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 3 Multiply both terms of the first by d, and of the second by a d b c b, they become and The denominators are now alike b d b d and the numerators may be added. 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 befh bdeh bdfh bdfh bdfh come and bdfg bd fh The answer is a dfh+bcfh+bdeh+hdfg bdfh 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. 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, cd g eb cf beeg beeg and 9 2 ac be and 3b f 2dg 5 am en and 2 r h 3 a 36 12. Add together and cbeg sign -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 contained in c? a 3. How many times is b Ans. is contained b c times in c, and is contained a Ъ Arith. Art. XXIII. Ans. If c is the part of some number, b part of the same number, and is will be α a 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 7 Solution. Reducing them to a common denominator, is 13, and is . is contained in 33 as many times as 24 is contained in 35; that is, or 11. Ans. 14. |
