Examples. 1. Reduce of of of to its equivalent simple fraction. Substituting the sign of multiplication for the word of, we get XXX. First, cancelling the 8 of the numerator against the 2 and 4 of the denominator, by drawing a line 1 3 $ 5 across them, we get x=x X Again, cancelling the 24 15 12. 3 and 5 of the numerator against the 15 of the denominator, 1 3 8 $ 1 we finally obtain X X 24.15 12 12° 2. Reduce of 1 of 3 of 4 of to its simplest form. First, cancelling the 7 and 5 of the numerator against 35 3 14.7 4 $ 7 35 8 9 11' of the denominator, we get X Again, can celling the 7 of the denominator against a part of the 14 of the numerator, and the 3 of the numerator against a part of 2 $ 14 7 4 $ the 9 of the denominator, we obtain X 3 Finally, cancelling the 2 and 4 of the numerators against 8 of the denominator, we getx 2 $ 1 11 33' 3 NOTE. We have written our fractions several times, in order the more clearly to exhibit the process of cancelling. But in practice it will not be necessary to write the fraction more than once. It will make no difference which of the factors are first cancelled; when all the common factors have in this way been stricken out, the fraction will then appear in its lowest terms. The student will find it to his interest to perform many examples of this kind, as this principle of cancelling will be extensively employed in the succeeding parts of this work. 3. Reduce of 1 of 2 of 33 of 3 to its simplest form. 4. Reduce of of 19 of 25 of 3 of of to its simplest + form. Ans. 5. Reduce of & of 18 of 1 of to its simplest form. 6. Reduce 18 Ans. T of 3 of 3 of 3 to its simplest form. 7. Reduce ofofofofto its simplest form. 8. Reduce of of of of of of of & to its simplest form. 9. Reduce of of 3 10. Reduce of 1 of 1 of to its simplest form. 21. To reduce fractions to a common denominator, we have this RULE. Reduce mixed numbers to improper fractions, compound frac tions to their simplest form. Then multiply each numerator by all the denominators, except its own, for a new numerator ; and all the denominators together for a common denominator. It is obvious that this process will give the same denominator to each fraction, viz: the product of all the denominators. It is also obvious, that the values of the fractions will not be changed, since both numerator and denominator are multiplied by the same quantity, viz: the product of all the denominators except its own. Examples. 1. Reduce,of,, and 3 of to equivalent fractions, having a common denominator. REDUCTION OF FRACTIONS TO COMMON DENOMINATORS. 33 These fractions when reduced to their simplest form are ,,, and . The new numerator of first fraction is 1×3×11×9=297. second fraction is 2 × 2 × 11 × 9=396. third fraction is 3×2×3×9=162. fourth fraction is 2 × 2 ×3×11=132. 66 66 66 66 66 66 66 66 66 The common denominator is 2×3× 11 × 9–594. Therefore, the fractions when reduced to a common denom. inator are 331, 386, 192, and 132. 297 5949 6 2. Reduce 3 of 3, of 1, and 41 to equivalent fractions having a common denominator. 22. To reduce fractions to their least common denomina. tors, we have this RULE. Reduce the fractions to their simplest form. Then find the least common multiple of their denominators, (by rule under Art. 10, or rule under Art. 11,) which will be their least common denominator. Divide this common denominator by the respective denominators of the given fractions, multiply the quotients by their respective numerators, and the products will be the new numerators. The correctness of the above rule may be shown in the same way as was the preceding rule. Examples. 1. Reduce of of 7 3 12' 20' and to equivalent frac 15' tions having the least common denominator. These fractions when reduced to their simplest form become 1 3 8' 20' 7 and The least common multiple of the denomina 15. tors 8, 20, and 15, is 120 common denominator. New numerator of first fraction is 120×1=15. Hence the fractions, when reduced to their least common denominator, become 1120, and 5. 15 18 6 120 2. Reduce of, 41, and to equivalent fractions having a common denominator. Ans. 1, 2, and. 20 3. Reduce of of 18 of 28, 1, and 7 to fractions having the least common denominator. Ans.,, and 150. 4. Reduce of 22 of 4 of 47, 61, and to fractions hav. ing the least common denominator. Ans. 4, 43, and . 33 5. Reduce of 2 of 2 of 55, of, and of 2 to fractions having the least common denominator. 6. Reduce, 114, and 75 to fractions having the least common denominator. 7. Reduce of 4 of 34 of 31, 32, and 10% to fractions having the least common denominator. 8. Reduce 17, 13 of 3, and 37 to fractions having the least common denominator. 3 9. Reduce 3, 51, 5, and 11 to fractions having the least common denominator. ADDITION OF FRACTIONS. 23. Suppose we wish to add and . We know that so long as these fractions are of different denominations they can not be added; we will therefore reduce them to a common denominator, we thus obtain =}; =28. Now, taking their sum we get +1+3= 15+28 35 Hence, to add fractions, we have this RULE. Reduce the fractions to a common denominator, and take the sum of the numerators, under which place the common denominator, and it will give the sum required. Examples. 3 1. Add the fractions of, of V, and 54. These fractions reduced to their least common denomina 4. Add the fractions 2, 3, 1, and 1. Ans.1337-21447. 5. What is the sum of, 3, and 43 ? |