Hence the least common multiple of any set of numbers may be found by the following RULE. Write the numbers in a horizontal line; divide them by the least number which will divide two or more of them without a remainder; place the quotients with the undivided numbers, if any, for a second horizontal line; proceed with this second line as with the first; and so continue until there are no two numbers which can be exactly divided by the same divisor. The continued product of the divisors, and of the numbers in the last horizontal line, will give the least common multiple. NOTE.-When there is no number which will divide two of the given numbers, their continued product must be taken for the least common multiple. What is a multiple of several numbers? Mention some of the multiples of 2, 3, 4, and 6. Are the number of multiples of any set of numbers limited? Repeat the Rule for finding the least common multiple of any set of numbers. When there is no number which will divide two of the given numbers, how is the least multiple found? EXAMPLES.. 1. What is the least common multiple of 12, 16, and 24? Hence, 2×2×2×3×2=48 is the least common mul tiple. 2. What is the least common multiple of 12, 15, 24? Therefore, 2×2×3×5×2=120 is the multiple sought 3. What is the least common multiple of 1, 77, 88? Ans. 616. 4. What is the least common multiple of 37, 41 ? Ans. 1517. 5. What is the least common multiple of 24, 60, 45, 180? Ans. 360. 6. What is the least common multiple of 2, 4, 6, 8? Ans, 24. 7. What is the least common multiple of 3, 5, 7, 9? Ans. 315. 8. What is the least common multiple of 2, 3, 4, 5, 6, 7, Ans. 2520. 8,9? 9. What is the least common multiple of 7, 14, 16, 18, Ans. 1008. 24? 10. What is the least common multiple of 1, 2, 3, 4, 5, 6, 7, 8, 9, 11? Ans. 27720. 42. We are now prepared to reduce fractions to their least common denominator. Let it be required to reduce to the least common de nominator the fractions F, %, and 4. If we take the least common multiple of the denominators 12, 16, and 24, which is 48, and divide it in turn by these denominators, we shall obtain the respective quotients 4, 3, and 2. Hence, if we multiply the numerator and denominator of each fraction by 4, 3 and 2 respectively, they will become, and . These fractions are equivalent to the original ones, and have their least. common denominator. Hence fractions may be reduced to their least common denominator by the following RULE. Reduce the fractions to their simplest form; then find the least common multiple of their denominators; (by Rule under ART. 41,) which will be their least common denominator. Divide this denominator by the respective denominators of the given fractions; multiply the quotients thus obtained by the respective numerators, and the several products will be the new numerators. Repeat the Rule for reducing fractions to their least common denominator. EXAMPLES. 1. Reduce, 1,, to equivalent fractions having the least common denominator. The least common multiple of the denominators 12, 15 24, is 120 common denominator. New numerator of first fraction 120 x 5=50. 24 Hence, the fractions, when reduced to their least common denominator, become 2. Reduce of of 1, 2, 7, to equivalent fractions having the least common denominator. Ans. %, 5%. Ans. 105, 130, 38. 309 3. Reduce 3, 4, 8, to equivalent fractions having the least common denominator. 4. Reduce,, 13, to equivalent fractions having the least common denominator. Ans. 168, 14, 117. 2025 3309 330 5. Reduce,, 6, to equivalent fractions having the least common denominator. Ans. 10, 3985. 6. Reduce,, 34, and, to equivalent fractions having the least common denominator. Ans. 30, 40, 125, 18. ᄒ. 7. Reduce to,,,r, to equivalent fractions having the least common denominator. Ans., 7, 21%, 21%. 0 8. Reduce, 4, 4, 5,, to equivalent fractions having the least common denominator. Ans. 48, 45, 48, 50, 27. 9. Reduce, 4, 5, 10, T76, to equivalent fractions having the least common denominator. Ans. 1, 1, 120, 128, 128, 170. 10. Reduce, 1, 1, 1, 1, 7, 1, 3, to equivalent fractions having the least common denominator. 315 Ans. 1898, 84, 8300, 5040, 520, 2520, 2540, 23%. 125 20 25 201 2520 ADDITION OF FRACTIONS. 43. Suppose we wish to add ✈ and 4. We know that to long as these fractions have different denominators, they cannot be added any more than pounds and yards can be added together. We will therefore reduce them to a comWe thus obtain mon denominator. Reduce the fractions to a common denominator, and take the sum of their numerators, under which place the common denominator, and it will give the sum required. NOTE.-The labor will be the least when we reduce the fractions to their least common denominator. EXAMPLES. 1. What is the sum of,, 1 1, and ? These fractions, when reduced to their least common denominator, are fa, fa, fa, and 2, the sum of whose numerators is 6+4+3+2=15. Hence we have NOTE: If any of the fractions are compound, they must first be reduced to simple fractions, (by Rule under ART. 39.) 7. What is the sum of of 4 of 4, 1 of 4, and ? |