COMMON DENOMINATOR. 167. A Common Denominator is a denominator common to several fractions, or a denominator to which all may be reduced. 168. Similar Fractional Units are those which are of the same kind; as 3 fifths and 2 fifths. 169. Dissimilar Fractional Units are those which are of different kinds; as, 3 fourths and 3 fifths. Principle. A common denominator of several fractions must be a common multiple of their denominators. MENTAL EXERCISES. 1. Reduce and to a common denominator. SOLUTION.-A common denominator for thirds and fourths is twelfths. In one there are 1, and in there are of 1, or, and in, etc. Reduce to a common denominator, 8. Describe the process of reducing two fractions to a common denominator. WRITTEN EXERCISES. OPERATION. 1, 1, 4= 128, 113, 198 1. Reduce,, and to a common denominator. SOLUTION. Since the product of the denominators of the fractions is a common multiple of their de nominators, 4×5×7, which equals 140, will be the common denominator. Then multiplying both terms of by 5x7 we have =108 (Prin. 5). Multiplying both terms of by 4×7, we have =113, etc. Hence the following Rule. Multiply both terms of each fraction by the de nominators of the other fractions. Reduce to a common denominator, Ans. 388, 388, 118. Ans. 1344 1664 6886 1538, 1838. 864 960 10 8.,,,, and . Ans. 57, 7882, 1882, 1882, 1988. 9. Show that the common denominator of several fractions is a common multiple of the denominators of those fractious 828176 LEAST COMMON DENOMINATOR. 170. The Least Common Denominator of several fractions is the smallest denominator to which all may be reduced. OPERATION. L. C. M.=24 Principle. The least common denominator of several fractions is the least common multiple of their denominators. 1. Reduce, , and to their least common denominator SOLUTION.-We find the least common multiple of the denominators to be 24, hence 24 is the least common denominator. Dividing 24 by 3, the denominator of, we find we must multiply 3 by 8 to produce 24; hence multiplying both terms of by 8, we have 4 (Prin. 5). Dividing 24 by 6, the denominator of g, we find we must multiply 6 by 4 to produce 24; hence, multiplying both terms by 4, we have =, etc. 2X8 3X8 一ˊ 5X4 =11 6X4 77X3 8X3 Rule.-I. Find the least common multiple of the denomi nators, for the least common denominator. II. Divide the least common denominator by the denomina tor of each fraction, and multiply both terms by the quotient. NOTE.-Reduce compound fractions to simple ones, mixed numbers to Improper fractions, and all to their lowest terms, before finding the least common denominator. To their least common denominator, 9. Reduce of 7, 1 of 33, 1111. Ans. 18, 18, H. Ans. 18, 18, 58. Ans. 15, §8, 88. Ans. ft, H, H.. 9.5 Ans. 126, 128, 12%. 21 Ans. 18, HI, HI, 117. 21 63 Ans. 1,82, 39, 48, 48, 48. 612 Ans. I, I, 2822. Ans. 1788, 22, etc. 252 ADDITION OF FRACTIONS. 840 171. Addition of Fractions is the process of finding the sum of two or more fractions. PRINCIPLES. 1 To add two or more fractions, they must express similar fractional units. 2. To add two or more fractions they must be reduced to a common denominator. 10. How then shall we add two fractions whose denominators are unlike? WRITTEN EXERCISES. OPERATION. 1. What is the sum of, g, and 7? SOLUTION.-Reducing the fractions to a common denominator that they may express similar fractional units, we have =,=41, 7=41; 18 twenty-fourths plus 20 twenty-fourths plus 21 twentyfourth equals 59 twenty-fourths. Hence the following Rule. Reduce the fractions to a common denominator, then add the numerators and write the sum over the common denominator. NOTES.-1. Reduce compound fractions to simple ones, and reduce each fraction and the sum to lowest terms. 2. To add mixed numbers, add the integers and fractions separately, and then unite their sums. 10. Find the sum of 21, 44, 34, 1%. Ans. 27488. Ans. 83-14. Ans. 2,870. Ans. 3. Ans. 3. Ans. 1218. SUBTRACTION OF FRACTIONS. 172. Subtraction of Fractions is the process of finding the difference between two fractions. PRINCIPLES. 1. To subtract two fractions they must express similar fractional units. 2. To subtract two fractions they must be reduced to a com mon denominator. MENTAL EXERCISES. 1. How many 12ths in the difference between and ? SOLUTION. equals and equals, and the difference between and is. 10. How then shall we subtract two fractions whose denominators are unlike? WRITTEN EXERCISES. 1. What is the difference between § and 7? SOLUTION. Reducing the fractions to a common denominator that they may express similar fractional units, we have 3 and 4: 56 seventy-seconds minus 45 seventy-seconds equals 11 seventy-seconds. Hence the following OPERATION. 7-f= Rule. Reduce the fractions to a common denominator, take the difference of the numerators, and write it over the common denominator. NOTE.-Reduce compound fractions to simple ones, and reduce each fraction and the difference to its lowest terms. 16. Subtract 84 from 124. SOLUTION.-We cannot subtract & from, so we take 1 from 12, which added to equals 14 or ; from 49 leaves, and 8 from 11 leaves 3; hence the difference is 35. 17. Subtract 8 from 12. 18. Subtract 57 from 10%. 22. Subtract 407 from 60%. WRITTEN PROBLEMS Ans. 31. Ans. 4. Ans. 98. Ans. 117. Ans. 98. Ans. 1918. IN ADDITION AND SUBTRACTION OF FRACTIONS. 1. A has $51, B has $44, and C has $63; how much money bave they all? Ans. $16. 2. A miller ground 7 bushels of corn for A, 94 for B, 10 for C; how much did he grind in all? Ans. 2717 bu. 8. A lady bought material for a wrapper costing $11, and buttons costing $; what change should she receive from a $5 bill? Ans. $31. 4. A lady went shopping with $100 and paid $12 for a bonnet, $32 for a dress, and $52 for a cloak; how much money did she bring home? Ans. $2.124. 5. A boy gave 12 cents for a slate, 183 cents for a knife, 37 cents for a grammar, and 62 cents for what did they all cost? an arithmetic; Ans. $1.311 6. A merchant bought two pieces of muslin, each containing 41 yd.; after selling 57 yd. from them, how many yards remained? Ans. 254 yd. 7. Mr. Weeks finds that his family burned last winter 1 tons of coal in December, 2 tons of coal in January, 2 tons of coal in February, and in March 18 tons; how much was burned during the four months? Ans. 7 tons. 8. Four loads of hay weighed upon the scales 49, hundredweight, 43, hundredweight, 39 hundredweight, and 458 hundredweight; what was the weight of the hay, the weight of the wagon being 15 hundredweight? Ans. 115 hundredweight. |