A COMMON DENOMINATOR. ART. 139. A common denominator of two or more fractions is a common multiple of their denominators. The least common denominator is the least common multiple. NOTE. Fractions have a common denominator, when all their denominators are alike. ART. 140. To reduce fractions to a common denominator. Ex. 1. Reduce, §, and to a common denominator. OPERATION. 160 Ans. 1, 182, 188. 3 × 6 × 8 = 144 new numerator for = 5 X 4 X 8 160" 66 66 =192 192 4 X 6 X 8 = 192 common denominator. We first multiply the numerator of by the denominators 6 and 8, and obtain 144 for its numerator. We next multiply the numerator of by the denominators 4 and 8, and obtain 160 for its numerator; and then we multiply the numerator of by the denominators 4 and 6, and obtain 168 for its numerator. Finally, we multiply all the denominators together for a common denominator, and write it under the several numerators, as in the operation. By this process the numerator and denominator of each fraction are multiplied by the same numbers, and consequently, both being increased an equal number of times, their relation to each other is not changed, and the value of the fraction remains the same. (Art. 133.) Therefore, Multiplying the numerator and denominator of a fraction by the same number does not alter the value of the fraction. RULE. Multiply each numerator into all the denominators except its own for a new numerator; and all the denominators into each other for a common denominator. NOTE. Fractions of this form may often be reduced to lower terms, without destroying their common denominator, by dividing all their numerators and denominators by a common divisor. QUESTIONS. Art. 139. What is a common denominator of two or more fractions? What is the least common denominator? When have fractions a common denominator? Art. 140. How do you find a common denominator of two or more fractions? Give the reason of the operation. What inference is drawn from it? What is the rule for finding a common denominator? How may fractions having a common denominator be reduced to lower terms? 4 Reduce,, and 1 Ans.,, or 12, 12. to a common denominator. Ans. 38, 73, 15. to a common denominator. Ans. 1, 2, 18. 5. Reduce, 12, 21 15 24 Ans. 3, 13, 14, or 38, 18, 38. 6. Reduce, 2, 3, and 240 9609 48 30 ART. 141. To reduce fractions to their least common denominator. Ex. 1. Reduce, §, and to the least common denomi 6 1 2 2X5: = 12 10 numerator for 12. 1x7 = 7 numerator for 3 × 2 × 2 = 12, the least common denominator. Having first obtained a common multiple, or denominator of the given fractions, we take the part of it expressed by each of these fractions separately for their new numerators. Thus, to get a new numerator for, we take of 12, the common denominator, by dividing it by 3, and multiplying the quotient 4, by 2. We proceed in this manner with each of the fractions, and write the numerators thus obtained over the common denominator. NOTE.The change in the terms of the fractions, in reducing them to the least common denominator by this process, depends upon the same principle as explained in the preceding article. RULE. - 1. Find the least common multiple of the denominators of the several fractions, and it will be their least common denominator. 2. Divide the least common denominator by the denominator of each of the given fractions, and multiply the quotients by their respective numerators, and their products will be the numerators of the fractions required. NOTE.-Compound fractions must be reduced to simple ones, whole QUESTIONS.- -Art. 141. How do you find the least common denominator of two or more fractions? Upon what principle does this process depend? What is the rule for reducing fractions to their least common denominator? What must be done with compound fractions, whole numbers, and mixed numbers? and mixed numbers to improper fractions, and all to their lowest terms, before finding the least common denominator. EXAMPLES FOR PRACTICE. 2. Reduce 2, 4, §, and 3. Reduce,,, and 9 4. Reduce 3,1%, and 72 to the least common denominator. 9 36 Ans. 18, 18, 20. 5. Reduce, 14, 1, and 5% to the least common denom inator. 3 5 20 6. Reduce 1⁄2, 1, §, §, 7, and 11⁄2 to the least common denominator. Ans. 12, 18, 21, 14, 21, 14. 7. Reduce, 3, 1, 1, 1, and 11⁄2 to the least common denominator. Ans. 18, 3, 3, 36, 36, 36. 8. Reduce,, and to the least common denominator. Ans. 38, 18, . 9. Reduce 7, 5, 7, and 8 to the least common denominator. Ans. 341, 244, 308, 352. 10. Reduce 2, 4, 5, 7, and 9 to the least common denomiAns. 2, 16, 20, 28, 36. nator. ADDITION OF VULGAR FRACTIONS. ART. 142. ADDITION of Vulgar Fractions is the process of finding the value of two or more fractions in one sum. ART. 143. To add fractions that have a common denominator. Ex. 1. Add 4, 4, 4, 4, and together. OPERATION. 1+2+4+5+6 Ans. 24. These fractions all being sevenths, that is, having 7 for common denominator, simply add their numerators = 1824. a together, and write the sum over the common denominator. we RULE.-Add together the numerators of the fractions, and place their sum over the common denominator, and reduce the fraction if necessary. Art. 142. What is addition of vulgar fractions? - Art. 143. What is the rule for adding fractions having a common denominator? Give the reason for the rule. EXAMPLES FOR PRACTICE. 2. Add, fr, T, fr, fr, and together. 18 37 5. Add 17, 19, 27, and 14 together. 6. Add 119, 134, and 1135 together. 7. Add 1978, 149, and 11 together. ART. 144. To add fractions that have not a common denominator. Ex. 1. What is the sum of 8, §, and ? OPERATION. Ans. 11. 2×3×2×2=24. Com. denominator 24 112. Having found the common denominator and new numerators, as in Art. 141, we add the numerators together and place their sum over the common denominator, and reduce the fraction. RULE. Reduce mixed numbers to improper fractions, and compound fractions to simple fractions; then reduce all the fractions to a common denominator; and the sum of their numerators, written over the common denominator, will be the answer required. NOTE. In adding mixed numbers, it is sometimes more convenient to add the fractional parts separately, and then to add their sum to the amount of the whole numbers. QUESTIONS. Art. 144. What is the rule for adding fractions not having a common denominator? How may mixed numbers be conveniently added? Ans. 570. 9. Add 1, §, 1, 4, 5, §, and 7 together. 10. Add 8, 1, 11, 12, 13, 14, and together. 1 Ans. 61443Ans. 11. Ans. 1. Ans. 1350 289 Ans. 17. Add 4 to 5. 18. Add 17 to 18. ART. 145. are a unit. To add any two fractions whose numerators Ex. 1. Add to . OPERATION. Sum of the denominators 459 Product of the denominators 4 X 5=20 Ans. 20 We first find the product of the denominators, which is 20, and then their sum, which is 9, and write the former for the denominator of the required fraction, and the latter for the numerator. The reason of this operation will be seen, when we consider, that the process reduces the fractions to a common denominator, and then adds their numerators. - Add together the given denominators, and place the sum over their product. EXAMPLES FOR PRACTICE. 2. Add to, to, to, to f, to f. 3. Add to, to, 4. Add to, to Tz, 5. Add to, to 12, 6. Add to, to, to b, 1 to 1. to,to,to,to,to,to TT: 7. Add to, to, to, to to, to TT, to 12. SUBTRACTION OF VULGAR FRACTIONS. ART. 146. SUBTRACTION of Vulgar Fractions is the process of finding the difference between two fractions of unequal values. QUESTIONS. Art. 145. What is the rule for adding two fractions when the numerators are a unit? What is the reason for this rule? Art. 146. What is subtraction of vulgar fractions? |