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LEAST COMMON DENOMINATOR. 170. The Least Common Denominator of several fractions is the smallest denominator to which all may be reduced.

Principle.—The least common denominator of several fractions is the least common multiple of their denominators.

1. Reduce &, k, and f to their least common denominator SOLUTION.—We find the least common multiple of the denominators to be 24, hence 24 is the least L. C. M.=24 common denominator. Dividing 24 by 3, the de nominator of %, we find we must multiply 3 by 8 to

3X8 produce 24; hence multiplying both terms of } by 8,

6X4 we have f=h (Prin. 5). Dividing 24 by 6, the

denominator of, we find we must multiply 6 by 4

6X4 to produce 24; hence, multiplying both terms by 4, we have f=ti, etc.

8X3 Rule.-I. Find the least common multiple of the denominators, 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, 2. Reduce , $, 7.

Ans. 48, H, H. 3. Reduce , 7, s.

Ans. 4, 44, 44. 4. Reduce , H, H.

Ans. 1, 8, H. 5. Reduce 44, 54, 74.

Ans. 11, H, H. 6. Reduce $, H, H.

Ans. 77, H8, Homo 7. Reduce $, H, tô, xh.

Ans. 48, 49, 114, 137. 8. Reduce 2], 54, 4, 2, 3. Ans. 4, 45, 46, 47, 4. 9. Reduce of, of 34, 1111. Ans. 44, 414, 42. 10. Reduce 1, 3, 4, 5, 5, 4, 5, $ Ans. 179% 14%, etc.

ADDITION OF FRACTIONS. 171. Additioz 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 nmilar fractional units .

2. To add two or more fractions they must be reduced to a common denominator.

MENTAL EXERCISES. 1. How many sixths in ; and f?

SOLUTION.-- is equal to f and į is equal to *; and are 3, which to equal to 11.

Find the sum 2. Of j and 1

6. Of and . 3. Of j and

7. Of f and f. 4. Of and .

8. Of $ and . 5. Of 4 and %.

9. Of j and 7. 10. How then shall we add two fractions whose denominatons are unlike?

WRITTEN EXERCISES.

OPERATION.

1. What is the sum of $, &, and } ? SOLUTION.- Reducing the fractions to a common denominator that they may express similar fractional units, we have t =*f, =lti 18

Hifit=lt. 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 sej arately, and then unite their sums. 2. Find the sum of f, g, is

Ans. 23. 3. Find the sum of 4, f, g.

Ans. 27 4. Find the snm of , H, H.

Ans. 24. 5. Find the sum of 4, 4, H.

Ans. 22 6. Find the sum of 24, 44, 14.

Ans. 844. 7. Find the sum of $, , }, H.

Ans. 3. 8. Find the sum of £, £, &, f.

Ans. 37. 9. Find the sum of £, £, Mo, .

Ans. 347. 10. Find the sum of 24, 44, 37, 11

Ans. 124. 11. Find the sum of 44, 75, 96, 736. Ans. 27486. 12. Find the sum of 214, 3511, 221, and 43. Ans. 8314 13. Find the sum of 175, 4915; 246, 1875. Ans. 10914. 14. Find the sum of off, of 4, of

Ans. 2010 15. Find the sum of 1, 1, , $, , , }, } to. Ans. 14H

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 f?

SOLUTION.equals to and f equals ts, and the difference between a and is the

Subtract
2. from

6. from
8. } from 1.

7. $from š: 4. { from .

8. * from 3. 5. f from

9. from 1. 10. How then shall we subtract two fractions whose denominatore are unlike?

WRITTEN EXERCISES, 1. What is the difference between f and f? SOLUTION.-Reducing the fractions to a common OPERATION. denominator that they may express similar fractional units, we have = and =*f: 56 seventy-seconds

44-4f=hl minus 45 seventy-seconds equals 11 seventy-seconds. Hence the following

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 traction and the difference to its lowest terms.

Subtract
2. | from Ans. 9. 9. H from H.

Ans 40 8. from Ans. H. 10. 8 from 18

Ans. 4. * from 7. Ans. 26. 11. 1 from of to

Ans. 17 6. from Ans. . 2. it from 4 of 3

Ans. 47. 6. if from ti. Ans. TÁT. 13. & from of 18. 7. & from H Ans. . 14. of H from f of 1. Ans. 8. 14 from 1 Ans. d. 15. off from of H. Ans.

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16. Subtract 84 from 124. SOLUTION.–We cannot subtract from £, so we take 1 from 12, which added to f equals 14 or 10; $from leaves 4, and 8 from 11 leaves 8; bence the difference is 34. 17. Subtract 86 from 124.

Ans. 31.

Ans. 41. 18. Subtract 5} from 10%. 19. Subtract 109 from 207.

Ans. 98. 20. Subtract 127 from 24%.

Ans. 1171 21. Subtract 20H from 30%.

Ans. 94. 22. Subtract 407 from 6012.

Ans. 1974.

WBITTEN PROBLEMS

IN ADDITION AND SUBTRACTION OF TRACTIONS.

1. A has $54, B has $44, and C has $6%; how much money have they all ?

Ans. $164. 2. A miller ground 7 bushels of corn for A, 94 for B, 104 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 $46; 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 $525 for a cloak; how much money did she bring home ?

Ans. $2.121 5. A boy gave 124 cents for a slate, 18 cents for a knife, 87} cents for a grammar, and 62 cents for an arithmetic; what did they all cost ?

Ans. $1.311 6. A merchant bought two pieces of muslin, each containing 41f yd.; after selling 57% yd from them, how many yards remained ?

Ans. 254 yd. 7. Mr. Weeks finds that his family burned last winter 11 tons of coal in December, 2 tons of coal in January, 2 tons of coal in February, and in March 1f tons; how much was burned during the four months? Ans. 777 tons.

8. Four loads of hay weighed apon the scales 4977 handredweight, 43 hundredweight, 39 40 hundredweight, and 451% hundredweight; what was the weight of the bay, the weight of the wagon being 15775 hundredweight?

Ans. 11538 hundredweight

MULTIPLICATION OF FRACTIONS. 173. Multiplication of Fractions is the process of inding a product when one or both factors are fractions.

174. There are Three Cases: 1st. A fraction by an integer; 2d. An integer by a fraction; 8d. A fraction by a fraction.

OASX I. 175. To multiply a fraction by an integer.

MENTAL EXERCISES. 1. How many are 5 times ts? SOLUTION.—5 times its are H, which reduced to its lowest terms equals f. Therefore, etc.

SOLUTION 20.—5 times to equal ts, or *; if 5 times to equal }, 5 times is equals 2 times or ; hence 5 times A equals f. How many are 2. 3 times f?

6. 3 times 5%? 3. 6 times 75?

7. 4 times 77? 4. 7 times it?

8. 8 times 64? 5. 8 times 11?

9. 9 times 477? 10. Since 5 times is equals }, how may this result be obtained by omitting the analysis ?

11. How then may a fraction be multiplied by omitting the Analysis ?

WRITTEN EXERCISES. 1. Multiply H by 4 SOLUTION.-Multiplying the numerator (Prin. 1), OPERATION. we have 4 times 11 equals t1; or, dividing the de

11x41 nominator (Prin. 4), we have 4 times th=", or 3.

11x4=4 Rule.- To multiply a fraction by an integer; multiply the numerator, or divide the denominator.

Multiply 2. #1 by 6. Ans. 51. 10. 73 by 5.

Ans. 391 8. H by 5. Ans. 41. 11. 187 by 3.

Ans. 564 4. 1} by 12. Ans. 111. 12. 284 by 5. Ans. 1434 b. 38 by 18. Ans. 91. 13. 174 by 7.

Ans. 125. 8. H by 14. Ano. 71 14. 2448 by 12.

Ans. 2971 7. He by 28. Ans. 284. 15. 464 by 13.

Ans. 6054 8. 144 by 32. Ans. 15. 16. 12844 by 18. Ans. 23104. 9. Ms by 144. Ans. 1977. 17. 4281,4 by 11. Ans. 47097

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