CONTRACTION S. I. When either the multiplicand, or multiplier, or both, have cyphers towards the right hand, reject the cyphers, multiply the fignificant figures as before, and to the product, annex as many cyphers as are in both factors. Ex 12. 97638000 x 467000=455969450000000 II. When the multiplier is an unit with cyphers to the right, as 10, 100, 1000, &c. the product will be equal to the multiplicand with so many cyphers annext as are in the multiplier. Ex. 13. 3674x 10 10000 | = | 36740 36740000 III. When the multiplier is a compofite number, multiply continually by the factors; i. e. multiply by one factor, and that product by the other &c. and the last product will give the answer. Ex. 15th. Multiply 3604670 by 35,5×7, 3604670×725232690, x5=126163450. Answer. Ex. 16th. Multiply 3704604040 by 56,8×7, 3704604040829636832320, X7=207457826240. § V. Simple DIVISION. IVISION is a compendious Subtraction; or of another, called the Dividend, as often as possible, in order to find a third. number,, called the Quote or Quo tient; because it fhews (quoties) how often the Divifor is contained in the Dividend, thus A General RULE. . Set down the Divifor and Dividend in the form above, confider if the Divifor be lefs or equal to the fame number of the left hand figures of the Dividend; if it is less or equal, fet the figure in the quote, expreffing the number of times it is contained in that part of the Dividend : but ift, take one place more of the dividend figures than are in the Divifor, and fet in the quote the number of times the Divifor is contained in this part of the Dividend, as before. 2. Multiply the Divifor by the quotient figure. 3. Subtract the product from the faid Dividend figures ;. and fet down the remainder. 4. Make a prick under the next figure of the Dividend, in order to mark it, and bring it down to the right of the remainder, then this number call the Dividual. 5. Seek how oft the Divifor can be had in the dividual, and fet the figure expreffing the number of times in the quote, multiply the Divifor thereby, fubtract the product. as before, and in this manner the operation must be continued till all the figures in the dividend are brought down one by one. And note, for every figure brought down,. a figure must be placed in the quote, except when the Dividual is less than the Divisor, and then write a cypher in the quote. Note. If the product of any quotient figure and divifor exceeds the dividual, the quotient figure belonging to fuch product must be leffened till the product is equal to, or lefs, than its dividual: again, if after fubtracting the product from its dividual, the remainder is equal to, or exceeds the Divifor; the quotient figure must be increasedi till the remainder be less than the Divifor.. zd. If there be a Remainder after divifion is finished, annex it to the quote with the divifor under it, with a small. line drawn between then the quote will be a mixt: number. To prove Divifion, multiply the Divifor and quote toge ther, adding the remainder, if any; and the Product will He equal to the Dividend if the work is right.. EXAMPLES. Divifor Dividend Quote Ift, 3) 8 1 2 4 (2 7 0 8. 6 21 21 · 24 24 EXPLANATION. First ask how oft 3 in 8, which is 2 times, then place 2 in the Quotient, and multiply 3, (the divifor) by 2, and the product is 6; which fubducted from 8 leaves 2; then prick under and bring it down to the right of 2. which is then 21 for a dividual; then ask how oft 3 in 21,, the answer is 7, which place in the quotient; then 7 times 3 is 21, which fubducted from 21 the dividual, leaves o. Then prick and bring down the next figure 2, and ask how oft 3 in two, the anfwer is o, which place in the quotient; then prick and bring down 4 to the right of 2, and the dividual is 24, then ask how oft 3 in 24, the answer is 8, which place in the quotient; then 8 times 3 is 24; which fubducted from 24, (the dividual) leayes nothing. Then 2708 is the quote. Proved thus: 2708 x3=8124 the dividend. Proof 3150×6023, +134,=18972584. De Demonftration of the RULE. In Ex. 1. Since 3 is contained twice in 8 therefore it is contained 2000 times in 8124, i. e. 2 must be in the 4th place. And fince 3 is contained 7 times in 21, it is contained 700 times in the whole remainder 2124, and therefore 7 muft occupy the third place in the quote; and as 3 cannot be had once in 2, it cannot be had 10 times in the whole remainder 24, therefore a cypher must occupy the tens place of the quote; but fince 3 is contained 8 times in 24, and 4 poffeffes the units place of the Dividend, 8 muft confequently poffefs the units place of the quote. Therefore the Divifor is contained in the whole dividend 2708 times. And in Ex. 2. fince 6023 is contained 3 times in 18972 it is contained 3.000 times in 18972584; and roo times in the remainder 903584; and 50 times in the next remainder 301284; and o times in the last remainder 134. Or the Divifor is contained in the whole Dividend 3150 times. CONTRACTIONS. 1. When your Divifor is 12 or less than 12, divifion may be expeditiously performed by multiplying and fubtracting mentally, and writing down only the quote below the Dividend. II. If the Divifor hath cyphers to the right of it, cut them off, then cut off fo many of the right hand places of the Dividend as there are cyphers in the Divifor, which annex to the remainder, when the operation is finifhed. III. To divide by an unit with cyphers as 10, 100, 1000, &c, cut off from the Dividend fo many places as the Divifor has cyphers, and the figures fo cut off on the left is the quote, and thofe on the right the remainder. Ex. Divide 7217367 by 100. The Quote is 72173 and 67 remainder. IV. When the Divifor is a compofite number, i. e. when it is the product of two or more small numbers, it is much easier to divide continually by thofe numbers, than by the whole divifor at once, i. e. divide the dividend by one of those numbers, and that quotient by the other, and fo proceed. Note. If there be any remainders after fuch divifions, mul tiply the last remainder by the preceding Divifor, and to the product add the remainder belonging to the fame Divifor; then multiply the fum by the next preceding Divifor, and to the product add its correfponding remainder and fo proceed through all the Divifors and Remainders, and the laft fum will be the true remainder as if the Divifion had been performed at once. |