2. Divide 5371 by 192. 4)5371 8)223-4 We find 192 equal to the product of 4 times 6 times 8, = 4 x 6 x 8 192. We therefore divide by these factors, as in the last example. To find the true remainder, we multiply the last remainder, 7, by the 27-7187 last divisor but one, 6; and to the product add the last remainder but one, 4; this sum we multiply by the first divisor, 4; and to the product add the first remainder, 3; and find the amount to be 187. RULE. I. To multiply by 25. -Annex two ciphers to the multiplicand, and divide it by 4, and the quotient is the product required. Rationale. By annexing two ciphers, we increase the multiplicand one hundred times, and by dividing this number by 4, the result will be an increase of the multiplicand only twentyfive times, because 25 is one fourth of 100. RULE. - Annex two ciphers to the multiplicand, and divide it by 3, and the quotient is the product required. Rationale. As in the last case, by annexing two ciphers, we increase the multiplicand one hundred times; and by dividing the number by 3, we only increase the multiplicand thirty-three and one third times, because 33 is one third of 100. RULE.-Annex three ciphers to the multiplicand, and divide by 8, and the quotient is the product. NOTE. — By annexing three ciphers, the number is increased one thousand times; and, by dividing by 8, the quotient will be only one eighth of 1000, that is, 125 times. 7. Multiply 12345678 by 125. RULE. OPERATION. 8) 12345678000 1543209750 Product. IV. To multiply by any number of 9's. ·Annex as many ciphers to the multiplicand as there are 9's in the multiplier, and from this number subtract the number to be multiplied, and the remainder is the product required. 8. Multiply 87654 by 999. OPERATION. 87654000 87654 By annexing three ciphers, we make the number one thousand times larger. If from this number, with the ciphers annexed, we subtract the multiplicand, we make the prod 87566346 Product. uct one thousandth part less; that is, the product will be only 999 times the multiplicand. Q. E. D. 9. Multiply 7777777 by 9999. 10. Multiply 5555 by 999999. Ans. 77769992223. NOTE. To multiply by any number of 3's, proceed as above and divide the product by 3; but if it be required to multiply by 6's, proceed as above and then multiply the product by 2, and divide the result by 3, and the quotient is the product. 3)175306846914 58435615638 Product, Ans. 14. Multiply 345678 by 6666666. Ans. V. When the multiplier can be separated into periods, which are multiples of one another, the operation may be contracted in the following manner. 15. Multiply 112345678 by 288144486. OPERATION. 112345678 288144486 674074068 the product by 6. 5392592544 = the foregoing product x by 8 for 48. 16177777632 32355555264 the last product x by 3 for 144. the last product x by 2 for 288. 32371787641631508 Product. SECTION VII. CONTRACTIONS IN DIVISION. I. To divide by 5. RULE.-Multiply the dividend by 2, and the product, except the last figure at the right, is the quotient. NOTE. The remainder will be tenths. 1. Divide 67895 by 5. Ans. 13579. RULE. OPERATION. 67895 13579,0 Quotient. II. To divide by 25. Multiply the dividend by 4, and the product, except the last two figures at the right, is the quotient. NOTE. The two figures at the right are hundredths. 2. Divide 8765887 by 25. RULE. OPERATION. 8765887 4 350635,48 Quotient. III. To divide by 33}. Ans. 3506354. Multiply the dividend by 3, and the product, except the last two figures at the right, is the quotient, and the last two are hundredths. 3. Divide 876735 by 331. OPERATION. Ans. 2630280. 876735 RULE. · Multiply the dividend by 8, and the product, except the last three figures, is the quotient, and these last three figures will be thousandths. OPERATION. This method differs from the 24)16294896(678954, Ans. common way by placing the right RULE. Let the dividend be divided into periods, consisting each of as many fig ures as there are 9's in the divisor, beginning the division into periods at the left hand. Then write the left-hand period under the second period, the second period under the third, and so on, omitting all the figures which would thus fall to the right of the original dividend, and let this be called a second dividend. Again, write the first period of the second dividend under the second period of the same, its second period under its third, und so continue writing successive dividends, omitting the figures at the right as above, till the first period of the first dividend falls under the last period of the same. Add together the several dividends thus formed, cutting off as many figures in the sum, as there are 9's in the divisor. Now observe how many units in adding were carried from the right to the left hand figure in the part thus cut off, and add this number of units to the whole. The figures on the left of the separatrix will be the quotient, and those on the right the true remainder. 1. What number multiplied by 1728 will produce 1705536? Ans. 987. |