131. To change two or more fractions to a common denominator. 1. Reduce and to a common denominator. OPERATION. 3 x 5 4 x 5 2x4 5×4 = = SOLUTION. -We multiply the terms of the first fraction by the denominator of the second, and the terms of the second fraction by the denominator of the first (124). This reduces each fraction to the same denominator, for each new denominator is the 20 product of the given denominators. 15 20 8 OPERATION. 1 × 1 = 1 1 × 13 = 12 1 × 1 = 11 2. Change,, and to a common denominator. SOLUTION. We multiply the terms of each fraction by the product of the denominators of the other two (124). This reduces each fraction to the same denominator, since each new denominator will be the product of the three given denominators. RULE.-Multiply the terms of each fraction by the denominators of all the other fractions. Mixed numbers must first be reduced to improper fractions. 3. Change and to fractions having a denominator. Ans. 13, 13. 210 210 224 4. Change, 12, and to fractions having a common denominator. Ans. 388, 318, 3885. Change,,, and to fractions having a common denominator. Ans. 144, 118, 114, 188. 6. Change, and to fractions having a common denominator. Ans. 113, 111, 18. 7. Change §, 21, 3, and to fractions having a common denominator. Ans. 144, 144 14†' 144° 8. Change 13,, and 4 to fractions having a common denominator. Ans. 150, 24, 380. 243 144 96 69 0 360 108 48 809 9. Change 11,76, and 9 to fractions having a common denominator. Ans. ff, ff, 540. 132. To change fractions to the least common denominator. The Least Common Denominator of two or more fractions is the least denominator to which they can all be reduced, and is the least common multiple of the lowest denominators. OPERATION. 2 6 8 12 4 6 2 3 3 222 3 ᄒ SOLUTION. The least common multiple of the given denominators is 24. This is the least common denominator to which the fractions can be reduced. We mul 2 × 2 × 2 × 3 = 24 tiply the terms of each fraction by such a number as will reduce the fraction to the denominator, 24. Reducing each fraction to this denominator, by 130 we have the answer. } = 2 Ans. √2 = 11 12 1. Change,, and to the least common denominator. 5. Change 51, 21, and denominator. RULE.-I. Find the least common multiple of the given denominators, for the least common denominator. II. Divide this common denominator by each of the given denominators, and multiply each numerator by the corresponding quotient. The products will be the new numerators. 2. Change, o, 3, and to their least common. denominator. 3 45 41 192 32 3369 Ans. 1, 1, 10, 180. 3. Change,,, to their least common denomiAns. 18, 18, 336, 336° nator. 4. Chauge, 21, 4, and 6 to their least common denominator. Ans. 5, 181, 1512. to their least common Ans. 44, 18, 11. 56 12 1 504 23 352 154 6. Change, 3, 4, and to their least common denominator. Ans. 14, 11, it, lt. 7. Change, 1, 4, 25, and 15 to their least common denominator. Ans. 128, 21, 1888, 178, 168. 8. Change,, 33, 9, and 7 to equivalent fractions having the least common denominator. 48 60 ADDITION. EXAMPLES. 133. 1. What is the sum of 1, §, §, and 7? OPERATION. } + 3 + § + } = 16 = 2, Ans. SOLUTION.-Since the given fractions have a common denominator, 8, their sum may be found by adding their numerators, 1, 3, 5, and 7, and placing the sum, 16, over the common denominator. We thus obtain 15= 2, the required sum. 5 2. Add 7, 3, 10, 10, and 1. 5 3. Add 12, 12, 12, 12, 12, and 1. 4. What is the sum of 5. What is the sum of 6. What is the sum of OPERATION. 5, 2, 2, 13, 18, and 11? 16 9 25 7. What is the sum of and ? 13 3, Ans. 21. Ans. 212. 63 71 120, 6, 12, 120, 12%, and 198? 120 76 140 5, 118, 1, and ? Ans. 2. SOLUTION. In whole numbers we can add like numbers only, or those 3+3=33+18=27+10=37, Ans. having the same unit value; so in fractions we can add the numerators when they have a common denominator, but not otherwise. As and have not a common denominator, we must first reduce them to a common denominator, which we find to be 45; } = 23 and = 19. Adding the numerators and placing the sum over the common denominator, we find the answer to be 33. RULE. -I. When necessary, reduce the fractions to their least common denominator. II. Add the numerators, and place the sum over the common denominator. If the amount is an improper fraction, reduce it to a whole or a mixed number. 8. Add & to . 9. Add to 11. 10. Add 4, 1, 4, and 12. 97 16 11. Add 1, 3, and 4. Ans. 140. RULE. — To add mixed numbers, add the fractions and integers separately, and then add their sums. If the mixed numbers are small, they may be reduced to improper fractions, and then added after the usual method. 16. What is the sum of 144, 31%, 13, and 18? 17. What is the sum of 3, 17, 10%, and 5? Ans. 1874. 18. What is the sum of 17, 185, and 26,4? 19. What is the sum of, 1, 11, 3, and 17? 20. What is the sum of 1254, 327-52, and 251? 21. What is the sum of 148, 7, 16, 18, and 188? 22. What is the sum of 3%, 21, 404, and 10? 23. I bought 3 pieces of cloth containing 1257, 96%, and 48 yards. How many yards did I have? 3209 24. If it takes 5 yards of cloth to make a coat, 31 yards for a pair of pantaloons, and of a yard for a vest, how many yards will it take for all ? Ans. 9 yards. 25. A farmer divides his farm into 5 fields; the first contains 26 acres, the second 401 acres, the third 519 acres, the fourth 593 acres, and the fifth 623 acres. How many acres are there in the farm? Ans. 2411. 26. A speculator bought 1753 bushels of wheat for $205, 3254 bushels of barley for $2963, 27011 bushels of corn for $20011, and 4377 bushels of oats for $15623. How many bushels of grain did he buy, and how much did he pay for the whole? Ans. 1209 bushels; $859. SUBTRACTION. EXAMPLES. 134. 1. From 7 take. OPERATION. 7- &=+= f, Ans. SOLUTION. Since the given fractions have a common denominator, 10, we find the difference by subtracting 3, the less numerator, from 7, the greater, and write the remainder, 4, over the common denominator, 10, thus obtaining, the required difference. 2. From & take §. 3. From 4. From 2 5. From 4 6. From take 12. 7. From take 18. 8. From take f OPERATION. 128 182 348 take 1. take 27. take 38. Ans. . Ans. Ans. 14. Ans. 43. Ans. 9. From take 3. 10. From take 3. 11. Subtract Ans. SOLUTION.-As 8-838-18=32732 = √2 = 1, Ans. in whole num 36 3 bers, we can sub from §. tract like numbers only, or those having the same unit value, so we can subtract fractions only when they have a common denominator. As and have not a common denominator, we first reduce them to a common denominator, 36, and then subtract the less numerator, 30, from the greater, 32, and write the difference, 2, over the common denominator, 36. We thus obtain = 1, the required difference. RULE. - I. When necessary, reduce the fractions to a common denominator. II. Subtract the numerator of the subtrahend from the numerator of the minuend, and place the difference over the common denominator. 6 |