are multiplied by the same quantity, viz: the product of all the denominators except its own. 5 EXAMPLES. 1. Reduce, of 3, 1, and 7 of, to equivalent fractions having a common denominator. These fractions, when reduced to their simplest form, are,,, and . 3. The new numerator of the first fraction is 1 × 3 × 11 × 9=297. The new numerator of the second fraction is 2×2× 11 x 9 396. The new numerator of the third fraction is 3×2×3× 9=162. The new numerator of the fourth fraction is 2×2×3 X 11=122. The common denominator is 2×3× 11 × 9=594. Therefore, the fractions, when reduced to a common denominator, are 21, 324, 162, and 132. 297 396 5947 6 594 2. Reduce of, of, and, to equivalent fractions having a common denominator. 3. Reduce, 1 of 3, and 41, to equivalent frac 37 3 ་་ tions having a common denominator. 23 31 4. Reduce 3, 3, 4, and 11 to equivalent fractions 17, having a common denominator. Ans. 24 € 3599 53 2048821 29 17 27546 3106277 3106277 3106277 3106277 5. Reduce,, and, to fractions having a com mon denominator. 662257 Ans. 656197 682007 710131 7151319 710131 6. Reduce, and, to fractions having a com 22. To reduce fractions to their least common denominators, 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 numerator. The correctness of the above rule may be shown in the same way as was that of the preceding rule. EXAMPLES. 1. Reduce of of,, and, to equivalent fractions having the least common denominator. These fractions, when reduced to their simplest form, become,, and . The least common multiple of the denominators 8, 20, and 15, is 120=common denominator. New numerator of the first fraction is 120×1=15. of the second fraction is 12x3=18 66 20 of the third fraction is 120x7=56. Hence, the fractions, when reduced to their least common denominator, become, %, and 1. 56 2. Reduce of 1, 41, and, to equivalent fractions having the least common denominator. Ans. 14,, and . 3. Reduce of of 1 of 28, 11, and 7, to fractions 18 26 having the least common denominator. Ans.,, and . 4. Reduce of 3 of 4 of 12, 64, and, to frac 8 tions having the least common denominator. 5. Reduce of 22 of of 3, to fractions having the least Ans. 1176120 Ans., 43, and . 22 33 5, of, and of common denominator. 7194487, 422576 76063689 76 0 6 36 81 7606368 6. Reduce, 115, and 735, to fractions having the least common denominator. Ans. 1, 2016, 1852. 363 7. Reduce of 4 of 4 of 31, 32, and 10%, to frac 30 489 tions having the least common denominator. Ans.,, 503. 56 8. Reduce 11, 13 of, and 37, to fractions having the 17 3 least common denominator. Ans. 3225, 244, 7838. 3995 9. Reduce 3, 51, 51, and 11, to fractions having the least common denominator. 4840 8008 8085 Ans. 14, 1998, 15, 1340 ADDITION OF FRACTIONS. 23. SUPPOSE we wish to add and . We know that so long as these fractions are of different denominators they cannot be added; we will, therefore, reduce them to a common denominator; we thus Now, taking their sum, we get 28 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. These fractions, reduced to their least common denominator, are, and 72; and their sum is 3. Add the fractions of, %, and 41. Ans. 5. 4. Add the fractions, 1, 1, and. 5. What is the sum of, 3, and 42? 240 Ans. 123-131. 6. What is the sum of 4%, 2%, and ? Ans. 1435375-2 75 289 680043 7. What is the sum of 4, 3, and ? 49 680043 Ans. 2. 95 = 8. What is the sum of of 41, 21⁄2 of 64, and ? 11. What is the sum of 11, 21, 31, 4}, and 5†? Ans. 16. SUBTRACTION OF FRACTIONS. 24. To subtract one fraction from another, we have this RULE. Reduce the fractions to a common denominator, and subtract the numerator of the subtrahend from that of the minuend; place the common denominator under the difference. These fractions, when reduced to their least common denominator, become. Therefore, ¬}= 93 88 41-341=34T. 2. From subtract 3. From 27 subtract Ans. 3 8 31 Ans. 321 11. |