CHAPTER VII. DIVISION. 149. Division is the operation by which, when a product and one of its factors are given, the other factor is found. With reference to this operation, the product is called the dividend; the given factor is called the divisor; and the required factor is called the quotient. 150. When the given factor is the multiplicand, the factor sought is the multiplier. In this case, the question is: What must we multiply the divisor by to get the dividend? To answer this question, it is necessary to find how many times the divisor is contained in the dividend, and the answer will be so many times. Thus, in the question, How many times are 6 cents contained in 30 cents? the factor sought is the multiplier, 5; 5 times 6 cents are 30 cents. When the given factor is the multiplier, the factor sought is the multiplicand. In this case, the question is: What must we multiply by the divisor to get the dividend? To answer this question, it is necessary to divide the dividend into as many equal parts as is indicated by the number in the divisor, and the answer will be so much to each part. Thus, in the question, How much will each boy receive if 30 cents be divided among 6 boys? the factor sought is the multiplicand, 5 cents; 6 times 5 cents are 30 cents. The arithmetical process and the numerical result are the same in both cases; but the name to be attached to the result depends upon the nature of the question. 151. Division is indicated by the sign÷, by the colon, :, or by writing the dividend over the divisor, and drawing a line between them; thus, each of the expressions, 15 15÷3, 15:3, means and is read, "Fifteen divided by three." SHORT DIVISION. 152. When the divisor does not exceed 12, the work may be written in the following manner: the divisor is placed to the left of the dividend, the quotient under the dividend, and each remainder is added as so many tens to the next figure of the dividend not divided. (1) Divide 2736 by 4. 4)2736 Wording. 4 in 27, 6, carry 3; in 33, 8, carry 1; 684 in 16, 4. (2) Divide 2736 by 9. 9)2736 304 Wording. 9 in 27, 3; in 3, 0, carry 3; in 36, 4. Answer, 304. (3) Divide 3696 by 12. 12)3696 308 Wording. 12 in 36, 3; in 9, 0, carry 9; in 96, 8. The pupil will observe that, the divisor being a whole number, each quotient figure is of the same order of units as the right-hand figure of the partial dividend used in obtaining it. 153. Divide 963 by 3; 846 by 2; 846 by 3; 846 by 6; 848 by 4; 52.05 by 5; 84.028 by 7; 13.31 by 11; 1.728 by 12. When the divisor is a whole number, the pupil will be careful to put the decimal point in the quotient as soon as the decimal point in the dividend is reached. 154. If the divisor is not contained in the dividend without a remainder, ciphers may be mentally annexed to the dividend, and the division continued until the required number of decimal places is obtained. If the remainder, when the last quotient figure is obtained, is greater than half the divisor, increase the quotient figure by 1. Divide 3.1 by 4, by 5, and by 7, carrying the division to the third place of decimals. 155. If the dividend and divisor are both multiplied, or both divided, by the same number, the quotient is not changed. Thus, 12÷4=3, and (when both dividend and divisor are multiplied by 2) 248-3. Again (when both dividend and divisor are divided by 2) 6÷2=3. Hence, if we have to divide 2.24 by 35, we may first divide by 7, and then by 5, as follows: 7)2.24 Again, 5).32 .064 Answer, .064. Dividing both dividend and divisor by the same number is called cancelling equal factors in dividend and divisor. Find the quotients, to five decimal places, of: 3÷7 411; 3.1714; 7.85÷21. 156. Since the quotient is not altered if the dividend and divisor are both multiplied, or both divided, by the same number, it follows that, If the divisor contains decimal places, we may remove the decimal point from the divisor, provided we carry the decimal point in the dividend as many places to the right as there are decimal places in the divisor. If the divisor is a whole number and ends in zeros, the zeros may be cut off, provided the decimal point in the dividend is carried to the left as many places as there are zeros in the divisor. Divide 78.52 by .008, by .8, and by 8000, first making the required changes of the decimal point. 8)78520 8)785.2 8).07852 157. From the last three examples it is seen that, If the first figure of the quotient is placed under the righthand figure of the first partial dividend, the decimal point in the quotient will be written directly under the decimal point in the dividend. If the divisor is not contained in the dividend without a remainder, carry the quotient to the fifth decimal place. LONG DIVISION. 159. The process of Long Division is precisely like that of Short Division, except that the work is written in full, and the quotient is written over the dividend. Divide 3.1415927 by 1.73. Suppress the decimal point in the divisor, and move the point two places to the right in the dividend. 1.81595 173)314.15927 173 1411 1384 275 173 1029 865 1642 1557 857 692 165 The one 173 is contained in 314 once. is written over the 4, and, by 156, the point is placed over the decimal point in the dividend. The process, which has been gone through with, mentally, in short division, is written out below the dividend. 173 is subtracted from 314; to the remainder, the 1 in the dividend is annexed; then 8 x 173 is subtracted; then to the remainder, 5 in the dividend is annexed, and so on. The fifth place in the quotient is 4, but written 5, because the remainder, 165, is more than half 173. Divide 23.13685 by 7.843. Suppress the decimal point in the divisor, and move the point three places to the right in the dividend. |