so that 5, and 28 are the new equivalent fractions with the common denominator 70; and the steps taken are comprised in the operations here subjoined: first, 1 x 5 x 7 = 35) 2 × 2 × 7=28 the new numerators: 3 x 2 x 5 =30] and 2 × 5 × 7 = 70, the common denominator: wherefore the equivalent fractions are 5, 48 and 8, as above: and hence we derive the following Rule. RULE. Multiply each numerator by all the denominators except the one placed under it, for the new numerator: and multiply all the denominators together for the common denominator. 85. If two or more of the denominators have a common measure, the equivalent fractions may be expressed in simpler terms than obtainable by the Rule, and still having a common denominator: thus, if the fractions be, and, we have from Article (54), the least common multiple of the denominators: then, with the least common denominator 12; and the new numerators are here obtained by multiplying those of the fractions proposed by the quotients arising from its division by their respective denominators. Fractions may be compared by means of this rule : and we see that mixed quantities, compound and complex fractions must be reduced to simple fractions, before it can be applied. Examples for Practice. (1) Bring and ; and ; and, respectively, to common denominators. Answers: 10 12 9 14 33 36 15' 15 ; (2) Reduce to, or express with, a common denominator,,and ; also,, and . (3) Express with a common denominator, the fractions,,,, and . (4) Change the forms of 11, 2 and 3 into those of improper fractions with a common denominator. (5) Reduce, 25 and 3 to fractions having a common denominator. (6) Put 7, §, 1010 and 264 into fractions of the same denomination. Answer: 4312 385 6720 16104 616' 616' 616 and 616 (7) Transform, and into equivalent fraetions, with the least common denominator. (8) What are the fractions having the least common denominator, which are of equal values with 14, and ? (9) Reduce 3, 3, and to the least common denominator. 9 (10) Exhibit, 4, and 25, as fractions having the least common denominator. (11) Reduce, and, so as to have the least common denominator. Answer: 140 105 80 1680 1680' 1680 and 28 1680* (12) Represent 1, 3, 4, and 6 in the fewest figures, so as to have a common denominator. (13) Express with the least common denominator, ,,, and . 3 (15) Find the greatest and least of the fractions, 2 (17) Compare the quantities 24, of 98 and 71 28 I. ADDITION OF FRACTIONS. 86. RULE. Express the fractions with a common denominator; add together the new numerators, and under their sum place the common denominator: and the resulting fraction, reduced if possible, will be the sum required. For, let 7 and 47 be the proposed quantities, which reduced to improper fractions are 51 and 2: then, since addition can be performed only upon quantities of the same denominations, these fractions must first be reduced to a common denominator; and their sum will be The process in this case may be simplified: for, the sum of the integers = 7 + 4 = 11 : and therefore the entire sum = = 11+1=12, as before; and this is much easier, particularly when the numbers are large: but attention to the following examples worked out, will suggest other abbreviations and forms of practical importance. whence, the required sum will be 11 21 23 22 105 23 150 + = + + + Examples for Practice. 15. (1) Find the sum of and ; of and §; of and, and of § and 14. (2) Add together 1 and 7; 24 and 13; 5 and 12, and 37 and 2413. Answers 8, 16, 1738 and 6215. 38 (3) What is the sum of 43 and 2; of 57 and is; of 2 and 49; of 17 and 11? 8 Answers: 39, 1314, 941, 421. (4) Add together, and ; }, % and 1; †, and;, and 1. |