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 1, &, and to a common denominator. Ans. 114, 192, 193. OPERATION. 60 3 X 6 X 8=144 new numerator for = = 152 4 X 6 X 8 192 common denominator. We first multiply the numerator off 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 in ference 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 ? 5. Reduce f, 12, 5 EXAMPLES FOR PRACTICE. Ans. 28, , or f. 12. Ans. 78, 73,98 4 Reduce 4, g, and to a common denominator. Ans. , 211, 218. and to a common denominator. Ans. 301, 36, 346, or 34, 36, 38. 6. Reduce , }, }, and to a common denominator. Ans. 0 68,885, $48, 48, or 3, 1980198, 12%. Art. 141. To reduce fractions to their least common denominator. Ex. 1. Reduce , ., and is to the least common denomi. nator. OPERATION. 33 6 12 1 2 common denominator, 21 2 4 3 4 X 2= 8 numerator for 6! 2 X 5 10 numerator for š= 12. 1 2 121 1X7= = 7 numerator for I = Tz. 3 X 2 X 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 s, 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 , , , and 3 to the least common denominator. 120, 125, 126, 126. 3. Reduce 1, §, f, and íí to the least common denominator. Ans. 1983, 1989 1988 1987 4. Reduce }, , and 7 to the least common denominator. Ans. 90 Ans. 25, 6, 40 310 5 5. Reduce 4, 14, 15, and 54 to the least common denominator. Ans. , 15, 1, 12 6. Reduce 1, 4, 6, , and is to the least common denominator. Ans. 14, 18, 38, 14, 21, 7. Reduce $, , }, }, }, and I'a to the least common denominator. Ans. 18, 38, 12, 46, 46 36 8. Reduces, f, and 72 to the least common denominator. Ans. 38, 16,5 9. Reduce 74, 501, 7, and 8 to the least common denominator. Ans. 241, 244, . 10. Reduce 4, 4, 5, 7, and 9 to the least common denominator. Ans. , 46, 48, 28, 36 ADDITION OF VULGAR FRACTIONS. Ans. 24. 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 7, 4, 4, 4, and together. These fractions all being 1+2+4+5+6 sevenths, that is, having 7 for 48= 24. a common denominator, 7 7 ng 7 7 simply add their numerators together, and write the sum over the common denominator. RULE. — Add together , the numerators of the fractions, and place their sum over the common denominator, and reduce the fraction if necessary. OPERATION. we QUESTIONS. - 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 , ÍT, IT, it, it, and is together. Ans. 311 3. Add to, and 17 together. Ans. 217 Ans. 225. 4. Add a'5, 285, 3s, and it together. 5. Add 77, 14, 37, and 11 together. Ans. 214. 6. Add 114, 117, and 15 together. Ans. 1114. 7. Add 1971, 1141, and 174 í together. Ans. 11441. Art. 144. To add fractions that have not a common de. nominator. Ex. 1. What is the sum of ã, , 2, and 7z? Ans. 131. OPERATION. 26 8 12 | 24 common denominator. 33 4 6 6/ 4 x 5=20 83 X3= 9 21 4 2 new numerators. 12 2 x7= 14 2 1 Sum of numerators 43 2x3 X2 X2= 24. Com. denominator 24 =111. 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. 527 1260 EXAMPLES FOR PRACTICE. 2. What is the sum of 5, 11, and 18 ? 3. What is the sum of zo, is, and ? 4. What is the sum of zi and 31 ? 5. What is the sum of i, á , and 12? 6. Add 4, it, and together. 7. Add 13, 1, and it together. 8. Add 75, 8, 74, and 1% together. Ans. 213. Ans. 224 Ans. 2268 QUESTIONS. - Art. 144. What is the rule for adding fractions not having a common denominator? How may mixed numbers be conveniently added ? 280 • OPERATION 9. Add }, , , , , 4, and together. Ans. 5,79 10. Add , 1o, it, it, tj, 13, and 14 together. Ans. 646436 11. Add şof i to ã of . Ans. 145 12. Add of to ij of z. Ans. 136. 13. Add į of to of 7o. Ans. 3396 14. Add of off to of $ of You Ans. % 15. Add 1 of 1 of 11 to 1 of g. Ans. 7 36 16. Add 3 to 444: Ans. 814 17. Add 4 to 54. Ans. 101 18. Add 171 to 1813. Ans. 365 Art. 145. To add any two fractions whose numerators are a unit. Ex. 1. Add 1 to . Ans. 20 We first find the Sum of the denominators 4+5= 9 product of the deProduct of the denominators 4 X 5 = 20 nominators, 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. Rule. - Add together the given denominators, and place the sum over their product. EXAMPLES FOR PRACTICE. 2. Add to , to, to }, } to, to . 3. Add į to lī, što , 1 to $, } to , f to It, što : 4. Add i to , to l', što , to l', to }, IT to 1. 5. Add į to , } to this, } to it, s to ], to }, } to . 6. Add {to }, 4 to , to }, # to , 4 to 7, 7 to itt: 7. Add į to 7, š to 3, 5 to š, s to to, što II, s to T's. 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 ? |