39. Multiply 7 by ab. 46. 3 ac 40. Multiply by 4 ab? — 460 17 -4bc 41. Multiply by 40%. 16 a 12 a3b4a 23 m - 13 42. Multiply 35 m* c d - 7 mo c + 42 m ac by 43. Multiply by 5. Dividing the denominator by 5 it becomes , or 3. multiplied by b is as 1 In fact = 1, and is being a times much as 1 must give a product a times as large, or a 6 times 1, which is a. Hence, if a fraction be multiplied by its denominator, the ques duct will be the numerator. in 13 ab 49. Multiply by 17 a'. 17 | 50. Multiply 15 a c +37bc by 10 ab- 2 c. 10 a b2c 47 a m' + 3b-C 51. Multiply by a x* — 3 am amt a 2 — 3 a' m +1 Two ways have been shown to multiply fractions, and two ways to divide them. To multiply a fraction, S the numerator the denominator To divide a fraction, divide S the numerator. To multiply a fraction, the denominator. Arith. Art. XVIII. Reducing Fractions to Lower Terms. a XVII. If both numerator and denominator be multiplied by the same number, the value of the fraction will not be altered. Arith. Art. XIX. For multiplying the numerator multiplies the fraction, and multiplying the denominator divides it; hence it will be multiplied and the product divided by the multiplier, which reproduces the multiplicand. In other words, signifies that a contains b a certain num 7 ber of times, if a is as large or larger than b; or a part of ontime, if b is larger than a. Now it is evident that 2 a will contain 2 b just as often, since both numbers are twice as large as before. So dividing both numerator and denominator, both divides and multiplies by the same number. 2 X 3 6 7 x 3 21 3 x 6 36 56 2 acd 6 26 2bcd 6 a b 36 x 2 a 36 x 30 3 c 5 x 6 ac 56 b c 2 a 9 b c Hence, if a fraction contain the same factor both in the numerator and denominator, it may be rejected in both, that is, both may be divided by it. This is called reducing fractions to lower terms. 15 a' c* — 25 a? 6. Reduce to its lowest terms. - 54 cca 7. Reduce to its lowest terms. 108 a + 81 x 90 m2 2003 8. Divide 35 a b m by 7an mo r. Write the divisor under the dividend in the form of a fraction, and reduce it to its lowest terms. 56 m xc Ans. an 15. Divide 17 a cx by 13 a c x. 16. Divide 28 a*cy by 14 ay%. 17. Divide 36 of my by 54 a'my. 18. Divide 75 a' by by 35 a* c* yo x. 19. Divide a +6 by 2c-d. 20. Divide 2 a c-7abc+ 15 a* cd by 13 a'cd. 21. Divide 18 am - 54 am + 42 am * by 30 am - 12 a cm. 22. Divide (a + b) (13 ac + bc) by (m -c) (a + b). 23. Divide 3 c* (a — 2c)" by 2bco (a — 2 c)". 24. Divide 36 b* c* (2a + d)' (76---d) by 126* (2 a + d)'(76—d)' (a -d). 5 Addition and Subtraction of Fractions. XVIII. Add together and į and This addition may be expressed by writing the fractions one after the other with the sign of addition between them; thus, +++ N. B. When fractions are connected by the signs + and - the sign should stand directly in a line with the line of the fraction. It is frequently necessary to add the numerators together, in which case, the fractions, if they are not of the same denomination, must first be reduced to a common denominator, as in Arithmetic, Art. XIX. 1. Add together and 1 2 2 을 Ans. 3+2 7 2. Add together 7 and Ans. " 3 3 3a 5 a c Ans. 2a + 5ab 3. Add together 3a+2a cd cd ab 3 cd 5. Add together and 4. These must be reduced to a common denominator. It has been shown above that if both numerator and denominator be multiplied by the same number, the value of the fraction will not be altered. If both the numerator and denominator of the first fraction be multiplied by 7, and those of the second by 5, the fractions become and They are now both of the same denomination, and their numerators may be added. The answer is 31 6. Add together and į Multiply both terms of the first by d, and of the second by b, they become b c and The denominators are now alike ፩ bd bd and the numerators may be added. The answer is ad + b c b d ad с e 7. Add together and g & 용 In all cases the denominators will be alike if both terms of each fraction be multiplied by the denominators of all the others. For then they will all consist of the same factors. Applying this rule to the above example, the fractions bead fh bcfh bdeh and bd fhbd fh' bdfh' The answer is a dfh +bcfh+bdeh +bdfg . bdfh 8. Add together and Ans. 15 ad + 4b c 2 b c 5ď 10bcd come bdfg bdfh 3 a 2 c |