Multiplying both dividend and divisor by the same number does not alter the quotient. 6th. If a given divisor is contained in a given dividend a certain number of times, one half the divisor will be contained the same number of times in one half the dividend; one third of the divisor will be contained the same number of times in one third of the dividend; and so on. Hence, Dividing both dividend and divisor by the same number does not alter the quotient. NOTE.-If a number be multiplied and the product divided by the same number, the quotient will be equal to the number multiplied; hence the 5th case may be regarded as a direct consequence of the 1st and 3d; and the 6th, as the direct consequence of the 2d and 4th. To illustrate these cases, take 24 for a dividend and 6 for a divisor; then the quotient will be 4, and the several changes may be represented in their order as follows: 117. These six cases constitute three general principles, which may now be stated as follows: PRIN. I. Multiplying the dividend multiplies the quotient; and dividing the dividend divides the quotient. PRIN. II. Multiplying the divisor divides the quotient; and dividing the divisor multiplies the quotient. PRIN. III. Multiplying or dividing both dividend and divisor by the same number, does not alter the quotient. 118. These three principles may be embraced in one GENERAL LAW. A change in the dividend produces a LIKE change in the quotient; but a change in the divisor produces an OPPOSITE change in the quotient. SUCCESSIVE DIVISION. 119. Successive Division is the process of dividing one number by another, and the resulting quotient by a second divisor, and so on. Successive division is the reverse of continued multiplication. Hence, I. If a given number be divided by several numbers in successive division, the result will be the same as if the given number were divided by the product of the several divisors, (95, I). II. The result of successive division is the same, in whatever order the divisors are taken, (95, II). CONTRACTIONS IN DIVISION. CASE I. 120. When the divisor is a composite number. 1. Divide 1242 by 54. OPERATION. 6) 1242 ANALYSIS. The component factors of 54 are 6 and 9. We divide 1242 by 6, and the resulting quotient by 9, and obtain for the final result, 23, which must be the same as the quotient of 1242 divided by 6 times 9, or 54, (119, I). We might have obtained the same result by dividing first by 9, and then by 6, (119, II). Hence the following 23 Ans. RULE. Divide the dividend by one of the factors, and the диа tient thus obtained by another, and so on if there be more than two factors, until every factor has been made a divisor. tient will be the quotient required. The last quo TO FIND THE TRUE REMAINDER. 121. If remainders occur in successive division, it is evident that the true remainder must be the least number, which, subtracted from the given dividend, will render all the divisions exact 1. Divide 5855 by 168, using the factors 3, 7, and 8, and find the true remainder. Dividing 1951 by 7, we have 278 for a quotient, and a remainder of 5. IIence, 5 subtracted from 1951 would render the second division exact. But to diminish 1951 by 5 would require us to diminish 1951 × 3, the dividend of the first exact division, by 5 × 3 15, (93, III); and we therefore write 15 for the second part of the true remainder. Dividing 278 by 8, we have 34 for a quotient, and a remainder of 6. Hence, 6 subtracted from 278 would render the third division exact. But to diminish 278 by 6 would require us to diminish 278 × 7, the dividend of the second exact division, by 6 x 7; or 278 × 7 × 3, the dividend of the first exact division, by 6 × 7× 3 = = 126; and we therefore write 126 for the third part of the true remainder. Adding the three parts, we have 143 for the entire remainder. Hence the following RULE. I. Multiply each partial remainder by all the preceding divisors. II. Add the several products; the sum will be the true re mainder. 14. Divide 386639 by 720 2 x 3 x 4 x 5 x 6. 719. 18. Divide 116423 by 10584 = 3 × 72 × 8 × 9. 10583. 19. Divide 79500 by 6125 = 53 × 72. 6000. CASE II. 122. When the divisor is a unit of any order. If we cut off or remove the right hand figure of a number, each of the other figures is removed one place toward the right, and, consequently, the value of each is diminished tenfold, or divided by 10, (57, III). For a similar reason, by cutting off two figures we divide by 100; by cutting off three, we divide by 1000, and so on; and the figures cut off will constitute the remainder. Hence the RULE. From the right hand of the dividend cut off as many figures as there are ciphers in the divisor. Under the figures so cut off, place the divisor, and the whole will form the quotient. 123. When there are ciphers on the right hand of second remainder 3. Multiplying the last remainder, 3, by the preceding divisor, 100, and adding the preceding remainder, we have 300+ 48348, the true remainder, (121). In practice, the true remainder may be obtained by prefixing the second remainder to the first. Hence the RULE. I. Cut off the ciphers from the right of the divisor, and as many figures from the right of the dividend. II. Divide the remaining figures of the dividend by the remaining figures of the divisor, for the final quotient. III. Prefix the remainder to the figures cut off, and the result will be the true remainder. |