$189. If the letters of the divisor are not found in the dividend, the divi-ion is expressed, as we have before shown, $177, by writing the divisor underneath the dividend, in the form of a vulgar fraction. EXAMPLES. 54. Divide 4y+7x, by a—b. Ans. 4y+7x a-b 1 §190. When the dividend is a compound quantity, the divisor may be placed underneath the whole dividend if we choose. It may also be placed under each term of the dividend, which is the same as dividing each term, according to §184. By this method the answer of the last sum would 3ac2 263 C be + + Answer the following by both me2c 2c 2c a 59. Divide 6a+ab—3b, by 2b. 60. Divide 2x+2y+3ax—2a3y, by 3ay. 61. Divide ax-bx-a2b2+ab, by 2b. §191. When we divide each term separately, we may use both methods of division; that is, we may actually divide such terms as we can by §180; and merely express the division in such terms as cannot be divided. 2ab 63. Divide ax+bx-2ab+2x, by x. Ans. a+b— 64. Divide 2am—3a2b+b3m-3a3m, by -ɑ. Ans. -2m+3ab b3m +3am. α 65. Divide 26-a®b+3b3c+a3b3-ac, by bo. In all of our divisions so far, it has been the case that either the divisor or the quotient is a simple quantity. For the method of dividing when both of them are compound quantities, see §213, &c. EXERCISES IN EQUATIONS. §192. It may be well now to generalize the questions in section 5 of Equations. And with this section, we will make our calculations more purely algebraical than in the preceding sections; as we shall take care to use no numeral quantities at all in stating the questions. 1. Two persons, A and B, lay out equal sums of money in trade. A gains c ($126,) and B loses d ($87 ;) and now A's money is m (two) times as much as B's. What did each lay out? See page 45. In sums of this kind, the only difficulty is to determine what quantity to divide by in the last step, to leave x alone. But this difficulty is easily overcome, by dividing mentally the left hand member by x, and observing the quotient. Of course, dividing the same member by that quotient will produce x; which is our only object. 2. The 2d sum on page 45, is performed as follows: §193. It is customary to represent those numbers which stand for times, by the letters m, n, p, q, &c. All the other questions in section 5th are to be performed in the foregoing manner. FRACTIONS. Reduction of Fractions to Lower Terms. §194. We showed in §84, that a fraction may be reduced to lower terms, without any alteration in its value, by simply dividing both terms by a number that will divide each without a remainder. Fractions that are expressed by literal quantities may frequently be reduced in the same axy manner. Thus, in the fraction both the terms may be 9 ab divided by a; and the fraction will then become These examples will remind the pupil, that, (because fractions are merely expressions of division,) when each term has its sign, then the whole fraction will have a sign according to §185 and 186. §195. When we divide a compound quantity by a simple quantity we divide each term, §184. Hence, in reducing fractions to lower terms, we must find for the divisor, a quantity that is a factor in every term both of the numerator and denominator. §196. By this principle, we may often simplify answers to questions in division. That is, we may put the divisor |