o 
 
 00 
 
 co 
 
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LIBRARY 
 
 01 THK 
 
 UNIVERSITY OF CALIFORNIA. 
 
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 Received G2,c3f~ . 
 
 Accession No. 7^-3 bO . Clots No. 
 

 ARITHMETIC 
 
 SHORT METHODS 
 
 * SWEET * 
 
OF THE 
 
 UNIVERSITY 
 
SWEET'S 
 
 Hand Book 
 
 OF 
 
 SHORT METHODS 
 
 Arithmetic 
 
 j. s. SWEET, A. M., 
 
 'Principal of the Santa Rosa IJusiness College. Santa Ho.-a, Cal. 
 
 formerly President of the Oix-^oii State Xormal School. 
 
 Ashland. Or., Author of Sxveet '- Sv~u-m of Act- 
 
 nal Hnsiness Practice, Element* of Geom- 
 
 etrv, lousiness t"orm~. Etc. 
 
 SANTA ROSA, CALIFORNIA 
 

 Entered according to Act of Congress, in the yi-;ir IS!)!!. 
 
 ByJ. S. SWEET, 
 In the Office of the Librarian of Congress, at Washington, D. C. 
 
 >. 
 
 J 
 
PREFACE. 
 
 The principal object of this little work is to place in 
 the hands of the student, in compact form, many of the 
 briefer methods of rapid calculations. ''Time is money," 
 and especially so to many of our young people who are 
 trying to obtain a business education in a brief time and 
 with limited means. 
 
 Hoping that many may profit by the suggestions here- 
 in contained, I most respectfully dedicate this little volume 
 to the young business people of America. 
 
 Santa Rosa, Calif., 1893. J. S. SWEET. 
 
6 XIIORT METHOD* 
 
 2. Slims Greater than 9. 
 
 54321543 
 
 56789678 
 
 6543 654 
 
 6789 789 
 
 765 76 87 
 
 789 89 89 
 
 8 9 
 
 9 9 
 
 .V. To Itetrtl fit Sight. 
 
 When a student sees the figures 1 and 3 written side by 
 side, he instantly recognizes "thirteen" or "thirty-one" ac- 
 cording to their positions. The same facility may be ac- 
 quired in regard to numbers in addition; thus, 4 over or 
 under 8, may be read "twelve" as readily as the figures 1 
 and 2 side by side. Ten minutes practice daily for one 
 month will accomplish the work. 
 
 4. Always add TWO or MORE figures at a time. Never 
 be guilty of adding single figures. Name the results of the 
 following as rapidly as possible : 
 
 246975634674-89 
 35323678988723 
 
 38765725475399 
 
 48797999888789 
 

 IX AltlTHMKTIC. 
 
 56854322462537 
 
 49987873878689 
 73776298897778 
 84698549739894 
 
 J. Nine added to any number is always ONE LESS in its 
 unit's place than the number. Thus, 
 
 8 9 - 7 in its unit's place. 
 36 9 5 
 
 6*. Eight added to any number is TWO LESS in its unit's 
 place than the number. Thus, 
 
 7 8 == 15, 15-8 = 23. 
 T. To Add btf TenM. 
 
 A good method is to add by 10's, carrying the EXCESS 
 in the mind, as in the following : 
 
 8" 7 2 
 
 9 5 
 63 95 
 7 6 
 
 30 27 
 
 Here the 3 of the 13 is carried to the 7 of the 17 mak- 
 ing three tens in all. Add in this manner the following: 
 
 3 
 
 9 
 
 6 
 
 5 
 
 9 
 
 8 
 
 8 
 
 8 
 
 8 
 
 8 
 
 7 
 
 5 
 
 5 
 
 7 
 
 9 
 
 9 
 
 9 
 
 5 
 
 9 
 
 6 
 
 4 
 
 3 
 
 4 
 
 4 
 
 5 
 
 6 
 
 4 
 
 9 
 
 8 
 
 6 
 
8 *1IORT MKTHOIt* 
 
 8. When the Columns are Long. 
 
 When there are two or more columns of consider- 
 able length, add each column separately as instructed, and 
 write the sum of each alone, then combine results into 
 one number, as follows : 
 
 32476 
 58976 
 76892 
 39428 
 73548 
 67943 
 28745 
 "^8 
 37 
 46 
 43 
 33 
 
 378008 
 
 This method is almost indispensable in book-keeping, as 
 an error can be located much more readily than when the 
 separate results are not known. 
 
 9. To Add Two Columns at a Time. 
 
 To add two columns at a time practice on the fol- 
 lowing, by adding the tens' column first, and by reading the 
 units' column, tell at a glance the number to carry : 
 
 23 
 36 
 
 72 
 49 
 
 35 
 44 
 
 66 
 27 
 
 38 
 79 
 
 38 
 44 
 
 59 
 71 
 
 88 
 64 
 
 39 
 
 89 
 
 88 
 26 
 
 86 
 49 
 
 94 
 
 87 
 
 75 
 
 89 
 
 85 
 94 
 
f.\ ARITHMETIC. 9 
 
 10. Proof* of Addition. 
 
 In long columns the best proof is to add them again, up 
 or down, the opposite of your first addition. In short col- 
 umns and several of them to add, you may prove the work 
 by casting out the 9's as shown below. 
 
 25189654 - 4 
 36972105 - 6 
 94375517 - 5 
 15155815 - 4 
 85310652 - 3 
 95315175 - 
 352318918 - 4 
 
 Casting out the 9's of the first nnmber, we have an e.< 
 of 4 ; of the second, 6 ; of the third, 5 ; and so on, finally 
 casting out the 9's of these results which gives an excess of 
 4. Also by casting out the 9's of the sum, we have 4, we 
 therefore conclude that the work is correct. 
 
 XOTK. This is not always a sure test, the answer mi_<rht he wrong 1 and 
 vet prove bv this method. 
 
SUBTRACTION 
 
 11. When the forty-five combinations treated of in Ad- 
 dition are thoroughly memorized, the process of subtraction 
 is a very simple one. This consists of being able to discern 
 at a glance the digit which will combine with one of those 
 given to produce the other. Thus, 
 
 8 
 3 
 
 are given, and the question is : what number combines with 
 3 to produce 8? The process is nearly the same as in ad- 
 ding, the only difference is that we must furnish one of the 
 numbers to the combination, the result already being known. 
 Read the differences as rapidly as possible : 
 
 98767896757988 
 43234454326325 
 
 15 16 17 14 13 12 18 
 
 8986789 
 
 Daily drills in both addition and subtraction should not 
 be neglected. The process of this method is very simple 
 and is readily learned. Practice, only, will perfect it and 
 give value to it. 
 
MULTIPLICATION 
 
 7V. With Multiplication we begin our Slinrt Methwls. 
 supposing the student to be sufficiently advanced to know 
 the multiplication table to the 12's. If not, he should learn 
 the following 
 
 MULTIPLICATION TABLE: 
 
 1 
 
 > 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 o 
 
 
 
 10 
 
 11 
 
 12 
 
 2 
 
 4 
 
 
 o 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 30 
 
 33 
 
 36 
 
 4 
 
 8 
 
 12 
 
 16 
 
 20 
 
 24 
 
 28 
 
 32 
 
 36 
 
 40 
 
 44 
 
 48 
 
 5 
 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 
 
 50 
 
 OD 
 
 60 
 
 6 
 
 12 
 
 18 
 
 24 
 
 30 
 
 
 42 
 
 
 :>4 
 
 60 
 
 66 
 
 72 
 
 , 
 
 14 
 
 21 
 
 28 
 
 
 42 
 
 4U 
 
 56 
 
 !63 
 
 70 
 
 77 
 
 84 
 
 - 
 
 16 
 
 
 Mi' 
 
 40 
 
 48 
 
 56 
 
 64 
 
 
 80 
 
 88 
 
 96 
 
 9 
 
 18 
 
 27 
 
 
 45 
 
 -")1 
 
 68 
 
 72 
 
 81 
 
 90 
 
 99 
 
 108 
 
 10 
 
 20 
 
 
 40 
 
 :,d 
 
 C)() 
 
 70 
 
 80 
 
 90 
 
 100 
 
 110 
 
 120 
 
 11 
 
 22 
 
 33 
 
 44 
 
 55 
 
 66 
 
 . i 
 
 88 
 
 99 
 
 110 
 
 121 
 
 132 
 
 1-2 
 
 24 
 
 36 
 
 48 
 
 60 
 
 12 
 
 84 
 
 96 
 
 108 
 
 120 
 
 132 
 
 144 
 
 //>. The following squares of numbers should also be 
 memorized : 
 
12 XI10RT METHODS 
 
 13 X 13 == 169 19 X 19 361 
 
 14 X 14 = 196 20 X 20 == 400 
 
 15 X 15 = 225 21 X 21 =- 441 
 
 16 X 16 256 22 X 22 == 484 
 
 17 X 17 == 289 23 X 23 529 
 
 18 X 18 = 324 24 X 24 576 
 
 25 X 25 = 625 
 
 14. To multiply any number const stint/ 
 of two digits by 11. 
 
 RULE. \Vrite the *inn of the digit* between 
 ilieiti, ilie number fltii* e.vpre^ned ?'s I lir product. 
 
 EXAMPLES. 11 times 24 = 264, 
 11 " 36 == 396, 
 11 " 57 == 627. 
 
 XOTK. \Ylu-ii tin-it- -tun is 10 or inure, carry one to tin- liutulivd's di-it. 
 
 EXERCISES. 
 
 15. 1. Multiply 45 by 11. 4. Multiply 75 by 11. 
 
 2. 38 by 11. 5. 96 by 11. 
 
 3. 92 by 11. 6. 88 by 11. 
 
 16. To multiply any number by 11. 
 
 RULE. Write the unit's iigurc; next, -write the 
 
 sum of the ^l-nits and tens, tJifit i lie ^nin of 1/ir fi-n* 
 and hundreds, etc., writing flic left IKI'IK/ figure 
 last, carrying i^lien uec:es?:r\'. 
 
 EXAMPLE. 11 times 12345 == 135795. 
 
 5 
 
 4 j 5 9 
 
 :i 4 - 7 
 
 2 3 5 
 
 1-2 3 
 1 
 
7.Y ARITHMETIC. 13 
 
 EXERCISES. 
 
 /;. 7. Multiply 663 by 11. 4. 6731 by 11. 
 938 by 11. .7. 9884 by 11. 
 734 by 11. 6. 72596 by 11. 
 
 18. To multiply by 22, 33, etc. 
 
 Rl'LK. Mil hi ply h\> 11 //>- <i}>,>r<>, ami rhru !>\' ?, .V, 
 <>r 4, etc. 
 
 EXAMPLE. 22 times 234 =. 2574 X 2 = 5148. 
 
 XOTK. The work should be done mentally, only results being; written. 
 
 EXERCISES. 
 
 W. 7. Multiply 64 by 22- 4. 374 by 55. 
 65 by 33. .J. 874 by 66. 
 46 by 44. V. 336 by 77. 
 
 20. To multiply by ant/ tuttnbet* betirecn 
 12 and 2O. 
 
 RULE. Multiply by the unifs figure only, i.'rir- 
 mg the r<\<it/r uJt>riln> innitlx>r <m<l <mr ~p}<ir t > r<> 
 tin' n'-hr, thru ,i,J<l. 
 
 .KS. 13 times 24 24 
 
 312 An*. 
 14 times 175 175 
 
 700 
 2450 An. 
 
 EXERCISES. 
 
 21. 1. Multiply 262 by 13 .7. 9624 by 17 
 
 382 by 14. 6. 32694 by 18, 
 
 497 by 15. 7. 27314 by 19. 
 
 .;. 1824 by 16. 8. 98794 bv 12. 
 
22. To multiple/ by 21, 31, 41, fil, etc. 
 
 RULE. Multiply by the tens only, i^n'ti'tig Hie 
 result render ilie ninuhe r <i n <1 our pluee 1o ilie left, 
 then add. 
 
 EXAMPLE. 31 times 24 24 
 
 72 
 744 
 
 EXERCISES. 
 
 23. 1. Multiply 35 by 31. k 728 by 51. 
 & 46 by 41. 5. 3824 by 61. 
 3. 245 by 21. 6. 8452 by 71. 
 
 24. To multiplt/ by lo. 
 
 RULE. Anne.\- one eipli.er /<> ////' number <nnl <i<I<t 
 its half. 
 
 EXAMPLES. 15 times 28 = 280 
 
 V 2 of 280 = 140 
 
 420 
 
 15 times 35 = 350 
 175 
 525 
 
 EXERCISES. 
 
 25. 1. Multiply 44 by 15. 4. 248 by 15. 
 
 87 by 15. 5. 7634 by 15. 
 3. 394 by 15. 6. 98768 by 15. 
 
 20. To multipJt/ by XI. 
 
 RULE. Take one-half the nmnber <tnd write it 
 
 two places to the left <nul (J<!. 
 
_ 
 
 flTNlVERSITY ) 
 
 IX ARITHMETIC. 15 
 
 KXAMPLKS. 51 times 72 = 72 
 V 2 of 72 36 
 
 4372 
 
 51 times 45 = 45 
 
 225 
 2295 
 
 EXERCISES. 
 
 1. Multiply 78 by 51. 4. 1384 by 51. 
 324 by 51. ,5. 4633 by 51. 
 723 by 51. 6. 78254 by 51. 
 
 28. To sqttftre ft n tun her tr/iose tntft fiy- 
 ttre is ,7. 
 
 RULE. Multiply rite /-;/>-' <ligjr />v i 
 <t 11 (1 ' 
 
 EXAMPLE. 25 times 25 = 625. 
 
 2 times 3 6, annex 25 625. 
 
 EXERCISES. 
 
 1. Multiply 35 by 35. -7. 75 by 75. 
 
 ' 45 by 45. 6. 85 by 85. 
 
 55 by 55. ?. 95 'by 95. 
 
 4- 65 by 65. 8. 105 by 105. 
 
 V0. To find the prodttct of two nttnthers 
 units' (I if/ its nre ,7's. 
 
 RULE. To rJir product of the tens add one-half 
 
 tln'ir :uin mxl <nnn\\- 25 if 
 
 XOTK. Fractions of one-half are dropped. 
 
X 6 SHORT 
 
 EXAMPLES. 25 times 45 -1125. 
 
 1/2 of (2 + 4) -h 2_X 4 = 11, annex 25 = = 1125. 
 
 25 times 35 =875. 
 1/2 of ( 2 -h 3) -f- 2 X 3 == 8, annex 75 875. 
 
 NOTK. - plus 1? is odd. 
 
 EXERCISES. 
 
 />'/. J. Multiply 25 by 65. 4. 45 by 35. 
 
 2. 25 by 85. 5. 65 by 35. 
 
 3. 105 by 25. H. 75 by 65. 
 
 32. To find the product of two numbers 
 whose tens 9 digits are identical and the sum 
 of the units 9 digits is 10. 
 
 RULE. Multiply the tens' digit by OTIC greater 
 and annexe the product of the, unit*' <l,igi1*. 
 
 EXAMPLE. 43 times 47 = 2021. 
 4X5 and annex 7X3 == 2021. 
 
 EXERCISES. 
 
 33. 1. Multiply 29 by 21. 5. 38 by 32. 
 2. 28 by 22. n. 37 by 33. 
 
 27 by 23. 7. 49 by 41. 
 
 4. 39 by 31. 8. 48 by 42. 
 
 34. To find the product of two numbers 
 whose tens 9 digits are consecutive,, anil the 
 sum of the units 9 digits is 10. 
 
 Rl'LK. To (lie product of tin* /<>ss ten* and one 
 more than the grfutcr, <nni<'.\- tin- complement of 
 the square of the greater number' * unit figure. 
 
 XOTK. Complement of ;i munher i- TOO les^ the numlu-r. 
 
 EXAMPLE 87 times 73 6351. 
 
 7 < 9 63; complement of the square of 7 = 51; 
 annex it to 63 6351. 
 
IX ARITHMETIC. 
 
 EXERCISES. 
 
 Multiply 47 by 33. 
 56 by 44. 
 64 by 56. 
 
 -9 by 71. 
 (>. 84 bv 76. 
 
 #6*. To jitttt the product of tiro numbers 
 tcheti their fats' (lif/its are the sattte. 
 
 Rl'LK. Ta,he the prmluct <>f tin' unir*, ne.\~r rhe 
 print ncr ;/s riling the sum <>f tli-e unit*, then 
 
 flu' pr>J i/ cr nf i/ir rcn*, always r,i ~rr\-i m? rli*' r'7;s, 
 if itii v. 
 
 EXAMPLE. 
 
 73 times 75 
 5 3 
 
 8" 7 
 7 7 
 
 5475 
 
 15 write 5, carry 1 
 56 carry 5. 
 4V) 
 5475 
 
 EXERCISES. 
 
 /. Multiply 74 by 72. 
 85 by 83. 
 67 bv 65. 
 
 4. 97 by 94. 
 .5. 88 by 89. 
 '/. 79 bv 78. 
 
 V#. To /r tt ft the product of two tttrtttbet's 
 trhett the units 9 digits are iflettfieftt. 
 
 Rl'LE. Talif r/i.,' pnnlitcr of the nn.it ^ 
 the sum <>f t lir ten* times the liuit*, mul the product 
 of the tens, currying i^'In'n necessary. 
 
 KXAMPLK. 44 times 74 3256. 
 
 EXERCISES. 
 
 / Multiply 46 by 56. 
 54 by 34. 
 43 bv 53. 
 
 4- 73 by 63. 
 
 6. 87 by 47. 
 n. 98 bv 28. 
 
 OF THE 
 
i8 XIIORT METHODS 
 
 40. To find the product of fntt/ two num- 
 bers consisting of two digit*. 
 
 RULE. Take the product of the units, th-e sum of 
 the products of each ten times the other unit, and 
 the product of the tens, carrying if necessary. 
 
 EXAMPLE. 47 times 36. 
 
 6 X 7 ~- 42 
 
 6X4 a <" 7 = 45 
 4X3= 12 
 
 1692 
 EXERCISES. 
 
 41. 1. Multiply 35 by 27. 4. 68 by 34. 
 2. 47 by 34. 5. 78 by 46. 
 
 52 by 46. 6. 39 by 35. 
 
 42. To find th e produ ct of numbers wh <> 1 t 
 one part of the multiplier is a factor of the 
 other. 
 
 RULE. Multiply by the factor, then tliit? product 
 by the quotient of the factor into tin- oilier part, 
 <i tnl 
 
 EXAMPLE. 231 
 
 183 
 
 Multiply by 3 = 693 
 
 " this product by 6 = 4158 
 42273 
 
 423 
 126 
 
 Multiply by 6 = 2538 
 
 " this product by 2 = 5076 
 
 53298 
 EXERCISES. 
 
 43. L Multiply 1247 by 255. 
 
 2. 792 by 279. 
 
 3635 by 1089. 
 
IX ARITHMETIC, 19 
 
 44. Jty tnnltijrtu ft// the factors of ft num- 
 ber. 
 
 RULE. Multiply by one factor <iml /// /.- 
 other. 
 
 EXAMPLE. 21 times 65 == 7 times 65 = 455 
 and 455 3 1365. 
 
 EXERCISES. 
 
 4X. 1. Multiply 73 by 42. 4. 97 by 14. 
 
 83 by 35. 5. 87 by 36. 
 
 123 by 27. 6. 79 by 49. 
 
 46. ToiHHltijrti/bt/ 1O, WO, 10OO, etc. 
 
 Annex as tna>ny cipln'r* <i* there ure in 
 1 lie in it It i pi ier. 
 
 EXAMPLES. 10 times 76 - 760. 
 
 100 times 125 12500. 
 
 4;. To nitdtiptt/ ft// ft nu multiple of 1O. 
 10O, WOO, etc. 
 
 RULJZ. Multiply by the digital number and tlieji 
 <t n n e.\- ciphers. 
 
 EXAMPLE. 400 times 123 49200. 
 2000 times 243 = 486000. 
 
 48. To inultinlt/ ft// .9, or fftit/ number 
 of ,9'x. 
 
 RTLK. Annex 11$ many ciphers as there &re 9* s 
 
 and finhtrarr thr nuni-her multiplied. 
 
 EXAMPLES. 9 times 435 = 4350 435 = 3915. 
 99 X 267 = 26700 267 == 26433. 
 
20 ^IIORT MKTIIOD* 
 
 EXERCISES. 
 
 49. I. Multiply 47 by 9. J t . 148 by 9. - 
 2 125 by 9. 5. 725 by 99. 
 
 238 by 9. ' 6'. 675 by 999. 
 
 50. To multiply btj any number endint/ 
 in 9. 
 
 RULE. Multiply by //// n<'.\-f greater number 
 and from the product subtract the number multi- 
 plied. 
 
 EXAMPLE. 382 times 49 = 382 X 50 382. 
 
 382 
 
 50 
 
 "19100 
 
 382 
 
 18718 
 
 EXERCISES. 
 
 ,7/. 1. Multiply 128 by 69. 3. 326 by 599. 
 2. 245 by 59. //. 262 by 499. 
 
 32. To multiply by ant/ number a little 
 less or a little greater than 100, 1000, etc. 
 
 HULK. JLnnex as ma-ny ciphers ^ liters a re fi^- 
 nrc* in th.c. multiplier <nn1 mihtruci <>r <nld, the pr<>- 
 <hn;t of tlic difference between 10O, 10()(),<>t<\, ami 
 (lie multiplier. 
 
 EXAMPLE. 423 times 996 423000 4 >< 423. 
 
 ._ 1(51)2 
 421308 
 
 EXERCISES. 
 
 ,-T.V. L Multiply 993 by 624. J. 9994 by 425. 
 
 0. 997 by 529. <i. 9998 by 827. 
 
 992 by 895. 7. 99993 by 963. 
 
 .',. 326 hv 104. 8 1003 by 724. 
 
": 
 
 IX ARITHMETIC. 21 
 
 34. To multiply by any multiple of 1), 
 t e.rceed i ny i)O. 
 
 RULE. ^Multiply b\' the multiple <>/ mi 
 
 tlnni. t he git 'ru nt-u fri pi irr, and iii>tr<'icr ?'rs 
 ne-tenth. 
 
 EXAM PI. K. 454 times 72 
 
 454 
 
 80 
 
 36320 product by 80 
 
 3632 " " 8 
 
 32688 " " 72 
 
 EXERCISES. 
 
 ,7,7. /. Multiply 4(5 by 18. 5. 288 by 54. 
 75 by 27. r>. 384 by 63. 
 82 by 36. 7. 772 by 75. 
 
 4. 144bv 45. 8. 1244 bv 81.5 
 
 OI 
 
 .76*. To multiply by complements. 
 
 P'roiu eitliLT number subtract the comf>li' 
 of the Other, <tinl <nnn\\- t/n> pr<nlucr <tf- t 
 com 
 
 XOTK. Tlie product should have ;i~ mail v figure- as are in both nuiii- 
 -S: sup]>ly oi]ihi-rs To maki- tht-m the same. 
 
 p]>ly oi]i 
 
 EXAMPLES. 94 comp. 6 999 comp. 1 
 
 97 comp. 3 999 comp. 1 
 
 9118 998001 
 
 A'd 
 
 EXERCISES. 
 
 ,7;. /. Multiply 92 by 87. 4. 996 by 995. 
 94 by 75. 5. 993 byi9\L. 
 99 by 93. n. 998 by 895. 
 
22 SHORT METHODS 
 
 58. To find the product of two tt a in hers, 
 each of which is a little over 1OO. 
 
 RULE. From the sit-in, of the numbers snl>rr<ict 
 100 (nid (.inn-cjc the product of tltc ejccesses. 
 
 EXAMPLE. 115 times 104 == 11960 
 115 -f 104 100 == 119 
 To 119 annex 15 X 4 == 11960. 
 
 EXERCISES. 
 
 39. 1. Multiply 114 by 105. 4. 144 by 107. 
 
 2. 122 by 103. .7. 160 by 106. 
 
 135 by 102. 6. 138 by 108. 
 
 XOTH. Applv The same principle to the following: 
 
 1. Multiply 1008 by 1007. 8. 1250 by 1003. 
 
 2. 1125 by 1004. 4. 1475 by 1002. 
 
 60. To find the prod act of two n ambers 
 one of which is more ami the other Jess than 
 100. 
 
 RULE. From, tlic srrm of tlir nnn^hcr^ *ubtr<ict 
 100, aimex two <~ip//rr* <ni<l ^ul>f-r<ict 1 lit- product of 
 the excess and 
 
 EXAMPLE. 108 
 98 
 
 8 excess. 
 2 complement 
 
 10600 
 16 
 10584 
 
 EXERCISES. 
 
 61. 1. Multiply 102 by 94. 4. 125 by 92. 
 
 & 103 by 97. 5. 112 by 99. 
 
 115 by 96. 6. 116 by 95. 
 
 XOTK. Apply The same principle to the following; 
 
 1. Multiply 1004 by 92. S. 1015 by 92. 
 
 2. 1008 by 95. 4. 1025 by 96. 
 
IX ARITHMETIC. 
 
 ALIQUOT PHRTS. 
 
 TABLE. 
 
 y 2 of 100 == so 
 
 Vs " - 33^3 
 
 V4 " = 25 
 
 Vs " = 20 
 
 Ve " - 16% 
 
 VT " - 
 
 Vs of 100 = 12% 
 
 1/9 " = 111/9 
 
 Vio " - 10 
 
 Vl2 
 VlG 
 
 of 100 = 37 tf 
 = 62*4 
 
 - 87y 2 
 
 - 66^3 
 
 - 83*-3 
 
 6 " ^ 1834 
 
 s/ 16 of 100- 31 H 
 
 7 /16 ' -4334 
 
 9 /i6 {> - 56M 
 
 11/16 " = 6834 
 
 13 /16 " 81J4 
 
 9334 
 
 1>2. To multiply by an rtfiquot part of 
 100. 
 
 Anin\\- ti.'o ciphers, </ 
 
 inn! multiply fry the ntnncrafor of t hr frac- 
 tional f<n~t It /.- of 100. 
 
 EXAMPLE-. 50 times 12 = 7200 -r- 2 = 3600. 
 162 3 times 84 8400 -*- 6 == 1400. 
 
 EXERCISES. 
 
 1. Multiply 48 by 25 
 33% by 24. 
 35 by 20. 
 
 4. 63 by 14- 1 - 
 
 5. 184 by 12i o. 
 
 6. 960 by 8V 3 . 
 
 7. 3603 by 
 
 8. 2560 by 
 
24 SHORT 
 
 1. Multiply 72 by 37V 2 . 4- 423 by 66%. 
 
 2. 56 by 12V 2 . -5. 144 by 83V 3 . 
 96 by 87V 2 . tf. 216 by 18%. 
 
 fl-/. To multiply by 10 times an aliquot 
 part of 100. 
 
 RULE. Annex three ciphers <m<l proered ^s In-- 
 fore. 
 
 Ex. 166% times 84 = 84000 4- 6 === 14000. 
 
 times 144 = 144000 - 12 - 12000. 
 
 EXERCISES. 
 
 6V>. J Multiply 125 by 48. 5. 112 by 62V 2 . 
 0. 1236 by 3331/3. 4- 192 by 83V 3 . 
 
 66. To multiply by (t little more or a lit- 
 tle Jess than- an aliquot part. 
 
 RULE. Multiply by the nearest ali<im>t parr, </.- 
 above, and add or subtract the difference 1 itm-* the 
 mi i iiber. 
 
 EXAMPLE. 13 1X 2 times 64 864 or 
 
 12 1X 2 times 64 = 6400 s- 8 = 800 
 1 times 64 64 
 
 864 
 
 EXERCISES. 
 
 6T. 1. Multiply 72 by 14 l - 2 . 4. 78 hy 
 2. 84 by 152/ 7 . . 123 by 
 
 54 by 17%. n. 144 by 84 ' i 
 
 6*8. To mult inly by WO an ft an aliquot 
 ^ art of 100. 
 
 Annex two ciphers ami <i <l <1 to the tntin- 
 f ii indicated by llic <t I if/u of part. 
 
IX AK1T1IMKT1C. 25 
 
 EXAMPLES. 125 times 128^ 12800 - % of 12800 
 
 = 16000. 
 1331/3 times 36 3600 - 1200 = 4800. 
 
 
 EXERCISES. 
 
 0.9. 1. Multiply 96 by 116^3. 4- 72 by 112 l _>. 
 120 by 137K. o. 84 by 1142/ 7 . 
 345 by 116%. 6. 106^4 by 144. 
 
 This same principle may be carried to more than 100 
 and an aliquot; to 200, 300, and even to thousands. The 
 student will find much in this field for original investigation. 
 
DIVISION 
 
 70. To divide by 5. 
 
 RULE. Multiply by 2 and cut off one fig it re. 
 EXAMPLE. 125 divided by 5 == 125 X 2 = 25.0. 
 
 EXERCISES. 
 
 71. 1. Divide 135 by 5. 4. 265 by 5. 
 & 145 by 5. 6. 325 by 5. 
 
 3. 175 by 5. 6\ 875 by 5. 
 
 72. To divide by 25. 
 
 RULE. Multiply by 4 and cut off two figures. 
 EXAMPLE. 125 divided by 25 == 125 X 4 = 5.00. 
 
 EXERCISES. 
 
 73. 1. Divide 275 by 25. 4. 875 by 25. 
 . 325 by 25. 5. 925 by 25. 
 
 475 by 25. 6. 975 by 25. 
 
 74. To divide by 125. 
 
 R ULE. Multiply by 8 and cut off three figures. 
 Ex. 375 divided by 125 = 375 X 8 ^ 3.000. 
 
L\ ARITHMETIC. 27 
 
 EXERCISES. 
 
 M. 1. Divide 500 by 125. S. 875 by 125. 
 625 by 125. 4. 1125 by 125. 
 
 ;6*. To dh'ide ht/ an aliquot part of 10O. 
 
 Mn hi ply by the denominator of the frac- 
 tion expressing the alf<{nt part, Divide In' tJie 
 numerator and a/r off'neo iigurcs. 
 
 EAMFLES. 240 -r- 5 = 240 X 20 48.00. 
 840 H- 25 := 840 X 4 33.60. 
 1200 *- 12^ == 1200 X 8 = 96.00. 
 1350 -f- 16^/3 == 1350 X 6 = 81.00. 
 
 EXERCISES. 
 
 ;;. Divide 245 by 25. 820 by S 
 
 268 by 20. 725 by 83^. 
 475 by 33^3 446 by 125. 
 
 <H. To rtiriflr Inj 1O< 1OO, 1OOO, etc. 
 
 RULE. Cur off <i^ iiKiny figures as there are 
 in t lie <lirf*or. 
 
 EXAMPLE. 1240 divided by 100 12.40, 
 
 7.9. To wilnce tlte dirt'sor 1o sotue tttun- 
 of tens, hnndwls, thousands, etc. 
 
 RULE. Multi-ply both divisor and dividend by 
 some nmnhcr that ?.'/'// make the divisor a, multiple 
 of tens, Jin ml reels, t it cnisa mis, etc., and divide as in 
 short division. 
 
 EXAMPLE. 15) 2365 
 
 2 ' 2 
 
 3.0)47370 
 
 157 and 10 rein. 
 
 XO'l'K. Divulo the remainder I'D hv '2 to find the true remainder. 
 
 EXERCISES. 
 
 80. 1. Divide 3845 by 35. 5. 8732 by 75. 
 *. 6492 by 45. .$. 6288 by 125. 
 
28 SHOUT MKT110IIX 
 
 DIVISIBILITY OF .NUMBERS. 
 
 81. To tell when a number is fit risible bt/ 
 2, 3, 4, 3, 6, 8, 9, 10, etc. 
 
 82. All numbers are divisible by 2 when they end in 
 0, 2, 4, 6, or 8. 
 
 83. By 3 when the sum of their digits is divisible by 3. 
 
 84. By 4 when the two right hand figures express a 
 number divisible by 4. 
 
 So. By 5 when they end in or 5. 
 86*. By 6 when divisible by 2 and 3. 
 
 87. By 8 when the three right hand figures express a 
 number which is divisible by 8. 
 
 88. By 9 when the sum of their digits is divisible by 9. 
 8,9. By 10 when they end in 0. 
 
 90. By 7 or 11 if they consist of four figures, the first- 
 and fourth identical and the second and third ciphers. 
 
 91. By any composite number if divisible by all of its 
 prime factors. 
 
 CANCELLATION- 
 
 92. Cancellation is a method of dividing by re- 
 jecting equal factors. 
 
 RULE. Cancel any or 11 factors common to 
 both dividend and divisor. Ih'cide tlta product of 
 those remaining in the dividend by (lie product of 
 those remaining in the divisor. 
 
 EXAMPLES. 42 X 36 ^ 24 X 14 == ? 
 Arrange the numbers as follows : 
 
 03 ;5 
 
 At 30 9 
 
 U X U 2 
 
 '1 
 
 EXERCISES. 
 
 9:$. 1. Divide 84 times 72 by 36 times 21. 
 
 144 times 216 by 56 times 128. 
 3. 512 times 1728 by 144 times 216. 
 
j FRACTIONS 
 
 . To a d ft ff(t ction* lt<iciny ft cot tun on 
 denominator. 
 
 RULE. Add their iiuim-rator* ami write tin 
 
 .^11 It oi'fr tJir r<->in iitoii denoin iitatr. 
 
 EXAMPLE. V 7 - 2/7 + % = %. 
 EXERCISES. 
 
 ,ai. /. Add 2 9 % T ( , . 
 
 f>6*. To rff7^ f*ro fractions 
 tton 
 
 RULK. ^Inltiplv tlir ^uin of the denominator s by 
 
 O'imnon ninii> : r<tror and ivripe the result over 
 of the denom imttin-f. 
 
 Ex. i 2 i 3 == (2 - 3) > 1 over 2X3 = 
 % % (3 5) X 2 over 15 = 16 /i 5 - 
 
 EXERCISES. 
 
 f>;. 1. Add 3 4 ^. -7 r> ii. 
 
 % - 3 /7- ^ 6 /7 + 6 /H- 
 
 5 -f % & 10 ia - 10 /7- 
 
30 SHORT METHOD* 
 
 98. To add fractions not having a com- 
 mon numerator nor common denominator. 
 
 RULE. Multiply each numerator into all the 
 denominators except its own for new numerators, 
 and take the prod^lct of all the denominators for a 
 common denominator, then add. 
 
 EXAMPLES. % + % = ^jt-" 19 /i5- 
 
 12 -r 1<> IS 
 
 EXERCISES. 
 
 .9.9. ^. Add % 4- y 7 . A V 2 % % 
 
 %-f 6 /n. 4- %-h 3 /7 r 8 /n. 
 
 NOTE. When .several fractions whose denominators are not prime to 
 each other are to be added, reduce them to their least common deuominaior 
 and add. 
 
 TOO. To add mixed numbers. 
 
 RULE. Add ivhole numbers and fractions sepa- 
 rately and tlien unite results. 
 
 EXAMPLE. 8% -f 12%. 
 
 8 r 12 20 
 
 % %-=' 1 % 5 - His 
 
 EXERCISES. 
 
 101. L Add 91/2+141/3- 4- 283/ 5 + 3 5 4/ 5 . 
 ^. 18%-J 252/7. 5. 431/5 72y 7 . 
 
 21%1275/y. ft 66%- 231/4+ 17y 5 . 
 
 102. To subtract fractions h rinr/ a co tn- 
 
 RULE. Take the difference of tli<> 
 <tinl write it over the common denominator. 
 
 EXAMPLE. % minus % = % - %. 
 
IX ARITHMETIC. 31 
 
 EXERCISES. 
 
 1. Solve : % _ 44 .; 13/ 15 __ li/ 15 . 
 
 10/ 13 __ 5/ 13 . ,J. 42/_ 3 __ 27/_ 3 . 
 
 To subtract fractions liarina a com- 
 mon 
 
 RULE. Multiply the (lijft'cn'iH-f </' the <lcntnninu-- 
 tors In' the common numerator <iiid -write the re- 
 *u It over the pr<l it ct of the >1 eu<n 7 7 ir/r ;-.-. 
 
 2 / > 
 
 EXAMPLE. % - 2/ 7 = 
 
 
 EXERCISES. 
 
 /OJ. /. Solve: % 3/ 7 . ,;. s/^ _ 8/ 15 . 
 
 % - % 5. 
 
 ~ 5 ll. 6'. 
 
 *. 7V> subtract fractions had it f/ neither 
 numerators nor comnton denom- 
 inators. 
 
 RULE. Mu I tiply ench numerator into the other 
 
 (If/loin iii'i r<>rs, take the difference un<l i^rite it over 
 t In- pnnl a rr of the denominators. 
 
 I.', l:-j 
 EXAMPLE. % 3/ 7 - _ 
 
 EXERCISES. 
 
 107. 1. Solve : 6/ ? _ 5/ 8 . .;. 
 
 . To subtract tni.red numbers. 
 
 RULE, ^rihrract u'h.ole numbers and fractions 
 ^l v, uniriu.g 
 
32 SHORT UKTUOUX 
 
 NOTE. If the traction of the .subtrahend is i>-n-aU-r than tliaT of the min- 
 uend subtract a unit from the minuend and add it to tin- traetion before 
 taking the difference. 
 
 EXAMPLE. 8% 5% 
 
 8 5 := 3 
 
 % ~ % Vlo 
 
 11 - 8 - 3 
 IVa ~ V 2 % 
 
 3%. 
 
 EXERCISES. 
 
 Solve: 22% 16%. 5. 89% - 35%. 
 75% _ 48%. ^ 9 5 i/ 6 __ 7434. 
 
 XOTE. A g\.od method is to take the complement of the diiference of th 
 fractions when the subtrahend fraction is the greater. 
 
 EXAMPLE. S 1 /^ 2% 
 
 42 2 
 
 % 2 VG write the complement % 
 
 EXERCISES. 
 
 110. 1. Solve: 8i/i - 51/3. S. 25% -- 17%. 
 
 . 2. 15% - 4%. 4. 44% - 313/ 4 . 
 
 . To fl-nt? the square of ff nti.rert num 
 ber whose fr et ion is 1/2. 
 
 RULE. Multiply the. integer by tin- next li-igliei 
 number a-nd a>miex %. 
 
 EXAMPLES. 2% X 2^ == 2 X 3 -f- ^ == 
 3^ X 3^ - 3 X 4 < = 
 
" 
 
 IX ARITHMETIC. 33 
 
 EXERCISES. 
 
 >. /. Multiply 4V 2 by 4V 2 . & 8V 2 by 8V 2 . 
 51/2 by 5%. ^. 9V 2 by 9%. 
 
 To /iwd f/if product of two wired 
 n anthers n'ltose fractions arc 17 2 . 
 
 Rl~LE. *To rlif pr<xl n ct /'/ lir iiii-i'gt T.-- */// 1 5 tln-ir 
 .<ti in (Did a n.ni'.\- L 4 . 
 
 Ex. 2i o C 4V 2 = 2 > 4- 3 X 14== 
 
 3V 2 >: 4V 2 = 3X4 - 3V 2 + ^4 = 15%. 
 
 XOTE. The fraction will be ooe-fparth if the sain of the two integers i 
 .-ii : if the sum is odd thi- fraction is three-fourths. 
 
 EXERCISES. 
 
 114. 1. Multiply 21/2 by 6V 2 . ',. Sifeby 
 3V 2 by 5V 2 . 4. 41/2 by 
 
 To find the ptoduct of two mixed 
 s whose integers are identical and 
 the sutn of whose fractions is a unit. 
 
 RULE,. ^Multiply the inrrgrr by the. nr.\-f In'glirr 
 TininbtT ami <iiuit'.\- rlir prodnrr oftlie fr<tct; 
 
 Ex. 2i 3 2% =23 I's X % 6%. 
 
 3# X 33^ = 3 > 4 ^ ( 3^ == 123/ 16 . 
 
 EXERCISES. 
 
 / /6\ /. Multiply 4% by 43/ 5 . ^. 9y 7 by 93/ 7 . 
 ?. 5% by 5i x 5 . 5. 12% by 10%. 
 
 63/ 8 by 6%. 6. 153/n by 158/ii. 
 
 I IT. To find the product of two n anthers 
 whose integers are consecutive and the snnt 
 of whose fractions is a unit. 
 
34 SHORT METHODS 
 
 RULE. Multiply the greater number increased 
 by 1, by the less; and for the fraction annex ilir 
 complement of the square of the fraction of tli-r 
 greater number. 
 
 EXAMPLE. 4/ 3 X 3^ = 5 X 3 -f- % ?= 15%. 
 
 XOTK. The .square of one-third equal.-* one-ninth, its complement is 
 eight-ninths. 
 
 EXERCISES. 
 
 118. 1. Multiply 514 by 4%. 4. 9% by 8%. 
 2. 63/5 by 5%. 5. 12% by 11%. 
 8. 83/7 by 7y 7 . . 205/12 by 19% 2 . 
 
 119. To find the prod net of tico iHi.red 
 numbers whose inteyei's are identical. 
 
 RULE. To the product of the Integers add tlic 
 product of the siim of the fractions times tJic com- 
 mon integer and the product of the fraction*. 
 
 Ex. 6y 2 X 6 l /3 = 6 X 6 + 6 X % + % X # - 
 36 + 5 -: V 6 =~ 41^. 
 
 EXERCISES. 
 
 120. 1. Multiply 8 1X 2 by S^i 4. 24% by 24%- 
 . 12 1/ 3 by 12%. o. 351/5 by 353/ 5 . 
 8. 142/7 by 146/7. 6. 45% by 45%. 
 
 121. To find the product of ttro mixed 
 numbers when the fractions are identical. 
 
 RULE. To tlic product of the integers add the 
 product of the sum of the integers times the com- 
 mon fraction and the product of the fractions. 
 
 Ex. 41/3 X 8i 3 = 4 X 8 X 12 X 1X 3 - 1X 3 X 17 3 
 32 4 i' 
 
 EXERCISES. 
 
 122. 1. Multiply 6^ by 18^. .1 36 17 8 by 44 1/ 8 - 
 3. 91/3 by 15%. 4. 721/9 by 36%. 
 
WO. 
 
 IX .\RIT1I MET 1C. 35 
 
 To multiply by (in aliquot pftrt of 
 
 RULE. AII.II.CJ; two ciphers to the multiplicand 
 
 nnd tube such a part of it as the multiplier is a 
 part of 100. 
 
 EXAMPLE. 24 < 16% 2400 ~6 400. 
 
 EXERCISES. 
 
 124. 1. Multiply 39 by 33V 3 . 4- 54 by 66%. 
 
 2. 48 by 12> 2 5. 72 by 37 1 ,> 
 
 64 by 8V 3 . fj. 144 by 83V 3 . 
 
 12Z. To Multiply a f fact to it by a frac- 
 tion. 
 
 RULE. Cancel all common factors in numer- 
 ators and denominators and divide the product of 
 those remaining in the numerator by the prod^lct 
 of those in the denominator. 
 
 EXAMPLE. 3 4 :\ :\ 
 
 _ \/ ___ \/ ___ N,/ ^__ -- _ 
 
 4 2 6 7 2 i0 2S 
 
 EXERCISES. 
 
 126. 1. Multiply % by % by % 5 . 
 2. % by 2i/ 25 by 27/ 32 . 
 
 7^7. To fit ride a fraction, by a fraction. 
 
 RULE. Invert the divisor and proceed as in 
 multiplication of fraction *. 
 
 EXAMPLE. % X % + 7 /io X 9 /i 6 = 
 
 2 
 
 3 4 10 1C) :-l-2 11 
 
 X X X ~ 1 
 
 4 $ 7 -03 -21 L>1. 
 
 EXERCISES. 
 
 128. L Solve : % X 7/ 10 X % -f- 2 V 2 4 X 15 / 28 . 
 
 ^. 6 /7 X H12 -*- 22 /49 X % X % 
 
PERCENTAGE 
 
 119. To find the percentage when the rate 
 is an aliquot part of 200. 
 
 RULE. Take such a part of the iimnbcr as tlie 
 rate is a part of 100. 
 
 EXAMPLE. 12M per cent of 64 == }i of 64 = 8. 
 
 EXERCISES. 
 
 13O. 1. Find 50 per cent, of 38. Of 346. 
 
 2. 33^ " " 42. Of 543. 
 
 8. 160 " " 96. Of 186. 
 
 4- 12# " " 128. Of 4168. 
 
 131. To find the percentage tc/teit the 
 rate is an aliquot part of WOO. 
 
 the number by 10, and ta-hr. *ucli a part 
 of it as the rate is a part of 1000. 
 
 Ex. 830 per cent of 144 = 17 i 2 of 1440 ^ 120. 
 
 EXERCISES. 
 
 132. 1. Find 333 ^ per cent of 27. Of 279. 
 
 & 1660 " " 66. Of 576. 
 
 3. 83V 3 " 96. Of 3612. 
 
 4- 62^ " " 288. Of 1624. 
 
fX ARITHMETIC. 37 
 
 133. To find the percentage when the 
 rate is ant/ number. 
 
 RULE. ^Multiply the base bv the rate and point 
 
 off tu'o places. 
 
 Ex. 12 per cent of $400 = 400 < .12 = $48.00. 
 
 EXERCISES. 
 
 134. 1. Find 15 per cent of 500- Of 1879. 
 
 22 " " 750. Of 4321. 
 
 18 " " 560. Of 8765. 
 
 4. 27 " " 1340. Of 9876. 
 
 135. To find the bff.se , the rate an<l per- 
 centftf/e briny f/iren. 
 
 KTLE. Divide the pereenrage !,y the rate. 
 
 EXAMPLE. Rate = 12 per cent, Percentage ~ 96. 
 96 -f- .12 800 Base. 
 
 EXERCISES. 
 
 13<>. 1. Rate 4 per cent, Percentage 52, Ba.^e ? 
 
 2 " 9 " " 144 " ? 
 
 .;. " 12 " 176 " ? 
 
 13 7. To find the rate, the percentage and 
 base behif/ f/iren. 
 
 KULE. Divide tlte percentage by rlie base. 
 
 EXAMPLE. Base = 400, Percentage = 36. 
 36 -T- 400 = .09, or 9 per cent. 
 
 EXERCISES. 
 
 138. 1. Base 500, Percentage 35, Rate = ? 
 . " 1200, " 72, " = ? 
 
 -;. " 1800, 144, " 
 
3 8 SHORT METHODS 
 
 139. To find the rate of loss or t/nhi. 
 
 RULE. Divide the loss or gain by the cost. 
 
 EXAMPLE. Cost == $250, Selling price = $300. 
 $300 $250 = $50, Gain, 
 $50 -^ $250 =* 20 per cent., rate of gain. 
 
 EXERCISES. 
 
 140. Find Rate of Gain or Loss : 
 
 1. Cost = $400, Selling Price, $500. 
 
 2. " ^$279, " $540. 
 2. " = $720, " $600. 
 
 14,1. The following formulas are a very good illustra- 
 tion of the problems of percentage : 
 
 FORMULAS OF PERCENTAGE. 
 
 Base X Rate == Percentage. 
 Percentage -~ Base = Rate. 
 Percentage -~ Rate == Base. 
 
 Amount -f- 1 Rate := Base. 
 Difference -=- 1 -- Rate Base. 
 
 By applying the formulas above to these applications, 
 problems of Percentage are very readily solved. 
 
IX ARITHMETIC. 
 
 39 
 
 ^ o 
 
 c 
 
 2. =r = = 
 
 X 
 
 R 
 
 7T 
 
 ^ 
 
 M 
 
 X 
 
 
 Cfl 
 
 
 ^ 
 
 n 
 
 o 
 3 
 
 3 c 
 
 I- ~. 
 
 7: 
 
 
 c ~ c -2. 
 
 
 
 
 r 
 
 
 
 
 
 D 
 
 
 i j-i a 
 
 
 
 
 -- 
 
 Efl 
 3' 
 
 ?' 7 
 
 
 
 3 or 5 . 
 
 
 
 
 
 
 . ' y. 
 
 r^ 
 
 ^-^- 
 
 ? r = 1" 
 
 
 
 
 
 ~ 
 
 -r 
 
 y 
 
 - 
 
 
 
 ^ 
 
 *""" ' 
 
 y 
 
 ^ 
 
 c 
 
 pQ 
 
 c ^1 
 
 
 
 \J 
 
 r|ii 
 
 - -. - - 
 
 i X'aliK' 
 
 X 
 
 ~ 
 
 l. 
 
 P 
 
 n 
 
 - 
 
 gj 
 
 o' 
 
 r 
 
 r 
 n 
 
 \ i I i 
 
 -. - ~ 
 
 : 1 -^ 
 
 " 
 
 ~ 
 
 PQ 
 rr 
 
 |o. ))i:>| 
 
 pr 
 
 - 
 pa 
 
 n 
 2 
 
 ij 
 
 ~ pc 
 
 ~' p^- 
 
 pa 
 
 s 
 
 H 
 
 1 
 
 CO 
 
 J= i-. 5 1" 
 
 O 
 
 __; 
 
 
 
 ~ . 
 
 ^ r-r ~ ". 
 
 
 
 
 .___ 
 
 
 
 o 
 
 - - 5 ^ 
 
 -r 
 
 
 *^n 
 
 ^ c 17 c 
 
 - r 
 
 ^ 2- -^ 
 
 E 
 b 
 
 | 
 
 - ' 
 - - 
 /. - . 
 
 c 
 
 0!~ 
 
 B 5 S' 
 
 - _ - 
 
 3 3 C 
 
 
 2 f r ^ 
 "* x ~ 
 
 C 
 
 
 = ?' El 
 
 ? 
 
 
 ^^ t f-* 
 
 * 
 
 
 
 ? 
 
 - L J 
 
 '* = 5 
 
 
 
 
 
 
 171 
 
 ". 
 
 - 
 
 i 
 = |: 
 
 > 
 
 1 
 
 c 
 
 
 " 5 ' 
 
 I = 
 
 W "-* 
 
 1 
 
 
 
 ? 
 
 < 
 
 
 __ 
 
 / 3 ^ 
 
 '~ 
 
 c* 
 
 
 * ? 
 
 5' 
 
 
 
 2 x 
 
 
 K 
 
 n 
 
 
 
 
 x 7' 
 
 
 h 
 
 
 
 
 
 
 
 
 
 
INTEREST 
 
 r . " 
 
 CANCELLATION METHOD. 
 
 '143. EXAMPLE. Find the interest on $420 for 30 
 
 days, at 7 per cent. 
 
 35 
 
 $00 30 days 
 
 .07 
 $2.45 interest. A us. 
 
 EXAMPLE. Find the interest on $540 for 7 months at 
 9 per cent. 
 
 45 
 
 7 months 
 .09 
 
 $28.35 interest. A-ns. 
 
 RULE. \Vrite the principal, rate and time </& / 
 dividend, and one year expressed in the same de- 
 nomination as the time gii'eii. as <i divisor, cancel 
 and reduce. 
 
 EXERCISES. 
 
 144. Find the interest : 
 
 /. Of $1200 for 42 days at 6 per cent. 
 
 2. Of $1800 for 33 days at 5 per cent. 
 
 3. Of $2250 for 60 days at 1 per cent. 
 
 4. Of $8400 for 5 mos. at 8 per cent. 
 
 5. Of $9600 for 9 mos. at 6 per cent. 
 
 6. Of $9636 for 1 year, 4 mos. at 7 per cent. 
 
7.V ARITHMETIC. 
 
 To find tin' interest tr/ten the tune is 
 in months ft no 1 flays. 
 
 EXAMPLE. \Vhat is the interest on $240 for 3 months, 
 12 days at 6 per cent. 
 
 3 months, 12 days 102 days, or, 
 
 3 months, 12 days 3.4 months. 
 
 3 $240 '-' 2 $240 120 
 
 10-2 days 12 :j.4 months 
 
 .06 2 .0* 
 
 $4. OS interest. $4. OS interest. 
 
 RULE. 7Vo'/ </ a-s //* cancellation method, rc- 
 hi rin ^ r//< time r<> </<M'>, or r< nnmrli.-- <nul tenths <>f 
 (t month. 
 
 XOTE. When the number of day* is ;t multiple of I* it shorten- th,- 
 vvork hv using 1 months ;\nd tenths of ;i month. 
 
 ABBREVIATED METHOD. 
 
 The cancellation method may be somewhat 
 shortened by omitting the rate and using instead of 360 as 
 a divisor the quotient of the rate into 360. Thus : 
 
 When the rate is 
 
 2 
 
 per cent. 
 
 use 
 
 180. 
 
 ; ( tt 
 
 3 
 
 
 
 (i 
 
 120. 
 
 . 
 
 4 
 
 ti 
 
 
 90. 
 
 a 
 
 5 
 
 
 tt 
 
 72. 
 
 .. 
 
 6 
 
 n 
 
 
 
 60. 
 
 a ti 
 
 8 
 
 (I 
 
 a 
 
 45. 
 
 tt it 
 
 9 
 
 tt 
 
 it 
 
 40. 
 
 a 
 
 10 
 
 tt 
 
 a 
 
 36. 
 
 a 
 
 12 
 
 
 
 
 30. 
 
 18 " " 20. 
 
4 a *1IORT METHODS 
 
 EXAMPLE. What is the interest on $720 for 33 days 
 at 5 per cent? 
 
 10 
 
 $:>.: : >() interest. 
 
 EXAMPLE. What is the interest on $1260 for 66 days 
 at 8 per cent ? 
 
 1$ \ 85 
 $1200 
 0022 
 
 $18.70 interest. 
 
 EXERCISES. 
 147. Find the interest : 
 
 1. Of $840 for 18 days at 6 per cent. 
 
 2. Of $960 for 27 days at 8 per cent. 
 
 3. Of $1240 for 36 days at 4 per cent. 
 
 4. Of $3260 for 63 days at 9 per cent. 
 
 BANKERS' METHOD. 
 
 148. EXAMPLE. What is the interest on $1344 for 
 75 days at 6 per cent ? 
 
 $13.44 = interest for 60 days. 
 
 3.36 interest for 15 days. 
 
 $16 80 = interest for ~75~days. 
 
 IiULE. Point off two places, ivhich -will give the 
 interest for the rate and corresponding time as 
 fol ioius : 
 
IX ARITHMETIC. 
 
 
 '2 per ceu i for ltf(. 
 
 
 3 
 
 
 
 
 4 
 
 
 90 
 
 
 5 
 
 
 72 
 
 
 6 
 
 
 60 
 
 
 - 
 
 
 45 
 
 
 .</ 
 
 
 40 
 
 
 10 
 
 
 36 
 
 
 12 
 
 
 30 
 
 
 IS 
 
 
 20 
 
 Then take such nHquot part* 
 v iiee<led f<' the gii<en time. 
 
 /</vs. 
 
 >f t/// 
 
 EXERCISES. 
 
 Find the interest : 
 
 /. Of $810 for 90 days at 4 per cent. 
 
 Of $648 for 45 days at 8 per cent. 
 
 Of $1232 for 36 days at 10 per cent. 
 
 Of $7200 for 37 days at 9 per cent. 
 
 Of $963.75 for 80 days at 6 per cent. 
 o. Of $2140.50 for 90 days at 8 per cent. 
 V. Of $5235.60 for 66 days at 6 per cent. 
 7. Of $4840.40 for 72 days at 10 per cent. 
 
 PROBLEMS I.N INTEREST. 
 
 1.5O* The following formulas are illustrative of the 
 four problems of interest. 
 
 4- Principal > Rate Time Interest. 
 
 .',. Interest -~ Principal Rate ~= Time. 
 
 .'. Interest : Principal Time = Rate. 
 
 /. Interest ~ Time ~ Rate Principal. 
 
 131. Applications of Percentage involving the ele- 
 ment of time are as follows : Interest, Discount. Partial 
 Payments, Insurance, and Stock Investments. 
 
44 
 
 XII ORT METHODS 
 
 To find the timv when t/te principal, 
 rate and interest is given. 
 
 EXAMPLE. Principal - $900; Rate = 8 per cent.; 
 Interest, $6.00; to find the Time. 
 
 300 
 
 $<>.00 
 
 $000 
 
 oO days, the tune. 
 
 EXAMPLE. Principal $720; Rate 6 per cent.; Interest 
 $25.20. Find the time. 
 
 n 00 
 
 (?) 
 .00 
 
 7 months. 
 
 RULE* Use the cancellation inetJiod </<s in reck- 
 oning interest, using the product of the interest 
 and one year expressed in the proper denomination 
 as a dividend and the product of the principal and 
 rate as a divisor. 
 
 133. To find the rate when the principal 
 time and interest are f/iveti. 
 
 EXAMPLE. Principal, $960; Time, 45 days; Interest, 
 $8.40. Find the rate. 
 
 3 ' 
 
 360 $000 ^ 
 7 10$ ^ days 1$ 
 Interest M0 : (?) 
 
 An*. 7 per cent., Rate 
 
 EXAMPLE. Principal, $1050; Time, 3 months: Inter- 
 est, $21.00. Find the rate. 
 
 4 12 
 
 21.00 
 
 3 months 
 
 Ans. 8 per cent., Rate. 
 

 IX ARITHMETIC. 45 
 
 KULK. Same as for 152, except that the product 
 of thr Principal and Time is used as a divisor. 
 
 . To find the pfincijtft/. the rate, time 
 and interest beiny f/iren. 
 
 300 [ ? ] 
 
 -5 60 days 
 
 Interest $5.23 .07 
 
 An*. *450, Principal. 
 
 3 
 
 12 
 Interest S45. 00 
 
 2.50 f, 00 
 
 9 month: 
 
 . *750, Principal. 
 
 RULE. Same as for 15'J, '.\->:i'pt that tJi>' 
 Tim? and Rut? is us?<i as a difisor. 
 
 L>,>. To find the Rank Discount of any 
 
 sum. 
 
 EXAMPLE.^; Find the bank discount of $840 for 63 days 
 discounted at bank at 10 per cent. 
 
 $?<40 70 
 
 03 21 
 300 10' 
 I 
 
 An*. $14.70 bank discount. 
 
 ItL'LK. Kititl tin 1 sfmpl? iii.r.?r?st. fr the given 
 
 tiii)<' an 1 1 r<t t r. 
 
To fin tl the True Discount of <tnt/ 
 
 EXAMPLK. -What is the True Discount and present 
 worth of a debt of 530, due in one year, discounted at 6 
 per cent? 
 
 #530 : 1.06 $500 the present worth; 
 $530 $500 $30 the true discount. 
 
 /vf'LK. Divide the aiivount of the debt by I />//<> 
 the rate for the given time t th<i* i^i-ll give 'tin-, pres- 
 ent worth; subtract the present luorth from- th< j 
 debt, tlie difference is f/ir t rue discoutit. 
 
 ANALYSIS. 
 
 The /i wtt step in analysis is to rp<hir<> to tin' 
 unit as follows: 
 
 If 4 hats cost $20, 1 hat will cost ^ of $20, or $5. 
 
 The second step is to reduce to a number: 
 If 1 hat cost $5, 7 hats will cost $35. 
 
 The third step combines the first and second: 
 If 7 coats cost $84, 1 coat will cost $12 ; 4 coats will 
 cost $48. 
 
 EXERCISES. 
 
 258. If 13 hats cost $39, what will 7 hats cost? 
 
 2. If 11 pairs of shoes cost $46.50, what will 7 pairs 
 cost? 
 
 3. If y* of a ton of hay cost $10, what will J^ of a 
 ton cost? 
 
47 
 
 15 & Reduce the following first to the fractional unit, 
 then to the integral unit, then to the required number of 
 fractions. 
 
 EXAMPLE. If % of a ton of hay cost $12, what will fe 
 of a ton cost ? 
 
 % of a ton cost $12, 
 
 l / 5 of a ton will cost $3, 
 
 % or 1 ton will cost $15, 
 r /6 will cost /s of 15 or *%, 
 ~> will cost 7'times i5/ 8 =i05/ 8=: 
 
 EXERCISES. 
 
 1(>0. /. If 73 of a bushel of wheat is worth 72 cents, 
 what are 10 bushels worth? 
 
 2. If 9/io of an acre of land cost $108, what will % O f 
 an acr*e cost at the same rate? 
 
 .}. If y?> of ^ of a cord of wood is worth $3.50, what is 
 of % of a cord worth ? 
 
 1(>1. To find interest on overdrafts. 
 
 EXAMPLE. Overdrafts for the week were as follows : 
 
 7. 1200 
 1500 
 
 Interest at 10 per cent. 
 IbOU 
 
 1600 
 1850 
 
 9500 -*- 360/ 10 $2.64. 
 
 RULE. Divide the sum of the daily overdrafts by 
 360 divided by tlie rate, and point off two decimal 
 places. 
 
48 
 
 102. How to find ert-ors s/towtt by a trial 
 balance. 
 
 I. See that your former baln-ncc of Imlnm'cs is in 
 balance. 
 
 ;'. Be sure that your additions are correct. 
 
 Find the exact amount out of balance, and look for 
 it and its one-half among the ledger items. 
 
 Jf. If the error is 9 or a multiple of 9, look for reversed 
 figures. 
 
 EXAMPLE. 65 written 56 would make a difference of 
 9 ; 57 written 75 would make a difference of 2 times 9, or 
 18 ; 63 written 36 would make a difference of 27, etc. 
 This may occur in any or all columns. 
 
 5. If there is an error of 1 in any column, look tor er- 
 rors in addition. 
 
 #. If the error is small, look for it in Interest or Dis- 
 count. 
 
 7. Examine the Bills Receivable and Bills Payable ac- 
 counts and note that the Debit and Credit entries are ex- 
 actly alike as far as posted. 
 
 8. See if your cash account in the Ledger or Cash 
 Book agrees with your Banking Ledger and cash on hand. 
 
 9. If the error is in cents column, it is not necessary to 
 add the dollars column. 
 
 10. If the above tests will not indicate to you the er- 
 rors, it will be necessary for you to re-check everything from 
 the previous balance of balances. Do not go over the work 
 without checking, you will .waste your time if you do. 
 
 J/ 
 

 
U. C. BERKELEY LIBRARIES