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Here I have supposed the divisor to be only increased or diminished by one, but the same may be applied with equal advantage with a greater excess or deficiency; for example, let 74367 be divided by 993; here by the substitution of 1000 we may readily perceive the quotient to be 74, with a remainder of 885; for in the division of the given number by 1000, having subtracted 74 sevens too many, by the multiplication of 74 by 7, and the addition of the product to the former remainder 367, we obtain the true one : again, let 37642 be divided by 1009, we, with the same facility discover the quotient to be 37, - with a remainder of 309; for here we have subtracted 9 times 37 too few, which being deducted from 642, the first remainder leaves 309 for the true One.*
When the divisor is a series of nines, division may be always performed with ease and certainty, let the dividend be ever so great, by dividing by an unit with as many ciphers annexed thereto as there are nines, and again dividing the quotient resulting by the same, and again, if necessary, until 0 be obtained for the integral quotient, preserving the several quotients and remainders; then adding each separately, and dividing the sum of the remainders, by the true divisor, and adding the quotient and remainder of this last division to the sum of the other quotients, the true quotient and remainder will be obtaincol.f
"It will not be advantageous in general. to employ this method, if the given divisor exceeds or falls short of that substituted by a greater excess or deficiency than. 12, or if the given dividend is so great as to give a quotient the digits of which exceed those of the divisorSee this subject treated at large in John Walker's Philosophy of . Arithmetic—pages 24 and 25. o
+ Whenever the divisor is not more than 12 less than a number consisting of an unit with ciphers annexed, division may be performed with nearly the same facility; but here it will be necessary to multiply the several quotients by the deficiency, and add the several quotients and remainders.
If the divisor be a series of threes by multiplying the dividend by 3 and employing the above method, and dividing the remainder by 3. we obtain the same quotient and remainder as would result by dividing by the given divisor—lf the divisor is a series of ones: multiply the of by 9, and divide as above directed, also divide the remainder thus found, by nine, for the true remainder. Es
215. 68447104 -i- 1111 216. 10546341 -- 1007 217. 8149136841 , . 9998 218. 8471084 ... 704
219, 674.14861 ... 9988 220. 96428494.13 . . 9999 221. 5.4341684 ... 99999 222. 714684217 . . .3333
Division affords several contractions in Multiplication, the most useful of which are as follow:—
When the multiplier is 5, annex a cipher and divide by 2, to which add a cipher for 50, two for 500, &c. When 25, annex two ciphers and divide by 4. When 125, annex three ciphers, and divide by 8. When 15, annex a cipher, divide by 2, and add the two results together. When 75, annex two ciphers, divide by 4 and subtract the result. When 35, add two ciphers, divide by 4, and add the quotient to the multiplicand, as if but one cipher was annexed. When 175, add two ciphers, divide by 2, also divide the quo- tient by 2, and add the three sums together. When 275, add 3 ciphers, and divide by 4, put down the quotient with a cipher less, and add the two sums together. The third of the product of a series of nines will give that of a series of threes, which doubled, will give that of a series sixes, or by subtracting the third of the product of a series of mines from itself, will give the same. The ninth of a series of nines will give that of a series of ones, which subtracted therefrom, will give that of a series of eights.
Cramples. Multiply 74194 by 5 Multiply 87643 by 25 2) 741940 4) 8764300 370970 2191075 Multiply 74324 by 125 Multiply 63815 by 15 8)74324000 319075 92.90500: 95.7225 Multiply 373874 by 75 Multiply 648.125 by 35 4)37387400 - 4) 648.12500 9346850 w 16203125 28040550 2268.4375T
By Multiplication we may also contract several operations in division, the most useful of which, that I have seen or practised, are the following:— - To divide by 5, double the dividend and cut of the last figure or cipher, the half of which is the remainder. To divide by 15, 35, 45, or 55, double the dividend and divide the result in the first case by 30, in the second by 70, in the third by 90 and in the last by 110, and in each case take the half of the remainder for the true remainder. To divide by 25, multiply by 4, and cut off 2 figures, one-fourth of the number expressed by them, will be the true remainder.” To divide by 125, multiply by 8, cut off three figures, and take one-eighth of the figures cut off for the remainder. To divide by 75, multiply by 4, and divide by 300, and take oné-fourth of the remainder for the true one. There are some others which the student may easily discover, in
eo they are only the converse of those given for multiplication £, lVision,
Çramples. Divide 1473421 by 5 . Divide 341742 by 25 294684+ - 13669,9. = 17 remr.
• To the others I think it unnecessary to give examples. This method of dividing by 25, is particularly useful in calculations of Interest, as will be more particularly noticed in another part of this work.
ADDITION OF DIVERS DENOMINATIONS,
Pi, Ace the numbers to be added, orderly under each other, having due regard to their several denominations. Then add the numbers of the lowest denomination together and divide the sum by the number of parts of this denomination which compose one of the next higher, set down the remainder, under the column added, and add the quotient as so many ones to the next higher denomination; proceed in like manner through all the denominations to the highest, the sum of which set down as in simple addition.
When there are two places of figures in any denomination, first add the units of each together, then the tens, as in simple addition; divide the total sum as before directed: the sum of the highest is to be wholly set down, unless there is a higher than is mentioned in the given sums to be added, in which case it will be sometimes best, to divide its sum by the number of parts of this denomination, which make one of the higher, and place the quotient to the left of the other denominations. The method of proof is the same as in simple addition.
NotE-Compound quantities are those which are of the same kind, but of different denominations.—to This rule is self evident,
In Ireland, accounts are kept in pounds, shillings, pence, and farthings, or parts of a penny.
4 farthings. . . . . . make ...... One penny......d. 12 pence. . . . . . . . . . ---- - - - - - - - One shilling....s. 20 shillings. . . . . . ...... . . . . . . . One pound......& One farthing............ is marked ..................
-oOne half-penny ............ ......................... t Three farthings ....................................... ; - The pound and shilling in this table are only imaginary, and until of late there was not a penny, as a current coin in Ireland; the farthing, though sometimes seen, is almost wholl in disuse, and bits of lead and other tokens substituted in the few instances where a farthing is required,
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