he receive for the work ? 4O3. When the rate is an aliquot part of 100, it is generally more convenient to use the equivalent fraction. Thus, = .16f=f 6J:% = .061 = T V. = .12i= i. 5% =.06 = ^. 25% =.25 = j, 10% =.10 = = .20 = EXAMPLES. 4O4. AVhat is What is What is 1. J of 1728- ? 5. 331% of $375 ? 9. 50% of $487.20 ? 2. 25% of 3472 ? 0. 12J% of $848 ? 70. 5% of $9742 ? #. .25 of 6418? 7. 2 % of $6480? 77. 10% of $1764.30 ? 4. T^t of 7264 ? #. 20% of $9875 ? 12. 37|% ({) of $875.60 ? 13. From a bill of goods amounting to $475.60, 5% is deducted for cash. What is the net amount of the bill ? OPERATION. 475 23 451 ANALYSIS. 5$ is ^. To divide by 20, divide by 2 and place the quotient figures one place to the right. 82 144 PERCENTAGE. [Art. 404. 14. A commission merchant pays $2375.40 for a quantity of grain, and charges %\% for his services. What is the total cost ? 15. Mr. B's tax is $175.60. If the collector is allowed 5f e additional, what is the total amount paid ? 16. The gross amount of a bill of tinware is $97.40. What is the net amount, if the trade discount is ?&\% ? 17. A house is sold for $16400, and 25% of the purchase money is allowed to remain on bond and mortgage. What is the amount of the mortgage ? 18. A house worth $7200 is insured for 62|-% (f ) of its value. What is the amount of the insurance ? What is its cost at \% ? 405. To find the base, the percentage and rate being given. Ex. 18 is 6^ of what number. OPERATION. ANALYSIS. The question "18 is 6% of what num- Kate. Percentage, fcer?" is equivalent to " 18 = .06 x what number?" If ____ 18 is the product of two factors and one of the factors is Base 300 *06> the other may be found by dividing 18 by .06. Or, since the Percentage = the Base x the Rate, the Base = the Percentage -5- the Rate. Or, if 18 is Q% of a certain number, \% is \ of 18, or 3 ; and the number, or 100^, is 100 times 3, or 300. NOTE. The student should remember that the Percentage = the Base x the Rate, and that when the Percentage and one of its factors are given, in the operation of finding the other, the Percentage becomes the dividend and the given factor, the divisor. If the relation between the given terms is indicated in the form of an equation, the student will have no difficulty in determining which number should be the divisor in solving the great variety of examples which occur in percentage and its applications. 406. RULE. To find the base, divide the percentage by the rate expressed decimally. EXAMPLES. 407. Find the unknown term in the following equations: 1. 48 6 times . 6. % of 380 = . 2. 48 = .06 x . 7. 72 is 4^ of . 8. 324 is Q% of . 8. 184 = .23 x . 4. 448 = .08 of . 9. 175 is .07 times , 5. .04 of = 375. 10. $576 x .06 = . Art. 4O7.] PERCENTAGE. 145 11. $324 = 8$ of . lo. $144 = | of . jf;2. $144 .16 x . 16. 12J$ of $475 = . 13. 52$ of $440 = . 17. $325 is 13$ of . 1. $875 = 25$ of . 18. $48.60 is 2^$ of . 19. The product of two factors is 75 ; if one of the factors is .03, what is the other factor ? 20. The percentage is 60, and the rate 2 of 300 is equivalent to " 18 = how many 300 ) 18.00 ( .06 hundredths times 300." If 18 is the product of two factors, and one of the factors is 300, the other may be found by dividing 18 by 300. To find the rate per cent., the quotient must be produced in hundredths. Or, since the Percentage = the Base x the Rate, the Rate = the Percentage -T- the Base, 18 -s- 300 = .06 (6$), the required per cent. Or, 18 is dfr or ^ of 300. / = T fo, or 6 % . 409. RULE. To find the rate, divide the percentage by the base. NOTE. In finding the rate, to produce a quotient of hundredths, make the decimal places of the dividend exceed those of the divisor by 2. EXAMPLE S. 410. -1. The product of two numbers is 375 ; if one of the numbers is 30000, what is the other number ? Express the answer in hundredths. Find the unknown term in the following equations : 2. 75 is what per cent, of 375? 6. $12.50 is what % of $1000? 3. 144 = .** times 1728. 7. $232.50 is what % of $3720? 4. 72 = .** x 3456. 8. $60.40 is what % of $2416? 5. 165 ft. is what % of 5280 ft. ? 9. $21.20 is what % of $1484? 10. The assets of a bankrupt are $27387, and his liabilities $82161 ; what % of his indebtedness can he pay ? \11. A merchant paid for goods $345 and sold them for $258.75; the loss is what % of the cost ? 12. If a paymaster receives $150000 from the treasury, and fails to account for $225 thereof, what is the percentage of loss to the government ? IS. If the rate is 20^ and the percentage 440, what is the base? 14. $640 being increased by a certain % of itself equals $720 ; required the rate %. 15. A person owing me $2092, pays only $1150.60. What % do I lose on the debt ? . Art,41O.] PERCENTAGE. 147 16. A house, insured for $16500 in several companies, is damaged by fire to the extent of $7260. What f c of its insur- ance will each company pay ? 17. A tax of $18480 is levied upon a township whose valuation is $3,696,000. What is the rate of the tax, and what tax should a man pay whose assessment is $8500 ? ^ 18. The annual interest on a mortgage of $7500 is $337.50. What is the rate per cent. ? 19. A merchant with a capital of $24000 gains $3840 in one year. His gain is what per cent, of his capital ? 20. A bankrupt has liabilities to the amount of $12600 and his assets are only $7087.50. What % dividend will he pay ? 21. The dividend of a manufacturing company, whose capital stock is $125000, is $6000. What % does it pay ? 22. A man's salary is $1600 per year and his living expenses $1300. What % of his salary does he save ? 23. A house cost $8000, and rents for $750 per year. If the taxes and other expenses are $230 per year, what % does it pay on the investment ? 24. The cost of insuring a cargo for $8500 is -$63. 75. What is the rate of the insurance ? 25. I meter = 1.0936 yards. The meter is what % greater than the yard ? 26. A man subscribes for 36 shares ($100 each) of a gas com- pany. He pays in $1980. What % is still due ? 27. The Avoirdupois pound (334) is what % greater than the Troy pound (336) ? 28. A merchant bought a quantity of goods for $425. Being damaged he sold them for $340. What % of the cost did he lose ? 29. A horse and wagon are worth $600. What is the value of each, if the wagon is worth 87^% as much as the horse ? SO. A man bought a watch for $160 and sold it for $180. His gain was what % of the cost ? 31. A tea merchant mixes 40 Ibs. tea at 45c. per pound with 50 Ibs. at 27c. per pound. He sells the mixture at 42c. per pound. What % profit on the cost does he make ? 32. A consignment of flour was sold for $3148, of which $3124. 39 ^ were the net proceeds. What was the rate % of the commission ? 33. Mr. A's house is worth $12500. He pays $30 for insuring it for | of its value. What per cent, does he pay ? 148 PERCENTAGE. [Art. 411 REVIEW EXAMPLES. 411. 1. What is 116$ of 1200 ? 2. 144 is 120% of what number ? 3. 375 is what % of 300 ? 4. Find 95$ of $1260. 5. Of what number is 275, 100$ ? 6. $187.50is2i$ of what ? 7. What will be the charge for insuring a house for $4500 atf$? 8. A merchant buys one gross jars for $36. At what price must he sell them apiece to gain 20$ on the cost ? 9. The assets of a bankrupt are $67850, which sum is 43$ of his debts. What are his debts ? 10. A tax collector, whose average commission was 3J$, received $892.08 for his services. How much did he collect ? 11. At what price must an article, which cost $4.80, be sold so as to gain 16$ of the cost ? 12. A clerk- spends 48$ of his income, and saves $598. What is his income ? 13. A's property is assessed at $7500, and the rate of taxation is $2.165 on $100. What is his tax, including a commission of 1$? 14. At 1|$, the premium for insuring a factory was $178.20. Find the amount of the insurance. 15. A bank collected a draft of $9375.16. What were the pro- ceeds, the charge for collection being \% ? 16. A consignment of cheese was sold for $375.60, of which $365.27 were the net proceeds. What was the rate of the com- mission ? 17. A commission merchant sold 24160 pounds of leather at 29 J cents a pound, paid transportation $60.40, cartage $20, his commission being 2|-$, and his charge for inspection $20. What were the net proceeds ? 18. A bankrupt who is paying 36$ of his debts, divides among his creditors $44442. What do his debts amount to, and how much does he pay a creditor whom he owes $3648 ? 19. A merchant buys a bill of dry goods, Apr. 16, amounting to $6377.84, on the following terms : 4 months, or less 5$ 30 Art. 411.] PERCENTAGE. 149 days. How much would settle the account May 16 ? The amount paid May 16, is what per cent, of the full amount of the bill ? The above discount is equivalent to what rate per cent, per an- num ? 20. Mar. 16, a merchant buys a bill of goods amounting to $2475 on the following terms : 4 months, or less 5% if paid in 30 days. Apr. 15, he makes a payment of $1000, with the under- standing that he is to have the benefit of the discount of 5%. With what amount should he be credited on the books of the seller ? How much would be due July 16, the expiration of the 4 months ? NOTE. As in Ex. 19, the amount paid within 30 days is 95$ of that part of the bill which it settles or cancels. 21. A bought a bill of merchandise July 24, 1879, amounting to $6287.45 on the following terms : 6 months, or less 4% 30 days. He paid on account Aug. 23, 1879, $5000, with the understand- ing that the payment would cancel an equitable amount of the bill. How much was due Jan. 24, 1880 ? 22. Paid for transportation $664.95 on an invoice of goods amounting to $8866. What per cent, was the value of the goods thereby increased ? What per cent, must be added to the invoice cost to make a profit of 20% on the full cost ? 28. What is 3% of 247 13s. 6d. ? OPERATION. ANALYSIS. Multiply the number of each de- s. d. nomination by .03, as in the margin, and then re- 247 13 6 duce the decimal parts to integers of lower denom- .03 inations (289). -P 71 4.1 3Q To ^ r ' re( l uce shillings and pence to the decimal of 1 ' a pound (see note, Ex. 12, Art. 342), take the re- quired per cent., and reduce the decimal result to S. 8|.59 lower denominations. Thus, 12 247 13s. Qd. = 247.675 d ~Y~26 247.675 x .03 = 7.43025 = 7 8s. Id. When the rate per cent, is an aliquot part of 100, use the equivalent fraction (4O3). Thus, 5% of 247 13s. 6d. = -fa of 247 13s. Qd. = 12 7s. Sd. 24. Find 3% of 384. 28. Find 4^ of 75 12s. Qd. 25. Find S% of 440 16s. 29. Find 10% of 37 8s. 9rf. 26. Find $% of 375. 80. 16s. is %\% of what ? 27. Find 2 of 64 16s. 81. 1 8s. 4d. is 4 of what ? 150 PERCENTAGE. [Art. 412. PROFIT AND LOSS. 412. Profit and Loss treats of the gains (profits) and losses which arise in business transactions. The profit or loss is always estimated on the cost price, or the amount invested. Discounts are reckoned on the marked or asking price. (See Art. 415.) 413. The difference between the cost of goods and the price at which they are sold is a profit or a loss, profit if the selling price is the greater, loss if the cost is the greater. EXAMPLES. 414. 1. A man purchased a horse for $250, and sold it at a gain of 16$. What was the gain ? (Gain = .16 x cost.) 2. A merchant sold goods that cost 1325 at an advance of 12$; what was the selling price ? (Gain = .12 x cost, and selling price = cost -f gain ; or, selling price = 1.12 x cost.) 3. Bought a farm for $3600, and sold it at an advance of 25$; what was the gain ? NOTE. If, as in the above example, the rate per cent, is an aliquot part of 100, it is more convenient to use the equivalent fraction (4O3). Thus, 25% = .25 = ; gain \ of cost. 4> Cloth is bought at $6 per yard, and sold at a loss of 20%. What is the selling price ? (Selling price = $ of cost.) 5. Bought a house for $3475 ; at what price must it be sold to gain 36$ ? 6. Purchased flour at $6.25 per barrel ; at what price must it be sold to gain 20$ ? 7. If I buy hats at $27^ per dozen, at what price must I sell them apiece to gain 33^$ ? 8. A factory which cost $8775 was sold at a gain of 16$. What was received for it ? 9. If silk costs $1.68 per yard, and is sold at an advance of 12-|% what is the profit per yard ? 10. A merchant purchased goods to the amount of $8735, and sold them at a loss of 12$ ; what was his loss ? 11. Bought 125 barrels of flour for $600. If sold at an advance of 15$, what was the profit per barrel ? Art. 414.] PROFIT AND LOSS. 151 12. A lot of dry goods was sold at an advance of 18%. If the gain was $436.50, what was the cost ? (Gain = .18 x cost ; hence, gain -f- .18 = cost.) 13. A farm was bought for $7200, and sold at a gain of $900 what was the gain per cent. ? (Gain = gain % x cost ; hence, gain % = gain -j- cost.) 14. A man paid for merchandise $875, and sold it for $1015 : what per cent, did he gain ? 15. A man paid for merchandise $1015, and sold it for $875 ; what per cent, did he lose ?. 16. Find the rate % of profit on goods bought for $324 and sold for $364.50. 17. A painting was sold for $2343, at a gain of 32% ; what was the cost? [Selling price = 1.32 (100% + 32%) x cost; hence, cost = selling price -?- 1.32.] 18. Find the cost of goods sold at an advance of 12-J-%, being a profit of $76. 19. How much was paid for a farm sold for $9878, at 12% below cost ? 20. What is the profit on iron cold for $4520, at an advance of 13% on cost ? 21. What is the selling price of tea which cost 32 cents per pound and is sold at a profit of 37% ? 22. Sold drugs for $168, at an advance of 75% ; what was the profit ? 23. A merchant sold for $2576 a lot of dry goods for which he paid $3360. What was the per cent, of loss ? 24. A mixture is made of 1 gallon of wine at 50 cents a gallon, 3 at 90 cents, 4 at $1.20, and 12 at 40 cents. What per cent, would be gained by selling the mixture at $1.60 a gallon ? 25. If, by selling tea at 47 J- cents per pound, 1 lose 5%, at what price must I sell it to gain 15% ? 26. If, by selling goods for $126, I lose 16%, what per cent, would I have lost or gained if I had sold them for $168? 27. A merchant's price is 25% above cost price. If he allows a customer a discount of 12% on his bill, what per cent, profit does he make ? 28. If cloth, when sold at a loss of 25%, brings $5 per yard, what would be the gain or loss per cent, if sold at $6.40 per yard? 152 PERCENTAGE. [Art. 414. 29. Goods that cost $168 are sold at an advance of 25% ; what is the selling price ? SO. At what price must ribbon be sold per yard so as to gain 20%, if 22 yards cost $6.75 ? 31. A merchant gave $25000 for seven houses. What per cent, does he gain by selling them at $7000 each ? 82. A woman buys a certain number of apples at the rate of 3 for 1 cent, and as many more at the rate of 2 for 1 cent. What per cent, does she gain or lose, if she sells them at the rate of 5 for 2 cents ? 38. Eggs are bought at 27 cents per dozen, and sold at the rate of 8 for 25 cents. What is the per cent, of profit ? 34. A merchant by selling goods for $364, loses 9%. For what ought they to be sold to gain 8% ? 35. A drover bought 160 sheep for $400, and sold f of them at $2.25 each. At what price must he sell the remainder so as to gain 10% on the whole ? 36. A merchant sells goods to a customer at a profit of 60%, but the buyer becoming bankrupt pays only 70^ on the dollar. What % does the merchant gain or lose by the sale ? 87. If a merchant adds to the cost price of his goods a profit of 12}%, what is the cost of an article which he sells for $7.20 ? 88. Sold a horse at a gain of 33J%, and with the proceeds pur- chased another horse, which I sold for $120, at a loss of 20%. What was the gain or loss ? 89. A merchant's retail price for boots is $4. 75 per pair, by which he makes a profit of 33J%. He sells to a wholesale customer at a discount of 20% from the retail price. What per cent, does he gain or lose, and what does he receive per pair ? 40. 40 head of cattle weighing 52770 pounds are purchased in Chicago at $4.80 per cwt., and are sold in New York at 10 \ cents per pound, to dress 56 pounds to the hundred- weight. What was the total cost ? The total selling price ? What is the gain per cent., making no allowance for transportation ? NOTE. The quantity bought or sold does not affect the gain or loss per cent. 41. A speculator sold two building lots for $4800 each. On one he gained 20%, and on the other he lost 20%. Did he gain or lose, and how much ? Art. 415.] DISCOUNTS. 153 DISCOUNTS. 415. It is customary in many branches of business for manu- facturers and dealers to have fixed price-lists of certain kinds of merchandise ; and when the value changes, instead of changing a long price-list, the rate of discount is changed. The fixed price is called the List Price, and the discount allowed the Trade Discount. Books are usually sold by publishers and jobbers at certain discounts from the retail prices. 416. Many kinds of merchandise are sold at " time" prices, subject to certain rates of discount if paid at an earlier period. 1. Thus, the following or similar announcements are usually found upon the bill-heads of wholesale dealers : " Terms, 4 months, or 30 days less 5%"; or, " Terms 60 days, or \% discount in 30 days, or 2% discount in 10 days. 5 ' 2. In the same business house, certain goods are sold on long credit, and others on short credit. 3. When no rate of discount has been offered, merchants are generally willing, when bills are paid before maturity, to deduct the interest on the amount of the bill for the remainder of the time at the legal rate per annum. Ex. The list-price of a scale is $80 ; what is the net price if a discount of 25% and 10% is allowed ? OPERATION. $80 List-price ANALYSIS. The first rate of discount is reckoned ~ ~ upon, and deducted from the list price, and the others '/> or ? . aie dg^uc^ f rom the successive remainders. 60 The result is not affected by the order in which the 6 10% or 1 . discounts are taken. A discount of 25% and 10% is the same as a discount of 10% and 25%. 54 Net-price. EXAMPLES. 417. 1. The gross amount of a bill of shoes is $82.68. What is the net amount, the rate of discount being 5%? (See Ex. 1, Art. 391.) 2. A stove is sold for $45 less 30%; required the net price ? NOTE. If the discount is not required, multiply by .70 (100% 30%); the product will be the net price. To multiply by .70, multiply by 7 and place the figures of the product one place to the right. 3. What is the value of 466 II. O.W. casing @ 45 cts. per pound, less 1^ per cent. ? 154 PERCENTAGE. [Art. 417. 4. The gross amount of a bill of mdse. is $100.36 ; what is the net amount, the rates of discount being 20% and 10$? 5. The gross amount of a bill of notions is $49. 75 ; what is the net amount, the rates of discount being 10$ and 10$? 6. What is the value of 12 pair shoes @ $1.60 per pair, less 5$? 7. What single discount is equivalent to a discount of 20$ and 10$ ? ANALYSIS. Represent the gross amount by 100 % OPERATION -j 00 (1.00). 20$ (|) of 100^ = 20$ (.20), which subtracted 'on on*-i from 100 # (1 ' 00) ' leaves 8 ^ ( ' 80) - 10 # ( A> f 8 ^ jfX** ~~* (.80)=8$ (.08), which subtracted from 80$ (.80), leaves .80 72$ (.72). 100$ -72 $=28$, the direct discount. Qg 1 of 80. By the following rule, a single discount can be cal- culated from two discounts mentally: From the sum of ** the discounts, subtract ^ of their product. The remainder . 28 will be the real discount. Thus, 20 % + 10$ =30$ . 30$ 2$ (20x10 -^100) =28$. When a third discount is given, combine it with the result obtained from the other two. When the sums of two or more discount series are the same, the series, in which the discounts are the most uniform, will produce the least single dis- count; and the series, in which the discount is most concentrated in one dis- count, will produce the greatest single discount. Thus, a discount of 10, 10, and 10$ is equivalent to 27 X V$ ; 20, 5, and 5 to 27f$ ; and 25, 2|, and 2$ to 28fg$. 8. What single discount is equivalent to a discount of 15$ and 10$? 45$ and 10$? 20$ and 12|$? 60$ and 10$? 75$ and 12 J$? 20$, 20$, and 10$? 60$, 20$, and 20$? 9. The net amount of a bill of goods is $74.20. What is the gross amount, the discount being 30$ ? 10. The net amount of a bill of files was $36.75; what was the gross amount, the rate of discount being 10$ ? 11. A is offered dress goods at 26 s cts. per yd., "4 months or less 6$ cash" ; how many yards can he purchase for $49.82 cash ? The net amount of a bill of hardware is $175.26 ; what is the gross amount, the rate of discount being 45$ and 10$ ? 13. What is the net value of one case prints containing 2273 yd., @ 4 3 cts., less 5$, cooperage 25 cts. ? 14. A bill of merchandise amounting to $442.38 was bought Aug. 18, 1879, on the following terms : "4 months or 5$ off 30 days." How much would settle the bill Sept. 16, 1879 ? 15. What is the net value of a bill of iron amounting to $1103.75, at a discount of 45, 10, and 2 per cent.? Art. 417.] DISCOUNTS. 155 16. What is the net value of 1 case prints containing 3039 2 yd. @ 5 cts. per yd., less a discount of 3%; cooperage $.25-? 17. What is the difference on a bill of $875 between a discount of 40% and a discount of 30% and 10% ? 18. A bill of tinware is sold at the following discounts: $74.20 at 20% and 10% ; $43.75 at 40% and 5% ; $69 at 33% and 10% ; and $49.17 net. What is the total net amount of the bill ? 19. A bill of dry goods amounting to $914.37 is sold, Aug. 19, on the following terms : " 60 days, or less 1% if paid in 30 days, or less 2% if paid in 10 days." How much would settle the bill Sept. 18 ? How much Aug. 27 ? 20. Of a bill of hardware, $61.51 are sold at a discount of 60 and 5%; $18.75 at a discount of 10%; $16.86 at a discount of 12 J%; $44.25 at a discount of 40 and 5%; $29.60 at a discount of 40, 12|, and 10% ; $28.04 at a discount of 55% ; $16 at a dis- count of 65, 10, and 10%; $18.70 at a discount of 50%; $19.75 at a discount of 20% ; $18.50 at a discount of 15% ; $307.55 at a discount of 75 and 12 J% ; $36.61 at a discount of 60 and 10%; and $218.25 net. What is the total net amount of the bill ? 21. Goods are bought at a discount of 30% from a list price, and sold at the list price. What is the gain per cent. ? ANALYSIS. Assuming $1 as the list price, the cost is 70c., selling price $1, and the gain 30c. 30c. is what % of 70e. ? 22. Books are purchased at a discount of 25% from the list price. What is the gain per cent, by selling at the list price ? 28. What per cent, is gained by selling pans at 21 cents apiece, that cost $2.56 per dozen less 20 and 12 \% ? 24. Plows are bought at a discount of 50% from the list price. What per cent, is gained by selling at the list price ? 25. A. merchant purchases goods at a discount of 25% from the list price. What per cent, is gained by selling at the list price ? What per cent, if goods are purchased at a discount of 33J% ? 35%? 25% and 5% ? 20% and 12J% ? 15% and 10% ? 26. A merchant buys goods at a discount of 40 and 20% from the list price, and sells at a discount of 30 and 10%. What is the gain per cent. ? ANALYSIS. Assume $1 as the list price, find the net cost and selling prices, and then the gain per cent. 156 PERCENTAGE. [Art. 417, 27. If a merchant buys goods at a certain price 10 and 5 off, and sells them at the same price, 5 off, what per cent, profit does he make ? 28. What per cent, profit does a merchant make who buys at a discount of 20, 10, and 12-|-$, and sells at the list price ? 29. What must be the marked price of goods costing $32, that I may deduct 20$ from it, and still gain 25$ on the cost ? ANALYSIS. First find the selling price, and then the marked or list price. If the cost is $32, the gain will be 25$ () of $32, or $8; and the selling price will be $32 + $8, or $40. If the goods are sold at a discount of 20$, the sell- ing or net price is 80$ of the marked price. $40 = .80 of $50. 80. What must be the asking price for books that cost $1.60, in order to abate 20$, and still make a profit of 25$ ? 31. What must be the list price of goods that cost $18, in order to make a profit of 33-J-$, if they are sold at a discount of ?. Find the list price of goods that cost $75 and are sold at a Discount of 60 and 10$, at a profit of 20$. 33. A manufacturer sells his goods at a discount of 30 and and thereby gains 12$. What is the list price, if the cost is $28 ? 84. A hardware dealer sells certain goods at a discount of 75 and 12J$, and gains 20$. What is the list price, if the cost is $2.80 ? 35. What per cent, must be added to cost price in order to give a discount of 25$, and make a profit of 20$ ? ANALYSIS. Assuming 100$ as the cost price, the selling price is 120$ of the cost. 120$ is 75$ (100^-25$), or f, of 160$. 160$ 100$ = 60^, the per cent, to be added to the cost price. 2, vi N^^S. What advance on cost would be necessary in order to give a discount of 20$, and still make a profit of 20$ ? 87. At what per cent, above cost must goods be marked, so that when sold at a discount of 5$, there would be a profit of 25$ ? 38. If goods are bought at a discount of 2 10's and a 5 from a manufacturer's list price, and sold at a discount of 12$ (^), what is the gain per cent. ? I purchase books at $2 each less 33^$, and 5$ for cash. What is the net cost, and what per cent, discount may be given on the list price to produce a net profit of 10$ ? Art. 418.] BILLS. 15? BILLS.* 418. A Bill is a detailed statement of merchandise sold, or of services rendered. Bills of merchandise state the place and date of the sale, the names of the buyer and seller, the terms of the sale, the quantity, price, and distinguishing marks and num- bers of the merchandise, and other details. The terms Bill and Invoice are used by many interchangeably. The term Invoice is applied more particularly to statements rendered by consignees to commission merchants, showing marks, numbers, values, and accrued charges of goods shipped; to bills rendered to jobbers; and to bills received from for- eign countries. EXAM PLES. 419. Copy and extend the following bills : Messrs. WM. DOLTON & Co., (1. Canned Goods.) WILMINGTON, DEL., Nov. 16, 1889. Bought of JAMES MORROW & SON. Cases. 2 1 1 2 1 1 Messr Int Doz. 3 Ib. Peaches - * - - - - Saco Corn 321* 180 400 400 BUFFAL 3CHOELI allow no 9 * * * * 00 ** ** ** ** 50 $** ** 4 2 2 4 2 2 s. DA erest ct 3 ' Tomatoes, B. & L. - - - 2 ' Col. Pears 24 ' Apricots ------ Ctg. - B Price per dozen. (2. Flour.) KIEL GROUSE & SONS, Bought of i arged on all accounts after 30 days. We o, N. Y., Dec. 6, 1888. KOPP & MATTHEWS. Expressage or Exchange. 20 25 25 25 15 5 Bbls. Flour " Sunlight " Sacks - Bbls. - "Victor" Sacks - Bbls. - "Dakota" Sacks - ' "Superior" Sacks - 2177 Ih S Mpfll $7.05 7.25 6.05 6.25 5.30 8.55 1.20* .56 b Be. per bt *** *** *** *** ** ** ** *** ** ** ** ** *# ** *** ** 20 bags 70 " 264& bu. Oats $1 .90 per hundredweight. b 5 she!. * It is suggested that a part of these bills bo reserved for review. One or two of them may be given each week as a general exercise. 158 PERCENTAGE. [Art, 419. (3. "Window Glass.) PITTSBURGH, May 14, 18S8. EUREKA GLASS Co., Bought of CUNNINGHAMS & Co. Terms SO days. If not promptly paid, interest will be charged from date of bill. 2 Bx's7x9 "A" 7^ *# 1 ^ 8v 10 750 *** ** 2 9x12 7^ ** 1 10x14 7^ * ** 19 v 9ft 850 ** 2 13x36 10^ ** ** Uv 90 fts o ** 1 ^ v SO Q7A ** ** *** Less 60 and 20^ - - - *** ** (4. Provisions.) CLEVELAND, 0., Oct. 9, 18S6. Messrs. L. C. MAGAW & SON, Bought of J. P. ROBISON & Co. Terms Net Cash. No goods sold on 30 days. 10 "Rhl CJ M Pnrlr 1 7 *** " Mess Beef - 10 76 ** ** 5 3 1 1 " Hams 90M376 b -98 ****< 14^ " Shoulders 58 744 -57 *** 9^ " Dr. Beef 33 241 -22 *** 14^ Tc. Lard 406 -60 *** ll/- *** ** ** ** ** ** ** ** *** ** Number of pieces. b Gross weight. c Tare, or weight of barrel or tierce. d Net weight. (5. Fish.) GLOUCESTER, MASS., Sept. 28, 1886. Messrs. DANIEL WEIDMAN & Co., Bought of CLARK & SOMES. Subject to sight draft without notice after thirty days. Htl UPW ftpn Prvl ^ 7^ ** ~ 1 10 10 2 10 5 3 Bbl. Ex. * 1 Mackerel 20.00 Kits 15 Ibs. Ex. *1 Mackerel - - - 1.80 " 20 Ibs. Bay* 1 " - - - 1.80 Bbls.2Shore " lg. - - 12.00 Kits 20 Ibs. ft 2 Shore " " - - 1.50 Halfs New Labrador Herring - - - 3.82 " Round Shore " - - - 2.95 Box ' 88 , ctg. in Boston <8 #* ## #* * -^ ** #* $*** /Art. 419.] r Day Book, 115-797. Messrs. EDWARDS & Co., BILLS. (6. Groceries.) 159 NEW YORK, Feb. 1, 1889. Bought of H. K. & F. B. THURBER & Co. M 14385 1 Cask Old Prunes 1544 - 134 = **** Ibs. - ** ** 3 Boxes Old Muscatel Raisins - _ _ _ , * *# 3 " New " " _ _ QIO * ** 4 " Layer " - _ _ 16 * ** 1 " Cream Tartar, foil - - 20 Ibs. - .39 * ** 2 " Yeast-Cakes, 3 doz. ea., - 6 doz. - .65 ** 25 IT, "\7_7V* j-vl rt 7}r\v\wrii .16 * 10 " Nutmegs fll - - - - _ _ _ . 100 ## 1 Box O. K. Mustard, 1's - - - 12 Ibs. - .25 * 1 i's - - -12 " - .25 * Cartage on all - - 1 **# ** 1st item "M $4385" is mark and number upon the cask ; 1544, gross wt. ; 134, tare or weight of cask. 5th item \ foil, put up in Ib. packages and wrapped in tin foil. (7. Groceries.) Messrs. HORTON, CRARY & Co., NEW YORK, Aug. 13, 1886. Bought of AUSTIN, NICHOLS & Co. W. B 1 100 00 QA 56 A*C99 A. 1 -20 K ______ Ld& fwO 131 21|- ou ** ** 1 Bbl.' 85 Roa. Java Coffee ^ - 100 25| ** ** 2 " -so Ri u 11222 221 10920 42 " *** 24 ** ** H. R. P. 1 Case Cone. Lye - _ 5 50 Union. 2 Boxes Yeast Cakes, ea. 3 - - - - * 65 * X* 25 Ibs. Spice, B Bcr 900 -IKl * *# - 10^ 5 214 26 * ** A. N. & Co. 1 Keg Gr. Mustard ______ 50 35 #-* ** 10 lh \^hifp f-i-lnp 40 * iv-S" J.I/ 25720 #*** 26920 ** A. N. & Co. 5 Bbls. X. C. Sugar _ .25621 25318 25320 ^SHt 1 ^ 11^ &#& ** 8134 1 " W. D. Syrup 47 *** 60^ ** ** 1114 1 " C. D. " 45* *** 50 ** ** Ctg.^- - - 1 50 $*** XX The small figures at the right of the words " bag " and " bbl" are the prices of the same. 3rd item 121 Ibs., gross wt., 21 Ibs. tare, 100 Ibs. net wt. 4th item 112 and 109, gross weights; 22 and 20, tare; 221, total gross weight; 42, total tare. 12th item ^, gallon allowance for leakage. 160 PERCENTA GE. [Art. 419. (8. Dry Goods.) NEW YORK, March 20, 1889. Messrs. MARSHAL FIELD & Co., Sought of H. B. CLAFLIN & Co. Terms Cash in 30 days less 5#, or 4 months' note delivered within 30 days, and payable at Bank in New York exchange. 2875 8039 3369 1290 1590 2179 2507 6515 2985 1650 Bale Boott M. Brown " Continental C. do. " Pequot A. 36 in. " Great Falls E. - " Atlantic H. - - - 1038 -.07 3 " Boott F. F. " Pepperell 600 Drill - - - Case Blaokstone A. A. - - - - " Dwight Anchor ----- " Great Falls Q. " Pearl River Ticking - - - Cooperage 800 800 967 1111 ** 800 622 1649 1139 1492 708 7 1 8 15 2 54 ** -><# #* *#* *** #** ** ** How L would settle the above bill April 19, 1889 ? (9. Dry Goods.) NEW YORK, March S3, 1888. Messrs. DAVIDGE, LANDFIELD & Co., Bought of TEFFT, WELLER & Co. yd. c. 2 Naumkeag Bl. Jean - - 47 - - - 95 9 8 55 4 Roll Cambric - - - - J5 J5 - " **** 5 2 * ** 3 47 3 Pepperell Drill - - - - ao 3 - - - #*#* 8 * ** 443 1 T ir* \i7oll 1 / "Rtvuvn 38 14 2 * ** 40 Q *** 72 * o 40 1 45 3 45 5 New Market N. - - - 45* 58 l - **** 6 1 ** ** 46 2 2 *** 9 * ** 2 Otis B. B. Dk Stripe -. - Jjf - - . . *** 10 ** ** 1 Hamilton 30 in. Tick 48 3 II 2 * ** 2 Thorndyke C ---- 583 --- **** 8 2 *# ** 2 Wamsutta C. Blea. - - jjgl - - - **** 12 ** ** 8 Anr1rnT, 52 52 49 51 ' Anaros LI. - 512 513 51 522 *** 73 ** ** 1 Pepperell 10 / 4 36 3 22 * ** *** ** 1st item 2 pieces Naumkeag Bleached Jean containing 48 and 47 yards respectively ; total, 95 yards at 9 cents per yard. Art. 419.] BILLS. 161 (10. Dry G-oods.) Book 174, Page 14$. NEW YORK, March 30, 188S. Mr. JAMES MORGAN, Milwaukee, Wis. Sought of H. B. CLAFLIN & Co. Terms : Net 6O Days, or \% discount in 30 days, or 2# ) discount in 10 days, N. Y. Funds. No Exchange allowed, f $4641 53 PC'S Gordon Prints (Job) 21 2 48 2 38 40 1 48 2 48 3 37* 48 48 44 49 2 44 3 48 2 49 2 49 3 49 2 42 56 48 2 49 1 28 2 49 1 49 48 3 49 1 28 48 s 37 33* 49 2 52 33 3 40 48 49 1 49 1 24 48 2 48 2 52 48 3 49 47 2 48 1 48 2 49 1 49 2 48 3 48 2 48 2 43 2 49 1 49* - ***** S2601 54 PC'S Do. 48 3 48 49 42 22 1 49 1 49 48 2 53 2 4g2 473 4 8 a 48 2 49 44 49 492 432 49 2 49 49 48 2 47 3 47 48 2 49 1 56 50 2 49 1 41 1 48 1 50 27 1 49 48 2 48 3 21 3 29 1 51 3 46 3 48 2 48 2 28 2 48 2 49 1 49 2 45 2 47 48 2 40 2 50 1 39 2 48 2 46 1 ***** $4765 61 PC'S Do. 30 2 49 2 42 49 2 32 48 46 48 2 46 2 42 3 47 2 22 1 33 46 48 49 2 48 2 48 42 42 48 28 48 1 49 2 48 2 49 49 49 2 48 2 28 2 49 2 43 49 1 48 2 49 2 48 38 2 29 25 26 3 49 1 49 3 49 1 49 48 2 343 433 45 49 491 492 431 36 4g 29 2 49 3 48 2 31 1 48 2 49 48 1 - - ***** ***** .04 2 *** ** How much would settle the above bill April 8, 1888? How much April 28, 1888? (11. Dry G-oods.) NEW YORK, March 20, 1888. Messrs. JORDAN, MARSH & Co. Bought of A. T. STEWART & Co. Job. 8 Cases Gordon Fancy J. U. S 4561 2810 S. B. R. 4157 2902 1 H. Z. 3473 278T 2 S. J. L. 4224 2880 2 G. Q. 2777 2821 1 J.B. 3504 2842 2 J. Z. 3970 2883 1 J. H. 4198 2863 1 - - ****** .05 **** ** Less % - ** ** **** ** 1st column, distinguishing number of each case. 2d column, number of yards in the several cases. 162 PERCENTAGE. [Art. 419, (12. Hosiery.) Claims for Damages or Errors must be made on receipt of Goods. ^EW YORK, June 28, 1880. Messrs. JOHN FORD, SONS & Co., Sought of JAMES TALCOTT. Net SO Days. Note to your own order payable at a Bank in New York City. 1789 35 Doz. 3458 Mixed > Hose - - .80 28 25 2032 Fancy " - - - .80 ** 12 853 Col'd - - - 1.00 ** 12 1691 Fancy - - - 1.00 #* 18 1759 " ... .75 ** ** 20 1713 " ... i.oo *# 16 1716 " - - - 1.10 ** ** 6 3438 Fch. mx. % ... .90 * ** 22 Job Misses - - - .75 ** ** |*** ** Shipped per P.R.E. & C.B. & Q.R.R. Number on margin (1789), number of case. Numbers 3458, 2032, eie. t manufacturer's distinguishing numbers (stock numbers). Mr. JOHN BERWOLD, Terms Cash. (13. Books.) CHICAGO, ILL., May 7, 1878. Sought of HADLEY BROS. 12 18 24 36 Randall's Arithmetics, Part 1 - .60 " 2 - .50 Smith's Primers (paper) - - - .06 QrplW<5 99 7 * * 20 ** ** 18 12 A 2d Readers .45 3d " ----- .70 4th " 11^ * * * ** ** i'3- 6 6 5th " ..... 1.35 Doz. Brown's Copy Books - - - 1.80 * ** ** ** Less m%% ** ** ** ** ** *< 6 6 6 6 Jones' Geographies * 1 - - - - .35 2 - - - - .63 " " 3 - - - - 1.10 4 - - - - 2.00 2 * * ** 10 ** ** Less 25^ - ** * ** ** ** ** 3 3 Boxes Chalk Crayons - - - - .18 Doz. Blank Copy Books - - .50 * ** ** $** ** Art. 419.] BILLS. 163 (14. Hardware.) PHILADELPHIA, PA., Aug. IS, 1889. Messrs. N. RUTTER, Sox & Co., Bought of BIDDLE HARDWARE Co. Terms 60 days. 24 Sets Wd Wh'l Bed Casters ft 1 2 in. - .18 * ** 50% - - * ** 1 Doz. Russell's S.B.Knives 14 in. ft 1540 - 11 2.40 2.55 3.15 3.20 200 Carriage Bolts % x 1 2^ 5% 5^ * *# 5.95 6.25 6.50 6.85 100 ** ** 7.15 7.45 100 ** ** 7.90 8.05 100 " % x 8^ 8% - - ** ** 7.25 7.75 9.25 100 " M x 2 2J 4 - - - *# ** 11.35 11.75 13.25 100 " ^ x 6 6J 8 - - - ** ** 75 & 12i% - *** ** M ** Y2 C. Machine Bolts % x 8 8.70 * ** 15.10 16.60 Y* " % x 6 7 - - - ** * 99 Ibs. " . " % x 11 .10% ** ** ** ** 60 & 10$ - ** ** ** ** 3rd item 200 bolts of each of the following sizes: \ in. thick x 1 in. long, in. thick x 2 long, Vit. thick x 5 long, ^ *n. thick x 5^ in. long. The numbers 2.40, 2.55, 3.15, and 3.20 represent the prices per hundred of the several sizes. (15. Watches and Jewelry.) NEW YORK, Mar. 7, 1887. Mr. CHARLES BABCOCK, Bought of A. S. GARDNER & Co. Terms: Net Cash 4 months, or less 5# 30 days, with Exchange on New York. H 658 1 18 k. Ancre full Engrd. &Enld. S. W. 90 20422 1 14 k. Russell flat C. B. 46 50 1 18 k. Plain Ring 3% dwts. - - - 1^ * #* 2 14 k. Guards with slides ?^, - l 1 ^- 222 208 #*# # 1 Pr. Solid Roman SI. Buttons 908 - - 10 50 **# *i* 1 i How much would settle the above bill, Apr. 2, 1887? The letters and numbers on the margin refer to the numbers of tho watches. 4th item numbers 222 and 208 refer to the style numbers (stock- numbers) of the guards (chains), and the numbers above (37f and -56) express the weights in pennyweights ; $1.15 per pennyweight. 164 PERCENT A GE. [Art. 419 (16. Tinware.) ROCHESTER, N. Y., Oct. 16, 1889. Messrs. MCCARTHY & REDFIELD, Bought of JOHN H. HILL. Terms 60 days. If paid in 10 days 2 per cent, discount. 2 Doz. ft 21 Pieced Dish Pans - - 8.25 ** ** % " 9 in. Wash Boilers - - - 36.00 ** 3 " Pieced Bread Pans 3x9x3- 2.00 # 3 "5x9x2- 2.00 * 3 " ft 13 Pieced Cups - - - - .90 * ## 2 " ft 25 Dippers ... - 1.75 * ** 6 Nests ft 021 Flaring Pis & Dippers 1 ,14 * ** ** ** 20 & 12%$ - *# ** ** ** 1 Doz. Champion Nutmeg Graters 1 75 1 " Nests ft 4 Fancy Cov'd Pails 6 00 1 " ft 4 Burnished Tea Pots - - 6 75 85 ft UK* - #* ** ** 2 Doz. ft 10 Pudding Pans - - - 4.25 " ft 200 Pressed Kettles - - 5.50 * * ** 37%% - * ** ** ** .76 .90 6 Enameled Kettles Ea. 45 qt. - * ** 1.10 1.30 12 " 68 qt. - ** ** 60% - ** #* ** ** ** N.Y.C. & H.R.R.R. 975 Ibs. @ \W -X-* ** What amount would be due on the above bill Oct. 26, 1889? (17. Wooden "Ware.) CHICAGO, July. 9, 1887. Messrs. OLIVER & BACON, Bought of JAMES S. BARRON & Co. Terms Cash, with exchange on Chicago or New York. 6 Oak Churns ft 1 1.60 * ** Less 20% - * #* * ** 2 Doz. 1 bu. Corn Baskets - - - 4.00 * " Potato Mashers - - - - 1.00 " 6 ft. Ladders 4 * 1 "8ft. " 5 4 jo ft " /> ** Less 50 & 10% - * ** * ** 2 " ft 10 Shoe Brushes - 2.25 * ** ** ** Art. 419.] BILLS. , '16o NOTE. In the preparation of bills in the usual form, from the following items, use your own name as the seller, the name of your teacher as the buyer, and the present date and place. '18. \gro. Table Knives and Forks @ $8.40. T V doz. Cheese Knives @ $9.60. | doz. Razors each #100 $9, #101 $10, #102 $10.50. \ doz. Pocket Knives each #337 $6, #427 $7.50, #204 $3.75. 6 sets Champion Irons @ $1.50. 1 doz. Tacks each #1 2#6'., #2 22c. 9 #3 25c., #4 21c. f doz. Panel Saws @ $20 less 20$. Box and drayage, 75/. Eice 40 19, 229 20, @ 6|c. 5 iJ7. "A" Sugar 319, 306, 288, 319, 306 II. @ 9|c. 5 JJ/. Yellow C Sugar 314-19, 319-20, 329 20, 311 21, 32819, @ 8}c. 1 W?. Cut Loaf Sugar 236-20 @ 10|c. Cartage, $1.25. (See Ex. 7, llth item.) 28. 8 lengths 8" Drive Pipe 129' 10" @ $2,25 (per foot) ; 44 lengths 5|" Casing 801' 8" @ 70c.; 233 lengths 2" 0. W. Tubing 4507' @ 216-., less 7J- and 2$. Less freight 29400 ^. @ 236'. per cwt. 24. 12 Pr. Women's Grain B't. #443 Shoes @ $1.25. 12 Pr. TTos. Kid B't. #407 @ $1.50. 12 Pr. Wos. Kid P7. #406 @ $1.75. 12 Pr. Misses Kid B't. #301 @ $1.50. 12 Pr. Misses Goat B't. #302 @ $1.60. 12 Pr. Wos. Goat 5V. #428 @ $1.75. 12 Pr. Children's Goat PV. #200 @ $1.30. 12 Pr. Oh. Grain 57. #202 @ $1.20. 12 Pr. Oh. Glove Kid 57. #222 @ $1.10. 1G6 PERCENTAGE. [Art. 42 O. COMMISSION AND BROKERAGE. 420. Commission or Brokerage is an allowance made to an agent for transacting business for another ; as, the sale or purchase of property, the collection or investment of money, etc. An additional percentage is usually charged by commission merchants for guaranteeing the payment of sales made on credit. 421. The party who transacts the business is called a Com- mission Merchant, or Broker ; and the one for whom he acts is called the Principal. NOTES. 1. Commission Merchants usually have possession of the subject- matter of the negotiation, and make sales and purchases in their own name. 2. Brokers do not have possession of the merchandise bought or sold, and generally make contracts in the name of those who employ them and not in their own. They simply effect bargains and contracts. The name broker is often erroneously applied to dealers in stocks, bonds, etc., who buy and sell on their own account only. 4:22. A Consignment is a quantity of merchandise sent by one party to another. The party who sends it is called the Con- signor; and the party to whom it is sent, the Consignee. 4:23. The Net Proceeds of a consignment is the balance due the consignor after all charges or expenses have been deducted. The whole amount realized from a sale is called the gross proceeds. Th-- commission is usually a certain per cent, of this amount. 424. An Account Sales is a detailed statement rendered by the Commission Merchant to the Consignor, showing the sales of certain goods, the charges or expenses attending the same, and the difference or net proceeds. The charges embrace freight, cartage, inspection, advertising, storage, insurance, commission and guarantee, etc. 425. An Account Purchase is a detailed statement rendered by the Commission Merchant to his Principal, showing the cost of certain goods bought, and the charges or expenses attending the purchase. 426. Commission or brokerage is usually computed afc a cer- tain per cent, of the amount realized or invested, or of the amount Art. 427.] COMMISSION AND BROKERAGE. 107 involved in the transaction. In such cases the general principles of percentage are applied. NOTES. 1. In buying and selling stocks, bonds, etc., the par value, and not the actual value, is taken as the base. 2. The commission for buying and selling some kinds of merchandise is usually computed at a certain price per unit of weight or measurement ; as, grain per bushel, cotton per bale, etc. EXAMPLES. 427. 1. A commission merchant sold goods to the amount of $864 ; what was his commission at 2 (J- of 10) % ? 2. A salesman sells goods at a commission of 2% ; what must be his sales, that he may have a yearly income of $5000 ? 3. What is the brokerage for selling 850 bales of cotton at the rate of $25 per 100 bales ? 4. A lawyer collected a note of $2375 ; how much did he pay to the owner of the note, his commission being 5% ? 5. My agent in Chicago purchases for me 600 barrels of flour lit $3.75 per barrel ; how much do I owe him, his commission for purchasing being 2% ? 6. An officer collected $17850, and deposited $17493 in the Treasury, retaining the remainder as his commission. What was the rate per cent, of the commission ? 7. Sent to a commission merchant in Toledo $2080.80 to in- vest in flour, his commission being 2% on the amount expended ; how many barrels of flour would be purchased at $4.25 per barrel ? 8. A commission merchant sells merchandise amounting to $3325 ; how much is paid to the consignor of the merchandise, the charges being, for transportation $117.50, for advertising $10, for storage $15, for commission 2% ? 9. My agent in Chicago buys for me 1187.76 centals wheat at $2.123 per cental. What is his commission at per cent. ? 10. A commission merchant purchased for me 9-S-& bushels of clover seed at $8.55 per bushel. How much should I send to him in settlement, if his commission for purchasing is 1 per cent. ? 11. A broker buys 8375 pounds of leather at 26 cents per pound. What is his brokerage at j-%, and what is the net amount received by the seller, the brokerage being paid by him ? 12. A freight broker procures transportation for 375 tons of merchandise at $3.50 per ton ; what is his brokerage at $% ? 1G8 PERCENTAGE. [Art. 427. 13. A collector deposits $28117, retaining 3% on the whole amount collected. What amount did he collect and what was his commission ? 14. A lawyer, collecting a note at a commission of 5% thereon, received $6.25 ; what was the face of the note ? 15. An agent sold 6 mowing-machines at $120 each, and 12 at $140 each. He paid for transportation $72, and, after deducting his commission, remitted $2208 to the manufacturer. What was the % of his commission ? 16. A merchant instructs his agent in Cincinnati to huy pork to the amount of $5000. The charges on the pork being $16, and the agent's commission 1$, how much must be remitted to settle the bill ? 17. What are the net proceeds of the sale of 12372 pounds of leather at 22 cents per pound, the charges being $31, and a com- mission of 2J$ being paid for selling and 2J% for guaranteeing payment ? 18. A real estate agent, who charged 2J% for making the sale, paid to the owner of a house and lot $42412.50; what was the value of the property ? 19. A commission merchant sells 240 bbl. of potatoes at $3.75 per 1)1)1., and 260 bbl. at $3.60 per libl. How much is due the con- signor, the commission being 12J cents per barrel ? 20. John Smith is a disbursing agent of the United States. Jan, 1, 1880, there is in his hands $11870.63. Feb. 1, he pays out $3220.34, on which he is entitled to a commission of \\%. Mar. 1, he receives $3750.87. May 1, he pays out $3795.01, on which he is entitled to a commission of 2J%. Make a statement of his account, showing balance due the United States. 21. A lawyer collected 75% of an account of $3416, charging 5% commission. What amount should he pay over ? 22. A, having a claim against the government of $10970, agreed to pay an agent 8 per cent, of the amount collected. The amount collected was 22 per cent, less than the amount of the claim. How much was received by A ? 23. B sends $2240. 70 to his agent in Cleveland, requesting him to invest in provisions after deducting his commission of %% for purchasing ; what was the sum invested ? 2Jf.. A broker received $62.50 for selling some bonds, charging \% brokerage. What was the par value of the bonds ? Art. 427.] COMMISSION AND BROKERAGE. 169 Copy the following account, and make the necessary exten- sions, etc. (25. Account Sales.) NEW YORK, Oct. 19, 1889. Sold for account of A. W. RANDOLPH & Co., By DAVID Dows & Co. 1880. Sept. (i H Oct. t< 12 18 30 14 18 100 Bbls. "Sunshine"- - - - 5.75 125 " " Pride of the West "- 6.25 150 " "Sunshine"- - - - 6. 75 " ' 'Pride of the West "- 6.50 50 " " " - 6.60 *** *** #** *** *** ** ** **** ** Sept. Oct. 10 10 19 1Q Charges. Transportation 500 Bbls. @ 27^ - - - Cartage 400 " @ 5^- - - Storage 400 " @ 3j*- - - *** ** ** # ** 19 Commission and Guarantee 5% - - - *## ** *** ** Net proceeds ##** ** 26. According to the above form, prepare an Account Sales of 10 III. Yellow C Sugar, 3031 Ib. 8c.; 10 bbl Standard A Sugar, 2957 Ib. @ 9^.; 10 bbl Soft A Sugar, 2839 Ib. @ 8|c.; 2 Tc. Lard, 713 Ib. @ 9|-c.; 1 Tc. Rice, 608 Ib. @ Ic. Charges as follows : Cooperage, $1.80 ; Cartage, $3.60 ; Commission, 1J^. Present date and place ; Student & Co., commission merchants ; and sold for account of Teacher & Co. 27. If an agent's commission is $145.20, when he sells $5808 worth of goods, how much would it be when he sells $7416 worth ? 28. A creditor receives on a debt of $1725, a dividend of 60%, on which he allows his attorney b%. He receives a further divi- dend of 25%, on which he allows his attorney 6%. What is the net amount that he receives ? 29. A gentleman left a sum of money to be divided equally among 7 persons, subject to an inheritance tax of 5%, which caused a deduction of $364 from the whole amount. What did each receive ? 30. An agent's commission for the month was $128.40. If his sales had been $864 more, his commission would have been $150. Find the amount of his sales. 31. A man allows his agent 5% on his gross rentals, and receives a net rental of $3488.40. If the gross rental is 6% of the value of the property, what is the value of the property ? ^ c &* PERCENTAGE. [Art. 427. (32. Account Purchase.) i TOLEDO, 0., Mar. 6, ltf$7. Purchased by A. L. BACKUS & SONS, For account and risk of L. A. & W. B. SHAW. 9 227 9JJL 9JLL Bags "Montauk" .21 Bu. Mammoth Clover Seed - - 9 9 " Clover Seed 8^ * ** ** ** #* ** 25 * H Charges. * Commission \% ------ Charge your % ------ *** ** NOTE. The small figures at the left represent pounds. See Art. 53S. 33. According to the above form, prepare an Account Purchase of 3 Half-Chests Gunpowder Tea, 165 Ib. @ 35c.; 2 Hfc. Oolong Tea, 86 Ib. @ 20e.; 20 bags Rio Coffee, 2388 Ib. @ 13c.; 2 mats Java Coffee, 133 Ib. @ 19|-c.; 1 Hhd. P. R. Molasses, 143 yal (, divide by 4 ; at 2f , divide by 3 ; at 5%, subtract ; at 7$, add ; at 8$, add i ; at 10$, divide by 6, and multiply by 10 by placing the point one place to the right ; at any per cent., divide by six (second time) and multiply by the rate. 3. If the principal is a multiple of the divisor (6 in the model example), time can be saved by performing the division first. Thus, to find the interest of $1200 for 113 days, divide 1200 by 6, and multiply the quotient 200 by 113, producing 22600. By pointing off three places, the required interest is $22.60. 4. When the time is expressed in years, months, and days, reduce it to days by regarding each year as 360 days and each month as 30 days. EXAMPLES. 445. Find the interest of > {i wil1 * * of $ 8 - 64 ' or $2.16. Hence the interest for the given time will be $95.04 plus $2.16, or $97.20. Ex. 3. What is the interest of $375.60 for 55 days, at % ? ' .. 7 ANALYSIS. 55 days = 60 days Jess $3 75.60 = int. for 60 da. 5 dayg> The inter j t for 6Q d y ays fa _^ " $3.756, for 5 days ( T V of 60), $.313, and 3 | 443 = " " 55 " for 55 davs > $3.443 ($3.756 - $.313). 454. Aliquot Parts of 60. 1 = V 5 2 = ^ ; 3 = ^ ; 4 = ^; 5 = T V; 6=. T V; 10 = i; 12 = i ; 15 = i; 20 = i ; 30 = i. NOTES. 1. To divide by 10, place the figures of the basis one place to the right. 2. To divide by 20, 30, or 60, divide by the first figure and write the quotient figures one place to the right. 455. Ex. 1. What is the interest of $976 for 26 days, at 6^? OPERATION ANALYSIS. As 26 is not an aliquot 76 = int. for 60 da. part (388) of 60, take 20, which is $ of pj-q __ (( (f ^7> 7 60, and 6, which is -^ of 60. Divide the basis which is the interest for 60 da. by 3 9 ? 6 ' _? cla ' to find the interest for 20 da. ($3.253) ; and 229 = ({ " 26 da. the same sum by 10, to find the interest for 6 da. ($.976). (See Art. 454, 1.) The sum of these two results will be the interest for 26 days. 180 INTEREST. [Art. 455. Ex. 2. Find the interest of $732.80 for 2 yr. 8 mo. 27 da., at 6% ? OPERATION. ANALYSIS. The interest 32.8 = int. for 2 mo., or 60 da. for 2 mo., forming the basis, 16 is $7.328. Multiply this by 16 to find the interest for 32 mo. (2 yr. 8 mo.}. 27 = 15 + 12. To find the interest for 832 = int. for 15 da. 15 da., divide the basis by 4 466 = " " 12 da. (15 = of 60) ; and the same sum by 5 to find the interest 546 -.= required int. for ia da< By adding these results, we have the interest for the given time at Qf c . 456. If the number of days given is not an aliquot part of 60, it will need to be so separated that the component parts will be aliquot parts of 60. 43 73 1 1 Numbers not aliquot parts of 60, with best divisions : 7 = 6 + 1; 8 = 6 + 2; 9 = 6 + 3; 11=6 + 5, or 10 + 1; 13=10 + 3; 14 = 12 + 2; 16 = 10 + 6; 17 = 12 + 5, or 15 + 2; 18 = 12 + 6. (The interest for 18 days may be found by multiplying the basis by 3, and placing the figures of the product one place to the right); 19 = 15 + 4, or 10 + 6 + 3; 21 = 15 + 6; 22 = 20 + 2(2 = T V of 20); 23 = 20 + 3 ; 24 = 12 + 12 (or multiply by 4 and place the figures of the product one place to the right); 25 = 20 + 5 (5 = | of 20); 26 = 20 + 6 ; 27 = 15 + 12 ; 28 = 12 + 12 + 4 (4 = i of 12), or 20 + 6 + 2; 29 = 12 + 12 + 5, or 20 + 6 + 3. 457. RULE. Draw a perpendicular line two places to the left of the decimal point ; the result will be the interest at 6% for 2 months, or 60 days, the dollars being on the left, and the cents on the right of this line. Multiply this result by one-half the total number of months. To this product, add that proportion of the interest for 60 days, which the given number of days is of 60. XOTE. The interest at any other rate per cent, may be found as in Art. 448. 458. The interest at 6% may be found for 6 days by placing the point three places to the left (453). In many examples, when the time is less than 100 days, the process is shortened by taking as the basis the interest for 6 days instead of 60 days. Ex. Find the interest of $375 for 8 days at 6#. $425 for 79 days. $500 for 47 days. Art. 458.] 1ST OPERATION. 375 6 da. 125 2 da. 8 da. 500 INTER E ST. $ 2ND OPERATION. 425 6 da. 525 78 da. (13x6) 071 1 da. i $ ED OPERATION. 500 6 da. 5 5 4 000 083 48 da. 1 da. 596 79 da. 3 917 47 da. EXAM PLES. 459. Find the interest of the following at 6$. (See Arts. 453 and 454.) 1. $864 for 60 $396 for 20 $290 for 72 $785 for 66 $636 for 62 $400 for 90 $525 for 61 $600 for 10 $728 for 65 $340 for 15 2. 3. 4. 5. 6. 7. 8. 9. 10. days, days, days, days, days, days, days, days, days, days. 11. 12. 13. 14- 15. 16. 17. 18. 19. 20. What is the interest of r 21. $375.60 for 8 mo. 20 da., at rt. 453.) 22. $1727 for 7 mo. -tfrfla., at ^ ? At 23. $449.38 for 1 yr. 4 mo. 12 da., at 24. $285 for 1 yr. 5 mo. 10 da., at 25. $432.65 for 2 yr. 2 mo. 6 da., 26. $1235 for 2 yr. 5 mo. 5 da., at'M ? #7. $445.25 for 5 mo. 4 o^., at 6^:? At 9$ 0. $2440.50 for 97 days, at 6/? At 1% '? 455.) #0. $3125 for 38 days, at 6; $247.50 for 69 days, at $512.45 for 5 mo. 11 Art. 455.) $1478 for 1 yr. 2 mo. 13 da., at $2810.60 for 9 mo. 24 da., at 6^ $944.50 for 1 yr. 10 mo. 22 da., $575 for 2 yr. 8 mo. \16 da., at ( $1275 for 50 days. $2345 for 30 days. $1728 for 63 days. $375.60 for 5 days. $414.80 for 54 days. $1024 for 59 days. $2375 for 90 days. $1000 for 57 days $2480 for 63 $5000 for ? At 5^ ? (See Ex. 2, #? (448) ? At 7$ ? At 5$? ? At 8^? At 4^? (See Ex. 1, Art. ? At 7 % ? At . at Q ? ? At 7^ ? (See Ex. 2, ? at ? At 8 At 6f ? ? ? At At INTEREST. [Art. 459. u/ 36. $1112 for 3 mo. 14 da., at 6% ? $5285 for 1 yr. 6 mo. 21 da., at 38. $7218 for 11 mo. 18 Find the amount of at 6^ ? At % ? At At At At 8< At' $416.75 for 8 mo. 17 da., at 6% $1235 for 2 yr. 1 mo. 19 da., at 6^ 41. $575.60 for 1 yr. 4 mo. 23 da., at 6< 42. $2214 for 4 mo. 25 da., at 6% ? At 4#. $6315 for 5 mo. 29 da., at 6% ? At 4 $4312 for 4 mo. 26 da., at 6% ? At 45. $384.30 for 2 mo. 28 da., at Q% ? At 40. $1296 for 1 yr. 11 mo. 27 da., at 6% ? 47. $4375 for 2 yr. 8 mo. 24 da., at Q% ? Find the interest of the following at %. (See Art. 458.) At At 48. $2000 for 6 dc 49. $1728 for 37 dc 50. $3485 for 92 dc 51. $1234 for 69 dc 52. $375.60 for 8 63. $3748 for 53 da. (54-1.) 64. $4126 for 89 da. 66. $1289 for 39 da. 66. $4000 for 17 da. 7. $2000 for 28 da. NOTE. Find the time in the following examples both in months and days, and in exact days (3 1C). J&1234 from May 10 to Dec. 4, at 5^ ? At 69. $444.40 from Jan. 13 to Nov. 2, at 4% ? At 1575.20 from June 5, 1882, to Feb. 4, 1883, at 0.7. $2375 from July 17, 1884, to Nov. 27, 1885, at (Exact time, 1 yr. 133 da.) $3212 from Aug. 24, 1881, to Jan. 20, 1884, at ? 63. $475.80 from May 12, 1882, to Feb. 1, 1884, at At At 64. Find the interest of $180 for 253 days, at Q NOTE. In many examples, labor can be saved by having the time and principal exchange places. In the above example, the interest of $180 for 253 days is the same as $253 for 180 days ($2.53 x 3). 66. Find the interest of $600 for 173 days at 9#. At 4#. 66. Find the interest of $3000 for 111 days at 12$. At 3#. Art. 459.] INTEREST. 183 Find the interest of 67. $1800 from Jan. 17 to Oct. 2, at 6#. At . $540 from May 11 to Dec. 18, at 5%. At 2. $3000 from Feb. 4 to July 13, at 4$. At 70. $2400 from July 13 to Dec. 1, at 5$. At 7.7. $600 from Aug. 16 to Nov. 24, at 6$. At 72. $1200 from May 19 to July 3, at 1%. At 75. $480 from March 13 to Sept. 3, at Sf c . At 74. $720 from Feb. 27 to May 15, at 9$. At 75. $2100 from Sept. 2 to Nov. 30, at 10$. At 76. If $9200 is loaned Sept. 18, 1882, at 6$, what is due May 9, 1885 ? (Time by C. S.) 77. What is a banker's gain in 1 year on $10000 deposited at 6$, and loaned 11 times at \\% a month ? 78. A note for $1421, with interest after 4 months, at 7$, was given Dec. 1, 1881, and paid Aug. 12, 1883. What was the amount due ? (C. S.) 79. Nov. 6, 1881, I bought a lot of grain for $753.20 ; Dec. 1G, I sold a part of it for $375.60 ; and, Dec. 31, I sold the remainder for $411.40. Money being worth 6%, how much did I gain by the transaction ? 80. A merchant marks his goods with two prices, the one for cash and the other for 4 months' credit. If the cash price- is $28, what ought the credit price to be, money being worth 10$ ? 81. May 27, $328 is loaned at 6$, and Aug. 16, $1000 is loaned at 5J-%. What is the total amount due Dec. 11 ? 82. A banker borrows $100000 at 3J$, and pays the interest at the end of the year ; he loans it at 5$ and receives the interest semi-annually. How much does he gain in one year, if he loans the semi-annual interest until the end of the year ? 83. A buys a bill of goods amounting to $2776.40, on the fol- lowing terms: "4 months, or less 5$ cash." He accepts the latter terms, and borrows the money at 6$ to pay the bill. How much does he gain ? 84- A person buying a building lot for $5400, agreed to pay for it in four equal semi-annual installments, with interest at 6% ; what was the total amount of money paid, the first payment being made at the time of the purchase ? 85. A banker borrows $10000 at 4J$, and lends half of it at 6$ and half at 8%. What does he gain in 2 yr. 4 mo. 26 da. ? 184 INTEREST. [Art. 46O. ACCURATE INTEREST. 46O. To find the accurate interest (365 days to the year) for any rate and time. (See Art. 437.) Ex. What is the accurate interest of $865, at 4$, from June 21 to Dec. 13 ? OPERATION. $865 Principal. ANALYSIS. From June 21 to Dec. 13, there are 175 days. The interest of $865 for 1 yr., at 4%, is $34.60. For 175 days, 34.60 Interest for 1 yr. m m of 1 yr., it is HI of $34.60 -- r $1 ' 59 - 365 ) 6055.00 ( 16.59 461. KULE. Multiply the principal by the rate per cent, expressed decimally. The result will be the interest for one year. Multiply the interest for one year by the number of days, and divide the product by 365. NOTES. 1. When the number of days is a multiple of 5, multiply by the number of days, and divide the product by 73. In the above example, $865 x .04 x 35 -f- 73 = $16.59. 2. To find the interest at any per cent., multiply by twice the rate as an integer, by the number of days, divide the product by 73, and point off 3 places. In the above example, $865 x 8 x 175 -*- 73000 = $16.59. 3. To find the interest at 5 % , multiply tbe principal by the number of days, divide the product by 73, and point off 2 places. From this result to find the interest at Qft>, add \ ; 4^%, subtract $ ; 4^, subtract \. 462. Accurate Interest from Ordinary Interest. The difference between ordinary interest and accurate interest for 1 day equals the difference between -^fa and ^ T of a year's interest. 1 1 _ 365 360 _ 5 J5_ _1_ J^ 360 ~~ 365 ~ 365 x 360 ~~ 365~x~360 ~~ 365 360 ~ 73 1 5 J> _1 l_ . J_ ~~ ~~ * 360 3651T366 ~~ 360 365 ~~ 72 365 The difference between the two methods is ^ of ordinary interest, or -j-% of accurate interest (437, Note 4). Therefore, from ordinary interest to find accurate interest subtract y 1 ^. Art. 462.] ACCURATE INTEREST. 185 ' In reckoning accurate interest, on account of the many short methods of ordinary interest, many accountants prefer to calculate ordinary interest first, and then make the necessary deduction. Since -fa is about \\%, the following approximate method may be used in reducing ordinary interest to accurate interest : From the ordinary interest subtract 1% and \% of itself. Ex. Eeduce $32.70 ordinary interest to accurate interest. NOTE. The exact result should be $32.252. The results by this method are too great by 1 cent for each $27 interest ; $.036 for each $100 interest ; $.36 for each $1000 interest. Where greater accuracy is required, the neces- sary correction can be made. EXAMPLES. 463. What is the accurate interest of L $43.32, at 6#, for 25 days ? 5. $292, at $%, for 140 days ? $6030, at 5$, for 141 days? 6. $438, at 6$, for 210 days? 8. $780, at 6#, for 90 days? 7. $350, at 4$, for 150 days? 4. $437.80, at 1%, for 63 days? 8. $500, at 4J& f r 100 days? 9. $3110.45, at 5J^, for 90 days? $$73.70, at 7$, from June 4 to Dec. 28 ? 11. $500, at Q%, from July 24, to Sept. 16 ? 12. $365, at 5%, from June 30 to Dec. 21 ? 13. $1080, at Q from May 9, 1878, to Jan. 30, 1879 ? 14. $1728, at 7#, from Jan. 6, 1878, to Jan. 21, 1880 ? N ./J. Eequired the exact interest on three U. S bopds of $5000 lr eaclr at 3J^, from July 1 to Aug. 11. 16. What is the interest on three U. S. bonds of $1000 each, at 4J^, from Sept. 1 to Nov. 15 ? .^ '-^ZL What is the interest on a $5000 U. S. bond, at 4$, from Oct. 1 to Dec. 16 ? 18. What is the interest on a U. S. bond of $1000, bearing Z\% interest, from May 1 to July 19 ? 19. What is the interest on a $500 U. S. bond, at 4$, from Apr. 1 to May 10.? f 20. What is Aae interest on a $5000 U. S. bond from Nov. 1, 1881, to Jan. 3, TJf%, %f^? 21. What is the difference between ordinary and accurate interest of $10000 for 219 days at 186 INTEREST. [Art. 464. PROBLEMS IN INTEREST. 464. To find the rate, the principal, interest or amount, and time being given. Ex. At what rate will $720, in 1 yr. 4 mo. 10 da., produce $44. 10 interest? OPERATION. ANALYSIS. The interest on a given principal for a given time is in proportion to the rate per cent. At one per cent. $720 will, in 1 yr. 4 mo. 10 da., produce $9.80 interest. To produce $44.10 interest, the required rate must be as many times \%, as $9.80 is con- tained times in $44.10, or 4 times. $7 57 _1 6 )58 20 _8 60 20 80 80 ) $44. 10 ( 4J Ans. Hence the answer is 465. RULE. Divide the given interest by the interest of the given principal, for the given time, at 1%. NOTE. When the amount is given, find the interest by subtracting the principal from the amount. 466. At what rate will f 1. $864 in 8 mo. 10 da. produce $42 interest ? 2. $1000 in 9 mo. 9 da. produce $54.25 interest ? 8. $852 in 1 yr. 7 mo. 16 da. amount to $935.21 ? 4. $1926 in 2 yr. 8 mo. 24 da. produce $263.22 interest ? v. 5. $375.60 in 1 yr. 10 mo. 22 da. amount to $425.41 ? ^6. $1872 in 7 mo. 17 da. produce $41.31 interest ? 7. $435.60 in 1 yr. 2 mo. 18 da. amount to $478 ? 8. $1338.72 in 6 mo. 27 da. produce $34.64 interest ? 9. $1728 in 8 mo. 21 da. amount to $1778.11 ? 10. $3456 in 5 mo. 8 da. produce $91.01 interest ? 11. $5280 in 11 mo. 11 da. amount to $5720.12 ? 12. $1234 in 8 mo. 22 da. produce $80.83 interest ? 13. $6975 in 3 mo. 28 da. amount to $7215.06 ? 14. $525 in 1 yr. 11 mo. 18 da. produce $309.75 interest? 15. $500 in 3 yr. 11 mo. 12 da. amount to $658 ? 16. $4680 in 2 yr. 6 mo. 11 da. produce $710.58 interest ? 17. $614.45 in 162 days amount to $633.805 ? Art.467.] PROBLEMS IN INTEREST. 187 467.To find the time, the principal, interest or amount, and rate being given. Ex. In what time will $426, at Q%, produce $59.427 interest ? OPERATIONS. $426 .06 5^56 ) $59.427 ( yr. 2.325 51 12 12 ~8 307 mo. 3.900 7 668 30 6390 da. ^7.000 5112 Or, $426 $25.56 ) $59.427 ( 2 yr. 51 12 8.307 12 $25.56 ) 99. 684 (3 mo. 76.68 23.004 J30 $25.56 ) 690. 120 (M da. ANALYSIS. The interest on a given principal at a given rate % is in proportion to the time. In one year $426, at 6fc, will produce $25.56 interest. To produce $59.427 interest, it will require as many years as $25.56 is contained timefe in $59.427, or 2.325 yr. 2.325 yr. equal 2 yr. 3 mo. 27 da. (289). 468. RULE. Divide tJ^e gimn^interest by the- interest of the given principal, at the given rate, for one year. The integral part of the qwotient will be years. Reduce the decimal, if any, to months and days (289). EXAMPLES. 469. In what time will 1. $3000, at 7$, produce $108.50 interest ? 2. $1728, at 6%, amount to $1872 ? 8. $3932, at 7%, produce $597.88 interest ? 4. $735, at 5%, amount to $742.66 ? 5. $1222.25, at %, produce $39.52 interest ? 6. $375.60, at 7#, amount to $425.41 ? 7. $1461.75, at Q%, produce $420.25 interest ? 8. $1200, at 3f %, amount to $1413 ? 9. $4500, at 5%, produce $181.25 interest ? 10. $276.50, at 10$, amount to $303.46 ? 188 INTEREST. [Art. 469. In what time will 11. $1020, at 6$, produce $89.25 interest ? 12. $6495, at 1%, amount to $7161.81 ? IS. $100, at 6%, produce $100 interest ? 14. $125, at 7$, amount to $375 ? 47O. To find the principal, the interest, time, and rate being given. Ex. What principal will produce $152.64 interest, in 1 yr. 5 mo. 20 da., at % ? OPERATION. $.088^) $152.64 (1728 _3 __ 3_ .265 ) 457.920 ANALYSIS. The interest on any principal 265 is as many times greater than the interest of $1, as that principal is greater than $1. One doUar, in 1 yr. 5 mo. 20 da., at 6% (447), will 1855 produce $.088^ interest. To produce $152.64, the 742 principal must be as many times $1 as $.088 is contained times in $152.64, or $1728. 2120 2120 471. RULE. Divide the given interest ~by the interest of $1 for the given time, at the given rate. EXAM PLES. 472. What principal will produce 1. $1235 interest, in 1 yr. 8 mo. 12 da., at Q% ? 2. $49.81, in 9 mo. 24 da., at 1% ? 3. $186.75, in 1 yr. 4 mo. 20 da., at 6% ? 4. $244.44, in 7 mo. 18 da., at 5% ? 5. $375.60, in 2 yr. 4 mo. 6 da., at 8% ? 6. $54.25, in 3 mo. 3 da., at 1% ? 7. $387.40, in 2 yr. 8 mo., at \% ? . $456, in 93 da., at 6% ? 9. $375, in 63 da., at 1% ? 10. $1000, in 1 yr. 18 d., at ?>% ? Ji $538.80, in 10 mo. 24 da., at 5% ? 12. $416.75, in 8 mo. 21 da., at 4% ? 7#. $645.39, in 4 yr. 8 mo. 10 da., at 4$ ? Art. 473.] PROBLEMS. 189 473. To find the principal, the amount, time, and rate, being given. Ex. What principal will amount to $1880.64, in 1 yr. 5 mo, 20 da., at % ? OPEBATION. $1.088$) $1880. 64 (1728. 3.265 ) 5641.920 ANALYSIS. -The amounts of different principals for the same time and rate % , are 23769 to eac h other as the principals. One dollar, 22855 in * y r - 5 mo - ^O da., at 6% will amount to $1.088$. To amount to $1880.64, the prin- cipal must be as many times $1 as $1.088$ 6530 are contained times in $1880.64, or $1728. 26120 26120 474. RULE. Divide the given amount by the amount of $1 for the given time, at the given rate. EXAMPLES. 475. What principal will amount to 1. $1272.254, in 6 mo. 6 da., at 6%? & $5538.72, in 8 mo. 12 da., at 7% ? 8. $3695.04, in 1 yr. 4 mo. 18 da., at 5^? 4. $442.71, in 2 yr. 2 mo. 24 da., at 8$? 5. $14794.31, in 3 yr. 3 mo. 3 da., at 6% ? 6. $1793.38, in 7 mo. 17 da., at 6$? 7. $1010.65, in 5 yr. 8 mo. 6 da., at 7% ? A $977.75, in 1 yr. 10 mo. 10 da., at 9. $1716.75, in 3 yr. 4 mo. 21 da., at 4 .70. $2808.08, in 2 yr. 8 mo. 12 da,, at 11. $4312.22, in 1 yr. 2 mo. 11 da., at 12. $6528.49, in 4 yr. 7 mo. 6 da., at 13. $1763.02, in 1 yr. 2 mo. 21 da., at 14. $2457.28, in 2 yr. 5 mo. 23 da., at 15. $5375.34, in 1 yr. 6 mo. 15 da., a 16. $3536.87, in 2 yr. 7 mo. 10 da., at 9 17. $4221.50, in 3 yr, 10 mo. 27 da., at 190 INTEREST. [Art. 476. PRESENT WORTH AND TRUE DISCOUNT. 476. The Present Worth of a debt due at some future time is its value now. Theoretically, it is a sum that, if placed at interest to-day for the given time, would amount to the face of the debt. 477. The True Discount is the diiference between the face of the debt and the present worth. This subject is an application of the principle illustrated in Art. 473, the face of the debt being the amount, the present worth the principal, and the true discount the interest. In actual business true discount is little used, banks and merchants generally using bank discount (496). True discount is the interest on the present worth for the given time, while bank discount is interest on the face of the debt. The difference is therefore equivalent to the interest on the true discount. For discount on bills, etc., when time does not enter as an element, see Art. 415. Ex. Mr. B owes me $212, payable one year from to-day with- out interest ; what is the present worth of the debt, the current rate of interest being ANALYSIS. Since $1 in one year, at Q%, amounts to $1.06, it would require as many dollars to amount to $212, as $1.06 is contained times in $212, or $200. The true discount is $212- $200, or $12. 478. EULE. /. To find the present worth, divide the face of the debt by the amount of $1 for the given time, at the given rate. II. To find the true discount, subtract the present worth from the face of the debt. EXAMPLES. 479. The current rate of interest being 6$, what is the present worth and true discount of 1. $1000, due 2 years hence ? 8. $600, due in 1 yr. 7 mo. ? . 2. $500, due in 2 yr. 4 mo. ? 4. $800, due in 9 mo. 24 da. ? 5. $325, due in 2 yr. 5 mo. 12 da. ? 6. $175, due in 1 yr. 4 mo. 16 da. ? 7. $800, due in 5 yr. 8 mo. 22 da. ? 8. $900, due in 6 yr. 8 mo. 14 da. ? Art. 479.] REVIEW EXAMPLES. 191 9. Mr. C. desiring to pay a bill of $1728 4 months before it was due, was allowed a discount equivalent to the interest on the face of the bill for the unexpired time at 6^ per annum (bank discount). How much greater was this discount than the true discount ? 10. Goods to the amount of $3750 are sold on a credit of 4 months. For how much cash could the merchant afford to sell the same goods, money being worth 10% per annum ? 11. If $10000 will be due me May 28, and $8000 May 16, what discount should I make on the two claims Apr. 1, money being worth S%? REVIEW EXAMPLES, 48O. 1. What is the interest of $375.60, for 1 yr. 10 mo. 16 da., at Q% ? 2. What is the amount of $1765 for 7 mo. 20 da., at 7^? 3. At what rate will $1234, in 2 yr. 2 mo. 26 da., produce $138. 14 interest ? 4. In what time will $585, at Q%, produce $67.08 interest ? 5. What principal will, in 1 yr. 8 mo. 14 da., at 6%, produce $176.22 interest ? 6. The semi-annual interest on a mortgage at 1% is $350. What is the face of the mortgage ? 7. Mr. B. invests $49500 in a business that pays him $594 per month. What annual rate of interest does he receive ? 8. Which is the better investment, and what per cent., one of $8400, yielding $336 semi-annually, or one of $15000, producing $1425 annually ? 9. May 18th, a speculator bought 1600 bushels of wheat, at $1.50 a bushel. He afterward sold the whole for $2472 cash, his' profit being equivalent to 8% per annum on the amount invested. What was the date of the sale ? 10. The par value of Mr. A/s bank stock is $9000, and he receives a semi-annual dividend of $315. What per cent, is the dividend per annum ? 11. Mrs. C/s son is now 16 yr. old ; how much must she invest for him at %, that, on arriving at age, he may have, with simple interest, $25000 ? 12. A bill of goods amounting to $4316.75 is due May 27; how- much would settle it May 1 at Q%? How much July 3 ? 192 INTEREST. [Art. 480, 18. A gentleman loaned 115000, at 6$. Jan. 1, 1880, interest and principal together equalled $20000. When was the money loaned ? 14. Find the interest on $3000, from Mar. 16 to Dec. 4, at 6%, by the following methods (437): 1, ordinary interest and compound subtraction ; 2, ordinary interest and exact number of days ; 3, accurate interest. 15. A man loaned another a sum of money, payable in 5 months, with interest at the rate of 6$,. and at the end of that time received $666.25 in return. How much did he loan ? 16. A speculator borrowed $10925 at 6$, May 16, 1882, with which he purchased flour at $6.25 per barrel. June 11, 1883, he sold the flour at $7.50 per barrel, cash. What did he gain ? 17. B bought 225 A. 24 sq. rd. of land, Aug. 18, 1882, at $4 an acre, borrowing the money to pay for it, at 5%. He sold the land April 7, 1886, at an advance of $299.40 on cost. If mean- while he paid $46.50 for taxes on the land, did he gain or lose, and how much ? 18. A speculator bought 9000 bu. grain at $1.80 per bushel, Mar. 18, 1875, the money paid for it being borrowed at 5}%. Dec. 12, 1875, he sold f of the grain at $2.00 per bushel, and the remainder at $1.90 per bushel. What was gained or lost by the transaction ? 19. A owes B 260 9s. Qd., with interest at 5%, for 143 days. He pays 25% of the amount due ; how much remains ? NOTE. In England, interest is usually computed on the basis of 365 days to the year, when the time is given in days. The legal rate in England is 5%. To calculate interest on English money, reduce the shillings and pence to the decimal of a pound (see Art. 342, Ex. 12, Note), apply any of the methods under Art. 461, and reduce the resulting decimal to shillings and pence. Find the accurate interest of 20. 425, from Aug. 4 to Dec. 28, at 5%. 21. 625 12s., from Jan. 12 to Apr. 1, at 4%. 22. 717 16s. Wd., from Mar. 3 to June 16, at \%* 23. 429 10s. Sd., from Sept. 16 to Nov. 30, at 3%. 24. 516 18s. 3d., from Aug. 1 to Oct. 18, at 3J#. 26. 612 6s. lid., from July 1 to Nov. 3, at 5%. * When the time is less than 1 year, and the rate is G% or less, reject the pence, if less than 6 ; add 1 shilling, if more than 6. The result will be sufficiently accurate. Art. 481.] COMPOUND INTEREST. 193 COMPOUND INTEREST.* 481. Compound Interest is interest not only on the prin- cipal, but also on the interest after it becomes due (4:33). 1. Interest may be compounded annually, semi-annually, quarterly, etc. 2. Interest upon interest due, or compound interest, cannot be collected by law, that is, payment cannot be enforced ; but such a payment is equitable, ' and the receiving of it, if the debtor is willing or can be induced to pay it, does not constitute usury in the legal sense of the word. In the State of Missouri, parties may contract in writing for the payment of interest upon interest, but it shall not be compounded oftener than once a year. Ex. What is the compound interest of $1000 for 3 years, at 6^? OPERATIONS. $1000.00 Principal. Or 60.00 Interest for 1 yr. 1060 Amount for 1 yr., or 2d principal. 63.60 Interest of $1060 for 1 yr. 1123.60 Amount for 2 yr., or 3d principal. 67.416 Interest of $1123.60 for 1 yr. 1191.016 Amount for 3 yr. 1191.016 1000 _ Original principal. 1000 191.016 Compound interest for 3 yr. 191.016 482. RULE. Find the amount of the given prineipal for the first period of time, and make it the principal for the second. Find the amount of the second principal for the second period of time, and make it the principal for the third; and so continue for the whole time. The last amount is the amount required. The last amaunt, less the given principal, will be the compound interest. NOTES. 1. When the time is not a multiple of the interest period, find the amount of the principal to the end of the last period ; then compute the simple interest on this amount for the remaining time, and add it to the last amount. The sum will be the required amount. 2. The work of computing compound interest may be shortened by using the tables on pages 194 and 195. * For Annual Interest, eee page 319. 194 INTEREST. [Art. 483 483. Table showing the sum to which $1 will increase, at compound interest, in any number of years not exceeding 45. Yrs. H. 2J*. ,., 8#. 4fo 4J*. 5*. 61 9ft Yrs. 1 1.0200 1.0250 1.0300 1.0350 1.0400 1.0450 1.0500 1.0600 1.0700 1 2 1.0404 1.0506 1.0609 1.0712 1.0816 1.0920 1.1025 1.1236 1.1449 2 3 1.0612 1.0769 1.0927 1.1087 1.1249 1.1412 1.1576 1.1910 1.2250 g 4 1.0824 1.1038 1.1255 1.1475 1.1699 1.1925 1.2155 1.2625 1.3108 4 5 1.1041 1.1314 1.1593 1.1877 1.2167 1.2462 1.2763 1.3382 1.4026 5 6 1.1262 1.1597 1.1941 1.2293 1.2653 1.3023 1.3401 1.4185 1.5007 6 7 1.1487 1.1887 1.2299 1.2723 1.3159 1.3609 1.4071 1.5036 1.6058 7 8 1.171? 1.2184 1.2608 1.3168 1.3686 1.4221 1.4775 1.5988 1.7182 8 9 1.1950 1.2489 1,3048 1.3629 1.4233 1.4861 1.5513 1.6895 1.8385 9 10 1.2190 1.2801 1.3439 1.4106 1.4802 1.5530 1.6289 1.7908 1.9672 10 11 1.2434 .3121 1.3842 1.4600 1.5395 1.6229 1.7103 1.8983 2.1049 11 12 1.2682 .3449 1.4258 1.5111 1.6010 1.6959 1.7956 2.0122 2.2522 12 18 1.2936 .3785 1.4685 1.5640 1.6651 1.7722 1.8856 2.1329 2.4098 13 14 1.3195 .4130 1.5126 1.6187 1.7317 18519 1.9799 2.2609 2.5785 14 15 1.3459 1.4483 1.5580 1.6753 1.8009 1.9353 2.0789 2.3966 2-7590 15 16 1.3728 1.4845 1.6047 1.7340 1.8730 2.0224 2.1829 2.5404 2.9522 16 17 1.4002 1.5216 1.6528 1.7947 1.9479 2.1134 2.2920 2.6958 3.1588 17 18 1.4282 1.5597 1.7024 1.8575 2.0258 2.2085 2.4066 2.8543 3,3799 18 19 1.4568 1.5987 1.7535 1.9225 2.1068 2.3079 2.5270 3.0256 3.6165 19 20 1.4859 1.6386 1.8061 1.9898 2.1911 2.4117 2.6533 3.2071 3.8697 20 21 1.5157 1.6796 1.8603 20594 2.2788 2.5202 2.7860 3.3996 4.1406 21 22 1.5460 1.7216 1.9161 2.1315 2.3699 2.6337 2.9253 3.6035 4.4304 22 23 1.5769 1.7646 1.9736 2.2061 2.4647 2.7522 3.0715 3.8197 4.7405 23 24 1.6084 1.8087 2.0328 2.2833 2.5633 2.8760 3.2251 4.0489 5.0724 24 25 1.6406 18539 2.0938 2.3632 2.6658 3.0054 3.3864 4.2919 5.4274 25 26 1.6734 1.9003 2.1566 2.4460 2.7725 3.1407 3.5557 4.5494 5.8074 26 27 1.7069 1.9478 2.2213 2.5316 2.8834 3.2820 3.7335 4.8223 6.2139 27 28 1.7410 1.99C5 2.2879 2.6202 2.9987 3.4297 3.9201 5.1117 6.6488 28 29 1.7758 2.0464 2.3566 2.7119 3.1187 3.5840 4.1161 5.4184 7.1143 29 30 1.8114 2.0976 2.4273 2.8068 3. -434 3.7453 4.3219 5.7435 7.6123 30 31 1.8476 2.1500 2.5001 2.9050 3.3731 3.9139 4.5380 6.0881 8.1451 31 32 1.8845 2.2038 2.5751 3.0037 3.5081 4.0900 4.7649 6.4534 8.7153 32 33 1.9222 2.2589 2.6523 3.1119 3.6434 4.2740 5.0031 6.8406 93253 33 34 1.9607 2.3153 2.7319 3.2209 3.7943 4.4664 5.2533 7.2510 9.9781 34 35 1.9999 2.3732 28139 3.3336 3.9461 4.6673 5.5160 7.6861 10.6766 35 36 2.0399 24325 28983 3.4503 4.1039 4.8774 5.7918 8.1473 11 4239 36 37 2.0807 2.4933 2.9852 8.5710 4.2681 5.0969 6.0814 8.6361 12.2236 37 38 2.1223 2.5557 3.0748 3.6960 4.4388 5.3262 6.3855 9.1543 13.0793 38 39 2.1647 2.6196 3.1670 3.8254 4.6164 5.5659 6.7048 9.7035 13.9948 39 40 2.2080 2.6851 3.2620 3.9593 4.8010 5.8164 7.0400 10.2857 14.9745 40 41 2.2522 2.7522 3.3599 4.0978 4.9931 6.0781 7.3920 10.9029 16.0227 41 42 2.2972 2.8210 3.4607 4.2413 5.1928 6.3516 7.7616 1.5570 17.1443 42 43 2.3432 2.8915 3.5645 4.3897 5.4005 6.6374 8.1497 12.2505 18.3444 43 44 2.3901 2.9638 3.6715 4.5433 5.6165 C.9361 8.5572 2.9855 19.6285 44 45 2.4379 3.0379 3.7816 4.7024 5.8413 7.2482 8.9850 13.7646 21.0025 45 To find the sum to which a given amount will increase, at compound interest, at any of the rates per cent, and number of years expressed in the above Table : Multiply the given amount by the sum to which one dollar will increase at the rate and for the number of years required, marking off as many decimals from the product as there are decimals in the multiplier and multiplicand. NOTES. 1. The amount for any number of years not given in the table may be computed by finding the product for any two numbers of years whose sum equals the given time. Thus, the compound amount of $1 at 6# for 55 years, may be found by multiplying $13.7646, the amount for 45 years, by 1.7908, the amount for 10 years. 2. If the interest is compounded semi-annually, to find the amount from the table, take twice the number of years at one-half the rate. Thus, the amount at 8#, compounded semi- annually, for 5 years, is equivalent to the amount for 10 periods of 6 months each, at 4# for each period, and is the same as the amount for 10 years at 4%. If the interest is compounded quarterly, take 4 times the number of years at one-fourth the rate. 3. The compound interest of $1 is $1 less than the amounts in the above table. Ait. 484.] COMPOUND INTEREST. 195 484. Table showing the sum to which $1, paid at the beginning of each year will increase at compound interest, in any number of years not exceeding 50. Yrs. 3. 8J*. 4%. 5%. 6*. *?rf 8%. 10*. Yrs. 1 1.0300 1.0350 1.0400 1.0500 1.0600 1.0700 1 0800 1.1000 1 2 2.0909 2.1062 2.1216 2.1525 2.1836 2.2149 22464 2.3100 2 3 3.1836 3.2149 3.2465 3.3101 3,3746 3.4399 3.5061 3.6410 3 4 4.3091 4.3623 4.4163 4.5256 4.6371 4.7507 4.8666 5.1051 4 5 5.4684 5.5502 5.6330 5.8019 5.9753 6.1533 6.3359 6.7156 5 6 6.6625 6.7791 6.8983 7.1420 7.3938 7.6540 7.9228 8.4872 6 7 7.8923 8.0517 8.2142 8.5491 8.8975 9.2598 9.6366 10.4359 7 8 9.1591 9.3685 9.5828 10.0266 10.4913 10.9780 11.4876 12.5795 8 9 10.4639 10 7314 11.0061 11.5779 12.1808 12.8164 13.4866 14.9374 9 10 11.8078 12.1420 12.4864 13.2068 13.9716 14.7836 15.6455 17.5312 10 11 13.1920 13.6020 14.0258 14.9171 15.8699 16.8885 17.9771 213843 11 12 14.6178 15 1130 15.6268 16.7130 17.8821 19.1406 20.4952 23.5227 12 13 16.0863 16.6770 17.2919 18.5986 20.0151 21.5505 23.2149 26.9750 13 14 17.5989 18.2957 19.0236 20.5786 22.2760 24.1290 26.1521 30.7725 14 15 19.1569 19.9710 20.8245 22.6575 24.6705 26.8881 29.3243 34.9497 15 16 20.7616 21.7050 22.6975 24.8404 27.2129 29.8402 32.7502 39,5447 16 17 22.4144 23.4997 24.6454 27.1324 29.9057 32.9990 36.4502 44.5992 17 18 24.1169 25.3573 26.6712 29.5390 32.7600 36.3790 40.4463 50.1591 18 19 25.8704 27.2797 28.7781 32.0660 35.7856 39.9955 44.7620 56.2750 19 20 27.6765 29.2695 30.9692 34.7193 38.9927 43.8652 49.4229 63.0025 20 21 29.5368 31.3290 33.2480 37.5052 42.3923 48.0058 54.4568 70.4027 21 22 31.45-29 33.4604 35.6179 40.4305 45.9958 52.4361 59.8963 78.5430 22 23 33.4265 35.6665 38.0826 43.5020 49.8156 57.1767 65.7648 87.4973 23 24 35.4593 37.9499 406459 46.7271 53.8645 62.2490 72.1059 97.3471 24 25 37.5530 40.3131 43.3117 50.1135 58.1564 67.6765 78.9544 108.1818 25 26 39.7096 42.7591 46.0842 53.6981 62.7058 73.4838 86.3508 120.0999 26 27 41.9309 45.2906 48.9676 57.4036 67.5281 79.6977 94.3388 133.2099 27 28 44.2188 47.9108 51.9663 81.3227 72.6398 86.3465 102.9659 147.6309 28 29 46.5754 50.6227 55.0849 65.4388 78.0532 93.4608 112.2332 163.4940 29 30 49.0027 53-4295 58.3283 69.7608 83.8017 101.0730 122.3459 180.9434 30 31 51.5028 56.3345 61.7015 74.2988 89.8898 109.2182 133.2135 200.1378 31 32 54.0778 59.3412 65.2095 79.0638 96.3432 117.9334 144.9506 221.2515 32 33 56.7302 62.4532 68.8579 84.0670 03.1838 127.2588 157.6267 244.4767 33 34 59.4621 65.6740 72.6522 89.3203 110.4348 137.2369 171.3168 270.0244 34 35 62.2719 69.0076 76.5983 94.8363 118.1209 147.9135 186.1021 298.1268 35 36 65.1742 72.4579 807022 100.6281 26.2681 159.3374 202.0703 329.0395 36 37 68.1594 76.0289 84.9703 106.7095 134.9042 171.5610 219.3159 363.0434 37 38 71.2342 79.7249 89.4091 113.0950 144.0585 184.6403 237.&412 400.4478 38 39 74.4013 83.5503 94.0255 119.7998 153.7620 198.6351 258.0565 441.5926 39 40 77.6633 87.5095 93.8265 126.8398 164.0477 213.6096 279.7810 486.8518 40 41 81.0232 91.6074 03.8196 134.2318 174.9506 229.6322 303.2435 536.6370 41 43 84.4839 95.8486 09.0124 141.9933 186.5076 246.7765 328.5830 591.4007 42 43 88.0484 100.2383 1 114. 4129 150.1430 198.7580 265.1208 355.9496 651.6408 43 44 9i.7199 104.7817 120.0294 158.7002 211.7435 2S4.7493 385.5056 717.9048 44 45 95.5015 109.4840 125.8706 167.6852 225.5081 305.7518 417.4261 790.7953 45 46 99.3965 114.3510 131.9454 177.1194 240.0986 328.2244 451.9002 870.9749 46 47 103.4084 119.3883 138.2632 187.0254 255.5645 1 352.2701 489.1322 959.1723 47 48 107.5406 124.6018 144.8337 197.4267 271.9584 377.9990 529.3427 1056-1896 48 49 111.7969 129.9979 151.6671 208.3480 289.3359 405.5289 572.7702 1162.9085 49 50 116.1807 135.5828 158.77-38 219.8154 307.7561 434.9859 619.6718 1280.2993 50 To find the sum to which a given amount, per annum, will increase at compound inter- est, at any of the rates per cent, and number of years expressed in the above Table : Multiply the given amount, per annum, by the sum to which one dollar per annum win increase at the rate and for the number of years required, marking off as many decimals from the product as there are decimals in the multiplier and multiplicand. NOTE. If the amount be payable semi -annually, and compound interest is to be allowed Bemi-annually, take the amount for double the number of years at one-half the rate per cent. Thus, for a semi-annual payment of $1 for 10 years at 10 per cent., take the amount of $1 for 20 years at 5 per cent. = $34.7193. For a quarterly payment, take the amount for four times the number of years at one -fourth the rate per cent. 19G INTEREST. [Art. 485. EXAMPLES. 485. 1. What will $450 amount to at compound interest, in 4 years, compounded annually at 4% ? At 3%? 2. Find the compound interest of $360, for 2 years, interest compounded semi-annually at 6%. At b%. 3. AVhat is the compound interest of $800 for 1 yr. 3 mo. at 8%, interest compounded quarterly ? 4- At compound interest, what is the amount of $1728 for 3 yr. 4 mo. 16 da., interest compounded annually at Z% ? At % ? NOTE. First find the amount for 3" years, and use this amount as the principal for the remaining time. 5. B holds a mortgage against A's property dated Apr. 1, 1881, for $20000, interest payable annually at 6%. The interest due Apr. 1, 1882, is not paid until May 26, 1882. How much is then due, A having consented to pay interest upon interest ? (See Note 2, Art. 481.) NOTE. In solving the following examples, use the tables in Art. 483- 484. 6. A gentleman deposits in a savings bank $100 when his child is one year old. How much will this amount to when the child is 21 years old, interest being compounded semi-annually at 4^? At 5^? 7. If, at the age of 25 years, a person places $2000 on interest, compounded annually at 6^, what will be the amount due him when he is 50 years old ? 8. What will $625 amount to at compound interest, in 36 years, compounded annually at 3^ ? At 4% ? 9. At the age of 20, and every year thereafter, a young man places $200 at compound interest at 6%. How much will he have at the age of 30 ? At the age of 40 ? (See Art. 484.) 10. How much will a gentleman have at the end of three years, if he places at compound interest at 5% $300 at the beginning of each year ? 11. Mr. B., whose life is insured for $4000, pays an annual premium of $114. How much would this amount to at 6% com- pound interest in 20 years ? 12. A lady deposits $50 in a savings bank Jan. 1 and July 1, of each year ; how much will be placed to her credit in 15 years, money being worth %, compound interest ? Art. 485.] COMMERCIAL PAPER. 197 IS. What sum must be placed at compound interest, at 6$, to amount to $1000 in 5 years ? .-^In compound interest, as in simple interest, the amounts are proportional to the principals ; hence the amount of any principal is as many times greater than the amount of $1, as that principal is greater than $1. To find the principal, divide the given amount by the amount of $1 for the given time and rate. In simple interest, the interest on a given principal for a given time is in proportion to the rate per cent., and at a given rate, in proportion to the time; but, in compound interest, such is not the case. If the rate or time be doubled, the interest is more than doubled. 14. How much, should a gentleman invest at compound inter- est, 6$, for his son who is now 6 years old, so that, when he be- comes 21 years of age, he may have $10000 ? 15. In the above example, how much should be invested at the beginning of each year to produce the same sum ? 16. A gentleman at his death left $7850 for the benefit of his only son, 12 years old, the money to be paid to him when he should be 21 years of age. How much did he receive, interest at 6$, compounded send-annually ? /' 17. How much must a person at the age of 25 years, place compound interest at 6%, so that the amount due him, when h is 50 years old, will be $20000 ? 18. In the above example, how much should he invest anntfe to produce the same sum ? COMMERCIAL PAPER. 486. Commercial Paper embraces notes, drafts, bil exchange, etc. 487. A Note (also called a Promissory Note) is a -. promise to pay a certain sum of money on demand or at a s time. 488. The Maker of a note is the person who signs it, and thus becomes responsible for its payment. The Payee is the person to whom, or to whose order, it is made payable. The Face of a note is the sum promised. In Note 1, Art. 495, Peter Cooper is the maker ; George Peabody is the payee; the face of the note is $1000. 198 INTEREST. [Art. 489. 489. A Negotiable Note is a note which is made payable to bearer or to the order of some person. (See Notes, Art. 495.) 1. A note is non-negotiable when it is payable only to the party named in the note. 2. A negotiable note made in New Jersey must contain the words " with- out defalcation or discount ; " in Missouri, the words ' ' negotiable and payable without defalcation or discount." 3. Negotiable notes payable to order may be sold or transferred by the payee writing his name upon the back of the note. He then becomes an indorser. 4. Negotiable securities are good in the hands of one who purchases in good faith and before maturity, although the seller may have found or stolen them. 5. Where no place of payment is specified, a promissory note is payable at the maker's place of business, or if none is known, at the residence of the maker. 6. A note or draft must be presented at the place where it is made pay- able. If at a bank, during banking hours ; if at a place of business, during business hours ; if at a residence, during family hours ; and if the maker, or some one for him, is not ready with legal tender currency to pay it, the holder need not call again. A check, even if certified, is not a legal tender, and may be lawfully refused. 490. The Indorser of a note or draft is the person who writes his name on the back of it, and by so doing guarantees its payment. If Mr. Erastus Corning desires to sell or transfer Note 3, Art. 495, it will be necessary for him to indorse it. If he writes his name only, it is called an indorsement in blank, and the note is then payable without further indorse- ment to any person lawfully holding the same. He may indorse it in full by making it payable to a particular person, thus "Pay to the order of Henry 11. Pierson, Erastus Corning." Before it can be again transferred, it will require the indorsement of Henry R. Pierson. For greater security, checks, notes, drafts, etc., are indorsed in full w r hen sent by mail. If an indorser does not wish to guarantee the payment of a note, draft, etc., he writes "Without recourse" over his name at the time of the indorse- ment. Sometimes notes and drafts are drawn to the order of the maker or the drawer (to the order of myself or ourselves) to facilitate their transfer without the indorsement of the holder. 491. A Draft, or Bill of Exchange is an order or request addressed by one person to another, directing the payment of a specified sum of money to a third person or to his order. Art. 492.] COMMERCIAL PAPER. 199 492. The Drawer of the draft is the person who signs it. The Drawee is the person on whom it is drawn. The Payee is the person to whom, or to whose order, it is made payable. In Draft 5, Art. 495, C. P. Huntington is the drawer; Drexel, Morgan & Co. are the drawees; J. & W. Seligman & Co. arc the payees. 1. The person in whose favor the bill is drawn is sometimes called the buyer, and becomes the " remitter." After the bill is presented and accepted, the drawee is called the acceptor, and the draft, an acceptance. The draft then has the same legal significance as a promissory note. 2. A person accepts or promises to pay a draft by writing the word " Accepted " and the date over his name across its face. 3. Drafts are sometimes accepted in the following form: "Accepted August 20, 1881, and payable at the National Park Bank, New York, G. B. Horton & Co." 4. In the State of New York, both by law and custom, the drawee of a draft may demand 24 hours consideration from the time the draft is presented for acceptance. When accepted, it must bear the date when first seen by him. 5. To "honor" a draft is to accept it or pay it on being presented. 493. A Protest is a formal statement made by a Notary Public, declaring that a draft or note has been presented for pay- ment or acceptance, and was refused. 494. Days of G-race and Maturity. The day of matu- rity is the day on which a note becomes legally due. According to the laws of most of the States, a note is not legally due until three days after the expiration of the time specified in the note, except the note contain the words "without grace/' These days are called days of grace, but they are of no advantage to the payer, since interest is charged for them as for any others. 1. California has abolished days of grace altogether. In Georgia, Ala- bama, and Kentucky, grace is allowed on promissory notes only in case they are made payable, or are discounted or left for collection, at a bank or private banker's. 2. The following is an analysis of the Holiday law of 1887 of the State of New York : Paper due on a holiday is payable the following business day. Paper due on Saturday, except when payable at sight or on demand, is payable the fol- lowing business day. Paper due on Sunday must be paid on the business day following it. If one of the men- tioned holidays falls on Sunday, paper due on that day must be paid the following business day. Paper due on Monday, where the preceding Sunday is a holiday, is not payable until the following business day. 200 INTEREST. [Art. 494 3. In nearly all of the States, excepting New York, paper due on Sunday, or on a legal holiday, must be paid the preceding business day. Thus, if a holiday falls on Thursday, all notes, etc., must be paid on Wednesday ; if a holiday falls on Monday, all notes due Sunday or Monday would be payable on Saturday ; if a holiday falls on Saturday, notes due Saturday or Sunday would be payable on Friday. 4. The legal holidays in the State of New York are New Year's Day (Jan. 1), Washington's Birthday (Feb. 22), Decoration Day (May 30), Independence Day (July 4), Labor Day (the first Monday of September), Election Day (the first Tuesday after the first Monday of November), Thanksgiving Day (the day appointed by the President of the United States and Governor of the State, usually the last Thursday of November), Christmas Day (Dec. 25), and every Saturday from 12 o'clock at noon until 12 o'clock at midnight (Saturday Half-Holiday). When a legal holiday falls on Sunday, Monday is, by the statute of New York, made a legal holiday. 5. A note made due at a fixed date in the future, carries 3 days' grace (unless the words "without grace" are used in the contract). Thus, a note stating that " on May 1, 1882, I promise, etc.," would carry 3 days' grace, and would be payable May 4, 1882. 6. When the time of a no,te is expressed in months, calendar months are used to determine the day of maturity ; when in days, the exact number of days is used. Thus, a note dated July 16, and payable two months from date, would nominally mature Sept. 16, and, including the three days of grace, would legally mature Sept. 19. A note having the same date, and payable sixty days from date, would nominally mature Sept. 14, and, including the three days of grace, would legally mature Sept. 17. 7. A note due in one or more months from date, matures on the corre- sponding day of the month up to which it is reckoned, if there are so many days in that month ; but if not so many, it then matures on the last day of said month, to which the usual grace must be added. Thus, notes dated Jan. 28, 29, 30, or 31, and payable one month from date, would become due Mar. 3 (Feb. 28 with 3 days' grace added). 8. When drafts are payable a certain time after sight, the date of accept- ance and the time of the draft determine the day of maturity. Thus, if a draft is dated May 16, accepted May 20, and payable sixty days after sight, it would mature or be due 63 (including 3 days of grace) days after May 20, or July 22. If payable 60 days after date, it would mature 63 days after May 16, or July 18. It is not necessary to present for acceptance drafts drawn a certain time after date, but as a courtesy to the drawee, it is usually done. 9. Days of grace are allowed on drafts according to the custom of the place where they are payable. The statute of New York forbids grace on all sight drafts, no matter on whom drawn, and on all time drafts which appear on their face to be drawn ' ' upon any bank, or upon any banking association or individual banker, carrying on the banking business under the act to authorize the business of banking." Art. 495.] FORMS OF NOTES AND DRAFTS. 201 495. FORMS OF NOTES AND DRAFTS. 1. DEMAND NOTE. $1000. NEW YORK, August 19, 1887. On demand, I promise to pay GEORGE PEABODY, or bearer, One Thousand Dollars. Value received. PETER COOPER. The above note is payable on demand, that is, whenever presented; is negotiable (payable to bearer) : and bears interest from date at the legal rate of the State in which it is made. If the words " or bearer" were omitted the note would not be negotiable. How much would be due on the above note, Jan. 1, 1888, finding the time by compound subtraction ? 2. TIME NOTE INTEREST-BEARING. $876jfo. CINCINNATI, OHIO, July 16, 1888. Six months after date, I promise to pay GEO. C. MILLER, or order, Eight Hundred Seventy-five and -ffa Dollars, with interest at eight per cent. Value received. ALEX. MCDONALD. The above note is payable 6 mo. 3 da. after its date, or Jan. 19, 1889 ; is negotiable (payable to order); and draws interest from its date at 8fo per annum. If the rate of interest was omitted, it would bear interest at the legal rate of the State for such cases, 6$. (See Art. 436.) How much would be due on the above note at its maturity ? How much, March 1, 1889 ? Supposing the rate of interest to be omitted in the note, how much would be due May 4, 1889 ? 3. TIME NOTE WITHOUT INTEREST PAYABLE AT A BANK. $6000. ALBANY, N. Y., December 4, 1889. Sixty days after date, I promise to pay to the order of ERASTUS CORNING, Six Thousand Dollars, at the Second National Bank. Value received. DAVID MURRAY. The above note is payable 63 days from Dec. 4, 1889, or Feb. "5, 1890. It is payable at the Second National Bank. No interest will be due at maturity (Feb. 5). If the note is not paid at maturity, it will bear interest from that date. Supposing the above note was payable 90 days from date, what would be its due date (311, Ex. 10)? The note as given not being paid at maturity, how much would be due Feb. 26, 1890, protest fees $2.10? 202 INTEREST. [Art. 495. 4. JOINT AND SEVERAL NOTE. $416ffo. WORCESTER, MASS., May 27, 1888. Four months after date, we jointly and severally promise to pay JOHK S. BALLARD & Co., or order, Four Hundred Sixteen -ffo Dollars, with interest from date, value received. T. K. EARLE. CHAS. W. SMITH. If the above note were written "we jointly promise, etc.," it would be called a joint note. The makers of a joint note must be sued jointly, each being responsible for one-half of the amount of the note. The makers of a joint and several note may be sued separately, either being responsible for the full amount of the note. How much would be due on the above note, Dec. 30, 1888 ? How much Sept. 30, 1888, the date of maturity ? 5. SIGHT DRAFT. $8000. SAN- FRANCISCO, CAL., May 1, 1882. At sight, pay to the order of J. & W. SELIGMAN & Co., Eight Thousand Dollars, value received. C. P. HUNTINGTON. To DREXEL, MORGAN & Co., New York. 6. TIME DRAFT. $5000. BURLINGTON, IOWA, June 18, 1887. At sixty days' sight, pay to the order of ADDISOK BALLARD, Five Thousand Dollars, value received, and charge to account of A. G. ADAMS. To BARTON & JONES, Chicago, 111. Drafts are sometimes drawn a certain number of " days after date." (See Art. 494, Note 8.) For Foreign Bills of Exchange, see Art. 555. If the above draft was accepted June 19, 1887, what was the date of maturity? 7. A sixty-day (63) day note given on Monday will mature on what day? 8. A note payable 90 (93) days from date and given on Thurs- day will fall due on what day of the week ? If payable in 30 (33) days and given on Friday, on what day will it become due ? 9. A note, dated July 22, and payable in 90 (93) days, would mature on what date ? Art. 496.] BANK DISCOUNT. 203 BANK DISCOUNT. 496. Bank Discount is simple interest of a note, paid in advance, for the number of days the note has to run. It may be computed by any of the methods given for simple interest. On notes without interest (the usual case of notes discounted at banks), bank discount is reckoned on their face, the amount due at maturity ; on notes with interest, it is reckoned on the amount due at maturity, or their face plus the interest for the full time of the note. 497. The Proceeds of a note is the amount received by the holder from the bank when the note is discounted. It is the amount on which the discount is reckoned less the discount. 498. Call Loans. Banks in the large cities loan large amounts of money upon stocks, bonds, negotiable warehouse receipts for grain, cotton, petroleum, etc., as collateral security, payable on demand or on giving one day's notice. Such loans are termed " call " or demand loans, and interest on them is paid at the end of the time. (See Art. 436, Note d.) 499. The time to be reckoned on a loan or note is exclusive of the day of date, but includes the day of maturity or payment. Thus, in discounting, a note in the City of New York, Apr. 4, which would mature Apr. 24, the discount would be calculated for 20 days. 1. In Philadelphia, Baltimore and some other cities it is the custom of banks in finding time to include both the day of discount and the day of maturity. Thus, the discount on the above note would be reckoned for 21 days. 2. Banks of the City of New York reckon discount both on the basis of 360 and 365 days to the year, the greater number on the former basis. 3. When notes are payable in other cities or towns, some banks charge interest for the time required for the collecting bank to remit the money, in addition to the interest on the note from the day of discount to the legal day of maturity. Thus, if a note, maturing June 10 and payable at a Chicago bank, is discounted at a bank in New York, the remittance in settlement would not be received before June 12 in New York, and the New York bank in discounting the note would be justified in charging interest for two days beyond the day of maturity. 4. Some banks charge a small fee for collection and exchange in addition to the interest in discounting notes which are payable in other cities or towns. 204 INTEREST. [Art. 5OO. EXAMPLES. 5OO. Find the date of maturity and proceeds of the following notes : (*) 610000. NEW YOKE, July 16, 1889. Four months after date, I promise to pay to the order of FISK & HATCH, Ten Thousand Dollars, at the first National Bank, value received. S. D. BABCOCK. Discounted July 16, 1889, at 6$. ANALYSIS. The note is due 4 months (494, 6) and 3 days (days of grace, 494) after July 16, or Nov. 19. From the day of discount (July 16) to the day of maturity (Nov. 19) there are 126 days. The interest of $10000 for 126 days at 6$, if reckoned on the basis of 360 days to the year, is $210, and the proceeds are $10000 less $210, or $9790. The interest on the basis of 365 days to the year would be $2.88 less, or $207.12, and the proceeds would be $9792.88. If the note was discounted Sept. 1, the interest or discount would be reckoned for 79 days (Sept. 1 to Nov. 19). $8000. BROOKLYN, N. Y., July 16, 1891. Ninety days from date, I promise to pay S. B. CHITTEKDEK, or order, Eight Thousand Dollars, value received. A. A. Low. Discounted Aug. 31, 1891, at %. ANALYSIS. The note is due 93 days (494) after July 16, or Oct. 17. Compute the discount for 47 days (Aug. 31 to Oct. 17) on $8000. If the note had been discounted July 16, the date of the note, the interest would have been computed for 93 days, the full time of the note. NOTE. The results of the following examples will be given on the basis of both 360 and 365 days to the year. No. Date of Note. Time. Face. Date of Discount. Rate of Discount. 3 Jan. 24 90 days $1200 Jan. 24 Q% 4 May 18. . 3 mo. $5280 May 18 6% 5 Aug 31 ... . 60 days $2560 Aug. 31 8% 6 June 4 4 mo. $3756 June 4 1% 7 Oct 16 30 days $6425 Oct. 16 % 8 Mar 13 6 mo $8375 Mar. 13 51% Art. 500.] BANK DISCOUNT. 205 No. Date of Note. Time. Pace. Date of Discount. Rate of Discount. 9 May 29. 8 mo. $4500 July 7. 10% 10 July 27 60 days $8240 Sept. 2 6% 11 Mar 28 90 days $4324 Apr. 14 51% 12 May 27 6 mo. $4885 Aug. 15 8% 13 Jan. 3 120 days $9000 Feb. 28 6% 14 Sept 12 4 mo $5000 Oct 14 7% 15 Nov 1 90 days $6000 Nov 28 5- 4 -<& U 5 7 16. What were the proceeds of Note 3,, Art. 495, if discounted Dec. 16, 1889, at the legal rate ? 17. Find the date of maturity and proceeds of a note of $5000 payable 60 days from date, dated and discounted at a Philadelphia bank, Aug. 3. (See Art. 499, 1.) 18. Find the date of maturity and proceeds of a note of $3750, payable 60 days from date, dated and discounted at a Maryland bank, Jan. 31, 1882. NOTE. In the following examples, the charge for collection and exchange is a certain per cent, of the face of the notes, without reference to the time they have to run (499, 4). No. Date of Note. Time. Face. Date of Discount. Rate of Discount. Rate of Collection. 19 May 5 90 days $7000 May 5 6% 1% 20 March 1. ... 4 mo. $9000 Apr. 30 *% -&% 21 June 18 6 mo. $5000 July 31 Sf k% 22 July 28.... 60 days $4500 Aug. 1 e# -h% 23 Sept. 3 90 days $9000 Sept. 5 5| u.::$27:x. 8. 144 yd.: 175 yd. :: $18: x. Art. 523.] SIMPLE PROPORTION. 215 9. If 19 yd. of silk cost $28.50, what will 37 yd. cost ? 10. If 64 yd. of carpet 36 in. wide will cover a floor, how many yards 27 in. wide will be required to cover the same floor ? 11. A cane 3 ft. 3 in. high casts a shadow 5^ ft. long ; how long a shadow is cast by the steeple of a church which is 234 feet high ? 12. If the freight of a long ton (336, 3) is 70 shillings, what is the freight of 16375 pounds ? 13. The net assets of a bankrupt are $27675, and the liabilities $138375. How much must be paid to Mr. A, whom he owes $4800 ? 14. A building is insured in several companies for $28000. During a fire the building is damaged to the amount of $13500. What is the loss of company A, whose risk is $5000 ? 15. A invests in business $8450, and B $7200, and the gain or loss is divided according to the investments. "What is each part- ner's share of gain, the total gain being $3474.30 ? 16. The U. S. gold dollar (181, 183) contains 23.22 (25.8 A) g ra i ns of P ur e gold, and the standard silver dollar 371.25 (412.5 3*5-) grains of pure silver. What is the relative value of pure gold to pure silver ? 17. The assessed value of the property of a certain town is $325000, and the total tax is $10238. How much is the tax of Mr. A, whose property is valued at $5700 ? 18. A company with a capital of $250000 divides $8750 among its stockholders. How much will be received by a stockholder who owns 36 100-dollar shares ? 19. If a long ton of coal is worth $4.25, what is the value of a short ton ? 20. If a farm valued at $4500 is taxed $26.24, what should be the tax on property valued at $23500 ? 21. A merchant gains $625 by selling $12000 worth of goods ; what amount must he sell to gain $8000 ? 22. How many feet of boards will be required for a fence 764 feet long, if 888 feet of boards are required for 288 feet ? 23. If one franc is worth $0.193, and one pound sterling, $4.8665, what is the value of the pound sterling expressed in francs ? 24. If 2175 yards of cloth are made from 458 pounds of yarn, how many pounds of yarn would be required to make 1200 yards of cloth ? 216 RATIO AND PROPORTION. [Art. 523. 25. If a railway train goes 412 miles in 9 lir. 30 min., how many hours would it require to go 900 miles ? 26. The railway fare from A to B, a distance of 228 miles, is $6.75. What should be the fare from A to C, a distance of 375 miles ? 27. A certain quantity of grain will last 92 horses 48 days. How long will it last 64 horses ? 28. A house and lot are worth $9600, and the value of the lot is to the value of the house as 5 to 11. Find the value of the lot. 29. A merchant failing, owes $11375, and has property worth $4425. How much will he pay a creditor whom he owes $2345 ? 80. The distance between two poles was measured as 48 yards, but the yard measure was \ of an inch too short. What was the actual distance ? SI. From a field of wheat containing 375 acres, 4850 bushels are harvested. How many bushels would be harvested from a field containing 344 acres of similar wheat ? 82. If the tax on property, valued at $6500, is $144, what should be the tax on property valued at $3800 ? 83. If a Troy ounce of standard silver is worth 85 cents, what is the intrinsic value of the standard silver dollar (112) ? 84* If the freight on 575 pounds is $1.84, what should be the freight on 975 pounds ? 85. If a railway company charges $13 for carrying one ton 480 miles, what should be the charge on one ton for 650 miles ? 86. The through rate from A to C, a distance of 900 miles, is $48 per car. What should be the portion of the A. & B. R.K. (425 miles), and the portion of the B. & C. R.R. (475 miles)? 87. If 48 men can do a certain piece of work in 60 days, in what time can 64 men do the same work ? 38. If a Troy ounce (337) of silver is worth $1.20, what is the value of an Avoirdupois ounce ? 89. The ratio of the diameter of a circle to its circumference is 1: 3.1416. What is the circumference of a circle whose diameter is 475 feet ? 40. If 276 long tons of coal last a manufacturer 21 months, how long would 276 short tons last him ? 41. A bankrupt can pay 48 cents on a dollar. If his assets were $1887 more, he could pay 65 cents. Find his debts and his assets. Art. 524.] COMPOUND PROPORTION. 217 COMPOUND PROPORTION. 524:. Compound Proportion is an equality of a compound ratio and a simple ratio, or of two compound ratios. Th -' 8 ITslVdays H * 10: * 4 ! 5 M * ! "" ^"^ proportions. Ex. If 12 men earn $60 in 4 days, how much will 10 men earn in 2 days ? ANALYSIS. Since the answer (fourth term) is re- 4-2 ( :: 60:rr - .ftct **.** ^**- ** **'** #**,** EXCHANGE. 545. Exchange is the system by which merchants in distant places discharge their debts to each other without the transmission of money. Suppose, for example, A of New York owes B of Chicago $1000 for grain, and C of Chicago owes D of New York $1000 for dry goods. The two debts may be discharged by means of one draft or bill of exchange without the transmission of money. Thus, B of Chicago draws on A of New York for $1000, and sells the draft to C of Chicago, who remits it to D of New York. D of New York presents the draft to A of New York for acceptance or pay- ment, and thus both debts are cancelled. There is in effect a setting-off or exchange of one debt for the other. The business of exchange is usually conducted through the medium of banks and bankers, who buy commercial bills and transmit them for credit to the places on which they are drawn. They also sell their own drafts on their correspondents in any amounts demanded. The greater part of the exchange in the United States is effected through the banks of New York, Boston, Philadelphia, Chicago, St. Louis, Baltimore, and San Francisco. The financial centres of Europe are London, Paris, Antwerp, Geneva, Amsterdam, Hamburg, Frankfort, Bremen, Berlin, and Vienna. 546. A Bill of Exchange, or Draft, is an order or request addressed by one person (the Drawer) to another (the Drawee), directing the payment of a specified sum of money to a third per- son (the Payee) or to his order. It is issued at one place and payable at another. (See Art. 495, 5-6.) For brevity, bills of exchange are frequently called "exchange." According to the laws of most States, drafts drawn in one State and pay- able in another, are termed foreign bills of exchange. For the purposes of this book, the term " domestic exchange" will be applied to bills drawn and payable in the United States. 547. Bills of exchange are of two kinds, Inland or Domestic, and Foreign. 548. A Domestic or Inland Bill of Exchange is ona which is payable in the same country in which it is drawn. 226 EXCHANGE. [Art. 549. 549. A Foreign Bill of Exchange is one which is payable in a different country from the one in which it is drawn ; as a draft drawn in the United States and payable in England. 05 O. When drafts sell for more than their face value, exchange is above par or at a premium ; when for less than their face, below par or at a discount. When Chicago owes New York the same amount that New York owes Chicago, exchange will be at par ; that is, drafts will sell at their face value. When Chicago owes New York more than New York owes Chicago, drafts on New York will sell at a premium ; there will be more buyers of exchange than sellers, and drafts will sell for more than their face value. When Chicago owes New York less than New York owes Chicago, the demand in Chicago for drafts on New York will be less than the supply, and drafts will sell for less than their face value, or at a discount. DOMESTIC EXCHANGE. 551o Domestic or Inland Exchange relates to drafts drawn at one place on another in the same country. 552. The domestic exchanges on New York at the places named were quoted as follows, May 7, 1881 : Savannah, -J- @ f premium; Charleston, -J @ J- premium; New Orleans, $1.50 @ $2.50 premium; St. Louis, 25 cents premium ; Chicago, 50 @ 75 cents premium ; and Boston, 25 cents discount. 1. At Savannah and Charleston the rates per cent, of the premium or discount are given. Thus, when exchange is quoted at -|- premium, a draft of $100 may be purchased for $100} ($100.25). 2. At New Orleans, St. Louis, Chicago, and Boston, the premium or dis- count per $1000 is given. Thus, a draft of $1000 at $2.50 premium may be purchased for $1002.50. $2.50 per $1000 premium is equivalent to \% premium. 3. The selling rates are about \% ($1.25) higher than the buying rates, and bankers' exchange is usually higher than commercial. 4. The rate of domestic exchange is limited by the cost of shipping gold or currency by express, and the premium or discount will not exceed this cost. Thus, if a merchant in Chicago is charged a premium of $10 for a draft of $10000, and lie can send the currency by express for $7.50, it will be to his advantage to remit by the latter method. The following appeared in a New York financial paper, May 8, 1881, the date of the above quotations : " The domestic exchanges at the West are sufficiently high to permit of a movement of funds Eastward, but at the East, Art. 552.] EXCHANGE. 227 New York funds are still at a discount and some shipments of gold and currency continue to be made to the Eastern cities." 5. The preceding quotations refer to sight exchange. Time drafts are dis- counted in the same manner as promissory notes. In certain cases bankers in discounting notes and drafts payable in distant places, charge interest for the time required for the return of tho money when the note or draft is paid ; and in the case of drafts drawn a certain number of days after sight, bankers sometimes charge interest for the time required for the acceptance of the drafts. Thus, if a draft was drawn in New York on St. Louis and payable 60 days after sight, it would require, in the ordinary course of the mails, 3 days for the acceptance of the draft. The draft would be paid in 63 days (including the days of grace), and 3 days would elapse before the money would be returned to New York. The banker would be justified in charging interest for 69 days, the interval between the day he advanced the money in New York, and the day it was returned to him again. If the draft was drawn on San Francisco, fully 19 days (8 days for the acceptance, 3 days of grace, and 8 days for the return of the money) would be added to the time of the draft. Between New York and San Francisco and other distant places, money is frequently transferred by telegraph. (See Art.- 499, 3.) EXAMPLES. 553. 1. What is the value in Savannah of a draft on New York for $8750 at \% premium ? 2. Find the cost in New Orleans of a draft on New York for $8375 at $2.50 premium. Find the Value of the following drafts : Face. Exchange. Face. Exchange. 5.15000, \% premium. 8. $4287.75, 15? discount. 4. $4375, \% discount. 9. 83416.33, 25? premium. 6. $8417, \% premium. 10. $2825.49, $1.25 discount. 6. $9873, \% premium. 11. $9873.62, $2.50 premium. 7. $5284, % discount. .2. $8412.75, 75^ discount. 13. A of Chicago buys cattle for B of- New York to the amount of $9858.07. How large a draft should be drawn on B, so that when sold at a discount of 5W (-$%}, the proceeds would be sufficient to pay the bill ? NOTE. To find the face of a draft, instead of dividing the value of the draft by the rate of exchange (in the above example, .99^f or .9995), business men and bankers calculate the premium or discount on the value of the draft, and subtract or add it to the value as the case requires. Thus, in the above example, the discount would be \ of -fa% of $9858.07, or $4.93, which added to the given proceeds would produce the face $9863. This method produces too small a result in all cases, the error being equivalent to the percentage of the premium or discount. In this example the error is less than \ cent. EXCHANGE. [Art. 553. For ordinary examples in business, the foregoing method is sufficiently- accurate. At \ % , or $5.00 (a very high rate for domestic exchange) on a draft whose value is $10000, the error would be only 25 cents. If greater accuracy is required, the necessary correction can be made by adding the percentage of the premium or discount. Thus, if the value of the draft is $10000, and exchange is \% discount, the face would be $10000 + $50 (\% of $10000) + $0.25 (\% of $50) = $10050.25. If at \% premium, the face would be $10000 - $50 + $0.25 = $9950.25. By the above method, find the face of the following drafts : Value. Exchange. Value. Exchange. 14. $1876.16, J# premium. 19. $7375, 25^ premium. 15. $2437.75, J^ discount 20. $9218, 50^ discount. 16. $3342.38, \% discount. 21. $6438, $1.00 premium. 17. $2238.42, J$ premium. 22. $9243, $1.25 premium. 18. $8175.50, \% premium. 28. $5280. 75^ discount. 24. A of New Orleans being indebted to B of New York $9316.75, forwards to him a check on a New Orleans bank for that amount, to cash which B is obliged to allow a discount of \%. How much does A still owe B, and for what amount should theicheck have been drawn to net B the amount due ? 25. What is the value of a draft on New York for $3000, payable in 60 days (63 days) after date (494, 7), exchange being J$ premium, and interest 6%? NOTE. From the face of the draft, subtract the interest, and to the result add the exchange. 26. Find the proceeds of a draft drawn at Chicago on New York for $12000, and payable 90 days after sight, exchange 50^ discount, interest 5$, and allowing 3 days additional for the acceptance of the draft. 27. A banker in New York discounts a draft for $8000, pay- able in San Francisco 60 days after sight ; what would be the proceeds, exchange being \% discount, interest 6%, and allowing 8 days for the acceptance and 8 days for the return of the money ? 28. A merchant paid $6920.64 in Charleston for a sight draft of $6912 ; what was the rate of exchange ? 29. A commission merchant sold 13475 pounds of leather at 26} cents a pound. If his commission is 5%, and exchange \% premium, how large a draft can he buy to remit to the consignor ? SO. How large a 60-days' draft must I draw, so that when sold it will produce $10000, exchange \% discount, interest Art. 554.] FOREIGN EXCHANGE. 229 FOREIGN EXCHANGE. 554. Foreign Exchange relates to drafts or bills of exchange drawn in one country and payable in another. 555. To secure safety and speed in the transmission of foreign bills of exchange, they are drawn in sets of two or three of the same tenor and date. The separate bills are sent by different steamers, and when any one of them is paid, the others become void. Some merchants send only the first and second, and pre- serve the third. SET OF EXCHANGE. (I-) EXCHANGE FOR 1000. NEW YORK, May 16, 1889. Sixty days after sight of this FIRST of Exchange (Second and Third unpaid), pay to the order of A. T. STEWART & Co., One Thousand Pounds Sterling, value received, and charge the same to account of No. 1738. BROWN BROTHERS & Co. To BROWN, SHIPLEY & Co., | London, England. j (2.) EXCHANGE FOR 1000. NEW YORK, May 16, 1889. Sixty days after sight of this SECOND of Exchange (First and Third unpaid), pay to the order of A. T. STEWART & Co., One Thousand Pounds Sterling, value received, and charge the same to account of No. 1738. BROWN BROTHERS & Co. To BROWN, SHIPLEY & Co., \ London, England. J (3.) EXCHANGE FOR 1000. NEW YORK, May 16, 1889. Sixty days after sight of this THIRD of Exchange (First and Second unpaid), pay to the order of A. T. STEWART & Co., One Thousand Pounds Sterling, value received, and charge the same to account of No. 1738. BROWN BROTHERS & Co. To BROWN, SHIPLEY & Co., ] London, England. J 230 EXCHANGE. [Art. 555. Foreign bills of exchange are usually drawn in the moneys of account of the countries in which they are payable. Thus, drafts on England are usually drawn in pounds, shillings, and pence; on France, Belgium, and Switzerland, in francs; on Germany, in marks; on the Netherlands (Holland), in guilders. Foreign bills of exchange are usually drawn at sight (3 days) or at sixty (63 days) days' sight. Sight drafts are frequently called " short " exchange, and 60 day drafts, "long" exchange. "Long" exchange is sold at a rate below that for " short " exchange, sufficient to equalize the difference in interest between the dates of maturity of the two classes of bills. 556. A Letter of Credit is an instrument issued by a- banker and addressed to bankers generally, by which the holder may draw funds at different places and in amounts to suit his convenience,, the total amount drawn not exceeding the limit of the letter of credit. A bill of exchange is payable at a certain place, at a certain fixed time, and for a certain amount, while a letter of credit is payable at different places, at different times, and in different amounts. A person who intends to travel in foreign countries, may procure a letter of credit by depositing either cash or securities with a foreign exchange banker for the amount of the letter. When the American banker is notified of the payment of the traveler's drafts in London, he debits the account erf the holder of the letter of credit with the amount drawn and the charges, at the current rate of exchange. A small rate of interest is sometimes allowed on the account, and a settlement is made on the return of the traveler. 557. The Intrinsic Par of Exchange is the value of the monetary unit of one country expressed in that of another, and is based on the comparative fineness and weight of the coins, as determined by assay. (See Art. 566.) 558. The Commercial Par of Exchange is the market value in one country of the coins of another. 559. The Commercial Bate of Exchange is the market or buying and selling value in one country of the drafts on another. 1. In giving quotations of foreign exchange, no reference is made to the par value, the quotations being given by means of equivalents. 2. Premium or discount for exchange cannot long exceed the transporta- tion charges and insurance of shipping coin; for, if a merchant can ship gold cheaper than he can buy a bill of exchange, he will choose the former method of paying his indebtedness. When sight exchange is 4.84, gold can be im- ported at a small profit; and when sight exchange is 4.89, gold can be exported at a profit. Art. 559.] FOREIGN EXCHANGE. 231 3. When exchange is above par, we are exporters of gold ; when below par, we are importers of gold. 560. Exchange on England (Sterling exchange) is quoted by giving the value of 1 in dollars and cents. Thus, when exchange is 4.84, a draft of 1 will cost $4.84; of 100, $484. The intrinsic par value of 1 is $4.8665 (566). 561. Exchange on Franqe, Belgium, and Switzerland is quoted by giving the value of $1 in francs and centimes (hundredths of a franc). Thus, when exchange is 5.27^, $1 will buy a bill of 5 francs and 27-|- centimes; a draft of 1000 francs will cost $189.57 (1000 -H 5.27|). The intrinsic par value of 1 franc is 19^- cents (566) ; of the equivalent exchange, 5.18J (1.00 -4- .193). In French, Belgian, and Swiss exchange, the higher the apparent rate, the less the value of the draft. Thus, when exchange is 5.13, a draft of 1000 francs is worth $194.93, and each franc is worth 19^ cents. When exchange is 5.26f, the same draft would be worth $189.98, and each franc 19 cents. 562. Exchange on Amsterdam (Netherlands) is quoted by giving the value of one guilder (gulden) or florin in U. S. cents. The intrinsic par value of 1 guilder is 40 T 2 TI cents (566). 563. Exchange on Germany is quoted by giving the value of 4 reichsmarks in cents. The intrinsic par value of 1 mark is 23 T % cents (566) ; of 4 marks 95^ cents. 564. Documentary Exchange is a bill drawn by a shipper upon his consignee against merchandise shipped, accompanied by the letter of hypothecation, the bill of lading " to order," and the insurance certificates covering the property against which the bill is drawn. 565. Exchange on London in the countries named, and at London on the same countries, is quoted as follows : United States, by giving the value of 1 in dollars and cents. France and Belgium, by giving the value of 1 in francs and centimes. Germany, by giving the value of 1 in marks and pfenniges. Austria, by giving the value of 1 in florins and kreutzers. Netherlands, by giving the value of 1 in guilders and cents. India, by giving the value of 1 rupee in shillings and pence. 232 EXCHANGE. [Art. 566. 566. FOREIGN MONEYS OF ACCOUNT. Country. Standard. Monetary Unit. Value in U S. Gold. Argentine Republic. Austria, Gold and silver. Silver Peso of 100 centavos . . . Florin of 100 kreutzers . .96,5 .359 Gold and silver. "Franc of 100 centimes. . .19,3 Bolivia Silver b Boliviano,100 centavos .72,7 Brazil Gold Milreis of 1000 reis. . .54,6 British America/ Gold Dollar of 100 cents . $1.00 Chili Gold and silver. Peso of 100 centavos . . . .91,2 Cuba Gold and silver. Peso of 100 centavos.. . .93,2 Denmark .... Gold c Crown of 100 ore .26,8 Ecuador. Silver b Sucre of 100 centavos . .72,7 Effvpt Gold Pound of 100 piasters 4.943 France Gold and silver. 'Franc of 100 centimes. . .19,3 German Empire Gold. . Mark of 100 pfennige 238 Great Britain Gold . . Pound of 20 shillings 4866i Greece Gold and silver. Drachma of 100 lepta .19,3 Hayti Gold and silver. d Gourde of 100 centavos .96,5 India . Silver Rupee of 16 annas 6 .34,6 Italy Gold and silver. tt Lira of 100 centesimi .193 Gold and silver. YenoflOOsen-S Gold " .99,7 Liberia . Gold ( Silver. Dollar of 100 cents . . .78,4 1 00 Mexico . . ... Silver Dollar of 100 centavos. 79 Netherlands Gold and silver. Florin of 100 cents 402 Norway . Gold. ... c Crown of 100 ore 26 8 Peru Silver b Sol of 100 centavos 72 7 Portugal Gold Milreis of 100 reis 108 Russia Silver Rouble of 100 copecks . 582 Spain Gold and silver. "Peseta of 100 centimes 193 Sweden Gold c Crown of 100 ore 268 Switzerland Gold and silver. *Franc of 100 centimes 193 Tripoli Silver Mahbub of 20 piasters 656 Turkey Gold Piaster of 40 paras. . 044 U. S. of Colombia. . Silver b Peso of 100 centavos 72 7 Venezuela Gold and silver. *Bolivar of 100 centavos 193 The above rates, proclaimed by the Secretary of the Treasury, Jan. 1, 1887, are used in estimating, for Custom House purposes, the values of all foreign merchandise made out in any of said currencies. ( a ) The franc of France, Belgium, and Switzerland, the peseta of Spain, the drachma of Greece, the lira of Italy, and the bolivar of Venezuela have the same value. ( b ) The sucre of Ecuador, the peso of United States of Colombia, the boliviano of Bolivia, and the sol of Peru have the same value. ( c ) The crowns of Norway, Sweden, and Denmark have the same value. ( d ) The gourde of Hayti and the peso of the Argentine Republic have the same value. ( e ) The anna contains 12 pies, Art. 567.] EXAMPLES. 233 EXAMPLES. 567. 1. Find the cost of a bill of exchange on London for 225 at 4.81}. (56O) ANALYSIS. If 1 costs $4.81f, 225 will cost 225 times $4.81. 2. What is the value of a draft for 324 16*. at 4.87J ? ANALYSIS. Write one-half of the greatest even number of shillings as tenths of a pound, and if there be an odd shilling write 5 hundredths. 324 16s. = 324.8. (See Art. 342, Ex. 12, Note.) The value of 324 16s. at 4.87 is found by multiplying $4.87 by 324.8. 3, Find the value of a draft on London for 379 12*. Id., at 4.86|. OPERATION. 379.6 4.861 ANALYSIS. If each penny be regarded as 2 cents, the result will be sufficiently accurate. For lid. the maximum number of pence in any example, and exchange at 4.91, the error would be only $ cent. $4.86f x* 379.6 = $1846.28. $1846.28 + $0.14 = $1846.42. To save one addition, add the 14 cents to the partial products as in the operation. 1846.420 Find the value of Find the value of 4. 500 at 4.81. 8. 512 13*. at 4.84|. 5. 775 at 4.85J. 9. 834 6s. Gd. at 4.1 6. 837 at 4.83-J. 10. 675 11*. Sd. at 4.87|. 7. 84 8s. at 4.85. 11. 225 7*. 5d. at 4.82}. 12. Find the cost of a bill of exchange on Liverpool, for 875 12s. Qd. at the par value. (56O) 18. What are the proceeds of a draft of 959 5*. 4d., sold through a broker, at 4. 79 J-, brokerage \% ? 14. An exporter sold a draft for 540 3s. on Manchester, pay- able in London, at 4. 84, brokerage \%. What were the proceeds ? 15. Find the proceeds of a draft on Newcastle-on-Tyne, at 60 days' sight for 1764 15*., payable in London, at 4.82, brokerage on exchange \%. 16. An importer purchased a bill of exchange on London, at 3 days' sight, for 488 16*. 6d., at 4.85J. What was the cost ? 234 EXCHANGE. [Arl. 567. 17. How much exchange on London at 4.81f will $821.99 buy ? ANALYSIS. $4.Slf will buy exchange for 1 ; hence, $821.99 will buy as many pounds as $4.81f are contained in $821.99, or 170.625. 170.625 = 170 12s. Qd. (See Art. 289, and Art. 342, Ex. 19, Note.) 18. What will be the face of a 3 days' bill of exchange on London that can be bought for $5964.13, exchange 4.86|- ? 19. The face of a bill of exchange was 875, and its cost was $4233.91. What was the rate of exchange ? 20. An exporter received $9063.22 for a bill of exchange that was sold through a broker at $4.86J; what was the face of the bill, the broker's commission being \% ? 21. Find the cost of a bill of exchange on Paris for 7000 francs at 5.21J. OPERATION. 5.21-J ) 7000 ANALYSIS. Since 5.21| francs cost $1, g g 7000 francs will cost as many dollars as 5.21 1 francs are contained times in 7000 francs. 41.75 ) 56000.0000 ( Find the value of Find the value of 22. 6000 francs at 5.16. 25. 8475 francs at 5. 19}. 28. 5000 francs at 5.18|. 26. 7216 francs at 5.17}. 24. 4000 francs at 5.21|. 27. 987.60 francs at 5.20|. 28. Find the cost of a draft on Antwerp at 3 days' sight, for 9640 francs, at 5.19f. 29. What is the value of a draft on London for 416 165. 3d., at 4.85|? SO. Sold exchange on Geneva, through a broker, for 8000 francs at 60 days' sight ; what were the proceeds of the draft, exchange being 5.20f, brokerage \% ? 31. What are the proceeds of a draft on Paris for 12420 francs, at 5.19|, brokerage on exchange \% ? 82. What will it cost to remit to Antwerp 8750 francs at the par value ? (561) 83. Sold through a broker a draft on Geneva for 7324 francs. What were the proceeds, exchange being 5.18|, brokerage \% ? 84. What will be the face of a bill of exchange on Geneva that can be bought for $15372, exchange selling at 5.22J ? 85. Paid for a draft on Paris $3460.32 ; what was the face of the draft, exchange being 5.19-f ? Art. 567.] FOREIGN EXCHANGE. 235 36. A merchant paid $6272 for a bill of exchange of 32512.48 francs ; what was the rate of exchange ? 37. Find the cost of a bill of exchange on Hamburg for 14400 marks (Reichsinarks) at 94|. OPERATION. 4 ) 14400 ANALYSIS. Since 4 marks cost $0.94|, 14400 marks will cost 3600 (14400 -i- 4) times $0.94i, or $3388.50. 3388.50 Find the value of 38. 7200 marks at 94. 41. 1237 marks at 93|. 39. 8416 marks at 93. J$. 9894 marks at 95|. 40. 3456 marks at 95J. 43. 6515 marks at 94J . 44- What is the cost of a bill of exchange on Frankfort for 16200 marks at 95 ? 45. Sold a bill of exchange on Hamburg for 13200 marks, at 94 ; what was the amount received, brokerage \% ? 46. An importer purchased a bill of exchange on London for 318 10s. 1d. 9 at 4.85J ; what did it cost ? 47. What were the proceeds of a draft, sold through a broker, f for 8748 marks, at 94, brokerage \% ? 48. An exporter sold a draft on Paris for 12275 francs, at 5.19| ; what were the proceeds, brokerage \% ? 49. What is the face of a bill on Hamburg that cost $816, exchange 94J ? ANALYSIS. Since $.94$- will buy 4 marks, $816 will buy 4 times as many marks as $0.94 is contained times in $816. 50. What is the face of a 3 days' draft on Bremen, that was purchased in New York for $3261.60, exchange 94| ? 51. The cost of a draft of 12320 marks was $2922.15 ; what was the rate of exchange ? 52. Find the cost of a bill of exchange on Amsterdam, for 7240 guilders, at 40. 53. Find the cost of a bill of exchange on Amsterdam, at GO days' sight, for 12480 guilders, exchange 39|. 54. An exporter received $1890.86 for a bill of exchange on Amsterdam; what was its face, exchange being 41, brokerage 236 EXCHANGE. [Art. 567. 55. At 40-f, how much exchange on Amsterdam will $2877.93 buy? 56. The value of a draft of 5280 guilders is $2145 ; what is the quotation ? 57. The dividends of the N. Y. C. and H. R K. Co., are paid in London at the rate of 49 pence to the dollar. What is the equivalent rate of exchange ? 58. Find the value in U. S. money of 16319 bushels of wheat at 45. tyd. per bushel, exchange 4.86J. 59. A merchant sent a messenger with a bill of exchange of 20000 francs to two bankers, A and B, with instructions to sell it to the best advantage. A offered 5.27 and B 5.27^. The messen- ger imprudently accepted the latter offer. How much did the merchant lose by the ignorance of the messenger ? 60. When United States 4 per cent, consols are quoted in New York at 114J, and sterling exchange at 4.83J, what should be the London quotation of the bonds ? What should be the London quo- tation of 4J per cent, bonds, the New York quotation being 113J ? NOTE. In London, all American securities are quoted on an assumed value of the pound sterling of $5, instead of the actual value of $4.8665, or, more definitely speaking, its commercial value determined by the rate of exchange. Multiplying the New York quotation by 5 and dividing by the rate of exchange, the result will be the equivalent London quotation. 61. When American railway stocks are quoted in London at 88, what is the equivalent New York quotation, sterling exchange being quoted in New York at 4.88J ? 62. What is the London equivalent of a New York quotation of 142, exchange being 4.83 ? 63. At Paris, what is the value of a draft on London of 550, exchange being 25.36J? 64. At London, what is the cost of a draft on Hamburg of 8000 marks, exchange being 20.45 ? 65. At Vienna, what is the cost of a draft on London of 625, exchange being 11.75 ? 66. At London, what is the value of a draft on Calcutta of 12000 rupees, exchange being quoted at Is. S^d. ? 67. A commission merchant wishes to remit $2475 to his principal in England. How large a draft must he purchase, exchange being 4.83 J ? EQUATION OF ACCOUNTS. 568. Equation of Accounts (called also Equation of Pay- ments and Averaging Accounts) is the process of finding the time when several debts due at different dates may be paid in one amount without loss of interest to either party. It is also the process of finding the time when the balance of an account having both debits and credits may be paid without loss of interest to either party. This time is called the equated or average time. NOTE. It is important that the commercial student be thoroughly drilled in the theory and practice of Equation of Accounts, as examples in this sub- ject are of frequent occurrence in many wholesale and commission houses. 569. To find the equated time when the items of the account are all on the same side, i. e., all debits or all credits. ANALYTICAL STEPS. By assuming a certain date as the time of settle- ment, we find what the loss or gain of interest would be to the payer if all the bills were paid by him on that date. We next find in how many days the total amount of the bills would produce a sum equivalent to this loss or gain of interest, and find the true day of settlement by counting forward or back- ward this number of days from the assumed date. Thus, if the sum of the several bills is $1000, and the loss of interest to the payer at the assumed date of settlement is $10 (the interest of $1000 at 60 days at 6%), it is evident that the true date of settlement, or the time when there would be no loss of interest to either party, must be 60 days after the assumed date. NOTES. 1. The interest on the bills paid after they became due would equal the interest on the bills paid in advance, the former being a gain to the payer, and the latter, a loss. 2. Any date may be assumed as the time of settlement. For convenience, the earliest or latest date is generally used. If the earliest date is taken, the estimated interest is a loss to the payer ; if the latest is taken, the interest is again. 238 EQUATION OF ACCOUNTS. [Art, 569. When the time is found by Compound Subtraction, or each mouth is regarded as 30 days, the last day of the month preceding the earliest item is the most convenient. (See second interest method.) In Equation Tables, Dec. 31 or Jan. 1 is taken for all examples. The assumed date is sometimes called the focal date. 3. Any rate of interest may be used in making the computations, 6 and 12 being the most convenient rates. 57O. Ex. At what date may the following bills of merchan- dise be paid in one amount without loss of interest to either party? Due Apr. 10, $114; due Apr. 26, $140; due May 22, $320 ; due June 6, $976. OPEBATION. PRODUCT METHOD. Due Apr. 10, $114 x = " 26, 140 x 16 = 2240 " May 22, 320 x 42 = 13440 " June 6, 976 x 57 = 55632 1550 ) 71312 ( 46 days after Apr. 10, or May 26. ANALYSIS. For convenience, assume Apr. 10, the earliest due date, as the time of settlement. If the first bill, which is due Apr. 10, is paid on that date, there will be no loss or gain of interest to either party. If the second bill, which is due Apr. 26, is paid Apr. 10, 16 days before it is due, there will be a loss to the payer of the interest or the use of $140 for 16 days, or $2240 for 1 day. On the third bill, there will be a loss of the interest of $320 for 42 days, or $13440 for 1 day. On the fourth bill, there will be a loss of the interest of $976 for 57 days, or $55632 for 1 day. If all the bills are paid Apr. 10, there will be a loss to the payer of the interest of $71312 for 1 day, or of $1550 for 46 days. Since the loss of interest to the payer is equivalent to the interest of the total amount of the bills for 46 days, it is evident that the day when there would be no loss of interest must be 46 days after Apr. 10, or May 26. The payer is entitled to defer payment 46 days after the assumed date as a compensation for the estimated loss. The gain of interest to the payer on the first three bills, which are paid after they are due, equals the loss of interest on the fourth bill, which is paid before it is due. PROOF. The interest of $114 for 46 days at 6^ is $0.874 " " 140 " 30 " " 70 " 320 " 4 " " .213 Total gain of interest to the payer 1.787 The interest (a loss to the payer) of $976 for 11 days is . 1.789 Art. 570.] EQUATION OF ACCOUNTS. 239 NOTES. 1. In finding the number of days from the assumed date to the other dates, instead of calculating from the assumed date each time, find the interval from one date to the next and add it to the last number of days. Thus, from Apr. 10 to May 22 is 42 days, and from May 22 to June 6, 15 days ; hence, from Apr. 10 to June 6 is 57 (42 + 15) days. (See Art. 31O, Ex. 3.) 2. To determine the due date, find the number of days in the operation nearest to the quotient, and add or subtract, as may be necessary, the differ- ence between it and the quotient, to its corresponding date. Thus, in the above example, the number of days in the operation nearest to the quotient is 42 ; hence the due date is 4 (46-42) days after May 22, or May 26. (See Art. 311, Ex. 10.) S. If the fraction of the quotient is less than -J-, disregard it ; if more than ^, add 1 day to the integral number of days in the quotient. 571. RULE FOR THE PRODUCT METHOD. Assume the earliest due date as the day of settlement for all the items. Multiply each item by the number of days intervening between the assumed date of settlement and the date of the item; and divide the sum of the several products by the sum of the account. Count fonvard from the assumed date the number of days obtained in the quotient. TJie result will be the equated time. 572. OPERATION. FIRST INTEREST METHOD. Days. Interest Due Apr. 10, $114 $.00 tt 26, 140 16 ( - 233 for 10 days. ( . 14 " 6 (i May 22, 320 42 ( 1 - 60 64 " 30 " 12 a ( 4 88 " 30 " tt Juno G, 976 57 2, 44 " 15 " 60) 15.50 (l. 952 " 12 (i .258 ) 11.885 ( 46 days after Apr. 10, or May 26. ANALYSIS. Assume Apr. 10, the earliest due date, as the time of settle- ment. If the total amount ($1550) of the bills is paid Apr. 10, the assumed date of settlement, there will be a loss of interest to the payer of $11.885. The interest of $1550 for 60 days at 6^ is $15.50, and for 1 day, $0.258. It will take $1550 to produce $11.885 interest as many days as $0.258 is contained times in $11.885, or 46 days. If, at the assumed date of settlement, there is a loss to the payer of the interest of $1550 for 46 days, the true day of settle- ment must be 46 days later, or May 26. 240 EQUATION OF ACCOUNTS. [Art. 573 573. OPERATION. SECOND INTEREST METHOD, Mo. Days. Interest. Apr. 10, $114 $0.19 j .466 for 20 days. /CO. J.4:U < f 1.60 " I mo. 1 May 22, 320 \ 1.067 " 20 days. .107 " 2 " 7.75 ) 14.306 ( 1 wo. 25 da. after Mar. 31, 7.75 or May 25. 6.556 30 7.75) 196. 680 (25 days. 1550 293 ANALYSIS. By this method, the last day of the month preceding the earliest due date is assumed as the date of settlement, and the time is found by Compound Subtraction, each month, being regarded as 30 days. The months are placed on the margin and the days correspond with the number of days in the given dates. Mar. 31, the assumed day of settlement, there is a loss to the payer of $14.306 interest, or the interest of $1550 for 1 mo. 25 da. The equated time is therefore 1 mo. 25 da. after Mar. 31, or May 25. Since this method regards all months as 30 days each, its results are not strictly accurate. The error in this example is 1 day. (See preceding results.) When this method is used, and accurate results are required, the necessary corrections may be made by adding to the intervals of time 1 day for each intervening month containing 31 days. If the month of February is included, 2 days should be subtracted in a common year and 1 day in a leap year. In counting forward to find the equated time, the opposite correction should be made. Thus, if the assumed date is June 30 and the quotient is 2 mo. 20 da., the equated time would be Sept. 18, 2 days being subtracted for July and August. The following is the corrected operation for the given example, 1 day being added to the time of the fourth item for the month of May. The result is the same as by the product and the first interest methods. Ait. 573.J EQUATION OF ACCOUNTS. 241 OPERATION. Mo. Days. Interest. Apr. 10, $114 $0.19 26 140 .466 for 20 days. ' " 6 " " I mo. 1 May 22, 320 1.067 " 20 days. " 2 " " 2 mo. 2 June 6 + l,_976 < .976 " 6 days. 2 ) 15.50 ( .162 " 1 " 7.75 ) 14. 468(1 mo. 26 r/0. after Mar. 31, 7.75 or May 26. 6.718 _30 7.75)201.540(26 days. EXAMPLES. 574. 1. At what date may the following bills be paid in one amount without loss of interest to either party ? Due Sept. 10, $145; Sept. 28, $144; Oct. 8, $75 ; Oct. 23, $512. 2. What is the equated time for the payment of the following bills? Due Mar. 28, $446; May 3, $212; May 15, $116; May 31, $475 ; June 12, $345. 3. What is the average due date of the following bills, each being due at the date given ? Jan. 5, $127.85 ; Jan. 26, $134.18 ; Feb. 5, $249.40 ; Feb. 23, $418.73 ; Feb. 28, $176.25. NOTE. The result will be practically the same if the nearest dollar is used in multiplying or in calculating the interest. Thus, in the above example, regard the amounts as 128, 134, 249, 419, and 176 respectively. When there are several items in the example, some accountants omit the cents and units of dollars, and use the nearest number of tens. Thus, if the above account were of sufficient length, the numbers might be regarded as 13, 13, 25, 42, and 18 respectively. In this example the result is the same, but in some examples, containing the same number of items, there would be a discrepancy of one or more days. 4. Sold a customer bills at the due dates and to the amounts specified : June 1, $152.73 ; June 15, $114.28 ; July 16, $247.84 ; July 25, $88.90 ; Aug. 18, $735.42 ; Aug. 29, $416.34. When may the whole indebtedness be equitably discharged at one payment ? 242 OF ACCOUNTS. [Art. 574. 5. Average the following account : NEW YORK, July 1, 1882. MESSRS. KICE, STIX & Co., To LORD & TAYLOR, Dr. 1882. Apr. 4 Mdse. 30 days per bill rendered. 1816 37 " 21 " 30 " " " . . 724 25 May 13 (f 3Q f( What is the equated time for the payment of the balance of tho following account ? Dr. A in account with B. Cr. 1882. Mar. 16 Mdse. 4 mo. 444 57 1882, July 1 Cash. . . 400 " 30 " 60 da. 376 82 " 20 t( 375 Apr. 20 " 30 da. 712. 19 Aug. 16 re 700 May 17 " 4 mo. 628 75 " 30 a 600 " 28 " 4 mo. 419 31 5. Average the following account. What will be the amount due Jan. 1, 1882 ? Dr. C in account with D. Cr. 1881. 1881. Jun3 16 Mdse. 30 da. 517 25 June 16 Note60(63)<7. 1000 " 28 " 60 da. 487 50 July 30 Cash. . . 375 July 5 4 mo. 816 75 Aug. 13 Mdse. 4 mo. 900 " 21 " 6 mo. 924 30 Oct. 5 Cash. . . 500 Aug. 12 " 4 mo. 317 65 6. When will the balance of the following account commence drawing interest ? How much would be due Mar. 1, 1883. Dr. ANDREW CARNEGIE, Pittsburg, Pa. Cr. 1882. Sept. 4 Cash 100 1882. Aug. 16 Mdse. 4 mo. 647 13 " 4 Note 4 mo. 900 " 29 " 4 mo. 322 85 Oct. 31 Cash 250 Sept. 4 4 mo. 412 90 Dec. 28 (( 600 " 17 4 mo. 588 33 " 17 30 da. 246 12 Nov. 4 4 mo. 683 45 Art. 582.] EQUATION OF ACCOUNTS SALES. 251 7. Find the equated time for the payment of the balance of the following account. . Dr. JAMES B. FARWELL, Chicago, 111. Or. 1881. 1881. Jan. 4 Mdse. 4 mo. 637 20 Mar. 16 Cash. 300 00 14 " 4 mo. 412 87 Apr, 20 u 400 00 14 60 da. 214 35 May 3 if 200 00 Mar. 16 " 4 mo. 298 60 3 Note 4 mo. 800 00 " 28 30 da. 973 25 8. Average the following account : Dr. ABNOLD, CONSTABLE, & Co. Or. 1882. 1882. Apr. 4 Mdse. 4 mo. 426 32 Apr. 25 Cash. 375 " 20 " Cash. 387 40 June 30 u 600 May 13 60 da. 622 39 July 31 Note 60 da. 600 " 27 " 30 da. 584 75 Aug. 15 Cash. 500 July 5 4 mo. 224 50 Oct. 31 tt 400 " 16 " 4 mo. 838 95 ' 583. To find the equated time for the payment of the net proceeds (423) of an account sales (434). 584. 1. The sales form the credit side of the account, and the charges and advances the debit side. 2. The charges for transportation, cartage, and other items paid by the commission merchant are considered due at the time of the payment of the same. 3. The commission and other after-charges of the commission merchant are considered due by some at the average due date of the sales ; and by others, at the average date of the sales. Since the commission is so small compared with the gross sales, in many examples, it makes no difference which date the commission is considered due. Certain merchants enter the commission at the date the account sales is rendered, and, by so doing, produce a result sufficiently accurate. 4. Many commission merchants, when the consignments are not separated and numbered, enter the sales and commission only on the account sales (See Ex. 4, Art. 586), and enter the advances 252 EQUATION OF ACCOUNTS. [Art. 584. and the general charges in the account current (See Ex. 6, Art. 594). Accounts sales, when the shipments are continuous, are rendered monthly to the manufacturers or consignors, and "sketches " weekly or whenever a sale is made. 5. With the exception of finding the date for the commission and other after-charges, the process of averaging an account sales is exactly the same as that of averaging an account containing both debit and credit items. 585. Ex. What is the equated time for the payment of the net proceeds of the following account sales ? NEW YORK, Dec. 1, 1881. Account sales of Seed For account of WILLIAM STEPHENS & Co. By FRANKLIN EDSO^ & Co. 1881. Nov. 4 45^- lu. Timothy Seed . 30 da. 13JL 79 53 tt 18 50 " Mammoth Cl. Seed 60 da. 9iiJL 450 t( 28 49JJ1 Clover Seed . . Cash. SASL 418 32 947 85 CHARGES. Oct. 31 Transportation 60 00 Dec. 1 Commission 5% as Dec. 22, 1881. . 47 39 107 39 Net proceeds due Dec. 26, 1881. . . 840 40 ANALYSTS. The average due date of the sales is Dec. 22, 1881, which is taken as the due date for the commission. The account sales to be averaged will now be as follows : Dr. Due Oct. 31, 1881, " Dec. 22, " Cr. $60.00 Due Dec. 4, 1881, $79,53 47.39 " Jan. 17, 1882, 450.00 " Nov. 28, 1881, 418.32 By averaging the above, we find the net proceeds, $840.46, are due Dec. 26, 1881. If the commission is considered due Nov. 21, 1881, the average date of the sales, the net proceeds will be due Dec. 28, 1881. NOTE. If the same assumed date, or focal date, be taken in finding the average due date of the sales as in finding the average due date of the net proceeds, the operation of the former will form the credit side of the latter operation. Art. 586.] EQUATION OF ACCOUNTS SALES. 253 EXAMPLES. 586. Find the net proceeds and equated time of the following accounts sales. (Unless otherwise stated, the commission is con- sidered due at the average due date of the sales.) 1. Sales of 400 bbls. flour received per N. Y. C. & H. R. E. K., for account of A. W. ARCHIBALD, Ottumwa, Iowa. 1881. May 11 125 bbls. " Kirkwood " cash, . . 6^. *** ** ({ 12 150 " "Iowa" 4 mo., . 6^ *** 18 125 " "Kirkwood "4 mo., . 7^ *** ***# ** CHARGES. May 3 Transportation and Cartage, . . . 425 tt 4 15 (( 18 Storage, , 45 Commission and Guaranty 5$, . . . ##* #* **# ** Net proceeds due per average, , 1881, *##* ** E. & 0. E. E. R LlVERMORE. NEW YORK, May 20, 1881. What would be the equated time for the payment of the above proceeds, if the commission and guaranty were considered due at the average due date of the sales ? At the average date of the sales ? If considered due May 18, the date of the last sale ? 2. Account sales of 900 sides hemlock sole leather by MAS- SET & JACKET, for account of GRANT & HORTON, Ridgway, Pa. 1881. Aug. ft a 14 18 21 Sides. 400" 300 200 Description. Terms. Weight. Price. w 27J 27i | *#** **** **** ** ** ** **** ** "Ridgway" #7 87 88 4 mo. 4 mo. 30 da. 9407 6875 4712 CHARGES. Aug. tt 2 3 Transportation $70, Cartage $9, . . Inspection, Commission and Guaranty 5$, . . Proceeds due , 1881, .... ** 9 *** ** *#* ** ** **** E. & 0. E. MASSEY & JACKET. PHILADELPHIA, PA., Aug. 22, 1881. EQUATION OF ACCOUNTS. [Art. 586. 3. Find the equated time for the payment of the net proceeds of Ex. 25, Art. 427, supposing that the merchandise was sold for cash, and that the commission was due at the date given. 4. Sales by JAMES TALCOTT, New York, for account of Phenix Mills, Cohoes, N. Y. March 31, 1882.* Date. Cases. No. Description. Time. Yards. Price. Amount. Mar. 1 2 7619 Fancy Cassimere. 30 da. 966 2 1.35 ****** " 10 4 3475 <( t( 10 da. 1994 1.70 ****** " 13 3 4157 te t( 30 da. 1506 1 2.30 ****** " 17 4 6283 (t (( 4 mo. 1936 3 1.65 ****** " 26 2 3971 (f t( Cash. 978 1.85 **** ** Less Commission 5$, Proceeds due , 1882, ***** ** *** ** ***** ** 5. Account Sales of merchandise by Jonif F. COOK, for account of Excelsior Packing Co., Cincinnati, Ohio. 1881. Oct. 16 50 Bbls. Mess Beef, . . Cash. HJLJL *** ** it 24 100 " N. M. Pork, . . 17JJL **** " 31 25 " Hams 6376 Ibs., . 10 da. 13** *** ** Nov. 9 25 " Shoulders 5717 Ibs., 60 da. 90 *** ** tt 18 75 " C. M. Pork, . . 4 mo. 13H **** ** **** ** CHARGES. Oct. 13 Transportation, 325 15 Cartasre, 37 50 to 15 Cooperasre, 15 15 Inspection, 13 75 Nov 18 Storage, 48 Commission 5^, *** ** *** ** E. & 0. E. NEW YORK, N. Y., Nov. 20, 1881. JOHN- F. COOK. * If the commission is considered due at the average due date of the sales, and since there are no other changes, the net proceeds will he due at the same date. ACCOUNTS CURRENT. 587. An Account Current is an itemized account of the business transactions between two houses, showing the balance or amount due at the current date. The amount due is sometimes called the cash balance. 1. An account current is a transcript of the ledger account with the addition of certain details taken from the books of original entry, and is arranged in a different form. 2. Interest is charged, or not, according to the custom of the business, or the agreement between the parties. This chapter treats only of accounts in which interest is charged. When inter- est is not charged, the balance due is the difference between the two sides of the account as originally entered in the ledger. The interest may be reckoned according to any of the methods of Art. 437. In the illustrative example the exact time in days is found, and the days are regarded as 360ths of a year. In the examples for practice, unless otherwise stated, the interest is reckoned on the same basis. 3. Accounts current are rendered by merchants, bankers, and brokers annually (Ex. 2), semi-annually (Ex. 1), quarterly (Ex. 3), or monthly (Ex. 6). Since the interest draws interest after the account is balanced, the oftener the account is balanced, or the interest is added to the account, the greater the amount due. Some merchants render partial accounts current monthly, but do not carry the interest to the main column until the end of the year (Ex. 11). The twelve partial accounts current make, when combined, the complete account current for the whole year. 4' There are three methods in common use for finding the amount due on an account, including interest, at a certain date, all of which are presented in the following illustrative example : 1. By interest ; 2. By products ; 3. By daily balances. 256 ACCOUNTS CURRENT. [Art. 588. 588. Ex. Find the amount due, including interest at 6%, on the following account Jan. 1, 1882. Dr. GEO. W. CHILDS in account with A. A. Low. Cr. 1881. 1881. Oct. 1 Balance. 1800 Oct. 31 Cash. 1000 " 16 Mdse. 30 da. 360 Nov. 16 NoteSOdo. 600 Nov. 27 30 da. 432 Dec. 4 Cash. 240 Dec. 18 BillofH.C.&Co. 420 " 26 tt 300 589. OPERATION. INTEREST METHOD. Due. Oct. 1, Nov. 15, Dec. 27, " 18, Dr. Amount. $1800 360 432 420 $3012 2140 92 47 5 14 Interest. Due. $27.60 Oct. 31, 2.82 Dec. 19, .36 " 4, .98 " 26, $31.76 13.05 Or. Amount. Days. Interest. $1000 600 240 300 $2140 62 13 28 6 $10.33 1.30 1.12 _^3p $13.05 872 + 18.71 = 890.71. ANALYSIS. First find the due date of each item of the account. Each item will draw interest from its due date until the day of settlement, or Jan. 1, 1882. The total interest on the debit side of the account is $31.76, and on the credit side, $13.05. The balance of interest, $18.71, is therefore in favor of the debit side, or is due Mr. Low. Since both the balance of the account ($872) and the balance of interest ($18.71) are due the same party, the net amount due Jan. 1, 1882, is $872 plus $18.71, or $890.71. If the balance of interest had been on the credit side of the account, the net amount due would have been $872 minus $18.71, or $853.29. NOTES. 1. It will sometimes happen that certain items will fall due after the day of settlement. The interest on such items should be transferred to the opposite side of the account. (See Ex. 8.) 2. If the account has been averaged, the amount due at a given date may be found by calculating the interest on the balance of the account from the time it is due to the date of settlement. If the date of settlement is earlier than the average date, subtract the interest from the balance of the account ; if later than the average date, add the interest. (See Art. 574, Ex. 7, Note.) 3. The interest method is generally used in business. Since it gives the interest on each item and is readily understood, it is more satisfactory to those to whom accounts current are sent than the product method. When interest tables are used, it is shorter than any other method. Art. 590.] ACCOUNTS CURRENT. 257 59O. The following is a common form of an account current including interest : Dr. GEO. W. CHILDS in % current with A. A. Low. Cr. 1881. bays. Interest. Amounts. 1881. Days. Interest. Amounts. Oct. 1 Balance. 92 27.60 1800.00 Oct. 31 Cash. 62 10.33 1000.00 " 16 Mdse. as Nov. 15. 47 2.82 360.00 Nov. 16 Note as Dec. 19. 13 1.30 600.00 Xov.27 " Dec. 27. 5 .36 432.00 Dec. 4 Cash. 28 1.12 240.00 Dec. 18 BillofH.C. &Co. 14 .98 420.00 " 26 " 6 .30 300.00 1882. 1882. Jan. 1 Bal. of Interest. 18.71 Jan. 1 Bal. of Interest. 18.71 1 " " Account. 890.71 1830. 31.76 3030.71 31.76 3030.71 Jan. 1 Balance. 890.71 591. KULE FOR THE INTEREST METHOD. First find the due date of each item of the account. Then find the inter- est on each item from the date it becomes due to the day of settlement. The difference between the sums of the debit and the credit interest will be the balance of interest. To find the net amount due, when the balance of interest and the balance of items are on the same side, take their sinn ; ivhen on opposite sides, talce their difference. 593. OPERATION. PRODUCT METHOD. Due. Dr. Am't. Days. Products. Cr. Due. Am't. Oct. Nov. 15, 360 x 47 = 27, 432 x 5 = 420 x 14 = Days. Products. Dec. IB, $3012 2140 ~872 165600 Oct. 31, $1000 X 62 = 62000 16920 Dec. 19, 600 X 13 = 7800 2160 a 4, 240 X 28 = 6720 5880 tt 26, 300 X 6 = 1800 190560 $2140 78320 78320 $872 + $18.71 = $890.71. 6 ) 112240 $18.706 ANALYSIS. By multiplying the number of dollars by the number of days, and taking the sum of the products on each side of the account, we find that the total debit interest is equivalent to the interest of $190560 for 1 day, and the total credit interest to the interest of $78320 for 1 day. The balance of interest is therefore equivalent to the interest of $112240 for 1 day. The interest of $1 for 1 day is of a mill (446), and of $112240, 18706 (i of 112240) mills, or $18.71. Since the balance of items ($872) and the balance of interest ($18.71) are both due the same party, the net amount due is their sum, or $890.71. 258 ACCOUNTS CURRENT. 593. OPERATION. BY DAILY BALANCES. [Art. 593. Date. Dr. Cr. Dr. Balances. Days. Dr. Products. Oct. 1 1800 1800 30 54000 " 31 1000 800 15 12000 Nov. 15 360 1160 19 22040 Dec. 4 240 920 14 12880 " 18 420 1340 1 1340 " 19 600 740 7 5180 " 26 300 440 1 440 " 27 432 872 5 4360 3012 2140 92 6 ) 112240 2140 18.706 872 + 18.71 = 890.71. ANALYSIS. Arrange the debit and the credit items in the order of their dates as in the operation. Find the balance of the items at each of the dates. There is a debit balance of $1800 for 30 days ; the interest of which is equiv- alent to the interest of $54000 for 1 day. The interest of the next balance, $800, for 15 days is equivalent to the interest of $12000 for 1 day, etc. The total balance of interest is equivalent to the interest of $112240 for 1 day, or $18.71. The net amount due is $872 plus $18.71, or $890.71. (See Art. 446.) NOTES. 1. If, at any time in the above operation, there had been a credit balance, it would have been necessary to have had additional columns for "Cr. Balances" and "Cr. Products." 2. The above method is used by bankers and trust companies that pay interest to depositors upon their " daily balances." EXAMPLES. 594. 1. Find the balance due on the following account, Jan. 1, 1889 ; interest being reckoned at 6%. Dr. HOWARD THORNTON. Cr. 1888. 1888. July 1 Balance. 1830 45' Sept. 13 Net Proceeds. 876 40 Aug. 24 Mdse. 448 00 Oct. 31 a ; *** ** 4258 42 Nov. 20 " 2000 #* * ## 468 75 1881. Jan. 29 " 9998 2368 28 29 Balance of Interest to debit. #*# *# *** *# #**# *# 8. What was the amount clue on the following account Feb. 13, 1881, the estimated due date of a sight draft drawn Jan. 29, 1881, for the balance, reckoning interest at 5% (365 days to the year) ? F. L. BRUCKMANN on account of Consignment #14. 1880. Dr. Days. Interest. Amounts. Oct. 25 Account Sales due Jan. 9, 1881 35 44 80 9344 82 Dec. 31 " " " Mar. 7, 1881 22417 M 1881. Feb. 13 Balance of Interest to credit. *** ** *** ** ***** ** 1880. Or. ifuTie 80 Freight due May 14, 1880 *** ** ** 1176 83 May G G Draft 60 days' sight " July 18, 1880 1 " 69 " " " 18, 1880 i *** **# ** { 8COO < 10000 Nov. 19 " 60 " " " Feb. 1,1881 ** * ** 1881 2000 Feb. 13 Interest Rm. 22417.54 " Mar. 7, 1831 ** ** ** u 13 Balance of Interest to credit. *** ** Jan. 29 Draft at sight to balance due Feb. 13, 1881 **** ** *** ** ***** ** Art. 594.] ACCOUNTS CURRENT. 261 NOTES. 1. The interest on all items falling due after the day of settle- ment should be entered in the interest column on the opposite side of the account. Some accountants enter these items of interest on the same side of the account in red ink so that they will not be added to the other items, and transfer the " red interest " in one amount to the opposite side. 2. The foregoing represents an account in German marks (reichsmarks) kept in an auxiliary book by a consignor of merchandise to a commission merchant at Hamburg, Germany. The due dates of drafts, accounts sales, and other items are obtained from the letters from the commission merchant and from accounts sales and memoranda rendered by him. The correspondiDg consignment account as entered in the books of the consignor is represented in Ex. 7. 9. What was the balance due Jan. 1, 1882, at 6$, on the account represented in Ex. 5, Art. 582. 10. Find the amount due Mar. 1, 1883, at %, on the account represented in Ex. 6, Art. 582. 11. Calculate the interest Jan. 1, 1883, in the following partial account current, and find the total amounts. (Interest 6$, 365 days to the year.) (See Art. 587, 3.) G. D. SLOCUM in account with W. B. 1882. Dr. Days. Interest. Amounte. May 1 Totals from statement of May 1. 1882. 1387 63 28765 72 6 Draft H. B. Claflin & Co. 240 50 71 1285 43 9 " Austin, Nichols & Co. #** ** ** 674 89 13 " W. H, Schieffelin & Co. *** ** ** 346 27 ; of Massachusetts, 4$>. 659. A Non-Forfeiting Policy is one which does not be- come void on account of non-payment of premiums. 1. According to the laws of the State of New York, after three full annual premiums have been paid, the legal reserve of the policy, calculated at the date of the failure to make the payments, shall, on surrender of the policy within six months after such lapse, be applied as a single payment at the published rates of the company in either of two ways, at the option of the assured. (1) To the continuance of the full amount of the insurance so long as such single premium will purchase term insurance for that amount, or (2) to the purchase of a non-participating paid-up policy. 2. According to the Massachusetts limited forfeiture law of 1880, after two full annual premiums have been paid, and without any action on the part of the assured, the net value (Massachusetts standard) of the policy less a sur- render charge of 8% of the present value of the future premiums which the policy is exposed to pay in case of its continuance, shall be applied as a single payment to the purchase of paid-up insurance. 3. Certain companies voluntarily apply all credited dividends to the con- tinuance of the insurance ; others voluntarily apply the legal reserve to the purchase of term insurance at the regular rates. 4. In some companies, all limited payment life policies and all endow- ment policies, after premiums for three (or two) years have been paid and the original policy is surrendered within a certain time, provide for paid-up assur- ance for as many parts (tenths, fifteenths, twentieths, etc., as the case may be), of the original amount assured, as there shall have been complete annual premiums received in cash by the Company. 660. The Surrender Value of a policy is the amount of cash which the company will pay the holder on the surrender of the policy. It is the legal reserve less a certain per cent, for expenses. 308 LIFE INSURANCE. [Art. 661, TABLE OF KATES. 661. Annual premium for an Insurance of $1,000, with profits. LIFE POLICIES. Payable at Death, only. ENDOWMENT POLICIES. Payable as Indicated, or at Death, if Prior. AGE. ANNUAL PAYMENTS. AGE. In 10 Years. In 15 Years. In 20 Years. AGE. For Life. 10 Years. 15 Years. 20 Years. 25 $19 89 $42 56 $32 24 $27 39 25 $103 91 $66 02 $47 68 25 26 20 40 43 37 32 97 27 93 26 104 03 66 15 47 82 26 27 20 93 44 22 33 62 28 50 27 104 16 66 29 47 98 27 28 21 48 45 10 34 31 29 09 28 104 29 66 44 48 15 28 29 22 07 46 02 35 02 29 71 29 104 43 66 60 48 33 29 30 22 70 46 97 35 76 30 36 30 104 58 66 77 48 53 30 31 23 35 47 98 36 54 31 03 31 104 75 66 96 48 74 31 32 24 05 49 02 37 35 31 74 32 104 92 67 16 48 97 32 33 24 78 50 10 t 38 20 32 48 33 105 11 67 36 49 22 33 34 25 56 51 22 39 09 33 26 34 105 31 67 60 49 49 34 35 26 38 52 40 40 01 34 08 35 105 53 67 85 49 79 35 36 27 25 53 63 40 98 34 93 36 105 75 68 12 50 11 36 37 28 17 54 91 42 00 35 83 37 106 00 68 41 50 47 37 38 29 15 56 24 43 06 36 78 38 106 28 68 73 50 86 38 39 30 19 57 63 44 17 37 78 39 106 58 69 09 51 30 39 40 31 30 59 09 45 33 38 83 40 106 90 69 49 51 78 40 41 32 47 60 60 46 56 39 93 41 107 26 69 92 52 31 41 42 33 72 62 19 47 84 41 10 42 107 65 70 40 52 89 42 43 35 05 63 84 49 19 42 34 43 108 08 70 92 53 54 43 44 38 46 65 57 50 61 43 64 44 108 55 71 50 54 25 44 45 37 97 67 37 52 11 45 03 45 109 07 72 14 55 04 45 46 39 58 69 26 53 68 46 50 46 109 65 72 86 55 91 46 47 41 30 71 25 55 35 48 07 47 110 30 73 66 56 89 47 48 43 13 73 32 57 10 49 73 48 111 01 74 54 57 96 48 49 45 09 75 49 58 95 51 50 49 111 81 75 51 59 15 49 50 47 18 77 77 60 91 53 38 50 112 68 76 59 60 45 50 1. The above table represents the maximum rates of the leading New York companies. Surplus premiums or dividends are returned annually com- mencing at the payment of the second premium. 2. Policies which do not share in the dividends of the company, are issued at fixed rates 15 to 20$ less than the above. 3. The above rates are for annual payments only. To obtain semi-annual payments, add k% and divide by 2. To obtain quarterly payments, add 6 % and divide by 4. Art. 662.] LIFE INSURANCE. 309 EXAMPLES. 662. 1. Find the amount of premium for an ordinary life policy (653, 661) of $5000, issued to a person 35 years of age. 2. What is the first annual premium of a life policy of $6000, issued to a person 30 years old, $1.00 being charged for the policy ? NOTE. The policy fee is added to the first premium only. 3. Find the annual premium for a 20-payment life policy (654, 661) of $4000, issued to a person 28 years old. 4. What annual premium must be paid for a 20-year endow- ment policy (655) of $8000, age of the insured at nearest birth- day, 40 years ? If the insured dies during the tenth year, how much more would have been paid than if he had been insured on the ordinary life plan ? 5. What is the average daily cost of a life policy for $1000, no allowance being made for probable dividends, insurance commenc- ing at age 25 ? At 35 ? At 45 ? 6. How much must a person, aged 35, lay aside weekly to secure a life policy of $1000, payable in 20 annual payments ? 7. When 40 years old, a person took out a 20-year endowment policy of $10000. He survived the endowment period. How much less did he receive than he paid as premiums, not reckon- ing interest ? 8. Mr. A. when 26 years old took out an ordinary life policy of $20000. He died aged 41 years 2 months. How much more did his heirs receive than had been paid premiums, no allowance being made for interest ? 9. In the above example, supposing money to be worth $% (simple interest), what was the net gain of the above insurance ? 10. The annual premium, without profits, on a life policy of $10000 at age 35 is $222. How much would it be necessary to invest at 6% interest to secure the payment of the annual pre- mium ? How much would the insured leave his family at his death ? 11. A gentleman, age 30, insures his life for $20000, ordinary life plan. How much must he place in trust so that the interest at 6% will be sufficient to pay the premiums on the policy ? At his death, how much does he leave his family? 310 LIFE INSURANCE. [Art. 6G2. 12. If a man 32 years old takes out a life policy for $5000 and dies just before reaching the age of 40 years, how much less will his total payments be than if he had taken a 20-year endowment policy for the same amount ? IS. Mr. C. when 25 years of age secured a 20-year endowment policy of $6000 ; when he was 30 years of age, he obtained an ordinary life policy of $4000 ; when 35 years of age, he took out a 20-payment life policy of $10000. What was the total annual premium after taking the last policy ? 14. Suppose Mr. C. had died at the age of 40^ years, how much more would his heirs receive than had been paid as premiums ? 15. A single premium for an assurance of $1000, without profits, for a person 32 years old, is $300. What would be the excess of the assurance over the amount produced by placing the money at compound interest (483) at 4$, supposing the insured to live 20 years ? 30 years ? What would be the excess of the sum produced by the money at interest at 5$, over the assurance in 30 years ? 16. Mr. B., age 40, has $10000 at interest at 6$, which he intends to leave his family. What will this amount to at com- pound interest (483) in 25 years at 6% ? How much will he leave his family if he takes out a life policy and pays the premium with the intesest on his investment of $10000 ? 17. Mr. A., aged 30, secures an ordinary life policy, annual premium $100. How much more would his heirs receive from the insurance company than from the money at compound in- terest (484) at 5#, should he die at the age of 32 ? Of 40 ? Of 50 ? At about what age would the amount received from the money at interest exceed the assurance ? 18. What is the semi-annual premium (661, 3) on a 20-year endowment policy for $6000, age 32 ? The quarterly premium ? 19. Mr. A., who will be 35 years of age July 1, takes out Apr. 1 a 20-payment life policy for $10000, premium payable semi-annu- ally. Mr. B. , of the same age, takes out Apr. 1 the same kind of policy for $5000, and Oct. 1, another policy of the same kind for $5000, premium payable annually. How much less does Mr. B. pay as premium each year than Mr. A ? (661, 3.) 20. An ordinary life policy issued at age 35 for $10000 has, at age 45, a 4% reserve of $1262.60. How much non-participating paid-up insurance will this amount purchase, the single premium rate per $1000 at age 45 being $475.44 ? Art. 663.] REVIEW EXAMPLES. 311 REVIEW EXAMPLES. 663. 1. Add 17J, 28f, 36-| , 44f , 89 T \, and 76 J ; multiply the sum by 87 ; subtract 1022| from the product ; and divide the remainder by 234f . 2. Divide eighty-three, and seventy-five hundredths by one hun- dred twenty-five ten-thousandths ; add to the quotient sixty-eight, and six hundred twenty-five thousandths ; and multiply the sum by three, and two-tenths. 3. How many minutes in the month of February, 1900 ? 4. Find the cost of 7312 pounds of meal at $2.25 per civt. 5. The difference in the local time of two places is 1 hr. 7 min. 13 sec.; what is the difference in longitude ? 6. Find the number of square yards of paving in a street, 3000 ft. long and 50 ft. wide. 7. What is the charge for packing, marking, and shipping 251 bales merchandise at 6s. 6d. per bale ? 8. If 46 T. 12 cwt. of coal are worth $174.75, what is the value of 37 T. 8 cwt. ? 9. How many square yards of linoleum would cover a floor 22ft. 6 in. by 15 ft. 4 in.? Find its value at 63^ per sq. yd. 10. What is the freight of 5 T. 9 cwt. 2 qr. 8 lb., at 70 shillings per ton (2240 lb.)? 11. Bought 280 cords of hard wood, at $6. 75, and 790 cords of soft wood, at $3.62|- per cord. Also, 750 bushels of corn, at 62 J cents, and 925 bushels of oats, at 37 \ cents per bushel. What was paid for the whole, and what was the average price of wood per cord, and of grain per bushel ? 12. Bought on contract 350 reams of foolscap paper, at $3. 83| per ream, 45J reams of which were returned as unsuitable, and 275 reams of letter, at $2.67|- per ream, 37-| reams of which were rejected. How much was paid for the remainder ? 13. Feb. 26, 1879, the Nevada Bank of San Francisco sold 100,000 ounces of pure silver to the United States, at $1.08| per ounce. At this rate, what is the intrinsic gold value of the stand- ard silver dollar ? 14. What is the value of 45000 tons of steel rails at 97s. 6d. per ton ? What is the value per ton in U. S. money ? Of total in U. S. money ? 312 REVIEW EXAMPLES. [Art. 663. 15. What will be the cost of painting the walls and ceiling of a room, whose height, length, and breadth are 12 ft. 6 in., 27 ft., and 20ft., respectively, at 24 cents per square yard ? 16. What is the total cost of 561 23 bushels oats at 43 cents per bushel, and 41 1 14 bushels corn at 46 cents per bushel ? 17. Find the total freight on 68 cu.Jt. mdse. at 35 shillings per ton (40 cu. ft.}, and 123 cu. ft. at 40 shillings per ton, plus 10% primage on each item. 18. What is the cost of 250 ft. 3-ply hose, at 60 cts. per foot, less 30 and 10%, and 5 sets couplings at $1.50 each ? 19. May 10, A buys a bill of goods amounting to $5000 on the following terms : 60 days, or \% discount in 30 days, or 2% discount in 10 days. May 20, he makes a payment of $2000, and June 9, of $2500. How much would be due July 9, the end 1 of the 60 days' credit ? 20. Oct. 16, B bought a bill of merchandise amounting to $2000 on the following terms : 4 months, or 5% discount in 30 days, or 6% discount in 10 days. Oct. 26 he made a payment of $1000. How much would settle the bill Nov. 15 ? 21. B bought a bill of merchandise May 16 amounting to $3416.72 on the following terms : 4 mos., or less 5% 30 days. He paid on account June 21 (6 days after the expiration of the 30 days) $3000, with the understanding that he should have the benefit of the discount by paying interest for the time elapsed, at 6% per annum. How much was due Sept. 16, no compound interest being reckoned ? 22. A commission merchant in Chicago sells for me 12 bales brown sheeting, each bale containing 800 yards, at 7 cts. per yard ; pays transportation and other charges amounting to $72 ; and invests the proceeds in flour at $4.80 per barrel. If he charges %\% for selling and \\% for purchasing, how many barrels of flour does he send me ? 28. A of Chicago, sends to B of New Orleans, 8000 bu. of wheat and 500 bbls. of flour with instructions to sell it and invest the proceeds in sugar. B pays freight and cartage $3420; sells the wheat at $1.60 per bushel and the flour at $5.25 per barrel ; charges 2J% commission on the flour and \$ per bushel on the wheat. How many pounds of sugar are pur- chased at 8J cents per pound, the commission for purchasing being 3% ? Art. 663.] REVIEW EXAMPLES. 313 24. Mr. B. purchased 3G150 pounds of hay at $16.50 per ton, and 16438 pounds of oats at 70 cents per bushel. He sold the hay at a gain of 16$, and the oats at a loss of 8$. What were the proceeds ? 25. If I purchase two building lots for $3750 each, and sell one for J more than it cost, and the other for 33 \% less, what is the gain or loss on the two lots ? 26. A speculator sells two farms for $6000 each ; how much does he gain or lose, if he sells one for 20$ more than it cost, and the other for -J less than it cost ? 27. Bought coal by the long ton at $3.64, and sold by the short ton at $4.25. What was the gain per cent ? 28. Mr. A oifered to sell his horse for 12$ more than it cost him, but afterward sold it for $504, which was 10$ less than his first asking price. How much did his horse cost him ? 29. Find the interest of $375.60 for 1 yr. 10 mo. 22 da., at SO. Find the interest of $4128 for 8 mo. 26 da., at 31. What is 2J$ of 159 13s. lOd. ? 82. Find the date of maturity and the net proceeds of a note for $5000, dated May 16, payable 4 months after date, and dis- counted July 21 at 6$. S3. When the above note became due, its maker had discount- ed at 6$ a new note, payable 90 days after date, whose proceeds were sufficient to pay the first note. What was the face of the new note ? 34> Apr. 1, a merchant buys a quantity of coffee on 90 days' credit, with privilege of discounting within 30 days from date of purchase at the rate of 6$ per annum for the unexpired time. Apr. 16, he makes a payment of $28000 on account, no actual in- voice having been rendered. May 1, he receives the invoice, amounting to $29215, and on the same date full settlement is made What amount is required to cancel the bill ? (Exact days, 360 days to the year.) 35. Divide $2000 in such a manner between two brothers, aged 16 and 19 years respectively, so that when they arrive at 21 years of age they will have equal amounts, money being worth 6$ sim- ple interest. 36. What would be the share of each if money is worth 6$ compound interest ? 314 REVIEW EXAMPLES. [Art. 663. 37. Find the interest on $5000 from May 18 to Sept. 28, at 4^: 1, Ordinary interest and compound subtraction ; 2, Ordinary in- terest and exact days ; 3, Accurate interest. 38. Find the amount due on the following note Jan. 1, 1883, by the United States and the Mercantile Rules : DAVENPORT, IOWA, May 1, 1878. On demand, I promise to pay EDWIN D. MORGAN, or order, Five thousand dollars, with interest at six per cent., for value received. E. H. CONGER. On this note the following payments were indorsed : Received Jan. 16, 1879, $400. Received Dec. 12, 1880, $150. Received Sept. 7, 1879, $100. Received Aug. 18, 1881, $850. Received May 1, 1880, $500. Received Apr. 23, 1882, $100. 39. How much would have been due on the above note at 10$ ? 40. What is the value of a draft on Hamburg of 17468 marks at 95f ? 41. C. of London owes me for goods sold on my account, 129 18s. Id. How much do I receive in payment, if I draw a bill of exchange for the amount and sell it at 4.85-| ? 4%. My agent in Paris buys an invoice of merchandise amount- ing to 12488 francs, at a commission of 2J%. What is the cost of the draft which I remit in payment, exchange being 5.17-f ? 43. An exporter sold the following bills of exchange through a broker : 10000 francs on Paris at 5.16J, 375 16s. Sd. on Lon- don at 4.83-|, 16480 marks on Hamburg at 94 J, 5287 guilders on Amsterdam at 41 -J. What were the proceeds, brokerage \% ? 44' A commission merchant at New York sells goods for A. of Havre to the amount of $3435.27, and charges a commission of 2 J% for selling. What is the face of the draft which he purchases and remits in settlement, exchange being 5.27 ? 45. My agent in London has purchased for me, at a commis- sion of %\% 9 375 dozen kid gloves at 49^. per dozen, and 636 yards silk at 9s. 6d. per yard. When exchange is $4.86|, what will be the cost of the draft which J. remit to him in settlement ? 46. Purchased in England, merchandise amounting to 324 10s. 7d., and paid freight and duties $487.34. How much per must I sell these goods to gain &\% on the full cost, and what must I charge for an article invoiced at 6s. 8d., exchange 4.88 ? 47. What is the cost of insuring $18000 at 75^ less 15% ? Art. 663.] REVIEW EXAMPLES. 315 48. Average the following account, and find the amount due Sept. 28, 1882, at 6$. $874.32 on 30 days credit. 518.65 " 60 " 373.78 " 4 months " 429.31 " 60 days 657.70 242.28 Mar. 16, 1882, " 31, " May 5, " " 21, " June 18, " July 3, " " 24, " Aug. 19, " Sept. 13, " " 30 " 60 983.75 716.30 4 months 4 " 49. Average the following account, due Jan. 1, 1883 ? 536.60 " 60 days " What will be the amount Dr. DANIEL S. LAMONT, Albany, N. Y. Cr. 1882, 1882. i July 16 Mdse., 4 mo. $876 14 ; Sept. 10 Cash, . . . $900 00 Aug. 4 " 60 da. 415 65 " 21 tt 700 00 Sept. 10 " 30 da. 797 38 Oct. 13 (C 500 00 " 21 " 30 da. 686 96 " 31 Mdse., 30 da. 322 16 Oct. 13 " 4 mo. 524 27 Nov. 2 Cash, . . . 400 00 " 31 " 30 da. 859 75 " 28 Note, 4 mo. . 800 00 Nov. 28 " 60 da. 263 31 Dec. 27 Cash, . . . 500 00 Dec. 1 " 60 da. 172 64 " 30 " 30 da. 938 52 50. Prepare an account current, including interest at Q% to Jan. 1, 1883, from the above ledger account, according to the form and method of Art. 59O. 51. Sold five $1000 bonds at 116f, and invested the proceeds in railroad stock at 92$, which I sold at 98. What was the gain on the stock, allowing usual brokerage ? 62. Sold Aug. 11, 1879, 500 shares Chicago & Alton, at 94J, and covered my short sale Aug. 16, 1879, at 91. What was my profit, allowing the usual brokerage ? 53. What annual income will be obtained by investing $9923. 75 in bonds, bearing 5% interest, and purchased at 116f ? 54. Bought stock at 116f and sold at 112f Loss, $1295. What was the par value of the stock ? 316 REVIEW EXAMPLES. [Art. 663. 55. The tax levied in a town, having a valuation of $1800000, is $22500. What is the tax on $1, and what is the tax of A, whose real estate is assessed $5000 and personal property $1500 ? 56. What is the duty at 60% on an invoice of silk amounting to 36475 francs ? 57. What is the duty on 50 cwt. 3 qr. 14 Ib. (Long ton table) of steel at 2Jc\ per pound ? 58. Find the duty at 25% on an invoice of mdse. valued at 243 2s. 3d. 59. What is the duty on a block of marble 2 x 3 x 7 ft., im- ported from Italy, dutiable value 3450 lire, and duty $1 per cubic foot and 25% ? 60. A, B, and C are partners in business, investing as follows : A, $4000 ; B, $6000 ; C, $8000. The partners are to share the profits and losses in proportion to their investments. Each is entitled to compensation for services at the rate of $150 per month, to be credited the first day of the following month. Interest is to be reckoned on the salaries and on the amounts drawn out at the rate of 6% per annum. At the end of the year B and C purchase the interest of A, and in the payment therefor, it is desired that the remaining members shall so invest that their interests shall be equal. It is mutually agreed that the "good will" of the business shall be valued at $3000 in the final settlement. It is also agreed that a discount of 5% r ?hall be allowed upon all uncol- lected accounts as a fund to meet i debts and costs for collecting. A statement of the business previous to closing shows the follow- ing results : merchandise, horses, wagon, office fixtures, and cash on hand, $12410; sundry debtors, $17030; sundry creditors, $4050; expense account (not including partners' salaries), $2400 ; profit on merchandise sold, $15290. A withdrew on account of salary Apr. 1, $450 ; July 1, $300 ; Oct. 1, $400. B withdrew Mar. 1, $400 ; Apr. 1, $150 ; June 1, $400 ; Oct. 1, $800 ; Dec. 1, $500. C withdrew Apr. 1, $600 ; July 1, $700 ; Oct. 1, $600 ; Nov. 1, $200. How much must B and each invest or pay A, and how should the books of the new firm be opened ? NOTES. 1. B and C, not desiring to have the new books encumbered with the contingent accounts of "good will" and "reserve fund," closed these accounts after a settlement was made with A. 2. The loss or gain may be found from a statement of resources and liabilities, or from the Loss and Gain account. APPENDIX. GREATEST COMMON DIVISOR. 664. The Greatest Common Divisor of two or more numbers is the greatest number that will divide each without a remainder ; hence it is their greatest common factor. Thus, 2, 3, 4, and 12 are common divisors of 36, 48, and 60; 12 is their greatest common divisor. 665. To find the greatest common divisor of two or more numbers. Ex. What is the greatest common divisor of 168, 252, and 420 ? OPERATION. ANALYSIS. Divide the given numbers by 4 ) 168, 252, 420 any number that will divide them all without IJF \ ^2 j^3 Jog a remainder, and divide the quotients in the same manner until the last quotients have no 3 ) 6, 9, IP common divisor. Since 4 will divide all the 235 given numbers, and 3 and 7 will divide SUCCP^ v ely the resulting quotients, their 4 X 7 X 3 = 84. prod [, 84, is a common divisor of the given numbers. Since the last quotients have no common divisor or factor, 84 is the greatest common divisor. 666. RULE. Divide the given numbers by any factor that will divide all of them without a remainder. In like manner divide the resulting quotients, and continue the division until the quotients have no common factor. The product of the several divisors will be the greatest com- mon divisor. EXAMPLES. 667. Find the greatest common divisor of the following numbers : 1. 108, 144, and 360. 5. 405, 243, and 324. 0. 144, 336, and 240. 6. 378, 126, and 252. 3. 165, 550, and 220. 7. 375, 625, and 250. 4. 792, 144, and 216. 8. 288, 720, and 864. 318 GREATEST COMMON DIVISOR. [Art. 668. 668. To find the greatest common divisor of two numbers when they are not readily factored. 669. PRINCIPLES. 1. A common divisor of two numbers is a divisor of their sum, and also of their difference. 2. A divisor of a number is a divisor of any multiple of that number. 670. RULE. Divide the greater number by the smaller, and divide the last divisor by the remainder ; and so con- tinue until there is no remainder. The last divisor will be the greatest common divisor. NOTES. 1. When the greatest common divisor of more than two numbers is required, find the greatest common divisor of the smallest two first, and of this greatest common divisor and the next greater, and so on, until all the numbers are used. The last divisor will be the greatest common divisor of all the given numbers. 3. If the remainder at any time is a prime number, and it is not contained in the last divisor, there is no common divisor greater than 1 ; it will there- fore be useless to further continue the division. Ex. Find the greatest common divisor of 391 and 437. OPERATION. DEMONSTRATION. Since 23 is a divisor of 46, it 391 ) 437 (1 is a divisor of 368, a multiple of 46 (Prin. 2). 39^ Since 23 is a divisor of itself and 368, it is a divisor - of their sum, 391 (Prin. 1). Since 23 is a divisor 46 ) 391 (8 O f 46 an a 391, it is a divisor of their sum, 437. 23 368 is therefore a common divisor of 391 and 437, the ~~ numbers. 23)46(2 . The greatest common divisor of 391 and 437, _ whatever it may be, is a divisor of their difference, 46 (Prin. 1) ; also of 368, a multiple of 46 (Prin. 2) ; also of 23, 391 368 (Prin. 1). Since the divisor of a number cannot be greater than itself, the greatest common divisor of the given numbers cannot be greater than 23. 23 is therefore the greatest common divisor. EXAMPLES. 671. Find the greatest common divisor of the following numbers : 1. 319 and 377. & 611, 799, and 987. & 259 and 629. 6. 744, 984, and 522. 3. 589 and 713. 7. 391, 667, and 920. 4. 903 and 989. 8. 451, 481, and 737. Ait. 672.] ANNUAL INTEREST. 319 ANNUAL INTEREST. 672. When a note contains the words "with interest annu- ally," the laws of New Hampshire and Vermont, if the interest is not paid when due, allow simple interest on the annual interests from the time they become due to the time of payment. ILLUSTRATION. A agrees to pay B $6000 in three years from Jan. 1, 1880, with interest annually at 6%. By this contract, $360 becomes due Jan. 1, 1881, and on the first day of January in each year thereafter, until paid ; this is the " annual interest." Suppose A does not pay any portion of this interest until Jan. 1, 1883, when the principal becomes due ; then A, hav- ing had the use of money that his contract required him to pay to B, and B having been deprived of its use, B is entitled to have simple interest added to the annual interest, from the time when the same became due to Jan. 1, 1883 ; BO that on Jan. 1, 1883, B would be entitled to the following sums as interest : First year's int. $360 + 2 yrs. simple int. thereon, $43.20 = Second" " 360 + 1 " " " " 21.60= 381.60 Third " " 360 + (paid when due) 00 = 360 $1080 $64.80 = $1144.80 Amount of annual interest $1080.00 Amount of simple interest accrued upon annual interest . 64.80 Total amount of interest due $1144.80 In calculating the simple int. upon the annual int., shorten the operation by finding the int. upon the annual int. for the sum of the several periods. Ex. What is the amount due on the following note July 1, 1885? $10000. CON-CORD, 1ST. H., January 1, 1882. Three years after date, for value received, I promise to pay A. B. THOMPSON, or order, Ten Thousand Dollars, with interest payable annually. C. A. DOWNS. OPEKATION. Face of note, on interest from Jan. 1, 1882 $10000.00 Interest from Jan. 1, 1882, to July 1, 1885, 3 yr. 6 mo 2100.00 3 items of annual interest ($600 each) are unpaid : 1st from Jan. 1, 1883, to July 1, 1885, 2 yr. 6 mo. 2nd from Jan. 1, 1884, to July 1, 1885, 1 yr. 6 mo. 3rd from Jan. 1, 1885, to July 1, 1885, 6 mo. Int. on the annual int. = int. on $600 for 4 yr. 6 mo 162.00 Total amount due July 1, 1885 $12262.00 320 APPENDIX. [Art, 673. 673. EULE. To the given principal and its interest to the date of settlement, add the interest on each annual interest from the time it is due to the date of settlement. Tlic sum will be the amount due at annual interest. EXAMPLES. 674. 1. At 6%, interest payable annually, how much would be due Oct. 1, 1884, according to the laws of New Hampshire, on a note of $8000, dated June 1, 1881, no payments having been made? 2. What amount would be due Jan. 1, 1886, at 6$, on a note for $4200, dated Concord, N. H., May 16, 1882, interest payable annually, and no payments having been made ? 3. A note for $10000 was dated Apr. 1, 1882, and payable four years from date without interest. Attached to this note were 8 notes of $400 each for the semi-annual interest due Oct. 1, 1882, Apr. 1, 1883, Oct. 1, 1883, Apr. 1, 1884, Oct. 1, 1884, Apr. 1, 1885, Oct. 1, 1885, Apr. 1, 1886. How much was due, at 8$, Apr. 1, 1886, nothing having been paid ? NOTE. It is the custom of certain corporations when making loans for long periods of time on collateral security or on bond and mortgage, to have a note or mortgage given without interest for the principal, and to have separate notes given for each sum of annual, semi-annual, or quarterly interest, due and maturing at the time the interest is payable. These notes draw interest after maturity like any other note, and may be collected without disturbing the original loan. 4. What amount would be due July 1, 1884, on a note of $5000, dated July 1, 1882, given for 2 years, with notes for quar- terly interest, no payments having been made ? 5. Kequired the amount due Jan. 1, 1883, on a note of $3600, dated Jan. 1, 1881, due in two years, notes for semi-annual interest from date, at 6%, having been given, and nothing having been paid. 6. Find the amount of $1200, at 6^, interest payable annually, from June 16, 1882, to Dec. 28, 1886, no interest having been paid except for the first year. 7. What must be paid, Oct. 16, 1885, in settlement of a note for $2500, dated Manchester, 1ST. H., May 6, 1880, said note promis- ing interest annually, and no interest having been paid ? Art. 675.] NX W HAMPSHIRE RULE. 321 NE\V HAMPSHIRE RULE.* 675. According to the laws of New Hampshire, when pay- ments are made upon a note, or other contract, by virtue of which interest is payable annually (672), they should be applied in the following order to the payment of 1. Any simple interest that may have accrued upon the annual interest. 2. The annual interest. 3. The principal. 676. EULE. Find the interest due upon the principal and the annual interest at the annual rest (the time when the annual interest becomes due from year to year) next after the first payment. To the payment or payments made before this rest, add interest from the dates when they ivere made to the date of the rest, unless there is no interest due upon the principal, excepting that which is accruing during the year in which the payment or pay- ments were made, and the payments together are less than the interest thus accruing, in which last case no interest is to be added to the payments. Deduct the payment or payments, with or without interest, as aforesaid, from the amount of principal, annual interest, and simple interest upon the annual interest due at the time of said rest, if such payment or payments equal or exceed the annual and simple interest then due ; if less than such annual and simple interest, but greater than the simple interest due upon the annual interest, deduct the same from the sum of the annual and simple interest, and upon the balance of such annual interest find simple interest to the time when the next payment or payments are applied ; if less than the simple interest due upon the annual interest, deduct the same from such simple interest and add the balance without interest to the other interest due at the time when the next payment or payments are applied. Proceed in like manner to the time of the first annual rest following the next payment, and to the end of the time required. * From Report of State Superintendent of Public Instruction (1877). 322 APPENDIX. [Art. 677. EXAM PLES. 677. 1. According to the law of New Hampshire, how much is due Jan. 1, 1886, on. a note dated Jan. 1, 1880, for $2000, with interest annually at 6%, the following payments having been made : July 1, 1882, $500 ; Oct. 1, 1883, $50. 15 OPERATION. First annual interest due Jan. 1, 1881, $120 + 2 yr. simple interest thereon, $14.40 Second annual interest due Jan. 1, 1882, $120 + 1 yr. simple interest thereon, $7.20 Third annual interest due Jan. 1, 1883, Principal First payment, July 1, 1882, .... Interest thereon from July 1, 1882, to Jan. 1, 1883, Balance of principal due Jan. 1, 1883, Fourth annual interest of $1866.60, due Jan. 1, 1884, Second payment, Oct. 1, 1883 (being less than the interest accruing during the year, it does not draw interest) . . Balance of fourth annual interest unpaid Fifth annual interest of $1866.60, due Jan. 1, 1885, Sixth annual interest of $1866.60, due Jan. 1, 1886, Simple interest on unpaid balance of fourth annual in.*, for % yr. . Simple interest on fifth annual interest for 1 year . Balance of principal Amount due Jan. 1, $134.40 127.20 120.00 2000.00 $2381.60 515.0ft 1866.60 112.00 50.00 62.00 112 112 7.44 6.72 1866.60 2166.76 Solve Examples 2, 4, 8, and 9, Art. 5O5, according to the New Hampshire Rule, at the legal rate (436), supposing each note to contain the words "with interest annually." VERMONT RULE. 678. The Vermont Rule for notes with interest is essentially the same as the United States Rule (5O4) ; and for notes "with interest annually," it is the same as the New Hampshire Rule, except that when payments are made on account of interest accru- ing but not yet due, they draw interest from the date they were made to the annual rest, whether they are greater or not than the interest accruing during the year. Thus, by the Vermont Rule, the payment of $50, in the above example, would draw interest from Oct. 1, 1883 to Jan. 1, 1884, or 3 months. The unpaid balance of fourth annual interest would be $61.25 ($112 $50.75). STORAGE. 679. Storage is keeping or storing of goods in a warehouse until they are required for use, sale, or transportation. Storage is also the name applied to the price or compensation for storing 1 goods in a warehouse. 680. Storage is usually calculated at a certain rate per barrel, bale, bushel, box, or other unit for a certain time. 1. The storage term is one week, 10 days, 20 days, or one month. 2. In some warehouses, storage for a part of a term is charged at the same rate as for a full term. (j CASH STORAGE. 681. When the storage is paid or estimated when the goods are taken out of store or the receipt is surrendered, it is sometimes called cask storage. EXAMPLES. 682. 1. What was paid for the following storage at 6 cents per barrel per month or part of a month, the calculation being made at each delivery ? Received Oct. 1, 1800 III. ; Nov. 15, 360 lll.\ Dec. 18, 420 III.-, Dec. 27, 432 III. Delivered Oct. 31, 1000 III.-, Dec. 4, 240 ML; Dec. 19, 600 III.-, Dec. 26, 300 III. OPEKATION. Date. Received. Delivered. Oct. 1 1$00 " 31 Nov. 15 "5^19, $00 1000 Dec. 4 ." 18 2 420 000 " 19 $00 " 26 On hand 432 872 1000 240 560 40 300 .06 .== .18' = 60.00 43.20 ,18 100.80 12 = 12 = 4.80 36.00 $244.80 ANALYSIS. All goods delivered are deducted from the oldest receipt on hand. By the system of cancellation indicated in the operation, it can be easily determined when the storage commences. The 1000 bbl. taken out 324 APPENDIX. [Art. 682. Oct. 31 were placed in store Oct. 1, and pay 1 month's storage. The 240 bbl. taken out Dec. 4 were placed in store Oct. 1, and pay 3 months' storage. Of the delivery of 600 bbl., 560 were placed in store Oct. 1, and pay 3 months' storage, and the remainder, 40 bbl., were placed in store Nov. 15, and pay 2 months' storage, Dec. 19. The lot of 300 bbl. withdrawn Dec. 26, were placed in store Nov. 15, and pay 2 months' storage. The separate calculations are placed at the right in the above operation. NOTES. 1. Certain warehouses render bills at the end of each month for all goods taken out during the month. Others render bills monthly for all storage dues, whether the goods have been withdrawn or not. 2. Storage on goods for which negotiable receipts have been issued, and in many other cases, is collected when the receipt is surrendered or the goods delivered. 2. What will be the storage at 5 cents per barrel per month on the following ? Received Aug. 1, 800 bbl. ; Aug. 15, 700 bbl. ; Aug. 26, 900 bbl. Delivered Aug. 12, 400 bbl; Aug. 20, 800 bbl ; Sept. 1, 400 bbl ; Sept. 8, 800 bbl 3. Find the storage due on the following account June 24, at 3 cents a bale per month or part of a month. Received Apr. 13, 400 bales ; Apr. 30, 800 bales ; May 16, 200 bales ; May 25, 400 bales ; June 19, 600 bales. Delivered May 10, 600 bales ; May 20, 100 bales ; May 28, 700 bales ; May 31, 400 bales ; June 24, 600 bales. 4. Complete the following storage bill, the rate being lOc. per bale per month or part of a month. Messrs. ARMSTRONG, CATOR & Co., BALTIMORE, MD., Aug. 31, 1887. To MERCHANTS' STORAGE Co., Dr. Marks and Numbers. When received. When delivered. Rate. Amount. 4 Bales AC $174-177 June 4 Aug. 2 20c. 80 1 A C$298 May 21 4 30c. 30 5 A. C. & Co. 121-25 July 26 7 lOe. 50 3 AC $170-1 72 June 4 16 ** ** 2 C $ 29-28 Aug. 17 20 ** ** 6 A. C. & 0^115-20 July 26 22 *# * *# 1 A C$173 June 4 24 ** #* 4 AC it 299-302 May 21 27 ** * ** 5 A. C. & Co. $26-30 July 26 31 H-* * * ** NOTE. In many cases (see above example), storage is charged for the time the particular packages withdrawn have been in store. 5. The following quantities of wheat were stored at Ic. a bushel per month or part of a month. What was the amount of Arc. 682.] STOBA G E. 325 storage due June 1, a full settlement being made on that date and a new receipt being given ? Received Apr. 1, 600 bushels ; Apr. 25, 400 bushels ; May 9, 400 bushels ; May 27, 300 bushels. AVERAGE STORAGE. 683. At some warehouses, in computing storage on grain, flour, etc., when there are frequent receipts and deliveries, it is customary to average the time and charge a certain rate per month of 30 days. The process is called average storage, or storage on account. 684. Ex. Merchandise was received and delivered at a ware- house as follows : Received Oct. 1, 1800 III. flour ; Nov. 15, 360 bbl; Dec. 18, 420 bbl; Dec. 27, 432 III. Delivered Oct. 31, 1000 bbl; Dec. 4, 240 III.-, Dec. 19, 600 bbl- 9 Dec. 26, 300 bbl Find the average storage due on the above Jan. 1, at 6 cents per barrel per month of 30 days. 685. OPERATION. PRODUCT METHOD. Date. Received. Days. Products. Date. Delivered. Days. Products. Oct. 1 1800 92 165600 Oct. 31 1000 62 62000 Nov. 15 360 47 16920 Dec. 4 240 28 6720 Dec. 18 420 14 5880 " 19 600 13 7800 27 432 5 2160 " 26 300 6 1800 3012 190560 2140 78320 2140 78320 On hand 872 30 ) 112240 3741^ x .06 = 224.48 ANALYSIS. Assuming that there was nothing withdrawn, the 1800 bbl. would be in store from Oct. 1 to Jan. 1, or 92 days. The storage of 1800 bbl. for 92 days is equivalent to the storage of 1 bbl. for 165600 days. The storage of 360 bbl. for 47 days is equivalent to the storage of 1 bbl for 16920 days. In the same manner, we find the total storage, if nothing had been withdrawn, to be equivalent to the storage of 1 bbl. for 190560 days. The storage on the goods withdrawn is equivalent to the storage of 1 bbl. for 78320 days, thus making the net storage I 'bbl. for 112240 days, or 3741 months. 6 cents multiplied by 3741 equals $224.48, the total storage. 326 APPENDIX. [Art. 686. 686. OPERATION. BY DAILY BALANCES. Date. Received. Delivered. Balances. Days. Products. Oct. 1 1800 1800 30 54000 " 31 1000 800 15 12000 Nov. 15 360 1160 19 22040 Dec. 4 240 920 14 12880 " 18 420 1340 1 1340 " 19 600 740 ry I 5180 " 26 300 440 1 440 " 27 432 872 5 4360 3012 2140 92 30 ) 112240 Jan. 1 Bal. on hand 872 3741J 3012 3012 3744 x .06 = 224.48 ANALYSIS. Arrange the receipts and deliveries in the order of their dates as in the operation. Find the number of barrels on hand at each of the dates. The 1800 bbl. are in store from Oct. 1 to Oct. 31, or 30 days. The storage of 1800 bbl. for 30 days is equivalent to 1 bbl. for 54000 days. The total storage is equivalent to the storage of 1 bbl. for 112240 days, or 3741 months. 6 cents multiplied by 3741^ equals $224.48, the total storage. EXAMPLES. 687. 1. Find by either of the above methods the average storage at 5 cents per month of 30 days, of the account given in Ex. 2, Art. 682. 2. Find the average storage, at 3 cents per month of 30 days, of the account given in Ex. 3, Art. 682. 3. Find the total charge for pasturing cattle per the following statement, at 30 cents a head per week : Received July 5, 18 head ; July 12, 10 head ; July 20, 30 head ; Aug. 2, 40 head'; Aug. 10, 20 head ; Sept. 1, 10 head ; Sept. 4, 24 head ; Sept. 17, 26 head; Oct. 2, 20 head ; Oct. 27, 18 head; Nov. 1, 6 head; Nov. 2, 16 head. Withdrawn July 7, 4 head; July 9, 8 head; July 14, 10 head ; July 17, 6 head ; July 23, 20 head ; Aug. 4, 20 head ; Aug 13, 20 head ; Aug. 21, 12 head ; Aug. 29, 4 head ; Sept. 8, 10 head ; Sept. 14, 30 head ; Sept. 21, 18 head ; Oct. 30, 20 head ; Nov. 5, 10 head ; Nov. 9, 20 head ; Nov. 16, 26. head. NOTE. When the account is long, the first method is preferable. ALLIGATION. 688. Alligation treats of mixing ingredients of different values to find the value of the mixture, or to produce a mixture of a given value. NOTE. Alligation is sometimes and more properly called Average. ALLIGATION MEDIAL. 689. Alligation Medial is the process of finding the average value of a mixture, the rates and quantities of the ingredients being given. EXAMPLES. 690. 1. A grocer mixes together 7 pounds of coffee at 26 cents per pound, 4 pounds at 27 cents per pound, and 10 pounds at 34 cents per pound. What is the value of a pound of the mixture ? OPERATION. 7 x 26c. = $1.82 ANALYSIS. 7 Ib. at 26c. are worth $1.82. 4 Ib. 4 x 27 C. = 1.08 at 27c. are worth $1.08. 10 Ib. at 34c. are worth 10 X 34c. 3.40 $3.40. Hence the total mixture containing 21 Ib. is worth $6.30, and 1 Ib. is worth $6.30 *- 21, or 21 Ib. worth $6.30 30c> 1 II. " .30 2. A wine merchant mixed together 10 gallons of wine at 40 cents a gallon, 15 gallons at 50 cents, and 25 gallons at 70 cents. What is the value of a gallon of the mixture ? 8. A grocer mixed 60 Ib. of tea at 25 cents a Ib., 75 Ib. at 30 cents, and 65 Ib. at 50 cents. What was the value of a pound of the mixture ? 4. A farmer mixes together 20 lu. of oats at 40 cents a bushel, 30 bu. of corn at 50 cents a bushel, and 50 bu. of rye at 70 cents a bushel. What is the value of a bushel of the mixture ? APPENDIX. [Art. 691. ALLIGATION ALTERNATE. 691. Alligation Alternate is the process of finding the quantities of different values required to produce a mixture of a given value. 692. The values of several ingredients being given, to produce a mixture of a given value. EXAMPLES. 693. 1. How much tea worth 24, 28, 36, and 42 cents a pound must be mixed together, so that the mixture will be worth 30 cents a pound ? OPERATIONS 30 (1) (2} (3) (4) (5) (6} (7) (8) (9) 24 * i 1 9 /v 3 | i ; 3 etc. 28 } 6 6 6 6 ] 12 12 etc. 36 i 1 1 2 3 I 1 ; 3 etc. 42 A 1 1 1 1 2 | 2 etc. ANALYSIS. If we sell 1 pound for 30 cents, that is worth 24 cents, we gain 6 cents, and to gain 1 cent, we take | of a pound. We must now take a kind that is worth more than the average price so as to lose one cent. If we take a pound worth 36 cents and sell it at 30 cents, we will lose 6 cents, and to lose one cent, we must take ^ of a pound. In the same manner, we find that if we take ^ of a pound of the 28-cent tea and mix it with ^ of a pound of the 42-cent tea, there will be no gain nor loss by selling at 30 cents a pound. I is to as 1 is to 1, and | (^) is to T V as 6 is to 1. Or, columns 3 and 4 may be found by multiplying columns 1 and 2 respectively by the least com- mon denominators of the fractions. Column 5 is the sum of columns 3 and 4. An unlimited number of answers may be found to examples of this kind by combining 1, 2, or 3, etc., times column 3 with 1, 2, or 3, etc., times column 4. 2. A grocer has sugar at 5^, 7^, 12^, and 13^ per pound. How much of each kind will form a mixture worth 10 cents per pound ? 3. A jeweler wishes to make a compound of gold that shall be 20 carats fine. He has gold of 15, 19, 23, and 24 carats fine. What quantity of each must he take ? 4- How much tea at 25 cents, 50 cents, 60 cents, and 80 cents per pound must be taken to form a mixture worth 55 cents per pound ? Art. 693.J ALL1G A T10N. 329 5. How much wine at 50 cents, 70 cents, 80 cents, $1.00, and $1.20 a gallon must be mixed together that the mixture may be worth 90 cents a gallon ? 694. When the quantity of one ingredient is given. EXAM PLES. 695. 1. How much coffee at 30, 34, and 44 cents per pound, must be mixed with 10 pounds at 36 cents a pound, to make a mixture worth 40 cents a pound ? OPERATION. (1) (2) (3) (4) (S) (6) (7) (8) 30 TV 2 2 34 i 2 2 44 1- t i 5 3 1 10 18 36 i 1 10 10 40- ANALYSIS. We find the relative quantities (columns 4, 5, and 6) as in Art. 693. In order to use 10 pounds of the 36-cent coffee, we multiply column 6 by 10, producing column 7. Column 8 is found by adding columns 4, 5, and 7. Other combinations may be found, by multiplying columns 4 and 5, find adding the results to column 7. 2. How much coffee at 15c., 17c., and 220. a pound must be mixed with 5 Ib. at 180. per pound to make a mixture worth 20c. per pound. 3. How much gold of 21 and 23 carats fine, must be mixed with 30 oz. of 20 carats fine, so that the mixture may be 22 carats fine ? 4- How much tea at 20 cents, 25 cents, and 45 cents a pound, must be mixed with 36 Ib. at 60 cents a pound, so that the mixture will be worth 40 cents a pound ? 5. How much wine at $1.25 and $1.75 a gallon, must be mixed with 15 gallons of water, so that the mixture may be worth $1 a gallon ? 6. How much tea at 30 cents, 46 cents, and 48 cents a pound, must be mixed with 12 pounds at 38 cents, so that the mixture may be worth 40 cents a pound ? 330 APPENDIX. [Art. 696. 2 2 } 3 3 \ x 5 = 3 2 5 J 5 5 10 ) 50 ( 5 696. When the total quantity of the ingredients is given. EXAMPLES. 697. -?. A grocer mixed tea worth 20, 25, and 35 cents a pound. The mixture consisted of 50 pounds, worth 29 cents a pound. How many pounds of each did he take ? (1) (2) (3) (4) (5) i t i ANALYSIS. We find columns 3 and 4 as in Art. O93. Column 5 is the sum of columns 3 and 4. The required amount is 50 = 5 times 10, the sum of column 5. Hence the quantity of each may be found by multiplying each number in column 5 by 5. NOTE. Many results may be obtained for examples of this kind. Thus, in the above example, 9 times column 3 plus once column 4, 8 times column 3 plus 2 times column 4, etc., would each produce correct results. 2. How much wine worth 50 cents, 60 cents, 90 cents, and $1.20 a gallon, must be mixed together so as to make a hogshead of 110 gallons at 80 cents a gallon ? 3. A man bought 20 barrels of flour for $120, paying $4J, $5, $6|, and $7 per barrel. How many barrels of each did he buy ? OPERATION. (1) () (*) (4) (5) (6) (7) (8) (9) (10) (11) (IS) *J ! 1 1 1 2 2 3 3 5 1 8 8 6 6 4 4 6} 2 3 3 3 6 6 9 9 7 1 8 8 6 6 4 4 2 4 20 20 20 ANALYSIS. Find columns 2 and 3 as in Art. 693. Since 6, the sum of columns 2 and 3, is not an exact divisor of 20, the required amount, we must take a certain number of times column 2 and a certain number of times column 3. By trial, we find that 8 times column 2 plus once column 3 equals 20. Therefore multiply column 2 by 8, producing column 4, and column 3 by 1, producing column 5. Column 6 is the sum of columns 4 and 5. In the same manner, we find the results given in columns 9 and 12. 4. A man bought 50 animals. for $50, paying for lambs $ .each, for sheep $1J each, and for calves $3J each. How many of each did he buy ? Art. 698.] SQUARE ROOT. 331 SQUARE ROOT. 698. The Square Root of a number is one of the two equal factors of a number. Thus, the square root of 25 is 5. 5 x 5 = 25. 699. To find the square root of a number. 700. RULE. Beginning at units* place, separate the given number into periods of two figures each. Find the greatest square in the left-hand period, and write its root at the right in the form of a quotient in divis- ion. Subtract this square from the left-hand period, and to the remainder annex the next period to foi*m a dividend. Double the part of the root already found for a trial divisor. Find how many times this divisor is contained in the dividend, exclusive of the right-hand figure, and write the quotient as the next figure of the root. Annex this quo- tient to the right of the trial divisor to form the complete divisor. Multiply the complete divisor by the last figure of the root, and subtract the product from the dividend. To the remainder annex the next period, and proceed as before. NOTE. When the given number is a decimal, separate the number into periods of two figures each, by proceeding in both directions from the decimal point. EXAMPLES. 701. 1. Find the square root of 1156. OPERATION. ANALYSIS. Beginning at units' place, separate the num- 11 '5 6 ( 34 ber into periods of two figures each. The greatest square 9 in the left-hand period (11) is 9, and the root is 3, which is 64 256 written in the quotient. By subtracting this square (9) from the left-hand period (11) and annexing to the remain- der (2) the next period (56), we form the dividend, 256. By taking twice the root already found (3), we have 6 as a trial divisor, which is contained in the dividend (25), exclu- sive of the last figure, 4 times. "Write 4 in the quotient, and also to the right of the trial divisor, forming the complete divisor, 64. Multiplying the complete divisor, 64, by 4, the last figure of the root, and subtracting the product (256) from the dividend (256), there is no remainder. 34 is the required root. 332 A P P E ND I X. [Art. 7O1. Find the square root of 2. 1089. 3. 14641. 4. 18225. 6. 46656. 6. 47524. 7. 65025. 8. 86436. 9. 97344. 10. 119025. 11. 1406.25. 12. 512656. 18. 232.5625. Z. 3976036. 15. 431.8084. 16. 7463824. 17. 387420489 18. Find the square root of fff . Of f|f|. Of NOTE. In finding the square root of a fraction, extract the square root of the numerator and denominator separately. 19. Find the side of a square field whose area is 412164 square rods. 20. Find the side (in feet) of a square whose area is one acre. 21. Extract the square root of 2 to 5 decimal places. CUBE ROOT. 702. The Cube Root of a number is one of the three equal factors of that number. Thus, the cube root of 8 is 2. 2x2x2 = 8. 703. To find the cube root of a number. 704. RULE. Beginning at units' place, separate the given number into periods of three figures each. Find the greatest cube in the left-hand period, and write its root at the right in the form of a quotient in division. Subtract this cube from the left-hand period, and to the remainder annex the next period to form a dividend. Multiply the square of the root already found by 300 for a trial divisor. Find how many times this divisor is contained in the dividend, and write the quo- tient as the next figure of the root. Add to the trial divisor thirty times the product of the last figure of the root and the other figures of the root, and the square of the last figure to form the complete divisor. Multiply the complete divisor by the last figure of the root, and subtract the product from the dividend. To the remainder annex the next period, and proceed as before. Art. 705.] CUSS ROOT, 333 E X.A M P L E S . 7O5. 1. Find the cube root of 39304. Trial divisor. 300 X 3 2 =r2700 30x3x4= 360 4= 16 Complete divisor, . . . 3076 39'304 27 34 12 304 12304 OPERATION. ANALYSIS. Beginning at units' place, separate the number into periods of three figures each. The greatest cube in the left-hand period (39) is 27, and its root is 3, which is written in the quotient. By subtract- ing this cube (27) from the left-hand period (39), and annexing to the remainder (12) the next period (304), we form the dividend, 12304. By multiplying the square of the root already found by 300 (multiply by 3 and add two ciphers), we form the trial divisor, 2700 (300 x 3 2 = 2700). The trial divisor, 2700, is contained in the dividend, 12304, 4 times. Write 4 as the next figure of the root. To form the complete divisor, add to the trial divisor (2700) 30 times the prod- uct of the last figure (4) of the root and the other figure (3) (30 x 3 x 4=360), and the square of the last figure (4 2 = 16). Multiplying the complete di- visor, 3076, by 4, the last figure of the root, and subtracting the product (12304) from the dividend (12304), there is no remainder. 34 is the required cube root. NOTE. When the given number is a decimal, separate the number into periods of three figures each, by proceeding in both directions from the decimal point. Extract the cube root of 2. 5832. 6. 551368. 10. 98611128. 3. 10.648. 7. 7529536. 11. 279.726264. 4. 314432. 8. 9.663597. 12. 435519.512. .474552. 9. 13651.919. 13. 676836152. 14. The product of three equal numbers is 551368. What are the numbers ? 15. A cubical block of marble contains 103823 cubic inches. What is its length ? 16. Find the cube root of }f Of AVVV Of ifMf NOTE. In finding the cube root of a fraction, extract the cube root of the numerator and denominator separately. 17. A cubical box holds 20 bushels. Find the length of its side. MENSURATION. 706. To find the area of a rectangle (319) or paral- lelogram. 707. RULE. Multiply the length by the width. (32O) 708. To find the area of a triangle. 709. RULE. Multiply the base by half the perpendicu- lar height. 710. To find the area of a triangle when the three sides are given. 711. RULE. From half the sum of the three sides sub- tract each side separately. Then multiply the half sum and the three remainders together, and extract the square root of the continued product. 712. To find the area of a trapezoid. 713. RULE. Multiply half the sum of its parallel sides by the perpendicular distance between them. 714. A Trapezoid is a four-sided figure having only two of its sides parallel. 715. To find the area of a circle. 716. RULE. Multiply the square of the diameter by .7854 ; or multiply the square of the radius by S.lJj-16. 717. To find the surface of a sphere. 718. RULE. Multiply the square of the diameter by 3.1416. 719. To find the solid contents of a prism or cylinder. 720. RULE. Multiply the area of the base by the height or length of the prism or cylinder. 721. To find the solid contents of a sphere. . RULE. Multiply the cube of the diameter by Art. 723.] -V E N S UR ATIO N. 335 723. GEOMETRICAL, CONSTANTS. CIRCLES. Circumference = Circumference = Radius = Radius Radius = Radius = Diameter = Diameter = Diameter Diameter = Area = Area = Side of inscribed square Side of equal square Diameter x 3.1416. Radius x 6.2832. Circumference -f- 6.2832. Circumference x .1592. A/Area" -^3.1416. A/Area x .5642. Circumference -f- 3.1416. Circumference x .3183. = A/Area -f-. 7854. = VArea x 1.1284. = Radius square x 3.1416. = Diameter square x .7854. = Diameter x ,7071. = Diameter x .8862. Side of inscribed equilateral triangle = Diameter x .8603. Surface Surface Volume Volume Diameter Diameter Radius Radius Circumference Circumference Side of inscribed Side of inscribed SPHERES. = Diameter square x 3.1416. = Circumference square x .3183. = Diameter cubed x .5236. = Circumference cubed x .0169, = A/Surface x .5642. = ^Volume x 1.2407. = A/Surface x .2821. = ^Volume x .6204. = A/Surface x 1.7725. = ^Volume x 3.8978. cube = Diameter x .5774. cube =: Radius x 1.1547. ROOTS, ETC. Diagonal of square = Side x 1.4142. Square root of 2 = 1.4142. Side of square = Diagonal x .7071. APPENDIX. [Art. 721. EXAMPLES. 1. What is the area of a rectangular piece of land 15.50 chains long and 12.25 chains wide ? 2. Find the area of a triangular piece of land whose base is 3.25 chains and perpendicular width 5.20 chains. 3. What is the area of a triangular piece of ground whose sides are 20, 30, and 40 rods respectively ? 4' A piece of land in the form of a trapezoid has two parallel sides, 1225 ft. and 750 ft. respectively, and the perpendicular distance between them is 1540 ft. What is its area ? 5. The circumference of a circular race-course is one mile. What is its diameter in feet ? 6. Find the circumference of a circle whose diameter is 150 feet. 7. A cow is tied to a stake by a rope 42 feet long. Upon how much surface can she graze ? 8. The diameter of a globe is 2G inches. What is the area of its surface ? 9. How many gallons of water will a cistern hold whose diameter is 10 ft. and depth 6 ft. ? 10. What are the solid contents of a globe whose diameter is 26 inches ? 11. A lake, whose diameter is 1000 ft., is covered with ice 8 in. thick. What is the weight of the ice in tons, if a cubic ft. of ice weighs 920 oz. avoirdupois ? 12. Find the length of the side of a cubical bin, whose con- tents are 100 bushels. 13. The area of a square field is 5 acres. What is the ?ength of a side ? 14. Find the solid contents of a log 24: ft. long and 2ft. in diameter. 15. In making a square pond whose side was 204 ft., 10000 cubic yards of earth were taken out. What was its depth ? 16. A room is 13 ft. 4 in. by 13ft. 6 in. How many yards of carpet, f of a yard wide, will cover it ? 17. Find the diameter and circumference (in rods) of a cir- cular field containing 10 acres. 18. How many acres in a field one mile in diameter ? GENERAL AVERAGE. 725. If, in time of danger or distress, any loss or expense is voluntarily incurred for common safety of vessel, freight, and cargo, such loss or expense is made good by a " General Aver- age ; " the amount or value of such loss or expense being assessed upon the value of all interests involved and benefited. All other losses and expenses are of a " Particular Average " nature, and are to be borne by the specific interests to which they apply. 726. The losses and expenses constituting general average are as follows : 1. Jettison, or throwing overboard of cargo to lighten the ship ; damage to cargo by water going down the hatches during jettison ; damage by chafing or breaking after jettison ; freight on cargo jettisoned. 2. Sacrificing ship's materials, as the cutting away of masts, spars, etc. One-third of the cost of repairs of ship's materials is a special charge on the ship, as the new work is considered better than the old. No deduction is made for anchors. 3. Expense of floating a stranded ship. 4. Expense of entering a port of refuge, either to repair damage which renders it dangerous to remain at sea, whether such damage were caused by accident or sacrifice ; or otherwise to avert a common danger. 5. Expense of discharging cargo for the purpose of making repairs, warehouse rent, reloading cargo, outward expenses, etc. 6. Wages and provisions of crew from the date of bearing up until ready for sea. 737. Contributory Interests and Values. The ship con- tributes on its full value at the time which is made the basis of contribution. 338 GENERAL AVERAGE. [Art. 727. The cargo contributes on its net market value at the port of destination, less freight and charges saved. The freight contributes on the full amount., less -J- for the wages,, etc., of crew. In the States of New York, Virginia, Cali- fornia, and some others, |- is deducted. The underwriters (Insurance companies) contribute to the general average such a part of the expense as the insured value is of the market value of the goods (542). If, for example, a cargo is insured for $10000 and is worth in the market $12000, the underwriters are liable to pay f of the general average expense. 728. To give rise to general average, it must be shown that there was an imminent common danger, that the sacrifice was voluntary and necessary, and that the act was prudent and suc- cessful. 729. An Average Adjuster is a person who is familiar with the general average laws of the leading commercial nations, and who adjusts and apportions the losses and expenses of a general average. The principal difficulty of an adjuster is to decide whether the loss should be made good by a general average or should be made a special charge (par- ticular average) upon some particular interest. After the general average charges are determined, the apportionment of the loss among the several con- tributory interests is a simple arithmetical problem. EXAM PLES. 73O. 1. The bark Liberty sailed from New York for Galves- ton with the following cargo : Shipped by A, $5600 ; by B, $8700 ; by C, $16308 ; by D, $8360. After two clays out the bark en- countered heavy gales and was damaged to the amount of $630. 14. On the fifth day the vessel began to take water, and for the safety of the vessel and the cargo the bark bore away for New York for repairs. The disbursements of the agent at New York were as fol- lows : Custom-house fees, pilotage, protest, towage, unloading and reloading cargo, wharfage, inspection, consul fees, $1369.43 ; bill of H. Robin & Co., shipwrights, etc., $436 ; bill of Joseph Patti, ceiling ship, $194.14. Agent's commission for advancing funds and paying above bills, 5% ; on value of cargo landed, $17388, \\%. Wages and provisions of seamen from point of deviation, $630.47. The gross freight was $8096, and seamen's wages, etc., \ of gross Art. 730.] GENERAL AVERAGE. 339 freight. How is the settlement to be made, the value of the ship being $10000 and the adjuster's fee $100 ? NOTES. 1. In a general average, extracts from the log of the ship, the testimony of its officers, a complete statement of all expenses incurred, with the vouchers for the same, and all papers having any bearing upon the case are presented to the adjuster. The total amount of each item is entered in a column at the left of his statement of charges, and the amount is also entered in its proper column at the right. In addition to the general average column, there are usually columns to the right for the special charges upon the ship, owners, or cargo. 2. After determining the general average loss, divide it among the con- tributory interests in proportion to their values, by any of the methods given in Ex. 13, page 293. STATEMENT OF CHARGES. General Ship and Average. Owners. 1369 43 Expense of entering harbor, landing cargo,etc. 1369 43 436 Bill of H. Robin & Co., shipwrights, etc. 436 194 14 " " Joseph Patti, ceiling ship. 194 14 Agent's commission for advancing funds and 99 98 paying above bills, 5%. 68 47 31 51 Agent's commission on value of cargo landed, 217 35 $17388, \\%. 217 35 630 47 Wages, etc., of seamen. 630 47 100 Adjuster's fee. 100 General average. 2385 72 3047 37 Ship and owners. 661 65 CONTRIBUTORY INTERESTS AND APPORTIONMENTS IN GENERAL AVERAGE. Ship, value 10000 @. .045 pays 450 Freight, 8098 Less, 4048 4048 .045 182 16 Cargo, A, 5600 @ .045 " 252 B, 8700 .045 " 391 50 C, 16308 @ .045 " 733 86 D, 8360 @ .045 376 20 38968 @ .045 " 1753 56 53016 @ .045 " 2385 72 $2385.72 -r- $53016 = .045. 340 APPENDIX. SETTLEMENT [Art. 13O. BALANCES. DR. CB. To pay. To receive. Vessel and Owners. Pay ship's proportion of Gen.Aver. 450 " freight's " 182 16 " owner's column. 661 65 Receive seamen's wages. 630 47 663 34 Cargo. Pay proportion of Gen. Average. 1753 56 1753 56 Agents of Vessel. Receive their disbursements. 1999 57 " " commission. 317 33 2316 90 Adjusters. Receive their fee. 100 100 3047 37 3047 37 2416 90 2416 90 2. The general average charges were $4375.86,, and the con- tributory interests $64325. What was the per cent, of loss ? What was the loss of Mr. B., whose goods were valued at $7250 ? 3. Suppose A's goods in Ex. 1 were insured for $5000, how much of the loss would be shared by the insurance company ? 4. The ship Amazon, from Aspinwall to New York, being in distress, threw overboard part of the cargo, cut away the masts, and finally bore away to a port of refuge to repair in order to com- plete the voyage. The cost of replacing masts and rigging cut away was $6000 (less | new for old) ; the cargo jettisoned was worth compared with sound cargo delivered at destination $2000 ; freight on cargo jettisoned, $200 ; expenses of entering port of refuge, discharging, storing and reloading cargo, $1000; wages of master and crew from time of bearing away until ready for sea, $600; provisions of master and crew for same time, $500; adjuster's fee, $100. The vessel was valued at destination at $20000 (deduct gross repairs and add amount made good) ; cargo, value on arrival, $40000 (add amount made good) ; freight collected, $4000 (add amount made good and deduct J). What was the per cent, of loss, and how was the settlement made ? 5. The cargo of the ship Amazon was insured for $36000. How much was the claim against the insurance company ? Art. 73O.] GENERAL AVERAGE. 341 6. The ship Union, in her passage from Liverpool to Boston, during a storm threw overboard cargo to the amount of $1580, and cut away masts and rigging. She then entered the port of Halifax for repairs. The cost of replacing the masts and rigging which were voluntarily sacrificed, was $4578 (less new for old) ; cost of repairing accidental damage, $568 ; freight on cargo jetti- soned, $314. 75 ; expense of entering port of refuge, discharging cargo, etc., $716.87; wages and provisions of crew, $608 ; adjuster's fee, $150. The value of vessel on arrival at Boston was $30000 (deduct gross repairs and add amount made good) ; value of cargo delivered, less freight and duty, $48475 (add amount jettisoned) ; total expected earning of freight, $16320 (less in Boston. See Art. 727). The cargo was shipped by the following persons : A $8519, B $20376, C $6875, and D $14285. The cargo jettisoned was a part of A's shipment. How ought the settlement to be made? 7. The ship Ocean Qmen, from Pernambuco to New York, sprang a leak off Cape St. Koque, and for the safety of the vessel and cargo, threw overboard part of the cargo and put into Maran- ham for repairs. The disbursements at Maranham by the master of the vessel, including commissions, were as follows : Expenses of entering harbor, discharging, storing, and reloading cargo, $648.75 ; caulking and painting ship, carpenter work, etc., $843. Value of cargo delivered at New York, $34310.24; of cargo jettisoned, $1580.76 ; freight on cargo jettisoned, $364 ; wages and provisions of crew, $304 ; adjuster's fee, $150 ; agent's commission for col- lecting amount in general average, 2J%. How shall the settle- ment be made, if the net value of the ship was $3157 (value on arrival $4000, less repairs $843), and the total expected earning of freight was $2516 (less ) ? 8. A vessel which put into a port of refuge for repairs was without funds. It being very difficult to obtain a loan on bot- tomry, or to negotiate a draft on the owners of the vessel, the mas- ter was obliged to sell part of the cargo to raise funds. Value of cargo sold compared with cargo delivered at destination, $4566.06 ; produced at sale, $2985.30 ; freight on cargo sold compared with freight on cargo delivered, $363.93. What was the cost of funds, and how much should be apportioned to each interest, the general average charges being $773.52, the special charges on ship $956.10, and on the owners $1181.06 ? 343 APPENDIX. [Art. 731. 731. FOREIGN WEIGHTS AND MEASURES. ARGENTINE CONFEDERATION. Metric system used in the assess- ment of duties. Old Spanish weights and measures (See Spain) in common use. AUSTRIA, (AS GERMANY.) BELGIUM, (METRIC SYSTEM.) BOLIVIA. The metric system is the legal sys- tem, but the law has not been rigidly enforced. Old Spanish weights and measures (see Spain) still in use. For coin weight the metric gram is used. BRAZIL, (METRIC SYSTEM.) Diamonds are permitted to be sold according to the old Portuguese outava (55.34 grains). Ships' freights are for the most part, settled according to the English ton (2240 lb.). CANADA, (AS GREAT BRITAIN.) CHILI, (AS BOLIVIA.) For custom purposes the metric system is enforced. 1 Tael 1 Catty 1 Picul 1 Chih 1 Chang CHINA. = 1 oz. av. = H U. av. = 133 Ib. av. = 14.1 inches. = 11.75 feet. COLUMBIA, (METRIC SYSTEM.) DENMARK. 1 Pound (I kilogram) = 1.102 Ib. av. 1 Centner (100 Ib.) = 110.23 Ib. av. 1 To'nde of grain = 3.948 U. S. bu. 1 To'nde of coal = 4.825 U. S. bu. 1 Fod (Foot) = 1.03 U. S. ft. 1 Viertel = 2.04 U. S. gal. 1 Alen (Ell) = .6864 yd. Coinage laws are metric. The in^ troduction of complete metric system is in prospect. ECUADOR, (METRIC SYSTEM ) EGYPT, (METRIC SYSTEM.) FRANCE, (METRIC SYSTEM.) The old French aune = 1^ yd. is still used to some extent in the silk industries of France and the U. S. GERMANY. Metric system with a few changes in subdivisions in general use. 1 Pound ( kilogram) = 1.1023 Ib. av. 1 Centner (100 pounds) = 110.23 Ib. av. 1 Wispel (metric ton) = 2204. 6 Ib. av. GREAT BRITAIN. 1 Imp. Gallon = 1.2 U. S. gal. 1 " Bushel = 1.03 U. S. bu. 1 " Quarter = 8.25 U. S. bu. 1 Ale or Beer Gallon = 1 .22 U. S. gal. 1 Cental = 100 Ib. 1 Quarter of Wheat ) XQn at London f = * lb ' 1 Quarter of Wheat at Hull ) KA/( and Newcastle. \ = ** 1 Quarter of Wheat at Dun- dee and other places. Metric system permitted by law of 1864. GREECE. Metric system with the common Grecian names in general use. In the Ionian Islands the English weights and measures have been legalized since 1829. 496 lb. Art. 731.] FOREIGN WE IGIITS AND MEASURES. 343 INDIA. 1 Seer = 16 chattucks. 1 Bombay Maund of 40 seers: 1 1 Surat 1 " 1 " 42 40 42 44 =29.4' 1 Bengal Factory Maund =74f " 1 " Bazaar " =82 " 1 Madras Maund =25 " 1 Bom'yCandyof20Maunds=560 " 1 Surat " " " =746f" 1 Madras " " " =500 " 1 Travancore " " " =660 " 1 Tola =180 gr. 1 Guz of Bengal =1 yard. 1 Gorge =20 units. 1 Gorge Pound =20 Ib. Metric system permissive. ITALY. 1 Palm = .555 cu. ft. Metric system in general use. JAPAN. 1 Picul = 133 Ib. av. For coinage, in part, the metric unit of weight is used. JAVA. 1 Amsterdam Pond = 1.09 Ib. av. 1 Picul = 133i " 1 Catty = li 1 Chang = 4 yards. MEXICO. Weights and measures are legally the metric, but the metric system is not generally in force, the old Spanish weights and measures (see Spain) being still employed. NETHERLANDS. Metric system with a change in names in general use. 1 Last (30 hectoliters) = 85.134 bu. NORWAY AND SWEDEN. 1 Swedish Skalpond = 0.93 Ib. av. 1 Swedish Centner =934- " 1 Norwegian Pund = 1.1 Ib. av. 1 Swedish Fot = 11.7 inches. 1 Norwegian Fod = 12.02 " In Norway the metric system is used to some extent. In Sweden, the coin weight and the medicinal and apothecary weight are metric. The complete metric system has been obligatory since 1882. PORTUGAL. Metric system compulsory since Oct. 1, 1868. The chief old measures are 1 Libra =1.012 Ib. av. 1 Almunde of Lisbon =4.42 U. S. gal. 1 Alquiere =.3928 U. S. bu. EUSSIA. 1 Pound =0.9 Ib. av. 1 Pood (63 to a ton) =36 1 Berkowitz =360 " 1 Chetvert =5.956 U. S. bu. 1 Vedro =3.25 U. S. gal 1 Arsheen =28 inches. 1 Ship Last =2 tons. Metric system partially in use. SPAIN, (METRIC SYSTEM.) In many of the South American States and in Cuba, the old Spanish weights and measures, principally Castilian, are used. They are as fol- lows : 1 Libra = 1.014 Z&. av. 1 Arroba (25 Libras) = 25.36 " 1 Quintal (100 Libras) =101.44 " 1 Vara = .914 yd. SWITZERLAND. Metric system used with some changes of names and subdivisions. Pure metric system optional. TURKEY, (METRIC SYSTEM.) URUGUAY, (AS ARGENTINE CONFED- ERATION.) VENE3UBLA, (METRIC SYSTEM.) DETECTION OF ERRORS TRIAL BALANCES. 732. The following hints apply to the detection of errors in trial balances, or in any operation in which errors are made in addition or subtraction, or in transferring numbers from one place to another. 1. Ascertain the exact amount of the error. Much time is sometimes wasted in looking for errors which do not actually exist. 2. Kevise carefully the additions of the trial balance before looking for the error in the ledger or other books. 3. If the error is in one figure only (as 2000, 100, 50, etc.), it is probably an error in addition or subtraction. 4. If an amount is entered on the wrong side of an account, or is added when it should be subtracted or vice versa, the error will be twice the amount. 5. If the digits of any number are written to the right or left one, two, or three places, and the error be divided by 9, 99, or 999 respectively, the quotient will be the number. Thus, if $427 be written $4.27, the error will be $422.73 ; which divided by 90 (by 9 and 11), the quotient will be $4.27. The number of 9's by which the number can be exactly divided is equal to the number of places which the number has been transferred to the right or the left. 6. If two consecutive digits of any number are transposed, the error will be a multiple of nine ; and the quotient obtained by dividing the error by 9 will express the difference between the digits transposed. Thus, if 437, be written 473, the error will be 36 ; which divided by 9 produces 4, the difference between 3 and 7. The same error, 36, will arise if the figures transposed are and 4, 1 and 5, 2 and 6, 4 and 8, or 5 and 9. 7. If the error contains a number of figures, it is probable that some account or item has been omitted. 8. Look for the error systematically, and not in certain por- tions of the work selected at random. ANSWERS. Art. 2O. 3. 40865. 5. 49374; 98748. 77. 5,761,888; 7. 1614. 4. 110547. 4. H0775; 195,249,432. 2. 1654. 5. 8495098. 295400. 70. 18,413,409; 3. 19380. 6. 853759. 5. 243580; 556,524.675. 4. 23243. 7. 999895. 438444. 13. 8,326,575; 5. 26162. 5. 1109975. 6. 817281; 173,434,110. 6. 35130. 9. 6419754. 726472. 14. 25,930,788; 7. 4566. 10. 72540. 7. 130240; 317,327,062. 5. 3722. 77. 57249251. 182336. 75. 93,309,006; 9. 53609. 10. 44601. 70. 10648519. 75. 113558829. 5. 1,578,246; 2,367,369. 889,602,580. 16. 4,428,648; 77. 50480. 14. 15562130. 9. 494268; 26,888,220. 70. 34914. 75. 74,299,273. 617835. 77. 744. 16. 5,654,786. 10. 4,690,158; 18. 43200. Art. 2?. 77. 90,119,023. 7,035,237. 79. 2,419,200. 4. 4915. 9. 435. 75. 122,882. 77. 3,336,072; SO. 5250. 5. 4857. 70. 508. 79. 921294. 2,919,063. 07. 506880; 6. 394. 77. 3642 SO. 19.212,939. 70. 4,072,384; 1,098,240. 7. 376. 70. 3645 21. 4745. 3,563,336. 22. 26376. 5. 321. 13. 3755 00. 64535. 75. 3,824,910; 23. 106515; 7^. 54877. S3. 45009. 5,737,365. 153720. 75. 44444. 24. 27369. 14. 5,240,172; 24. 576. 16. 41568. 05. 41976. 3,742,980. 25. 4608. 77. 36311. 26. 12464. 75. 58080. S6. 5016. 75. 84839. 27. 62645. 16. 2016. 79. 139059. S8. 10514. 77. 24256. Art. 47. 20. 10078521. 29. 3211. 75. 63360. 1. 144000; 21. 561. 26. 3657. SO. 5821. 79. $194.40. 1080000. 00. 3691. 07. 6822. 57. 4004. 20. 1296. 0. 138240; 23. 1404. 05. 1711. 50. 5038. 864000. 24. 7921. 09. 1440. 33. 1235. Art. 44. 3. 241920; 25. 297. 30. 7529. 54. 11,594,495. 1. 63936; 831168. 1,451,520. 57. 14152. 55. 193,941,760. 0. 75218; 4. 185500; 50. 442,254,988. 36. $93,309,621. 1,463,858. 1,335,600. 33. 433. 57. 3025. 3. 70272; 5. 120000; 54. 1771. 55. 6558830. 2,436,096. 3,200,000. 35. 4653. 39. 3850814. 4. 209387; 6. 252000; 55. 39247. 40. 388904. 1,915,125. 1,036,000. 57. 16098. 41. 1106. 5. 358661; 7. 81600; 55. 813210. A m f> ** 1,264,432. 272000. 59. 6399. Art. 35. 6. 544375; 5. 84000; 40. 1,177,761,723. 1. 1107.90. 4,606,875. 2,100,000. 41. 5,302,516. 0. 317.26. 7. 720408; 9. 9,680,000; 4S. 324,423,840. 5. 6622.70. 7,213,316. 67,760,000. 43. $9858.94. 8. 661982; 10. 18,500,000; 44- $419360.87. Art. 41, 6.961,968. 92,500,000. Art. 33. 1. 164192; 187648. 9. 6,896,064; 87,772,352. 77. 7,407,000; 44,442,000. 1. 4337. 0. 340236; 10. 5,847,408; 70. 11,760,000; 0. 907823. 226824. 195,035,421. 131,600,000. 346 AJVS WERS. [Art. 47. 13. 6,698,000; 10. 88375; 94500. Art. 74. //. 921300; 114,260,000. 11. 95472; 96408. 2. 7623; 7161. 919450. 14. 8,019,200; 12. 91208; 92962. 3. 8232; 7980. 12. 868000; 98,808,000. 13. 77280; 80224. 4. 6768; 6912! 868875. 15. 67,200,000; 14. 15876; 14847. 5. 35991; 386613. 7& 838530; 614,880,000. 15. 40040; 41195. 6. 39520; 413504. 836836. 16. 86,400,000; 16. 82215; 80649. 7. 49104; 523776. Art. 90. 460,800,000. Art. 5O. 17. 58422; 57876. Art. 62. Art. 78. 1. 11872; 12432. 2. 10506; 10608. 3. 264; 176; 198; 352; 473; 363; 792; 891; 407; 484; 1012; 957; 1023; 704; 385; 396; 517; 187; 209; 528; 627. 2. 3724; 2964. 3. 2523; 8613. 4. 2655; 3105. 5. 5928; 27768. 6. 16653; 33733. 7. 23925; 56925. Art. 65. 1. 3136; 2304; 4136. 2. 4875; 3219: 1656. 3. 4851; 6375; 816. 4. 1739; 3819; 3825. 5. 8125; 4536; 3. 13176; 12810. 12412; 12992. 5. 15515; 16240. 6. 19536; 19008. 7. 1,010,024; 1,011,028. 8. 1,134,000; 1,138,500. 2. 1206; 1809. 12936. Art. 93. Art. 53. 3. 1224; 1530. G. 5776;' 1296; '1. 10379; 1J0165. 2, 2695; 3806; 4. 4158; 4851. 12996. . 10752; 10304. 3575; 4576; 5. 6048; 6804. 3. 10904; 11368. 8624; 5687; 6. 12420; 15525. Art. 81. 4. 9828; 10692. 9625; 10098; 7. 10206: 40824. 1. 7134; 7047; 5. 91455; 93465. 46398; 80564; 8. 13986; 32634. 4095. 6. 95665; 97679. 79398; 19008; 5. 39096; 58644. 2. 2068; 2912; 7. 100188; 93104, 48125; 92136. 3306. S. 95692; 97728. Art. 68. 3. 5548; 5925; Art. 56. 2. 600; 900; 925: 4284. ^4r. 99. 2. 52704; 35424. 1225; 1550; 4. 1892; 2860; 1. 39,456,174; 3. 22227; 50907. 9675; 11200; 4221. 26,304,116. 4. 9387; 36207. 12800; 18650; 5. 13572; 11235: 2. 24,413.116; 5. 283745; 10600; 20425; 15250. I6,275,410f. 260295. 23425; 13600; 3. 3,265,524; 6. 378216; 17925; 7950: Art. 84. 2,721,270. 600696. 8100; 6400; 2. 624; 7225; 4. 19,517,701; 7. 341284; 13900; 230600; 15616. 11,152,972. 174804. 209450; 3. 221; 9024; 5. 2,057,613; 8. 112875; 132000; 43200; 13216. 1,371,742. 150375. 141200. 4. 1224; 1221; 6. 197,730,864; 9. 85425; 42925. 11024. 123,581,790. 10. 281869; Art. 7O. 6. 625; 2021; 7. 58,642,209; 234969. 3. 83763; 96929. 21021. 26.063,204. 11. 338583; 4. 126936; 6. 1225; 3024; 8. 178,606,127; 386883. 293088. 24016. 51,030,322. 12. 75576; 338776. 5. 43344; 42656. 9. 49,377,285; 13. 263375; 6. 310148; -4?'. 87. 27,431,825. 350875. 137592. 2. 9603; 9118. 10. 31,025,988; 7. 47775; 88725. 5. 8008; 8360. 24,131,324. Art. 59. 8. 170556; 4. 8277; 8544. 11. 51,525,354$; 2. 10205; 13345. 863964. 5. 7275; 7350. 20,610,141$. 8. 5292; 6048. 9. 203912; 6. 9016; 8556. 12. 71,387,270; 4. 7830; 9918. 597376. 7. 8084; 8170. 35,693,635. 5. 6768; 6016. 10. 288834; ** ***- 5. e. ^F; W; *F. 7. H*; W; W- A V; W; W- 2. $16. 5. 93f ; 52. 4. 69; 1251 5. 57f ; 884. 7 36* f 111 8\ 32l|; 78. 9. 49|; 16J| 10. 24A; 13f 11. 20$i; 15| 12. 27|; 14. 64| is- BH; 7f&. 16. 22^; 62$. 17. 18H; 22H. 15. 463 T V. m 704-. ^1. 270f. ^. 388H- 25. 89^. ^. 472 T V. 27. 88*. 0. 126 3 . ^9. 299 1 . 30. 121 2 . 217 22 bu. 3. $364. $3684. 5. $21.22^. < S f/* 5 5. .00540625; 50. 396f. Carriage, 5. I 76. T %. .8455375. Art. 236. $440f. 50. $20031. 5. | 75. !? 9. 11.208704; .0100672. - 7. Ij. 27. 26. 2. 20. 22. 71. 57. $157.67. 52. 501. 7! 5hr. 2o! X! o 16 $) 1 3 70. 5.7054831; 34.01345|. 5. 371. ^- 15 - 55. $9.46. - T2T* *# ^6U* Q 9 O 5 77. 28.6480831; 4. 63. 24. 21. 5. 1171. 25. 10. 34. 123 T V gal. 55. Widow, Y/l 64 j^^ 1 21.984375. 72. .288; 44.0928. 5. 11. 26. 22f 9. i*. 27. 121 $2876.12; Each child, S: J: S: MA. 75. 93.056831; 4.02031. 10. 11. 25. 16f $1438.06. ' - 1 2 5' 14. 115.6666f; 77. 6|. 29. 18f 56. 14031. ? TF * 500.40291. 72. 9. 50. 21g 57. $192. ^5. byg-^. 75. .51153; 75. llff. 57. 15. 55. $4600. Art. 262. 3.85331. 14. 9. 52. 91. 75. 6ft. 55. 13. 76. 2yg-. 54 46. 77. 21. 55. 13. 75. If. 56. 23. 19. ft. 57. 331. 39. Lost $0.381. 40. 20191 f; 24348|. 41. 31963f; 185517 V 42. 6224f; 1. 492.319787. 2. 7462.31526. 3. 476.338U807. 4. 2.6591587. 5. 9710.27879. 6. 1.83586255. 76. 82.0166f; 1061.1796|. 77. 576; 432; 216; 345.6. 75. 170845.86. Art. 271. 7. 1764.06. 39. 131. 43. 19479| ; 1. .048. 40. ft; A; 5 H- 44. $198.Ji. 45. 19744. 9. 215.2741JV 70. 21.9026730|. 2. 250. 3. 104; 8.625. 4. 1.914; 2.82. 43. 2; l|. 46. $84.24. Art. 265. 5. .875; 100.8. 44. 24-; 2. 47. 27.80. 1. 3.9803. 6. 481.5; 385.2. 45. 6351; 1- 48. 76.66. 2. .26971. 7. 4.25; 6.2. 49. 320 rods. 3. 8999.1. 5. 15.24706; Art. 23^. 50. 6 days. 4. .4648. 2.25. 7. ^. 57. Gained 2 cts. 5. 16.6736. 9. .49; 82.6875. 2. -ff. 52. $629.30. 6. .010102. 70. .5694; 39. ^ -V . 55. $5487.98. 7. $86.17. 77. 18.66; 4. 1718|. 54. 110 bu. 5. 2.126155. 10.30152 + . 5. 1931|.. 55. $35.46. 9. 1.728-J. 12. 2722.02; 42. 6. 862ft. 56. $136.99. 70. $121.141. 75. 86.40; 69.12; 7. 31. 57. $115.30. 77. $1727.93^. 51.84; 138.24; 5. 3023564,. 55. $316.74. 72. .924f. 25.92. 350 A NS WERS . [Art. 271. 14. 1800. 77. T V 77. 2934| sq. yd. Art. 292. 15. 3720. 7/2. 176.27$. 18. 3200 A. /-Pi 70'. 12. 13. 2. 19. 354 da. ' 3"S i"? * | oz. 77. 2.525. 74. 576. 20. 2160 cu. ft. 5. A HI 18. 293.040015. 75. 5. 21. 42885d. " 9 <> - 1 * 1 * / 3 HI. 19. .03|; .12$. 16. 3i 22. 982 da. Tr" 320 *** SO. .16* ; .40. 77. 331.2. 23. 27782 Ibs. 0". 2. 21. .06*; .06. 18. $56.16. 24. 7583d. ^40* 7. -1 v. 2. .021; .03i. 79. $1575. 8 XT 5. .02^; .oof 24. .14?; .41 1. 20. $108.99. 21. $143.06. ^r. 2S5. 7. 35 6s. 3d. * tf * 9. J-r bu. 70. |. #5. $361.60; $452. 23. $42.07; & 75cd.83cu. ft. 26. $10800; $61.91. 3. 117 bu. 2 pk. Art. />. 4 / Jt 00. $1612.50; 26. 7200. ,1. 246.06 yd. ; Art. 389. $2700.40. 07. $196. 8858.25 in. -t db~t o nn . 23. $2325.38. 28. $15360. 0. 9.6558 Km. 1. $18.99; 24. $84. 09. $91500. 5. 259.008 H. ; $20.34 ; 26. $0.94. 50. $65500. 25900.8 A. $16.55. 07. $6.56. 31. $564. 4. 32808.3 ft. ; 0. $2.06. 28. $14.71. 50. 400. 6.2137 mi. 5. $12.16. 29. $3.96. 33. $930. 5. 828.04776 Ib. 4. $13.50. 30. $79.13. 5 $444. 6. 204.12. 7. 26.73 grams; 27.216 grams. 8. 1762 HI. 9. 668.9375 cu. m. 5. $5.42. 6. $12.16. 7. $16.39. 5. $32.89. 9. $3.44. 51. $53.83. 50. $1476.40. Art. 4O4. 55. $324. 56. $6210. 57. $456.80. 38. $505. 39. 1425 boxes. 10. 1308 cu. yd. 10. $40.25. 1. 432. 40. $59062.50. 11. 6540.48 1. 12. 291.824 sq. yd. 2626.416 sq. ft. 11. $54.07. 12. $76.48. 15. $245.80. 2. 868. 5. 1604.5. 1816. 41. $6464. 13. 6237 g.; 6.237 Kg. Art. 391. 5. $125. 6. $106. 1. .OH. 2. $179.80. 7. $162. ' QJ?' Art. 387. 3. $49.50. 3.48. 46. $1450.87; $51.91; 12. 1 y. 5 m. 18 d. / ^ fid. $1605.74. $38.93. 13. 16 y. 8 m. ty. $o.O'. 5. $6.57. 47. $5092.50; 76. $10658.20. 14. 28 y. 6 m. 26 d. 6. $6. $5032.71. 77. $1050. 7! $5^34. 48. $2. 78. $1556.66. Art. 472. 8. |X. 49. $10.66. 79. $27.84. 1. $12107.84. . $7.89. JO. $0.85. 50. $53.44. 51. $14.19. 80. $28.93. 81. $1356.18. 2. $871.31. 3. $2241. 11. $10.63. 52. $0.50. 82. $1562.50. 4. $7719.16. 12. $11.73. 53. $33.11. 83. $86.07. 5. $1997.87. 75. $18.14. 74. $0.31. 54. $61.20. 55. $8.38. 84. $5643. 85. $601.39. 6. $3000. 7. $3228.33. 75. $3.73. 56. $11.33. 8. $29419.35. 16. $10.07. 77. $35.63. 57. $9.33. 58. $34.96; $33.21. Art. 463. 1. $1.79. 9. $30612.25. 10. $31746.03. 18. $9.50. 79. $26.04. $35.65 ; $33.87. 59. $14.27; $18.73. 2. $116.47. 3. $11.54. 11. $11973.33. 12. $14370.69. 00. $46.67. 21. $16.28; $13.57. 22. $64.76; $14.47; $18.99. 60. $26. 73; $19.09. $27.29; $19.49. 4. $5.29. 5. $3.92. 6. $15.12. IS. $3436.99. Art. 475. $32.38. 61. $193.96; 7. $5.75. 1. $1234. 23. $36.85; $42.99. $113.14. 8. $6.16. 2. $5280. 24. $24. 70; $20.58. $195.15; 9. $42.18. 3. $3456. 25. $56.68; $75.57. $113.83. 10. $14.83. 4. $375.60. 26. $180.10; 62. $309.07; 11. $4.44. 5. $12375. $120.07. $347.70. 12. $10.44. 6. $1728. 27. $11.43; $17. 14. $310.14; 13. $39.35. 7. $723,01. 28. $39.45; $46.03. $348.90. 14. $246.89. 8. $879.54. 29. $19.79; $23.09. 63. $57.27; $81. 81. 15. $58.97. 9. $1511.67. 30. $2.85; $2.37. $57.82; $82.60. 16. $27.74. 10. $2309.28. 31. $13.75; $16.04. 64. $7.59; $10.12. 17. $41.64. 11. $3770.52. 32. $106.66; 65. $25.95; $11.53. 18. $7.58. 12. $5307.72. $142.22. 66. $111; $27.75. 19. $2.14. 13. $1642.31. S3. $137.72; 67. $76.50; $60.56. 20. $30.21. 14. $2138.94. $114.77. $77.40; $61.28. 21. $5. 15. $5063.11. 356 ANSW&KS. [Art. 475. 16. $2863.86. 3. $83.26. 9. Sept. 1 ; 39. $1523.25. 17. $3590.09. 4. $1909.63; $4430, or 40. $3081.09. $2104.72. $4430.96. Art. 479. 5. $1211. 10. Sept. 28; Art. 505. 6. $220.80; $8204.29, or 1. $678.54. ' $107! 14.' 2. $438.60; $268.51. 7. $8583.80. A $1811.44; $8204.78. 11. June 29 ; $4276.08, or 2. $242.17; $148.16. 3. $1102.69; 3. $547.95; $52.05. $2564.94. 9. $2794.32; $7798.54. $4276.73. 12. Nov. 30; $4768.85, or $1184.37. 4. $1327.21; $1410.94. < 'Qfy Qry ' 10. $993.03. $4770.44. 5. $835.74; 5. $283.35; $41.65. 6. $161.64; 11. $4445.17. 12. $2450.13. 13. $747.27. 14. $4172.57. 13. May 6; $8899.50, or $8900.88. 14. Jan. 15; $924.38. 6. $898.88. 7. $3073; $3363.56. 7. $595.39; $204.61. 8. $641.79; $258.21. 9. $0.68. 10. $3629.03. 11. $204.29. 15. $405.34. 16. $13363.84. 17. $4659.94. 18. $343.90. Art. 495. 1. $1022. $4909.58, or $4910.82. 15. Feb. 2; $5936.20, or $5937.07. 16. $5949. 17. Oct. 5; $4946.67. 8. $517.82; $716.62. 9. $3260.23; $4539.19. Art. 5O9. 2. $440; $447.16. Art. 480. 1. $42.32. 2. $1843.93. 2. $911.04; $919.21; $917.46. 3. Mar. 7, 1890; 18. Apr. 4; $3710. 19. Aug. 6; $6882.75. 3. $223.31; $214.37. 4. $651.97; $753.30. 4. ly.' 10m. 28 d. $6022.10. 4. $431.10; 20. July 4; $8909.75. Art. 513. 5. $1722.02. 7. Uf#. 8. Latter \\% better. ' $42485. 6. Aug. 21, 1887. 7. Monday. 8. Saturday ; 21. Dec. 21 ; $4838.61. 22. Sept. 29 ; $4451.25. 2. $262.24; $161.49. 3. $1108.57; $1192.53. 9. Oct. 3. Wednesday. 23. Dec. 5; 4. $1324.75; 10. 1%. 11. $19230.77. 9. Oct. 23. $8870.37. 24. Dec. 8; $1406.47. 5. $833.87; 12. $4298.04; $2970.75. $920.94. $4342.65. Art. 5OO. 25. June 24; 6. $897.77. 13. June 11,1874. 14. $129; $131.50; $129.70. 15. $650. 16. $1483.98. 17. $89.17. 18. $606.60. 19. 199 3s. 8d. 2. $7937.33, or $7938.19. 3. Apr. 27; $1181.40, or $1181.65. 4. Aug. 21; $5196.40, or $5197.55. 5. Nov. 2* $98)2.71. 26. Oct. 11; - $5894.25. 27. May 13; $5897.87. 28. Sept. 6; $8603.70. 29. $7837.33. 30. $8808. 7. $3067.14; $3347.31. 8. $549.89; $764.05. 9. $3260.51; $4594.82. Art. 523. 1. 12. 20. 8 10s. #1. 5 8s. 4d. 22. 9 5s. lOd. 23. 2 12s. lid. 24. 3 17s. 4d. .95. 10 9s. 8d. ^ir. 485. ' $2524.16, or $2524.65. 6. Oct. 7; $3664.71, or $3665.96. 7. Nov. 18; $6395.55, or $6395.96. 31. \%. 32. $15.87. 33. $3012.09. 84. Aug. 30; $3737.21. 35. May 19; $1641.17. 3. 160. 4. $36. 5. $50. 6. 240 Ib. 7. $111. 8. $21.875. 9. $55.50. 10. 85|yd. 1. $526.44; $506.48. 8. Sept. 16; $8135.73, or 36. Apr. 15; $882.22. 11. 396 ft. 12. 25 11s. 9d. . $45.18; $37.37. $8139.01. 38. $1523.62. 13. $960. Art. 523.] A N S WE R S. 35? 14. $2410.71. 7. $22.50. Art. 567. 62. 147. 15. A, $1875.90; 8. $56.88. 1. $1083.94. 63. 13950.75 fr. B, $1598.40. 16. 15.9883:1. 9. $71.43. 10. $57.60. 2. $1583.40. 4. $2407.50. 64. 391 4s. 65. 7343.75 fl. 17. $179.56. 18. $126. 11. $4880. 12. $28.56. 5. $3760.69. 6. $4050.03. 66. 1009 7s. 64 67. 512 3s. 2d. 19. $3.79. 20. $137.03. 13. $42. 14. $9000. 7. $409.34. 8. $2483.15. Art. 574. 31. $153600. 15. $9000. 9. $4076.72. 1. Oct. 10. ? 2355| ft. 16. 40%. 10. $3290.93. 2. May 11. 23. 25.215 fr. 17. $16.20; 11. $1087.98. 3. Feb. 11. V4. 252f f yd. 85. 20 hr. 45 min. $27. 18. $1440. 12. $4261.23. 13. $4593.93. 4. Aug. 4. 5. June 5, 1882. 9 sec. 15. $61.25. 14. $2611.06. 6. June 20. 26. $11.10. 27. 69 da. 20. $792. Z. M, $1761.36; 15. $8495.46. 16. $2373.24. 7. July 18; $1694.90; 28. $3000. P, $1409.09; 18. 1225 18s. 6d. $1686.16. 29. $912.23. T, $880.68. 19. 4831 8. Feb. 15, 1882. 30. 47i yd. 22. A, $454.54; 20. 1864 6s. 4d. 9. Oct. 6, 1881 ; 31. 4449 T V bu. C, $568.18. 21. $1341.32. $2403.88 ; 32. $84.18. 23. 2.767%. 22. $1162.79. $2367.55. 55. $0.73. '24. $2627.78. 23. $965.02. 10. May 31 ; 34. $3.12. 25. $315.33. 24. $767.20. $2480.32; 55. $17.60. 26. $28.03. 25. $1631.38. $2492.72. 36. $22.67; $25.33. 27. $155.70. 26. $1393.72. 11. July 14. 37. 45 days. 27. $189.60. 12. Nov. 15. 38. $1.09$. 59. 1492.26ft. Art. 553. 28. $1856.08. 29. $2023.10. 13. Sept. 5. 15. Oct. 26, 1882. 46>. 18| mo. 30. $1534.69. 16. Sept. 25 ; 41. $5328, assets; $11100, debts. 1. $8782.81. 2. $8395.94. 31. $2386.62. 32. $1688.75. $2425.16; $2437.33. Art. 526. 1. $105.63. 3. $5006.25. 4. $4358.59. 5. $8427.52. 6. $9922.37. 55. $1411.11. 34. 80318.70 fr. 35. 17972.04 fr. 36. 5.18|. 17. July 8. 18. Mar. 21, 1882. 19. Feb. 13, 1882. 20. April 12, 1883. 2. 6 hr. 5. 126 A. 7. $5270.79. 8. $4287.11. 38. $1692. 39. $1967.24. Art. 582. $4665.60. 9. $3417.18. 40. $822.96. 1. Jan. 17, 1889. 5. 4hr. 10. $2821.96. 41. $289.54. 2. May 25. 1888. 6. 125. 11. $9898.30. 42. $2359.10. 3. Dec. 3, 1882. 7. I day. 12. $8406.44. 43. $1543.24. 4. May 13, 1882. 8. 4 days. U. $1871.48. 44- $3867.75. 5. Sept. 6, 1882; 9. 15 hr. 15. $2443.86. 45. $3102.24. $276.53. 10. 9 days. 16. $3346.56. 46. $1547.25. 6. Feb. 15, 1883; 11. $64. 17. $2227.28. 47. $2061.40. $1053,23. 7,?. 80 days. 18. $8144.96. 48. $2359.33. 7. Feb. 8, 1881. 13. $66.13. 19. $7373.16. 49. 3467.73 marks. 8. Aug. 25, 1882, 14. $1493.33. 15. 32 days. 20. $9222.61. 21. $6431.57. 50. 13824 marks. 51. 94|. Art. 586. 16. 16320 Ib. 22. $9231.46. 52. $2905.05. 1. $2074.06; 23. $5283.96. 53. $4976.40. Aug. 28; Art. 544. 24. $23.29; $9340.10. 54. 4562 guilders. 55. 7128 guilders. Sept. 2; Sept. 2. 1. $93.75. 25. $2972.25. 56. 40f . 2. $5324.48; 2. $68.75. 26. $11834. 57. 4.885. Nov. 28. 3. $354.50. 27. $7854.67. 58. $17366.99. 3. $2751.14; 80^. 28. \% premium. 59. $3.60. Sept. 27. 5. $3600. 29. $3420.05. 60. 118.4; 117.1. 4. $12505.70; 6. $281.25. 30. $10118.89. 61. 85.98. Apr. 26. 358 A NS WER S. [Art. 586. 5. $4043.09; 33. $20.80. 19. $15000. 11. A, $6470.24; Dec. 10. 34. $45,066,444.72 20. $1.1692. B, $3235.12. 35. 100. 21. 1.65%. 12. $1510. Art. 594. 56. $66,000,000; 22. A, $177.63; 14. E, $2380.83; 1. $431.37. $14,000,000. B, $305.42. F, $3333.17; 2. $986.02. 37. 80. 23. $5887. G, $3809.33; 3. $3361.51. 5& 8%; 6%; 51%. H, $4761.67. 4. $1694.89. 39. $176; 33. Art. 628. 15. A, $2692.68; 5. $518.53. 40. 8%. 1. $9412; B, $2468.29; 6. $44955.75. 4&. Latter yV% $5647.20. C, $1884.88; 7. $400.91. better." 2. $97.02. D, $2154.15. 8. Rm. 9997.87. 43. 7y\ % . 3. $80.49. 16. A, $375; 9. $276.54. ^. Chatham $60 ^. $816.25. B, $318.75; 10. $1053.22. greater. 5. $85.40. C, $225; 11. $1513.77; 45. 166f. C. $135.21. D, $187.50; $32067.54; 46. 160; 1331. 7. $723.45; E, $131.25. $1704.29; .^7. 6|-%; 125; 621 $748.80. 17. A, $1533.46; $47288.32. 48. $2425. 8. $950; B, $1922.54. 12. Rm. 3869.18. ^5. $7,725,574.22; $609.60. 18. C, $819.97; 13. $52,23. 41.48%. 9. $1267.50, D, $745.43. 14. $856.19. 50. $754482; 10. $116.94. 19. A, $540; 15. $622.42. 75.6%. 11. $807.80. B, $560; 16. $4016.22. 5.Z. $3,957,320; 12. $1004; C, $600. $49,466,500. $794.25. 20. A, $2311.63; Art. 611. 52. $519.27. 13. $639.03. B, $3581.40; 1. $8750; $56. 53. $1128.34. 14. $839.40. C, $4106.97. 2. 8% ; $200. 3. $2,500,000. 54. $1682.91. 55. $4030.29; 15. $1623.80. 16. $183.90. 21. J, $1558.97; K, $1385.75; 4. $20,000,000; $4030.29. 17. $338.24. L, $1190.88. $2.000,000; 56. $475. 18. $11060. 22. A, $1529.98; 1%. 57. $1450. 19. $43.75. B, $1185.74; 5. $500. 58. $1625. 20. $208.50. C, $1070.98; 6. $61250. 60. 124.59; 21. $439.88. D, $940.30. 7. $36.745,000. 128.80. 22. $351. 23. $1750. 8. $60000. 62. 4%. 23. $1959. 24. $11287.62. 9. $23325. 24. $57.05. 25. R, $1925; 10. 550 shares. Art. 617. 25. $37242; S, $1425; 11. $240,000; 1. $7,690,418.82. $197.75. T, $1125; $185,237.50. 2. $11,615,280. 26. $2483.60. U, $925. 12. $23100. 3. 31 mills; 27. $487.27. 26. X, $500; 13. $36412.50. $38,666.37; 28. $156.30. Y, $200; 14. $226675. $11,986.57. Z, $700. 15. $17560. 4. Rate 5.8 mills ; Art. 639. 27. 65%. 16. $29043.75. $42.34. 3. A, $1960; A, $1235; 17. $68625: 6. $37.49. B, $2960. B, $3250; 18. $3277.50. 7. $57.85. 4. A, Or. $1623. 17; C, $1950; 19. $16200. 20. 8. 8. $231.39. B, Dr. $164.71. D, $3965. 21. $8230. 9. $60.48. 5. $833.33, bro. ; 28. 55%; 22. $80800. 10. $597.85. $3366.67. $9020. 23. $775. 11. $104.98. 6. C, $11431.88: 29. 42% ; 24. 500 shares-. 12. $65.28. D, $11279.75; F, $1764; 25. $8000. 13. $5284.88. E, $11190.75. H, $1050; 26. $117645. ' 14. $232.31. 8. M, $18529.25; K, $4956. 27. $28494.67. 15. $2138.05. N, $6389.75. 30. A, 60 ft, ; 28. $23544.58. 16. $393.42. 9. A, $26666.67; B, 80 ft. ; 20. $325. 17. A, $335.34; B, $27091.67; C, 100 ft. 30. $48600. B, $558.90; C, $2166.66. 31. $554.68. 31. $960. C, $465.76. 10. A, $5004.24; S2. P, $10229,71 ; 32. $34137.50. 18. $66500. B, $2502.12. Q, $10245.54. Art. 639.] ANSWERS. 359 S3. A, $3847.56; 10. $636.58; 17. 10. Os. 9d. Cr. B, $12695 B, $4902.44. $637.45. 18. $102. Cr. C, $12695 34. A, $12346.82; 19. $433.93. Cr. Sundry B, $58.77.67. Art. 662. 20. $889.36. Creditors, 35. A, $2505.74; B, $17307.58; C, $16723.84. 36. A, $17527.74; B, $20323.76; C, $6310.43 ; Will lose $140 37. A, $7331.20; B, $4950.50. 38. A, $4932.38; B, $3497.25; C, $6570.37. 1. $131.90. 2. $137.20. 3. $116.36. 4. $1638.40. 5. $0.054; $0.072; $0.104. 6. $0.655. 7. $356. 8. $13472. 9. $10469.12. 10. $3700; 21. $261.83. 22. 120 bbl. 23. 135458 Ib. 24. $676.77. 25. 0. 26. Loss, $500. 27. 30}-f%. 28. $500. 29. $35.58. 30. $137.26. 31. 3 19s.. lOd. 32. Sept. 19; $4950. $4050. Art. 667. 1. 36. 5. 81. 2. 48. 6. 126. 3. 55. 7. 125. 4. 72. S. 144. Art. 671. 1. 29. 5. 47. 2. 37. 6. 6. 3. 31. 7. 23. 4. 43. 8. 1. Art. 645. 1. 72^; * $5760. 2. $702000; 11. $9080;' $29080. 12. $996.80. 13. $717.68. 33. $5078.72. 34. $851.96. 35. $925.62; $1074,38. Art. 674. 1. $9715.20. 2. $5187.21. $35100. S. $35625. 4. $77400; $64000; 14. $12379.12. 15. $342.67; $26.98; $296.57. ' $1087.17. 37. $72.22; $73.89; ^72 88 S. $13648. 4. $5631.50. 5. $4051.44. 6. $1474.27. $86000. 5. $21375; $475000. 16. $42919; $29169.33. 17. $4190.03; 38. $4166.30; $4160.47. 7. $3426.67. Art. 677. 6. $3.338,100. 7. 26.26^ : $23,944,096.50 $1,213,591.50. $3084.60; $933.35; 54 yr. 18. $152.79; ' $5181.19.' 40. $4165.03. 41. $630.97. 42. $2474.07. 2. $261.58. 4. $1327.61. S. $549.51. 9. $3260.51. 8. $1,433,831.70. 9. $1047.54;' $427.84. 10. $2313.65. 11. 12^; $77.86. 19. $13.63. 20. $2655.65. Art. 663. 43'. $9825.03. 44. 17651.29 fr. 45. $1889.22. 46. $7.18; $2.39. 47. $114.75. Art. 682. 2. $120. 3. $72. 4. $6.20. c (917 C.' r/ 1. 104J-. 48. Aug. 29; O. p/*f. A 64 / ' 2. 21659.6. $5359.35. Art. 687. 4. 4j|jg-/0. S. 40320. 49. Jan. 14,1883; Art. 650. 4. $164.52. 5. 16 48' 15". $1409.40. 50. $1409.37. 1. $50.17. 2. $36.60. 1. $374.60; 6. 16666| sq. yd. 51. $320.31. 3. $lo9. $374.93. 7. 69 Os. 6d. 52. $1625. irt 60O 2. $669.35; 8. $140.25. 53. $425. ff) KQs* $671.70. S. $25.14. 9. 381-sq.yd.; $24.15. 54. $37000. 55. $81.25. Z. OoC. 3. 35c. / PtO^ 4. $755.28; 10. 19 3s. 6d. 56. $4224. ^. OoC. $757.38. 11. $5567.50; 57. $128.21. ^r. 603. 5. $462.10; $4.443, wood; 58. $295.75. $464.38. $.486, grain. 59. $208.50. 2. 2,1,5,1; 6. $190.91; 12. $1803.07. 60. B, $5011.83; 3, 2, 3, 5. $191.04. 13. $.8392. C, $1794.53; 3. 3, 4, 5, 1 ; 7. $96.15. 14. 219375; Dr. Mdse.,etc., 4, 3, 1, 5. 8. $328.60; $23.72; $12410; 4- 1,5,6,1; $329.37. $1067588.44. Dr. Sundry 5, 1, 1, 6. 9. $557.31; 15. $45.73. Debtors, > 5. 1, 1, 3, 6, 1; $557.86. 16. $430.72. $17030; 3, 3, 1, 1, 6. 360 AXSWEKS. [Art. 695. Art. 695. 70. 716. 75. 5. 5. S3. 18. 502.656 A. 2. 10, 10, 45. IS. 15.25. 77. 35.03 in. 3. 30, 90. 4. 12,48, 48; 36, 36, 108. 5. 15 15. 7^. 1994. 75. 20.78. 16. 2732. 77. 19683. JLr. 724. 1. 18.9875 A. Art. 730. 0. 6.8027^; $493.20. 6. 9,15,3; 16, 4, 20. Art. 697. 2. 5, 60, 15, 30; :&&,* 00. 208.71 ft. 07. 1.41421. -4*. 705. 0. 1.625 A. 3. 290.47 sq. rd. 4. 34 A. 145 sq. rd. 25 sq. yd. 8 scj. ft. 108 sq. in. 3. $225. 4. Vr. $3581.16; C p. $3681.16; Adj. rec. $100. 5. $4869.57. 6. Ship receives 8, 52, 24, 26. 0. 18. 5. 1680.6ft. $1888.08; 4. 33, 12, 5. 3. 2.2. 6. 471.24ft. Ar. $964.23- Art. 701. 4. 68. 5. 78. 7. 5541.78 sq. ft. 8. 14.7 sq. ft. Bp. $1472.82; C p. $496.94; 2. 33. 6. 82. 9. 3525 gal. D p. $1032.55; 3. 121. 7. 196. 70. 5.32 cu. ft. Adj. rec. $150. 4. 135. 8. 2.13 77. 15053 tons. 7. C p. $1200.79; 5. 216. 9. 23.9. 70. 59.9 in. S rec. $974.60: 6. 218. 70. 462. 13. 466.69 ft. Adj. rec. $150; 7. 255. 77. 6.54. 14. 75.3984 cu. ft. Agent receives 8. 294. 70. 75.8. 75. 6 ft. 6 in. $76.19. 9. 312. 13. 878. 16. 32 yd. 8. G.A., $516.81; 70. 345. 14. 82. 77. 45.14rd.; S, $638.79; 11. 37.5. 75. 47 ill. 141.8 rd. O, $789.09. TESTIMONIALS. HENRY C. and SARA A. SPENCER, Spencerian Business College, Washington, D. C.lt is an admirable text-book ; analytical, logical, practical, and accu- rate. We use it. CURTISS & CHAPMAN, Curtiss Business College, Minneapolis. We have been using your book from the first ; and the more we use it, the better we like it. It is the best Arithmetic of which we have knowledge. JOHN R. CARNELL, Prin. Alb. Bus. College, Albany, N. Y. We find the Packard Arithmetic in every way adapted to our requirements. GEO. W. SPENCER, Prin. Bus. College, Providence, R. T. We have used the Packard Arithmetic from the first issue, and have obtained better results than with any other book. W. I>. MOSSER, Prin. Bus. Col., Lancaster, Pa. We have used it from the first, and have found it satisfactory in every detail. W. T. WATSON, Prin. Bus. Cot., Memphis, Tenn. We have used it since its first appearance, and could not suggest an improvement. T. f7. CA\TON, Prin. Cotn. College, Minneapolis, Minn. We are fully con- vinced that it is the best work in print for Commercial Colleges. M. MacCORMICK, Prin. Bus. Col., Guelph, Ont.l have no hesitation in pronouncing it the best Commercial Arithmetic that has come under my notice. J. A. & M. H. HOLT, Oak Ridge, N. C.We are using the Packard Arithmetic with great and entire satisfaction. C. T. MILLER, Prin. Bus. Col., Newark, N. 7. We have used your series of Arithmetics since their first publication. The results have been satisfactory. They cover everything desired in business calculations. G. W. BROWN, Business Colleges, Jacksonville, Peoria, Decatur, and Grtlesburg, III. The Packard Arithmetic is used in all my colleges with very satisfactory results. 0. P. DE LAND, Prop. Bus. Col., Appleton, Wis.It is the best Commercial Arithmetic I have ever used. W. K. MULLIKEN, Prop. Bus. Col., St. Paul, Minn. I have used it since its first edition, and consider it unequaled. It is, without doubt, the best book for high schools and colleges that has yet appeared. 1. E. SAWYER, Luther, Mich. It is the best Arithmetic in the market. It is devoid of all puzzling questions and problems, has the shortest and most comprehensive methods, and is simple and direct. B. F. MOORE, Bus. Col., Atlanta, Ga. I regard it the best work published, both for the class room and the counting-room. A. E. MACKEY, Bus. Col., Genera, N. Y. It has been our favorite since its first publication, and we have no desire to change. Aside from its merits as a class-book, it con- tains a vast amount of information for the accountant and business man. 11 TESTIMONIALS. HfcKAY & FARNEY Bus. Col., Winnipeg, Man. Have used it since ils first publication. It is just suited to business college work. ALBERT C. BLAISDELL, Prin. Commercial College, Loivell, Mass. I have used many kinds of Arithmetics in the last ten years, but all were found wanting except Packard's. This just fills the bill. In my judgment it is without an equal. S. S. GRESSLY, Prin. Business College, McKeesport, Pa. We have used the Packard Arithmetic since its first appearance, and believe it to be decidedly the best book published. The revised edition cannot be too highly commended. C. IF. ROBBINS, Central Business College, Serial ia, Mo. We have used your book since its revision, and think it the best Commercial Arithmetic published. CHAS. FRENCH, Prin. Business College, Boston, Mass. The longer we use it, the better we like it. GEO. S. BEAN, Prin. Business College, Peterborough, Ontario. We use your book altogether, and find it very complete. I consider it the best work I have seen. E. C. A. BECKER, Worcester, Mass. For the last ten years I have been on the alert for a book that would give the best results, and can find nothing better than the " Pack- ard." It is all business simple, complete, and free from mathematical puzzles. W. N. FERRIS, President Industrial School, Big Rapids, Mich.-A.fev/ years ago I examined every Commercial Arithmetic that had been published in America ; as a result, adopted the "Packard." I do not hesitate to say it leads them all. MESSRS. EATON & BURNETT, Business College, Baltimore, Md.We use your book because we think it the best book in the market. 7. T. MURFEE, Prin. Military Institute, Marion, Ala. We use your Arith- metic, and regard it superior to any we have ever seen. E. M. HUNTSINGER, President Business College, Hartford, Conn. I have used the Packard Arithmetic since its first issue. It meets the demands of the aspiring busi- ness mathematician as does no other work. Our pupils are delighted with it. J. T. JOHNSON, President Business College, Knoxville, Tenn. I have used the Packard Arithmetic since its first publication, and like it better than any other Arithmetic I have ever examined. It is plain, practical, and concise. WILLIAMS & BARNES, Commercial College, Iowa City, Iou'a.We adopted your Arithmetic several years ago, because we considered it the best book published for commercial colleges. We expect to continue its use. E. L. McILRAVY, Pres. Business Unirersit?/, Kansas City, Mo We use the New Packard Arithmetic because we consider it the best work of its kind. JESSE SUMMERS, Pres. Normal College, Abingdon, III. We are using your book, and are well pleased with it. T. D. GRAHAM, Business College, Nashville, Tenn. I have used your New Commercial Arithmetic since it was first issued, and can speak authoritatively of its merits It is a most excellent book. T. R. BROWNE, Prin. Business College, Brooklyn, N. Y.It is not simply a little better than any other book, but beyond comparison with other books. I have used it from the beginning. C. A. FLEMING, Prin. Northern Business College, Owen Sound, Ont.We have used your Arithmetic since its first publication, and have found it in all respects com- plete. It is just the book for the business college student. E. 7. GANTZ, President Normal College, Humestown, Iou'a.l have used your Arithmetic since its first appearance, in the Commercial Department of this institution, and have seen none that I would exchange it for. I consider it the best book in use. TESTIMONIALS. Ill A. J. WARNER, President Business College, Eltnira, N. Y.l regard your Arithmetic the best in the market. You have made a great hit in its preparation. THOS. J. STEWART, Prin. Stewart & Hammond Business College, Tren- ton, N. J.l regard the New Packard Commercial Arithmetic as far superior to any similar work now published. The longer we use it, the better we like it. G. M. NEALE, Pres. Commercial College, Fort Smith, Ark. We use the Packard Arithmetic in our school because we think it the best book to be had. C. W. BUTLER, Supt. I'nblic Schools, Defiance, Ohio. Your Commercial Arithmetic has been used in our High School since its first publication. It is a practical, common sense book. I know of nothing better. A. H. HINMAN, Prin. Business College, Worcester, Mass. I use the Packard Commercial Arithmetic, like it, and heartily commend it, because its work is so com- prehensive and practical, and so admirably prepared for use in business schools. SHANNON & BISSON, Proprietors Business College, Muskegon, Mich. We are using your Arithmetic, and think it the best book to be found for universities, business colleges, and high schools. It is thoroughly practical and comprehensive. IT. D. McANENET, Pt-in. Business Department, Drake University, T>fs Moines, la. We have used the Packard Arithmetic since it was first published. For the gen- eral purposes of business schools it surpasses any other work of the kind that I have seen. TEMPLE & HAMILTON, Business College, San Antonio, Texas. We have adopted your book as the most thorough and teachable text-book in the market. It is a work of superior merit. J. M. ME HAN, Prin. Business College, Des Moines, Tow ft. The New Pack- aid Arithmetic has been in use in my school since its first publication, and it fully meets all wants. It is admirably graded, practical in all ways, and fully up to the times. 7. R. GOODIER, Prin. Business College, Port Huron, Mich I have used the New Packard Arithmetic since it was first issued, with the very best results. From the first page to the last the good of the student is kept constantly in view. BENNETT & GREEK, Proprietors Morrell Tnstittite, JTohnstown, Penn. We have used your book in our school for over three years, and know of no other that so fully meets modern requirements. We especially commend its short and easy methods and its practical examples. RICKARD & GRTJMAN, Minneapolis School of Business, Minneapolis, Minn. The Packard Arithmetic cuts a prominent figure in all our calculations. It teaches everything from round numbers to square root. No student can go through it and come out a cipher. It adds to his capacity, multiplies his faculties, and divides his merits. In its whole number of pages, we find no fraction of waste, and its decimals are always to the point. Our interest in this book is compounded every year. ROHRBOUGH BROS., Proprietors Commercial College, Omaha, Neb We have used your Arithmetic since its first publication. Did we not believe it to be the first in the market we should not use it. It should be introduced into every first-class commercial college in the country. E. H. FRITCH, Prin. Business College, Wichita, Kansas. We are using the New Packard Commercial Arithmetic, and believe it to be the best work of its kind ever published. VARNUM & BENTON, Proprietors Business College, Denver, Colo. We have used your book for the last four years, and consider it in all respects the best book before the public. J". W. HALEY, Fort Edtvard, N. Y. Your New Arithmetic is in every respect complete, and just what is needed in every business college. IV TESTIMONIALS. E. L. ELLIOTT, Waterloo, Iowa. The New Packard Commercial Arithmetic is a jewel. A. J. RIDER, Prin. Easiness College, Trenton, N. 7. It is decidedly the best work extant. W. II. ROGERS, Cashier Nassau Sank, New York. it is the best book on the subject that I have ever seen. O. G. NEUMANN, Prin. Business College, Austin, Texas. We have seen and examined many other books, but this takes the lead. T. J. rRICKETT, Prin. College of Commerce, Philadelphia, Penn.The use of your book for the past six years has only strengthened my opinion that it is the best work of its class. My teachers, without exception, are enthusiastic in their praise of the "PACKARD." HO WELL B. PARKER, Prin. Academy, Hampton, Ga. The New Packard Commercial Arithmetic has been severely tested in my school, and grows better every day. Having made a specialty of Arithmetic for twenty years, I can say that yours is the best that 1 have ever used. T. C. STRICKLAND, Prin. Acadwny, East Greenwich, R. I. It is admi- rable throughout, and worthy of special attention for its treatment of interest, equation of accounts, and partnership settlements. G. R. RATHBUN, Business College, Omaha, Neb. It is practical and fully adapted to business college work. No book could take its place. -8. A. DRAKE, Clark's Btisiness College, Erie, Penn.It fully meets the requirements of all grades of commercial classes. *7. E. GUSTITS, Prin. Business College, Rock Island, III. It is the most com- prehensive and tnorough treatise upon the subject of commercial calculations ever published. E. A. HALL, Prin. Business College, Logansjtort, Ind. I have taught com- mercial arithmetic for over twenty-five years, and consider the New Packard Commercial Arithmetic superior to any other book I have used. It is modern, progressive, and full of common-sense problems such as are used in every-day business affairs. THOS. H. SHIELDS, Prin. Business College, Troi/, N. T. We have used your Arithmetic in our school since it was first published, and consider it the best in the market. E. G. G UION, Prin. Business College, Washington, Penn.We use your book, and consider it a most valuable text-book. The absence of catch questions has left room for the practical and useful, which are well supplied. H. I. MATHEWSON, Milford, Conn. The New Packard Commercial Arithmetic fully meets my best expectations. W. F. L. SANDERS, Supt. Schools, Connersville, Ind. We know of no better Arithmetic than the New Packard Commercial. The school that adopts it will keep it. MISS E. A. TIBBETTS, Business College, Salem, Mass. I have found it a most comprehensive and satisfactory book from beginning to end. L. R. WALDEN, Prin. Business College, Austin, Texas. I know of no work that so nearly meets with my idea of a text-book on this subject. C. BAYLESS, Prin. Business College, Dubuque, Iowa. We have used it from the beginning. It is a work of superior merit. ,7. A. McMAHON, Prin, Business College, Beaver Falls, Pa. In my opinion it is the best Arithmetic of the kind that is published. L. A. GRAY, Prin. Business College, Portland, Me. The New Packard Com- mercial Arithmetic has been used in this college since it was first published. It gives entire satisfaction, and seems to increase in favor the longer we use it. YC 2244! M98496 THE UNIVERSITY OF CALIFORNIA LIBRARY