116. Fractions may always be reduced to a common denominator by the preceding rule; but if the denominators have any common factors, it will not be the least common denominator. The least common denominator of two or more fractions must be the least common multiple of their denominators. 5b Suppose it is required to reduce the fractions and to 2a 3x2 4x equivalent fractions having the least common denominator. The least common multiple of the denominators is 12x2. Mul 12x2 tiply both terms of the first fraction by or 4, and both 12x2 terms of the second fraction by or 3x, and we shall have > 4x which are equivalent to the given fractions, and have the least. common denominator. Hence we deduce the following RULE. Find the least common multiple of all the denominators, and use this as the common denominator. Divide this common denominator by each of the given denominators separately, and multiply each numerator by the corresponding quotient. The products will be the new numerators. 117. The denominator of a fraction shows into how many parts a unit is to be divided, and the numerator shows how many of those parts are to be taken. Fractions can only be added when they are like parts of unity; that is, when they have a common denominator. In that case, the numerator of each fraction will indicate how many times the common fractional unit is repeated in that fraction, and the sum of the numerators will indicate how many times this result is repeated in the sum of the fractions. Hence we have the following RULE. Reduce the fractions to a common denominator; then add the numerators together, and write their sum over the common denomi nator. If there are mixed quantities, we may add the entire and fractional parts separately. EXAMPLES. 1. What is the sum of and ? Reducing to a common denominator, the fractions become 2 It is plain that three sixths of x and two sixths of x make five sixths of x. ma-b 9. What is the sum of and na+b2 10. What is the sum of 11. What is the sum of 12. What is the sum of 13. What is the sum of and -1? m+n m + n x+y+z'x+y+z 3y2-2 1+x 1-x 1-x+x2 Subtraction of Fractions. Ans. 9. 1+x+x2 1-x2 118. Fractions can only be subtracted when they are like parts of unity; that is, when they have a common denominator. In that case, the difference of the numerators will indicate how many times the common fractional unit is repeated in the difference of the fractions. Hence we have the following RULE. Reduce the fractions to a common denominator; then subtract the numerator of the subtrahend from the numerator of the minuend, and write the result over the common denominator. and it is plain that ten fifteenths of x, diminished by nine fif teenths of x, equals one fifteenth of x. It must be remembered that a minus sign before the divid ing line of a fraction affects the quotient (Art. 111); and since a quantity is subtracted by changing its sign, the result of the subtraction in this case is which fractions may be reduced to a common denominator, and the like terms united as in addition. Multiplication of Fractions. 119. Let it be required to multiply a α by وابع First let us multiply by c. According to the first princi ас ple of Art. 109, the product must be b. But the proposed multiplier was ; that is, we have used a d multiplier d times too great. We must therefore divide the result by d; and, according to the second principle of Art. 109, we obtain ас C ас bd・ d; that is, &q=bd. · bd' Hence we have the following RULE. Multiply the numerators together for a new numerator, and the denominators for a new denominator. Entire and mixed quantities should first be reduced to fractional forms. Also, if there are any factors common to the numerator and denominator of the product, they should be canceled. 2 |