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Additional comments:/ Commentaires supplementaires: This item is filmed at the reduction ratio checked below/ Ce document est f ilme au taux de reduction indique ci-dessouIPPLIED IIVMGE inc 1653 East Main Street Rochester, New York 14609 USA (716) 482 - 0300 - Phone (716) 238- 5989 -Fox ■- 'I' THE WONDERS M' OF AKI7HMETIC OR The Art of resolving, usinfr only one figure, or by simple Addition, all rules of Interest or Dis- count, simple or complex, without having to divide, acquired in ten minutes study. STOCK OPERATIONS, The Four Boles proved by a simple AdditioOi as quick as thought. BY J. M. DANIAX7D. fc'^S' MONTREAL : BUSllBE SENfiCAL, PRINTER and PUBLISHER, 6, 8 and 10 St. Vincent Street, 1880 Entered according to the Act of Parliament of Canada, in the year one thousand eight hundred and eighty m the office of the Minister of Agriculture by J. M 0«^4j^4.^8 'Canada, d eighty, by J. M. CHAPTER I. Showing how all operations of Interest or Discount can be made without a Division. If you want to find the interest on any sum, for one year, you shall multiply the capital by the rate, either simple or complex, that is with decimal fractions. The rate is called simple when it is not followed by any fraction, as 1,2, 3, 4, 5, 6, 7, 8, 9, etc. The rate is called complex when it is followed by fractions, as 1.25, 1.75, 2.20, 2.25, 2.75, etc. When you have multiplied the capital by a simple rate, you point off two decimals, by which you have divided by 1 00. The dollars stand to the left of the point and the cents to the right. If you have a complex rate, you point off four deci- mals and the dollars stand to the left of the point, as above, and the cents to the right. What is the interest on $4842 at 6 o^o for one year ? Example : $4842 6 o;o $290.52 Answer: $290.52 cents. What is the discount on $542 at 4 o^o for one year ? Example : $542 4 0^0 $21.68 Answer : $21.68 cents. What is the interest on $6424.10 at 5o?oforone year? Example: $6424.10 5 ojo 321.2050 Ans.: $321.20 cts. 50 on i. What is the interest on $248 at ^ o/o for one year ? Example : $248 4.50 0/0 992 11.1600 Answer: $11.16 cents. V — 4 — $23,223 8 0/0 1857.84 Answer: $1857.84 cents oneVay ? '"""" °'' "''"""' "" «3«2 at i o;o for , Example: $3422.00 •art.^, ■ ... ■ ^80221 Answer $0.38 cents 471332 21 —days 471332 942664 B you $9.9 oper not Proc will call a«; «q Qn^l^^^? • ^"^wer : $9.89 79 that I Prooi by Division : 4242 4 a/o 16968 21 — days 16968 33936 356328 I 360 3232 1 « 35':8 9.898 2880 0000 2r Yc oddi Pr Answer : 9.90 cents. Oe — 5 — X, can be one after 1 it, begin- m right to each add- s if every It 4 0/0 for Its. il days? 79 that I usandths But as you llnd yourself with more than i cent rest, yo" may add a unit to the cents and have the result : $9.90 cents. This result is the same as by the other operation which is much shorter and in which you do not have to divide. What is the interest on $548.10 at 4 o^o for 4 days ? Example: 54810.00 6089998 4 •-days ^ . . ^ 24.35.99.92 Answer : $0.24 cents. Proof by Division : 548.10 4 0/0 219240 4— days 876960 1 360 1569 ' 1296 2436 2160 ^ . 0000 Answer : $0.24 cents. 2nd example : What is the interest on $4842 at 4 o;o for one dav? 4842.00 ' 537998 Answer : $0.54 cents. You strike off, as before, the thousandths, and you add a unit to the cents. Proof by Division: • 4842 4 0/0 19368 1 1368 1 ooon 0000 360 CO o OO.O Answer : $0.54 cents. 5ents. On the same capital at 4 o/o for 15 days y .^ — 6 — Example: 4842.00 537998 15 — days 2689990 537998 \ Proof by Divi^fonT' ^"^^^^ ^ ^^.07 cents. 4842 4 0/0 19368 15— 'days 96840 19368 290520 } 360 2520 I 0000 8.07 Answer: On the same capital at 6 o;o for one dav • Example : 4842,00 * 537998 268999 .07 cents. ^ _ 806997 /.nswer Proof by Division : 4842 6 0/0 .81 cents. 29052 J 2520 1 0000 360 80.7 Answer: $0.81 cents. Same capita! at 6 o/o for 20 days. Example : 4842.00 537998 268999 $0699? 20— jours 16.139940 Answer: $16.14 cents. Pre San Proof by Division : — 7 — 4842 60/0 29052 20 — days 581040 2210 504 1440 Other proof by Division : 4842 20— days 96840 36 8 24 360 16.14 Ans. : $16.14 cents. 16.14 What is the interest on Example : 490.00 54443 108886 45^days 544430 435544 Answer: |M4 cents, at i 0^0 foi 45 days ? 4.899870 Answer: $4.90 cents. Same capital at 5 0/0 for 120 days : Example : 490.00 54443 27221 13610 5 0^0 68050 120— days 136100 68050 8.166000 Answer: $8.17 cents. ~8 — Proof by Division; 490 8 0/0 3920 45— days Proof by Division: 490 5 19600 15680 2450 120 49000 2450 360 ' 176400 I 3240 I 0000 4.90 Answer : $4.90 cents. Two more examples : Capital ;— . 6848.00 IoS???~"* o?o interest for 380443—2 070 190221—1 oyo 95110— io;o 47555— i 070 23777-i 0/0 360 (t (( 294000 I 600 I 2400 8.166 2400 240 Answer: 8.17 cents. Capital : — 20000.00 one day— 2222222— 4 o70 1111111-2 070 555555-1 070 277777-1 0/0 138888 — J oJO I' 69999— J ojo ih^iiS^^? ^""^ represent the interest for one day on the respective capitals of $6848 or 20000. ^ „n !hl operation shows what is the result of addini? ttre^Cesfn "«"' " '«"' ^' ™ny «™ "a! '^^'ttiJ^^L'^^l 3m« Sus7?KS flrqt- iqn99i"^T?'' "' " '^^"' ^'^ ^^ isonehaliofthe T ' ,^^"''21 must represent 1 070. In the example on 20000.00, there being only one y Division; I ) J. 17 cents. .00 i22— 4 070 i 11-2 0/0 >55— 1 o;o ^77-1 (^0 188 — i 0/0 '99— i 0/0 le day on of adding *■ times as J capital, decimals ied off, to 3 right of up from ^presents . then re- alfofthe aaly one figure, 2, you cannot do «,• mise than to write 2 down under the ciphers and .vuu have 2222222, for the interest at 4 o/o for one day ; ! 1,1 1 1 1 1 for the interest at 2 mo, and so forth. It follows that when you have added up all the fi- gures of a capital sum, as many times as there are figures, and adding to the figure next following on the left hand the amount carried from each addition, the result will he the interest at 4 o/o for one day, one half of such result will be the interest at 2 o/o, and soforth. ffl.oiL*^®" y°^ ^^°* *^ ^'"^ o^t ^^6 interest at 4 o;o on |20000 for 50 days, multiply the result of the addition from right to left of 20000 by 50 days and the product will give you the interest, after you have pointed off six decimals, the dollars standing tc the left of the pomt and the cents to the right. Example : $20000.00 50 jours 2222222 4 0/0 for one day 111. 11110 Answer: $111.11 cents. The following is the way in which this adding up of the capital should be done to find out the interest at 4 0/0 for one day : Suppose you have to add upthe capital 4785. You begin by putting down the two necessary ciphers • then you add up all the figures in the capital, one after fi?^, 5'.,.*^^^"^ ^^^^ *° ^^^ *o '^e following figure to the left, the amount carried from each addition and the sum will give you the interest at 4 o/o for one day. Example: *4785,00 531664 — 4 0/0 for one day The sum 531664 has been obtained as follows : 5 and 8 are 13 and 7=20 and 4=24, 1 set down 4 and carry ^ ^^i^^l^*^^ '° *^® ^' I say : 2 and 5 are 7 and 8 = 15 and 7=22 and 4=26, 1 set down 6 and carry 2 which I say : 2 and 5 are 7 and 8=15 and 7=22 and 4=26 1 set down 6 and carry 2; 2 carried and 8 are 10 and /«17 and 4=21, 1 set down 1 and carrv 2; 2 carried — 10 — X dW, there wSh/ .1 ««»'« only, the result for one jnultiply th^whole t'm; 4heuevTf ZT' ^^^ " than one day's interest. ^"^^^^^ yo« have more »TOTHBR EXPLANATION OP THE ADOITION. it .?n"ecra'X"StU°t.^f:7';£ 'i-^-P""". on account of the iw„„C *'?.^* '" "'e last figure la the same manner wLnfc'^°,"''^'°8.'''« « body of numbers, you must eohJr.^^P''"''' '" '"« the number of cvohern ft.. %?'''' ""'=* """"s 'ban them on the left Cd siSe. ^™ "^^^ '''""""•'8 2o524'2!'l wUl put dodi° ^ ""'•"l-'P «■« '•""•wing: Kt?s^s-E£S??v"^^^^^^^^^ I add the flgi-es 2 I A I' iv k''* T^"*^ *« »Pi«al, aught and clrrv I '.'.:' ^' y'"'=i'„*''^ '«■ I set down ana -^ are 3 and 2=5 and '4-9ar,rt T'Vi ' i ' ,*=*"''««* •»'' carry 1 ; , carried andYa^sUliis'aZ \ id carry 1 ; l res in the ca- 4 0/0 interest .53 and some iber of days in one day. ?andths must on by num- the number 5st, because esult for one ' for a consi- necessary to have more TION. the capital, 3 last figure the capital, ters in the more than t following (following: er the ca- ?oing back one of the ^hich it is the num- he capital, set down 1 2=5 and 1 canied set down 2=5 and — 11 — il7?'fK^®*o1?,!^'V^.'^.°*^ ^*^^ *"^^*5 I ^egln again with the 2 (the first figure of the capital) and 1 say: 2 &na I are 4, 1 set down 4 and carry aught ; as I have nothing more to add for the ciphers, I set down 2 for each of them and also lor the first figure of the capital. The sum of this addition is 22249110. And if I point off SIX decimals, I will have 22.24, which I will put down as $22.25 cents interest for one day, leaving off the thousandths unless I should multiply by the num- ber of days my capital has been bearing interest. 11 1 want to find out the interest at 6 070 on this ca- pital which gave me as interest at 4 070 : 22249110 I shall take one half of that number, and add it up to the same and that will give rae the interest at 6 070 : it will then only bo necessary to multiply by the number ot days stated and having pointed off six decimals, I will have the $ to the left and the cents and thou- sandths to the right of the point. Example: 22249110— 4 070 11124555 33373665 70 — days 2336.156550 Answer: $2336.16 cents. «io^« i«*^® }^ ^^^^ ^^y' ^^ ^^^ interest asked for *i . :u 2^^^^^ ^^^ ^°°^® thousandths which are left off in the final result of the operations. It is well to strike off the thousandths, and add one to the cents, (as I said before) because in the final re- sult only cents need be accounted for. The addition of the capital can be made in various ways ; it can be made by putting down the capital five times echellons by respecting every figure,putting down fave times that representing the units, five times that ♦u^"i:'''"i'"5 ^^''^ iciiiiis, live limes liaai representing the hundredths, etc., etc. The sum of the addition will every time represent the mterest at 4 070 for one day. — 12 — What is the interest on $845 at 4 070 for 40 days ? Example: 845 845 845 845 845 9388795--4 o;o for one day 40— days 3.7555. iToo ^ Answer : $3.75.55 i. e. $3.76 cents. 55555 44444 88888 9388795—4 070 interest for one day 40— days ' same 3.75551800 Answer : $3.76 cents. hn^if!/"'"" '' *^® '^°*® for both operations. With both these methods, it is necessary to point off ei^ht certfth: ^^T' '' '^^« *^« ^ '« tL leVan t cents to the right of the point. Here is the proof of the two proceeding rules by addition of the capTtel Example; $ 845.00 ^^^fn~"i °^° ^^^ °°® ^*^y ■IV — -uuys 3.755480 Answer : $3.76 cents. for 40 days ? day } of the same )r one day $3.76 cents. lions. With nt off eight left and the proof of the pital. '3.76 cents. Proof by division : — 13 — 845 4 0/0 3380 40 — days 135200 I 360 2720 ' 2000 3.75 W41.41, *!. . J^^^ Answer : $3.76 cents. With the method of adding up the capital, the oper- ations are considerably shorter ; and when you have acquired the habit of adding up the capital as we have just been doing, you will be able to calculate interest or discount a? quick as thought. We have seen that the operations can be made by adding up all the figures of the capital in order to find the inteirtst at 4 o/o for one day. We propose now to show how the interest can be found for the number of days the capital has been bearing interesi. If you niultiply the capital by the number of days and if you add up the figures of the result, in the same way as above, you will have the interest for the number of days given. Example of the proceeding rules ; •845 Proof: 845.00 40 days 93887 liiSo !!"^"y' 3.7654 Answer : $3.76 cents. 3.755480 ..,, , . , Answer : $3.76 Other example with a capital of 3.241 at 4 070 for 12 days : ' 3241 Proof: 12 — days 6482 3241 38892 addition. 4.3210 Answer : $4.32 cents. 3241.00 3601.10 12~days 720230 360110 4.321320 Answer : 4.32 — 14 — Proof bv division ; 3241 4 0/0 25928 12964 155568 I 360 1156 I 768 4.32 48 Answer : $4.32 cents. the adSn multiplication takes place before Example : 845 40 days 33800 33800 33800 33800 33800 3.75551800 Answer $3.76 cents. rivld «t '^r^^t'jilP!™"""' the sam'e result is ar-' cessary" to point oFeieMdP^t™'^'?'-'"*' '} '^ ""^y^ "«- * .0 th^e 4 a"„Vti.^irsrft"gh''tirtii: xr" other example — 15 — 845 40 days 33800 00000 88888 33333 33333 4.32 cents. Jr of days, four 1 the result, on > foregoing re- ary to addsthe 5s place before result is ar- s always ne- ' to have the the point. 3.75551800 Answer : $3.76 cents. The result remains the same, after pointing off eight decimals. AH these rules give the interest at 4 o;o ; and when the interest at 4 o/o on any capital is once arrived at, it is easy to get at every rate. Take one half of 4 o/o you have 2 o;o ; take one half of 2 o/o you have I o»o. When you have the 1 op rate, you multiply by the rate given and thea by the number of days ; and you get the interest asked for after pointing off four deci- mals, if you have not added the two ciphers. Every time you add those two ciphers to the right of the capital, you must point off six decimals. But if the rates are complex, you must point of^two more decimals in the result. If you want to operate on any rate of interest, you can do so by getting first the * o;o interest which is i of the 4 070 i. e. of the sum of the addition from right to left, which represents the 4 o;^o rate. Example : 3421.00 h 5 oio for 21 days 380110 190055 95027 5 0^0 475135 21— days 475135 950270 9.977835 Answer : $9.98 cents. \ — J« — rate; takeTof'the Lsuft a^add ?,'^"^'' ^^^" ^^ '^' as there are figuresfn u • TatfJre "^ f ^ •'"''"y *'°»es sum and you will have th^«Tfn^^^^ to the right of the pofnt * ^ ^^" ^^'^ ^^« ^^^^s What is the intere^f on 8422 at S n-i« fn„ aq i Example : 8422 cSpiiar "^^^^ ^ 48— days 67376 33688 404256 I 3 o;o 1212768-totaI. , . 606384 to be added u p 303192 -^i of the total, 33.6878 Answer : 33.69 ^ Proof: 842200 935776^4 o;o 467888 233944— unit or i 3 0/0 701832 48 — days 5614656 2807328 33:68:78: 'a"S7s' added Ul"'.''^'''?*''' '.""^ ''°'<»>'>' sandths left out. ° "■* "*'«' and the thou- 'ay: multiply J, then by the 8 many times icimals in the md the cents r 48 clays ? )9 liplied by ■esult has addition um. fiirn- e amount the thou- - 17 — Other example on the same capital. 8422 Other example : 48 8422 48 67376 33688 404266 3 1212768 1347517 673758 33.6879 0^0 67376 37688 404256 449171 224585 112292 33.6877 Answer : 33.69. Answer : 33.69. In this operation, the capital has been multiplied by the number of days ; then by the rate, and the result was a total sum of 1212768. This result has been added up from bright to left, and the answer has been one fourth of the sum of this addition. Proof by division : 8422 3 0^0 25266 48 days 202128 101064 1212768 1327 2476 3168 2880 0000 360 33.688 Answer : 33.69. The rule to be followed as to the number of decimals to be pointed off is the following : For the capital, two, for the simple rate, two, for the two ciphers that fol- IffW ihf\ npnitfll iwn nnH if iVtCk Y>afo ia rtnmrAav f/\w the decimals in the rate, two, making all told eight decimals, if are combined in the operation all these conditions. — 18-. Sj^ me»thod of operattng by the unit, sample. 7424 d 7.50 for 50 Jays. 7424.00 824887—4 o;o 412443—2 o!o 206221-1 0/0 or unit. 7.50 1031 loT" 1443547 154665750 50-— days , 77.33287500 Answpr • 77 qo ..„# We Will give two examples of that below. What IS the interest on 743 at 5.25 for 43 days? oJr"" 632300 41277 ^^2554 20638-.unit 175638-unit 43-days 9 ^^^ Pro< Oth( Wh Ans 61914 ^2552 887434 5.25- -rate 1580742 22— days San Exa 4437170 1774868 44371 rn 3161484 3ioi484 34.776324 Answer : 34.78 cents. I 4.6590?8o. Answer : 4.66. Ans Proof by division — 19 — 5.25 3715 1486 3715 390075 43— days 1170225 1560300 77.32 cents, lit, let the )8l for one by number I represent lapital has 'equired to days? nit ays .78 cents. I 16773225 I 360 2373 I 2132 4.6592 3322 825 105 Answer : 4. 66. Other examples : What is the interest on 4842 at 4 ojo for one day ? 484200 Proof by division : 0.537998 4842 Answer : 0.54 cents. 4 o;o 19369 1360 1368 I 2880 538 0000 Answer : 0.54 cents. Same capital at 4 o?o for 15 days : Example : 4842.00 Proof by division : 537998 15 — days 4842 4 0/0 2689990 537998 19368 15— davs 8.069970 Answer : $8.07 cents. 96840 19368' 290520 — 20 — 290520 I 360 2520 0000 8.07 Answer : $8.07 cents* Same capital at 6 o/o for 20 days Example: 484200 Proof by division. 0^7998 — 4 o;o '"'" 268999—2 o/o 4842 20 — days 806997 20— days 16.139940 \ Answer: $16.14 cents. 96840 I 6 36 I 8 16.14 24 Answer : 16.14 cents. What is the interest on $490 at 8 o/o for 45 days? Example : 490.00 54443—4 o;o 108886— double or 8 o;o. 45— days 544430 435544 ^ 4.899870 Answer : $4.90 cents. Proof by division : 490 8 0/0 3920 45 — days 19600 15680 irromn l nnn 3240 I 0000 4.90 Answer : 4.90 cents. »20 I 360 .20' 00 8.07 r : $8.07 cents. hy division. 20 — days 14016 I6T4 24 : 16.14 cents, or 45 days? 0. ats. rs : 4.90 cents. — 21 — Same capital at 5 070 for 120 days. Example: 490.00 54443 13610 — unit or both together 5 0/0. 68053 120~days 1361060 68053 8.166360 Answer : $8.17 cents. Proof by division : 490 5 o;o 2450 i20-~days 4900 2450 360 294000 600 2400 8.166 2400 _., , , ,^ . , 240 Answer: 8.17 cents. What is the interest on 2333 a 5 010 for 22 days ? Example : 2333.00 259221—4 070 129610—2 o;o 64805—1 op or unit; 5 0/0 324025 22— days 648050 648050 7.128550 Answer: $7.13 cents. Proof; 2333 5 0^0 11665 22— days 23330 23330 256630 I 360 1030 7.128 3100 220 Answer: $7.13 cents. V — 22 — What is the interest on 745 at 7 o70 for 15 days? Example : tt)§3 ^ 745.00 id 5 & 82776—4 070 ' ^«^ 41388-2 0/0 ^ 5 20694—1 o/o 144858 15 — days 724290 144858 5215 15— days 2172870 Answer: 2.17 cents. Other example at 48948.20. 26075 5215 78225 622 2625 105 360 2.17 Ans. 2.17 cents. 0/0 for 20 days on a capital of «50|3 48948.20.00 "" - I 543868885—4 o/o 271934442— 2 070 135967221—1 o/o 951770548 20— days ^7i o CO <»^ Proof; 48948.20 7 0/0 34263740 20— days 190.35410960 Answer: 190.35 cents. Ans. 685274800 3252 1274 1948 1480 400 $190.35 cents.— 40 360 190.3541 You may see that the result arrived at is always the same as with a division, and the operation is much shorter as it is easier and more convenient to add un and to multiply than to divide. ^ With this method you will save much time, because you won t have the trouble to find out how many times „„.. , ^^^.j.. ^jjjj ^^^^^^ you have ac- quired the habit of adding up the capital from right to left, you Will be able to do all rules of interest or dis- count as quickly as you can write down the figures \ >r 15 days? 17 s. : 2.17 cents. n a capital of roof; 8.20 7o;o 1740 20— days 800 I 360 190.3541 8 80 400 -40 is always the tion is much 3nt to add up Lime, because V many times you have ac- from right to erest or dis- he figures. — 23 — RULE AT 6 o;0. In all rules at 6 op, where the number of days is divisible by 6 without leaving a fraction, the capital may be multiplied by the figure obtained as quotient. Example :— What is the interest on 745 at 6 o?o for 42 aays ? 745 745 for 12 days 7 2 1.490 5.215 Answer : 5,21 Aus. : 1.49 cents. What is the interest on 4323 at 6 o/o for 48 days ? Example: 4323 48 Proof: 34,584 4323.00 480332 240166 Answer : 34.58 Proof: 745.00 82776 41388 720498 48 5763984 2881992 34.583904 Answer: 34.58 Proof by division ; 745 6 0/0 4470 42 — days 8940 17880 124164 12— days 248328 124164 1.489968 Answer : 1.49 cents. By divisor 6. 745 42— days 360 187740 774 I 540 5.21 180 Answer ; 1490 2980 31290 12 9 6 5.215 30 Answer : 5.21 cents. 5.21 — 24 — The same rule applies to the 4 o;o interest where the Sw^ ^^ ^^ divisible by 9. Examples are given It is clear that, as 42 divided by 6 gives 7, 1 multiply 745 by 7 and pomt off three decimals in the result in order to have the $ to the left of the point and the cents to the right. What is the interest on 745 at 4 o70 for 36 davs? Example: 745 ^ 4 2.980 Answer : 2.98 Other example at 4 o?o for 72 days on a capital of 3248 i 8 Proof by addition of the capital : 745.00 8277b 36 — days 496656 248328 25 984 Answer: 25.98 Proof by division 745 4 0^0 2980 36 — days 2.979936 Answer 2.98 17880 8940 107280 J 3528 I 2880 0000 360 2.98 R«foa^o„ , u . Answer: 2.98 cents. Rates can also be computed with the capital. What is the mtorest on $840 at 6 o;o for 70 days ? Example: 840— 4o70) . . ^ 420—2 op / ^^^^°fif together 6 o/o. 1260 Tn A , 88200— to be added up 9.7998 Answer : $9.80 cents. Brest where the iples are given IS 7, 1 multiply n the result in point and the ir 36 days? a capital of f by division : 45 4 op 80 36 — days 80 30 j 360 ?0 2.98 30 2.98 cents. ipital. •r 70 days ? fether 6 o/o. — 25 — Proof by putting down the capital in echellons : Proof: 1260 1260 1260 1260 1260 13999860 70- 9.79990200 840.00 93332—4 6?n 46666—2 0^0 139998 70— days -days Answei ; 9.80 9.799860 Answer: 9.80 Other example at 7 o/o on a capital of 840, for the same number of days : Capital— 840— 4 o/o ) Half— 420— 2 o?o }• Together 7 o/o Quarter— 210— 1 o/o J 1470 70— days Proof : 102900— to be added up 11.4332 Answer: 11.43 cents. 840.00 83332—4 0/0 ) 46666 — 2 070 > together 7 o70 23333—1 0/0 ) 163331 70— days 80 cents. 11.433170 Answer: 11.43 cents. In the 6 o/o rule we have taken one half of 840, which we have added to the latter sum ; and we have multiplied the total 1260 by the number of days given, the result was 88200, we have added up the figures of this result as many times as there are figures in it; we — 26 — V have pointed off four decimals (as the two ciphers ?'?QOQ"?i Jd^ied) and we have had for an answer : y. /y98 that we called 9.80 leaving out the thousandths and adding one to the cents. r.„?f"' ^® ^°°,^ *^^' *h« addition can be made by puttipg down the capital in echellons in the manner we r^h^f 1. /"^S^fi?- r^^'^ '^ tantamount to adding up from right to left the figures in the capital. © i' "^ Some examples will be given here : EXAMPLES BY ECHELLONS. What is the interest on 840 at 7 o?o for 70 days ? 840—4 0^0 ■) 420—2 o?o (.together 7 070 210—1 (^oj ' 1470 1470 1470 1470 1470 16333170 70— days 01 Otl 11.43321900 Answer: 11.43 cents. Pre Other example : 00000 77777 44444 . mil Proof: 840.00 93332—4 070 46666-2 0/0 23333 — 1 0/0 16333170 70 — days 163331 70 — davs 11.43321900 Answer: 11.43 11.433170 Answer: 11.43 cts. V the two ciphers i for an answer : i the thousandths ;an be made by in the manner we ) adding up from i. for 70 days? ' 7 0^0 cents. Proof: 0.00 5332—4 0/0 3666 — 2 0/0 J333 — 1 0/0 1331 70 — days !170 wer: 11.43 cts. — 27 — Other proof: 840— Capital 70 — days 58800— to be added up and struck off 65331 — 4 0/0 32665—2 0/0 16332—1 0/0 11.4328 Answer: 11.43 cents. Other example at 8 o/o for 41 days on a capital of 42231.00 4692332 2— Multip. or doubling the sum 9384664 41 — days 9384664 37538656 384.771224 Answer: 384.771224 Proof by division: 42231 8 0/0 337848 41— days 337848 r 51392 13851768 3051 kill 2776 2568 360 48 Answer: 384.77 cents. — 28 — In all these rules by putting down the capital by echellons it is necessary to point off eight decimals in order to have the $ to the left of the point and the cents to the right. OPERATION BY ONE FIGURE ONLY. All operations can be made by one u^ure only either with the addition or with the division. What is the interest on 840 at 4 o/o for 70 days ^ \ Example : 840 7— regulating figure 5880— sum to be struck off 6531— addition 1680 —double capital 23331 4 0/0 rate 93324 70 — days 6.532680 Answer : 6.53 cents. Other example a 8 o?o on the same capital for 50 days. 840 7— regulating figure 5880— sum to be struck off 6531 — addition 1680 —double capital 23331— unit 50 — days 1166550 9.332400 Answer : 9.33 cents. Ihe capital by fht decimals in I point and the FLY. e iigure only, sion. 3r 70 days ? 'e koff )r : 6.53 cents, capital for 50 — 29 — Proof by addition: 840 70 — days 58800 6.5331 Answer : 6.53 cents. 0/0 cents. Proof by addition of the figures in the capital, at 8 840.00 93332 186664 50 days 9.333200 Answer : 9.33 cents. Other proof : 840 50 — days 42000 4.6666—4 o;o. 2 9.3332—8 0/0. Answer : 9.33 cents. cJhe'Sil^li^T ^' r^"Ja«"g figure, because it can De made use of m all operations In fart if fn». wJj], ? ":?" ''™'=* <"f «'>'l 'h« sum of th^aZmon pUa.r2 i? ^S^C^ ^S"" -«P'-«<>- of '^-■ ine poiui auu ihe cents to the richt "" "" "' '"" "' Otiier example on the same for 82 days . • capital of840 at 5.25 0/0 — 30 — 840 7— regulating figure 5880— to be struck off 6531— addition 1680 —double capital. 23331— unit 5.25 — rate 116655 46662 116655 12248775 82-.days 24497550 97990200 10.04399550 Answer : 10.04 cents. PROOF. 84000 93332—4 0/0 46666—2 0/0 23333—1 0/0 5.25 — rate 840 5.25 U6665 46666 116665 12249825 82— days 24499650 97998600 10.04485650 Answer : 10.04 cents. 4200 1680 4200 441000 82— days 882000 3528000 36162000 J 360 1620 I 180 10.04 Answer : 10.04 cents. — 31 — ♦v.}T® must here point off eight decimals on account of the two decimals on the rate with this method. Ifota. — Whenever the rate is simple, six decimals are pointed off, and eight decimals whenever the rate IS complex. .r.^^ *^u 5"'? ^' ,^ °/o '8 of the most frequent occur- ence with traders, I will give some more examples of it. What is the interest on 4800 at 6 o/o for 50 days ? Example: 4800.00 533332--4 o;o 266666—2 o;o 799998 50— days 39,999900 Answer : $39.99. .^^u""^ r.?^" ^*" ^^^' leaving out the thou- sandths and adding one to the cents. What is the interest on 2323 at 6 op for 16 days ? Example : 232300 258110— 4 0/0 129050—2 0/0 387160 16— days 2322960 387160 Proof; 2323— capital. 16 — days 13938 2323 37168 — to be added up 41295—4 070 20647—2 0/0 6.194560 R6ponse: $6.19 cents. R 1Q/.9 Answer : 6.19. The 6 0/0 rule can be made in the following manner : — 32 — ^^^^fn ' ^" * ^'^P^^^^ °^ ^^^ ^^'^ ^2 days. ^fi „, Proof by division: ^—0/0 840 5040-to be struck off __82-d8y8 5599--addition leso o40 — capital 6720 13999 82 — days 27998 111992 68880 8 28 48 . Answer : 1 1 .48 cents. 1147918 Answer ; 11.48 cents. The operation has been done this way : the canital has been multiplied by 6, the result was 5040, ^this ire fl'.urrinTt -^Z' '^^'^ "L?.?."^^"^ times as there are ngures m it; the sum was 5599, under this we hnvA put down the capital, beginning inder the te^l^^II we have seen done with the regulating figure 7 with this difference that this time we do not double tCca- pital ; then we have multiplied hy the number of davs given and we have pointed off five decimals (this is f general rule for this method) ^ The same rule by the divisor 7 is given below : Example: 840 p^oof : 840 I 932—4 o;o 5880 _466-2o/o ifi«n^ ^398 _^ 82-days 23331 6 o;o 139986 82 279972 1119888 •11.478752 2796 11184 11,4636 Answer: 11.46 cents. Answer: 11.48 cents. lays. of by division ; 840 82— deys 680 20 880 I 6 18 r: 11.48 cents. y: the capital as 5040, this times as there ' this we have the tenths, as Sgure 7, with louble the ca- imber of days als (this is a i below : 40 32 — 4 o;o 36 — 2 0/0 )8 J 2 — days )6 (6 : 11.46 cents. 11.48 cents. — 33 - nn!^!!^i!^'^ kind of proof, it would be necessary to add ord.r»n^'?T.f "'" "^^^'^'^" ^''^"^ '•ight to left, in given befow "'""^ '^^''^^' ^^^le examples are What is the interest on 450 at 6 o;o for 50 days ? Example : 450 7 3150 3499 900 — double capital 12499 6 0/0 Proof: 450 499—4 0/0 74994 50— ^ays 3.7500 Answer : 3.75 cents. 3.749700 Answer : 3.75 cents. For the modus operandi see first rule by the divi- fo ]^^^^ F^°°^' ^J?! a,<^^ition has been made from right i nin J""!? S?n ^^^^^ ' ^ ^*d i" this way 500, as the LTi.f 7.n^?' ?n ^ T ^"t'^^est which gives 750 ; X tW J^^.^y ^^^°^ ^ ha^« for an answer 3.75. f«n*c ?? r?^/"^® ^^ generally unsed by accoun- tants; It 18 to hud out the 6 0/0 interest for 60 days on any capital, by pointing off two decimals. What is the interest on $450 for 60 days at 6 o;o. Answer : 4.50 cents. j a.* u u^u. Proof by division : 450 60 ■li{j\j\} 30 I 4.500 Answer : 4.50 cents. Other proof: — 34 — 45000 49999 24999 74998 60 4.499880 Answer : 4.50 cents. What is the interest on 450 at 4 o;o for 90 days ? Answer: 4.50 cents. Proofby division : /»50 )0 40500 45 4.500 Answer : 4.50 cents. This, as you may see, gives a correct answer, but these two last rules, by pointing off two decimals, caii only be applied to interest for 60 or 90 days. When the mterest at 6 om for 60 days is found, you mav easily find it for 36 days, for 15, for 7i, etc. If for more than 60 days, you may add the 7i, 15, or 30 days in- lerest, etc. r.J^J}^Ju^° ^^^ ^ ^t^ ^"'^s, three decimals are Pf ihi,ri^° k' ^|J*"^L*.^^P*^6^ *s left out in the divisor; U should be 60 or 90 instead of 6 or 9 ; to equalize the matter it should be necessary to strike off one figure in the Gividend, if this has not been done, there is one decimal more m the quotient, three instead of two. It follows tbat the capital had better be multiplied «y.?!>"S???.«!: S^ .^^.y^ and divided by 6, if it is at V -„Yu, \ji uiviuuu vy y, 11 it ia ^i 4 o«q^ This last method may be applied to any number of — 35 — : 4.50 cents. 30 days ? : 4.50 cents. nswer, but cimals, cari lys. When , you may If for more 30 days in- icimals are -he divisor ; qualize the le figure in here is one of two. multiplied 6, if it is at number of BULE FOR MAKING ALL OPERATIONS WITH THE DIVISOR 9. What is the interest on $840 at 3 o;o for 50 days ? Example: 840 Proof: 31500 3 0/0 3.4999 Answer : 3.50 cents. 2520 50 — days total.— 126000 half.— 63000 quarter.— 31500 I 9 45 I 3.500 Answer : 3.50 cents. OTHER PROOF. 84000 840 93332 3 070 46666 ' 23333 2520 3 0/0 50— days 69999 50 — days 126000 1800 0000 3.499950 Answer : 3.50 cents. Answer : 3.50 cents. In these operations with the divisor 9, the result of the multiplication by the rate and by the number of days, should be divided first by 4 and then by 9. Remark.— The addition of the capital from right to left, as we have previously done, is equal to the nine- tieth part; if this ninetieth part is divided by four, we have the three hundred and sixtieth part, or the unit, or the interest for one day. If we should multiply the capital by the number of days before dividing by 9, we will have as quotient the interest at 4 o?o, for the number of davs the cauital has been multiplied by. If the capital is divided before the multiplication by the number of days, we shall follow the division up to three decimals ; and after the multiplication by the — 36 — number of days, we shall point der to have the $ to the left and of the point. What is the interest on 450 at Example : 450 50 22500 45 Answer : 2.50. 2.500 What is the interest on 5555 a Example : 5555 80 Answer : 5555 80 444400 84 34 70 70 7 49.38 cents off six decimals in or- the cents to the right 4 070 for 50 days ? Proof : 45000 49999 50 2.499950 Answer : 2.50 cents. 4 0/0 for 80 days ? Proof : 555500 617220 80 49.377 49377600 Answer : 49.38 cts. 444400 49.3776 Answer : 49.38 cents. Proof by addition. 5555 5555 5555 5555 5555 61721605 80 Answer : 49.38 cents.— 49.37728400 What is the interest on 752 at 4 ojo for 50 davs ? "^^^ 50 83554 50 37600 4.1776 Answer : 4.18, 50 I 16 > - — 4.177700 70 4.177 Ans. 4.18 cts. 70 Ans. 4.18. nmals in on- to the right )0 days ? 45000 49999 ' 50 2.499950 2.50 cents. iO days ? 555500 617220 80 49 377600 : 49.38 cts. — 37 — It will be seen from the proofs we have made of the proceeding rules, that the addition from right to left, is equivalent to division by 9. Everybody knows that the general rule is to com- Eute interest for a certain number of days ; it is only y exception that it is computed for a month or for i, i or I of a year. It is therefore necessary to use a division in order to make all operations of interest or discount. With this system, division is abolished, and advan- tageously replaced by the addition from right to left ; it is always easier and more convenient to add up than to divide. CHAPTER II. iddition. 555 55 5 605 80 iOO days? addition. 00 54 50 00 4.18 cts. RULES FOR THE STOCK OPERATIONS MOST IN USE. If the 5 o;o bonds are at $75, what capital may be necessary to acquire an income of $650 ? Example: 6500 9100 9750,0 Answer: 9750. This operation is done by doubling the income xirartiaA nnA «V.^» -^..itl-^i..: i jt. _ "...-^ '"Ir"-'" ■=■■"■- liSwii !iiui;ip:jr'i:;j^ i?y iiio tjUulSuuti. One decimal is pointed off in the result, which is di- viding by ten, and the $ stand to the left and the cents to the right of the point. 2 — 38 — Proof by the rule of three. Example : 5 : 75 : : 650 75 X 3250 4550 48750 37 25 Ai 9750 Answer: 9750. In this operation, the 650 income has been muItiDlied by the quotation, 75, and the result divided by 5 this means that as $5 is the income derived from '$75 what capital is the $650 to be derived from ? ' Tho above operation is made by the rule of three simple, and it reads thus : 5 is to 75 as 650 is to X. We will now reverse the preceeding rule and ask : 11 7.-? give 5 per cent, income, how much will 9750 giv e? Example : 75 : 5 : : 9750 : X 5 48750 I 75 375 I 000 650 ; Tf -re • ij o ,. Answer: $650. U 75 yields 8, how much will 5625 yield ? Example : 75 : 8 ; : 5625 : X 8 45000 I 75 000 ' 8 : 75 : : 600 : X 75 Snnn 4200 45000 600 Ans. : 600. To thef sum only ber. add dowi ^adde whin up ai of al must — 39 — : 9750. lultiplied >y 5 ; this 'om $75, ! of three md ask : ^50 giv e? 1650. ;oo. / 45000' 8 50 I 20 5625 40 00 Answer: 5625. PROOF OF THE FOUR RULES. Addition: 324—9 Subtraction; 84216—3 632—2 784—1 1740—3—3 Multiplication : 432—9 17—8 32214 52002—3 3024—9 432 7344—9 Division =.3 =4787 I 34 = 7 138 I «. =3 4787 270 14079=3 14 320 14 4773 PROOFS EXPLAINED. To prove the addition, it is necessary to add up all the figures of every number once, and reduce each sum to one figure. The first number gives 9, as 9 is only one figure, we will put it down opposite the num- ber. The second number is 632, which gives 11 : we add up tbis two figures and have 2, which we put down opposite The third number is 784, the figures added up give 19, we reduce to one figure and have 1, ■' "•-• i--«.-- Yi-Fvciw. iug inree uKures », ",c. i add up and reduced of all the figures roust also give the figure 3 „_, , .-, - added to one figure give 3. And the addition in the sum, after reducing to one, y — 40 — This proof may also be made by adding up the figures of every number as if they were written in one horizoi^tal line and forming but one number, passing over the figure 9 whenever it is found and reducing the sum to one figure. The proof of the subtraction is made by adding up the minuend and reducing to one figure, as for addi- rion, we have therefore 21, which reduced to one figure, gives 3. Then the subtrahend is added up with the remai-'^er and we find again 21 which reduced to one figure, .daves again the same figure 3, the figure being tljie same the operation is proved to be correct. In the multiplication, the multiplicand 432 is added up and gives 9, which is only one figure and needs no reducing ; the multiplier is also added up and gives 8, I multiply by 8 and I have 72, I reduce lo one figure and I have 9. The result of the multiplication is then added up and gives 18, which reduced to one figure gives 9, the two figures being equal, the operation is proved to be correct. In the division, the dividend is added up and if there is a remainder it is deducted from the dividend. Example 4787, the remainder 14 deducted, we have 4773, add up once every figure in the latter number, you have 21, reduce to one figure, you have 3. Add up the divisor 34, you have 7, add up also the quo- tient 14079, you have 21, reduce to one figure, you have 3, multiply 7 by 3, you have 21, which reduced to one figure, gives 3. The figure given by the dividend being the same as the one given by the multiplication of the two others numbers, the operation is correct. It is not necessary to divide the sum of additions by divisor 9, to prove an addition : all thai is wanted is to reduce to one figure, that being equivalent to the remainder after a division by number 9. Example : add up every figures of 789654, you have 39, reduce to one figure, you have 3. 1 2 3 4 5 6 7 8 9 10 g up the en in one ', passing reducing riding up i for addi- 1 to one \ up with iduced to he figure correct. is added needs no 1 gives 8, ne figure n is then le figure ration is p and if lividend. we have number, 3. Add the quo- :ure, you reduced same as others iddilions J wanted nt to the 354, you — 41 — By division, if vou divide 39 by 9, you have 4 • 4 limes 9 is 36, which deducted from 39, leaves 3. Thus, without division, the same result is arrived at, and operations can be proved as quick as thought. In adding up, the figure 9 may be passed over ; that IS to say, it is not necessary to add it up with the other figures. Multiplication Table. 2... 3... 4... 5... 6... 7... 8... 9... 10 4... 6... 8. - 10... 12... 14... 16... 18... 20 6... 9... 12... 15... fS... 21... 24... 27... 30 8... 12... 16... 20... '..... 28... 32.. 36... 40 10... 15... 20... 25... 30... 35... 40... 45... 50 12... 18... 24... 30... 36... 42... 48... 54... 60 14... 21... 28... 35... 42... 49... 56... 63... 70 16... 24... 32... 40... 48... 56... 64 .. 72... 80 .n I i?- ^^- ^^- ^^'" 5^- «3... 72... 81... 90 10 1 20... 35... 40... 50... 60... 70... 80... 90... 100 1 2 3 4 5 6 7 8 9 At 4 0/0 Multiply the capital by the number of days, add up from right to left and point oflTfour decimals. At 5 010 Multiply the capital by the number of days add up from right to left, add one fourth. At 6 0/0 Multiply the capital by the number of days add up from right to left, add one half. At 7o;o Multiply the capital by the number of days, add up from right to left, add onne half and one fourth. At 8 0/0 Multiply the capital by the number of days, ade up from right to left, multiply by 2. At 9 0/0 Multiply the capital by the number of davs. add up from right to left, multiply by 2, add the At 10 0/0 Multiply the capital by the number of days, add up from right to left, multiply by 2, add one — 42 — Ex.— 250— -4 0/0 30- -days 7500 0,8332 At 5 o;o--250 30 At 7 o;o— 250 30 7500 8332 4166 2083 7500 8332 2083 10415 At 6 0/0— 250 30 7500 8332 4166 12498 14581 At 8 o;o— 250 30 7500 8332 At 9 0/0 --250 30 7500 8332 16664 2083 18747 Table of numbers represewting the rates. hJ^ilT^ multiply the numbers representing the rat^^ tin finW?KP'^-*l ^"^ ^^^'" l>y the UmJt)ero^f days von If you want to find out interest at 9Z «-•« toi^^ .i. number 5556, to wdich add 94 qn in J ^?' !^^® ^^® and you^have the mtereV^'aS '"''" '^'"'"""^ — 43 — 1 Hates. i 0/0. i o;o. i 0/0. i 0/0. 3 0/0. I 0/0. ' 0/0. 0/0. H 0/0. , H 0/0., II 0/0.. 2 0/0.. 2io/o.. 2J 0/0.. 2| 0/0.. 3 0/0.. 31 0/0.. 3i 0/0... 3i 0/0... 4 0/0... H 0/0... 4i 0/0. . The SIGN 0/0 means per cent. numbers. .... Ul 1041 694 1389 ..... jHfoO 2430 1736 2778 3472 4166 .... 4861 .... 5556 .... 6250 .... 6944 .... 7639 .... 8333 .... 9028 .... 972'> ...10416 ...lllll ...11805 ...12500 Hates. 43 0/0. 5 0/0, 51 0/0. H 010. H 0,0. 6 0/0. 61 0/0. 6i •JO.. 63 0/0., 7 0/0 .. 71 0/0.. 7i 0/0.. 0/0.. 0/0... 0/0... 0/0.. 0/0 .., 0/0... 0/0 ... 0/0 71 8 81 Si 81 o H 9i 91 0/0. numbers. 13194 13889 14583 15277 15972 16667 17361 18055 18750 19444 .....20139 ....20833 ....21528 ....22222 .. 22916 ....23611 ....24305 ...25000 ....25694 .. 26388 ...27083