111. Fractions may be changed to others of equal value, with a common denominator, by multiplying each numerator by every denominator except its own, for the new numerator; and all the denominators together for the common denominator. edb bdf' a C e Let b' d' ƒ be the proposed fractions; then adf cbf bdf' bdf' are fractions of the same value respectively with the for mer, having the common denominator bdf. For cbf с ; and bdf d bdf edb e adf a bd f = b ; (Art. 101); the numerator and deno minator of each fraction having been multiplied by the same quantity, viz. the product of the denominators of all the other fractions. 112. When the denominators of the proposed fractions are not prime to each other, find their Greatest Common Measure; multiply both the numerator and denominator of each fraction by the denominators of all the rest, divided respectively by their Greatest Common Measure; and the fractions will be reduced to a common denominator in lower terms than they would have been by proceeding according to the former rule. 113. To obtain them in the lowest terms each must be reduced to another of equal value, with the denominator which is the Least Common Multiple of all the denominators. [If the denominators be simple algebraical quantities, and in certain other cases also, their Least Common Multiple will be found at once by inspection; but if they be compound quantities, the following rule is usually needed.] 114. To find the Least Common Multiple of two quantities, or the least quantity which is divisible by each of them without remainder. Let a and b be the two quantities, a their greatest common measure, m their least common multiple, and let m contain a, p times, and b, q times, that is, let m = pa = qb; then and since m is the least possible, p and q are the least a possible; therefore is the fraction in its lowest terms, and a p consequently q = -; hence m = qb x b [The rule here proved may be thus enunciated: Find the G.C.M. of the two proposed quantities; divide one of them by this G.C.M.; and multiply the quotient thus obtained by the other quantity. The product is the Least Common Multiple required. Ex. Required the Least Common Multiple of aa — x1 and a3 — a2x - a x2 + x3. The G. C. M. of these two quantities (See Art. 105, Ex. 2), is a2 - x2; and (a1 — x1) ÷ (a2 − x2) = a2 + x2. Therefore the Least Common Multiple required 115. Every other common multiple of a and b is a multiple of m. Let n be any other common multiple of the two quantities; and, if possible, let m be contained r times in n, with a remainder s, which is less than m; then n - rm = 8; and since a and b measure n and rm, they measure n- rm, or s (Art. 104); that is, they have a common multiple less than m, which is contrary to the supposition. 116. To find the Least Common Multiple of three quantities a, b, c, find m the Least Common Multiple of a and b, and n the Least Common Multiple of m and c; then n is the Least Common Multiple sought. For every common multiple of a and b is a multiple of m (Art. 115); therefore every common multiple of a, b, and c is a multiple of m and c; also every multiple of m and c is a multiple of a, b, and c; consequently the Least Common Multiple of m and c is the Least Common Multiple of a, b, and c. [Ex. Required the Least Com. Mult. of x3-ax− a x2+a3, - a1, and ax3 + a3 x — a2x2 - a1. Here aa3x — a2x2 - a1 = a (x23 + a2 x − a x2 - a3); + - .. to find the G. c. M. of this quantity and the first, reject the . x. a is the G. c. M. of the first and last of the proposed quantities; and their least com. mult. is (a x23 + a3 x − a2x2 - a1) (x2 - a2)......(1). The other quantity is (¿x2 + a2) (x2 - a2).....................(2). The G. C. M. of (1) and (2) ax3 + a3 x − a2x2 − a* and x2+ a2. former quantity, -- is (x2 – a2)× the G. C. M. of Rejecting the factor a in the .. the G. c. M. of (1) and (2) is (x2 + a2) (x2 — a2) ; .. least com. mult. required is (ax3 + a3 x − a2x2 — a1) (x2 — a2),. 117. A more expeditious method of finding the Least Com. Mult. of any number of quantities in certain cases is that of resolving each quantity into its component factors, as follows:-taking the last Example, (1) x3- a2 xa x2 + a3 = x2 (x − a) — a2 (x − a) = (x2 — a2) (x − a). (2) x1 — a1 = (x2 + a2) (x2 — a2) = (x2 + a°)(x−a)(x+a). (3) 4 ax3 + a3 x − a2x2 — a2 = a x2 (x − a) + a3 (x − a) = a (x2 + a2) (x − a). Now the G. C. M. of (2) and (3) is (x2 + a2) (x − a); .. least com. mult. of (2) and (3) is a (x1 — a1).....(4). Again, the G. C. M. of (1) and (4) is x2. a2; .. least com. mult. required is a (x1 − a1) (x − a), ADDITION AND SUBTRACTION OF FRACTIONS. 118. If the fractions to be added have a common denominator, their sum is found by adding the numerators together for a new numerator and retaining the common denominator. a с Thus + = down in Art. 99. a + c This follows from the principle laid 119. If the fractions have not a common denominator they must be transformed to others of the same value, which have a common denominator (Art. 111...113), and then the addition may take place as before. Here a is considered as a fraction whose denominator is 1. |