CHAPTER V. DIVISION. SECTION I. Simple Quantities. 1. If you divide 15 cents equally among 3 boys, how many cents will each boy have? ANS. 5 cents. When any given quantity is to be separated into a certain number of equal parts, the value of one of those parts is determined by Division. Thus, if we were to count off 15 cents into three equal piles, we should find that each of those piles would contain 5 cents; that is, 3 is contained 5 times in 15. The quantity to be divided is called the Dividend; the quantity, denoting the number of equal parts into which the dividend is to be divided, is called the Divisor; and the value of one of those parts is called the Quotient. Thus, in the above example, 15 is the Dividend, 3 is the Divisor, 5 is the Quotient. As Division is the reverse of Multiplication, the divisor and quotient being multiplied together, will reproduce the dividend. Thus, in the question above, 3 x 5 = 15. Indeed, the dividend may be regarded as a product, of which one of the factors, the divisor, is known; and the whole object of Division is, to find the other factor, namely, the quotient. abc. a b c. 2. 6 a ÷ a = 6; for 6 × a = 6 a. 3. a b÷b = a; for a x b = a b. 4. abc abc; for b c xa 5. abcba c; for a cx b = 6. a b c c = a b; for a bx c = a b c. 7. 16 a4 4 a; for 4 a X 4 = 16 a. 8. 20 a b5 a 4 b; for 4 b × 5 a = 20 a b. 9. 32 abc8ab4c; for 4 c × 8 a b = 32 a b c. In each of these examples, our object has been, to find some quantity, which, being multiplied by the divisor, would produce the dividend. The mode in which this may be done, is evident, namely: Divide the coefficient of the dividend by that of the divisor ; and omit in the quotient all those letters which are common to the given terms. 10. Divide 18 a bmx by 9 a b x. 11. Divide 27 a b c my by 3 b 12. Divide 9 h m n p by 3 h m. 13. Divide a bx by a b x. 14. Divide 39 a z by 13 z. C y. 15. Divide 108 a b c h m n x by 12 a c m x. E 16. Divide 24 q x y z by 12 x y z. 17. Divide 36 a x y by 9 a x y. 20. Divide 56 a x y z by 7 x y z. 21. Divide a by b. ANS. Here, the given quantities being dissimilar, the division cannot be performed, but only represented; we therefore write the divisor under the dividend, in the form of a fraction. 22. Divide x by y. 23. Divide a b by c x. 24. Divide 4 a b c by 5 x y. 25. Divide 17 a b x by 21 c d y. 26. Divide a b by a x. ab This may be written and then reduced, as in ах, the answer, the two a's being cancelled. Let a = 4, reduce the result by dividing both terms by 7. 34. Divide 16 a b by 4 x. 35. Divide 20 a b by 4 a x. 36. Divide 15 Ꮖ y by 5 axy. 37. Divide 25 a b c m by 10 a b x. 38. Divide 18 abfm by 6 a bfmnx. 42. Divide a bf g h by a fhx. 43. Divide 49 a b c x y by 7 a b c d mx. 44. Divide a b m x by 4 a b m y. 45. Divide 12 (ab+9) by 8 (a−b + 9). 46. What is the quotient of a b (19+ xy) divided by b (19+ x − y)? -- 47. Divide 12 a b (x − y + z) by 3 a (x − y + z). 48. Divide 4 (a b-10+x) by 7 (a b'— 10 + x). 49. Divide 3 (a y 12 b) by a (a y — 12 b). SECTION II. Signs in Division. The process of dividing one algebraic quantity by another, consists of two parts: the first is, to ascertain the proper expression of the quotient, in letters and figures; the other is, to determine the character of that expression, either as positive or negative. In the last section, the mode of dividing one simple quantity by another, was considered alone, without any regard to the signs. The mode of determining these, will form the subject of the present section. 1. Divide a b by b. ANS. a. In this example, both of the given quantities are positive. And the divisor, b, being +, the quotient, a, must be also; for, multiplied together, they must give the product + ab, that is, the dividend. But if we suppose the quotient to be have ab, which will give a b. + divided by + gives + in the quotient. a, we shall Hence, ANS. α. Here, the divisor, b, being, the quotient, a, must be also; for a x b will give a b, and not +a b, which is the dividend. Hence, + divided by a b by b. 7 amp. 16 a y. ANS. a. In this example, as the dividend, a b, is a-quan tity, and the divisor, b, is +, the quotient must be —; for ab gives —a b, the dividend. -divided by +, gives — in the quotient. 12. Divide 16 a x by 4 a x. 13. Divide. 21 a b m n by 7 am n. 15. Divide 32 a b c my by 16 a b m y. |