as much as + must give a product a times as large, or a In fact + multiplied by b is # = 1, and + being a times times 1, which is a. Hence, if a fraction be multiplied by its denominator, the producivili: o: , the p 45. Multiply #. by 5 b d. 46. Multiply # by 3b c. Multipl 18w 2 47. Multiply 4 b m” by 4b m”. sm, 48. Multiply 12 moyo by b d n” ar. - - 13ab — in * -> - b 17 a”. 49. Multiply 17 a” y 0. Two ways have been shown to multiply fractions, and two ways to divide them. To multiply a fraction, } multiply { the numerator To divide a fraction, the denominator. Arith. Art. XVIII. Reducing Fractions to Lower Terms. XVII. If both numerator and denominator he multiplied by the same number, the value of the fraction will not be altered. o 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 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. 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. 1. Reduce 2. Reduce 3. Reduce 4. * 5. Reduce 6. Reduce 7. Reduce Write the divisor under the dividend in the form of a fraction, and reduce it to its lowest terms. 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. 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 #4 and 44. They are now both of the same denomination, and their numerators may be added. The answer is #}. 6. Add together £ and 5. b d Multiply both terms of the first by d, and of the second by # and # The denominators are now alike b, they become 7. Add together +, o, ., and #. ogether +, +, f and # 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. |