It was shown in Arithmetic, Art. XXII, that a common denominator may frequently be found much smaller than that produced by the above rule. This is much more easily done in algebra than in arithmetic. cg Here the denominators will be alike, if each be multiplied by all the factors in the others not common to itself. If the first be multiplied by e g, the second by c2g, and the third by bce, each becomes b c e g. Then each numerator must be multiplied by the same quantity by which its denominator was multiplied, that the value of the fractions may not be altered. The fractions then become aeg cbe eg c2 dg ebcf beeg beeg The answer is aeg + c2 dg+bce f 10. Add together 11. Add together 12. Add together 13. Add together 14. Add together 15. Add together 16. Add together bceg and sb f 2dg e c 36 3 a and 2m n 3 mp 17. Add together 3 m n3r and 13 c d. 2 m3 n and 2 a c-5 b. was changed to +. See Art. VI, example 6th XIX. Division of whole numbers by Fractions, and Fractions by Fractions. How many times is contained in 7? Ans. is contained in 7, 35 times, and is contained as many times; that is, 35 or 11 times. 2. How many times is contained in a? Ans. is contained in a, 8 a times, and is contained as many times; that is, . 8 a 3. How many times is contained in c? Ans. 1/ is contained b c times in c, and is contained as many times; that is, bc α a ъс α Arith. Art. XXIV. Hence, to divide a whole number by a fraction, multiply it by the denominator of the fraction, and divide the product by the numera tor. How many times is contained in Solution. Reducing them to a common denominator, is 24, and is 2. 4 is contained in 3 as many times as 24 is contained in 35; that is, 3 or 14}. Ans. 11. Solution. If is the part of some number, b 1 part of bc ad ad Hence, to divide a fraction by a fraction, multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor. Or more generally, when the divisor is a fraction, multiply the dividend (whether whole number or fraction) by the divisor inverted. Arith. Arts. XXIII. and XXIV. |