EXAMPLES FOR PRACTICE. 2. Reduce f and to common denominators. Ans. 1. 3t, or 2, 13. 3. Reduce ], $, and to a common denominator. Ans. 48, 47, 48. 4. Reduce #, , and it to a common denominator. Ans. If it, is 5. Reduce s, is, and to a common denominator. Ans. 338, 331, or 3%, 16, 14. 6. Reduce }, , }, and to a common denominator. Ans. 368, 385, $45, $48, or 25, 12, 138, 12 141. To reduce fractions to their least common denom. inator. Ex. 1. Reduce }, }, and 12 to the least common denominator. OPERATION. 11 2, least common denominator. 66 8 6 1 2 3 4 X 2= 8, new numerator. 6 2 X 5=10,5 1 2 1 X 7 7, TE 12, least common multiple, and common denominator. Having first obtained the least common multiple of all the denominators of the given fractions, we assume this to be their least common denominator. We then take such a part of it as expressed by each of the fractions separately for their respective new numerators. Thus, to get a new numerator for $, we take of 12, the least common denominator, by dividing it by 3, and multiplying the quotient 4 by 2. We proceed in like manner with each of the fractions, and write the numerators thus obtained over the least common denominator. In this process the value of each fraction remains unchanged, as both terms are multiplied by the same number. (Art. 140.) RULE. - 1. Find the least common multiple of the denominators for the least common denominator. 2. Divide the least common, denominator by each given denominator, and multiply the quotient by the corresponding numerator, for the new numerators. NOTE. Compound fractions must be reduced to simple ones, whole and 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 ? mixed numbers to improper fractions, and all to their lowest terms before finding the least common denominator. EXAMPLES FOR PRACTICE. 360 2. Reduce 1, $, &, and { to the least common denominator. Ans. 1920, 1980, 128, 123. 3. Reduce , ', $, and ft to the least common denominator. Ans. 146, 1980, 1980, 1980° 4. Reduce }, io, and 74 to the least common denominator. Ans. 36, 16, 263400 5. Reduce 4, it, it, and 53 to the least common denominator. Ans. , 38, 6. Reduce 1, 1, , }, and to the least common denominator. Ans. 27, 31, 34, dt, 31, 3. 7. Reduce $, }, }, }, }, and I to the least common denominator. Ans. 16, 36, 38, 396, 366 36 8. Reduce , $, and Iz to the least common denominator. Ans. 18, t. 9. Reduce 73, 511, 7, and 8 to the least common denominator. Ans. 34, 44,00 . 10. Reduce d, 4, 5, 7, and 9 to the least common denominator. Ans. , 6, 28, 29, 3 ADDITION 142. Addition of Fractions is the process of finding the sum of two or more fractions. Fractions can only be added when expressing fractional units of the same kind. OPERATION. 143. To add fractions having a common denominator. Ex. 1. Add +, 4, , 4, and .. Ans. 24. The fractions all being 2 4, 5 6 sevenths, we, add their numertott + L = 24 ators, and write their sum, 18, 7 7 7 7 over the common denominator, 7; and thus obtain = 24, the required sum. That is, we Write the sum of the numerators over the common denominator. 142. What is addition of fractions ? - 143. How are fractions having a common denominator added ? Give the reason. EXAMPLES FOR PRACTICE. 2. Add A, PT, IT, ir it, and H. Ans. 311. 3. Add it, I't17, 197, and 14. Ans. 24. 4. Add 25, 285, , and it. Ans. 2.45 5. Add 17, 18, 7, and A+. Ans. 21. 6. Add 134, 137, and they Ans. 1134. 7. Add 1971, Hit, and itir: Ans. 11961. 144. To add fractions not having a common denominator. Ex. 1. What is the sum of , j, and Ya? Ans. 114. OPERATION. 26 8 12 2 4, common denominator. 2 4 6 6 4 x 5 : 20 2 3 8 9 3 X 3 new numerators. 12 2 X 7=14 Sum of numerators, 43 -= 14%, Ans. 2X2X2 X3= 24. Com. denominator, 24 We reduce the given fractions to equivalent ones having a common denominator, that they may express fractional units of the same kind; and then we add the nuinerators, and write their sum over the common denominator, and reduce the fraction. RULE. — Reduce the given fractions to a common denominator. Add the numerators, and write their sum over the common denominator. Note 1. - First reduce mixed numbers to improper fractions, and compound fractions to simple fractions, and each fraction to its lowest terms. Note 2. In adding mixed numbers, the fractional parts may be added separately, and their sum added to the amount of the whole numbers. EXAMPLES FOR PRACTICE. 2. What is the sum of s, 11, and 1? Ans. 275. 3. What is the sum of , 1), and ? Ans. 1130 4. What is the sum of 31 and 34 ? Ans. 1577 5. What is the sum of i, s, š, and 15? Ans. 2 6. Add , T, 31, and L. Ans. 1384 7. Add 7, 8, and yt. Ans. 113. 8. Add 5, 18, 75, and 15. Ans. 2363 144. The rule for adding fractions not having a common denominator ? How may mixed numbers be added ? Ans. 28.9 Ans. 36 9. Add 1, 3, 3, , 6, 4, and y, together. Ans. 5235 10. Add , 1o, ti, 12, 13, 17, and 14. Ans. 634436 11. Add of 1 to $ of . Ans. 145. 12. Add off to 1) of t. Ans. 196 13. Add f of to g of jo. T350 14. Add of of toof of 10 Ans. is 15. Add of it of 11 to 4 of g. 16. Add 3 to 411 Ans. 814 17. Add 43 to 5%. Ans. 1045. Ans. 36. 18. Add 177 to 1835. 145. To add two fractions having 1 for their numerator. Ex. 1. Add to $ Ans. ਉਹ We first find the Sum of the denominators, 4+59 product of the denom inators, which is 20, Product of the denominators, 4 X 5 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. By this process we reduce the fractions to a common denominator, and then add their numerators. Hence, to add two fractions of this kind, Write the sum of the given denominators over their products EXAMPLES FOR PRACTICE. OPERATION. 2. Add 1 to $, 4 to $, 1 to }, } to t, 1 to $. SUBTRACTION. 146. Subtraction of Fractions is the process of finding the difference between two fractions. Note. — One fraction can be subtracted from another only when both express fractional units of the same kind. 145. How can you add two fractions when the numerators are a unit? The reason for this ? — 146. What is subtraction of fractions ? Ans. g. OPERATION. 147. To subtract fractions having a common denominator. Ex. 1. From } take . The fractions both being ninths, we subtract the less }-= $ numerator from the greater, and write the difference, 5, over the common denominator, 9; and thus obtain & as the required difference. That is, we Write the difference of their numerators over the common denominator. EXAMPLES FOR PRACTICE. Ans. Pr: 2. From ti take it. Ans. is 3. From 11 take 1o. Ans. 37. 4. From 37 take 47. Ans. 11 5. From tf1 take it. Ans. 21 6. From my take us. Ans. io. 7. From take o Ans. 8. From 7 take 100. 148. To subtract fractions not having a common denominator. Ans. 5. Ex. 1. From 13 take 7z. OPERATION. } 411 6 1 2 48, common denominator. new numerators. 1 1, difference of numerators. 48, common denominator. We reduce the given fractions to equivalent ones having a common denominator, that they may express fractional units of the same kind, and then we subtract the less numerator from the greater, and place the difference over the common denominator. RULE. — Reduce the fractions to a common denominator, then write the difference of the numerators over the common denominator. Note. - If the minuend or subtrahend, or both, are compound fractions, they must be reduced to simple ones. 147. How do you subtract fractions having a common denominator? – 148. The rule for subtracting fractions not having a common denominator? If the minuend or subtrahend is a compound fraction, what must be done ? |