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2. The value of a fraction is not altered by reducing it to its lowest terms. (Art. 116.)

3. When the terms of the fraction are small, the former method will generally be found to be the shorter and more convenient ; but when the terms are large, it is impossible to tell whether the fraction is in its simplest form, without finding their greatest common divisor. 2. Reduce 86 to its lowest terms.

Ans. 3. Reduce '

4. Reduce 5. Reduce 16

6. Reduce is 7. Reduce 4

8. Reduce 15 9. Reduce 63 105 10. Reduce 55

121: 11. Reduce 240

12. Reduce 126 13. Reduce 423.

14 Reduce 435

957 15. Reduce 385

16. Reduce 1740 17. Reduce 594

18. Reduce 64 65 2142

312

162•

2900

7335

CASE II.

19. Reduce to a whole or mixed number. Suggestion. The object is to find a whole or mixed number, whose value is equal to the given

Operation. fraction. But the value of a fraction is the quo 5)17 tient of the numerator divided by the denominator. (Art. 110.) Hence,

33 Ans.

121. To reduce an improper fraction to a whole, or mixed number.

Divide the numerator by the denominator, and the quotient will be the whole, or mixed number required. 20. Reduce to a whole or mixed number.

Ans. 93. Reduce the following fractions to whole or mixed numbers.

QUEST.--121. How is an improper fraction reduced to a whole or mixed number?

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31. Reduce the mixed number 15to an improper fraction.

Operation.

153

4
603 Ans.

Obs. In 1 there are 4 fourths, and in 15, there are 15 times as many. 4X15=60, and three fourths make 63 fourths. Hence,

122, To reduce a mixed number to an improper fraction.

Multiply the whole number by the denominator of the fraction : to the product add the given numerator. The sum placed over the given denominator, will form the improper fraction required.

Obs. 1. Any whole number may be expressed in the form of a fraction without altering its value, by making 1 the denominator.

2. A whole number may also be reduced to a fraction of any denominator, by multiplying the given number by the proposed denominator; the product will be the numerator of the fraction required. Thus 25 may be expressed by 9, 190,0

&c., for 25=y=100=400&c. So 12 = 1 ===%, for the quotient of each of these numerators divided by its denominator, is 12.

32. Reduce 8.1 to an improper fraction. Ans. Reduce the following numbers to improper fractions :

or

400
16,

QUEST.–122. How reduce a mixed number to an improper fraction. Obs. How express a whole number in the form of a fraction. How reduce it to a fraction of a given denominator.

33. Reduce 93 34. Reduce 165 35. Reduce 237 36. Reduce 45'2. 37. Reduce 6415 38. Reduce 5643. 39. Reduce 3041. 40. Reduce 7251 41. Reduce 45 to fifths. 42. Reduce 72 to eighths. 43. Reduce 830 to sixths. 44. Reduce 743 to fifteenths.

CASE IV.

45. Reduce of to a simple fraction.

Suggestion. of, is 2 times as much as 1 third of . Now of is ; for, multiplying the denominator divides the value of the fraction. (Art. 113.) And 2 thirds is 2 times, which is equal to ji, or. (Art. 120.) The answer is 4

Obs. This operation consists in simply multiplying the two numerators together and the two denominators. Hence,

123. To reduce compound fractions to simple ones.

Multiply all the numerators together for a new numerator, and ull the denominators together for a new denominator. 46. Reduce of gof; to a simple fraction.

Ans. 344, or : 47. Reduce of 19 of 1 to a simple fraction. 48. Reduce of of to a simple fraction. 49. Reduces of 1 of i} to a simple fraction.

124. In reducing compound fractions to simple ones, the operation may often be shortened by canceling the factors which are common both to the numerator and denominator.

Obs. 1. This, in effect, is dividing the product of the numerators

Quest.-What is of ? How does it appear that multiplying the denominator by 3 gives l' third of the fraction ? 123. How then are compound fractions reduced to simple ones ? 124. How may the operation be shortened? Obs. How does it appear that this method will give the true answer?

and the product of the denominators by the same number, and consequently does not alter the value of the fraction. (Art. 116.)

2. This method not only shortens the operation of multiplying, but at the same time reduces the answer to lower terms. A little practice will give the learner great facility in its application.

50. Reduce of off to simple fractions.

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of of of

Operation.

First we cancel the factors 2 and

3, which are common to the nu

5 or of

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merators and denominators, and then multiply the remaining fac

tors together for the answer. 51. Reduce of 19 of } to a simple fraction. Operation.

First, we cancel the 4 and 3 2

in the numerator, then the 12 in 12 3

the denominator, which is equal of of

*= Ans. 12

to the factors 4 and 3. Final

ly, we cancel the 5 in the denominator and the factor 5 in the numerator 10, placing the other factor 2 above. We have 2 left in the numera'tor, and 7 in the denominator.

Ans. 52. Reduce of g of 19 to a simple fraction. 53. Reduce of of 1 of 1 to a simple fraction. 54. Reduce ã of of of jo to a simple fraction. 55. Reduce of 1 of 39 to a simple fraction.

10 56. Reduce 4 of 1 of j of zo to a simple fraction. 57. Reduce of 19 of 13 of to a simple fraction. 58. Reduce ia of , of of į to a simple fraction. Note. For reducing complex fractions to simple ones, see Art. 143.

CASE V. Ex. 1. Reduce ] and } to a common denominator. Note.-Two or more fractions are said to have a common denominator, when they have the same denominator.

QUEST.-Obs. What is the advantage of this method ? Note. What is meant by a common denominator ?

Suggestion. The object of this example is to find two other fractions, which have the same denominator, and whose values are respectively equal to the values of the given fractions, į and {. Now, if both terms of the first. fraction }, are multiplied by the denominator of the second, it becomes, and if both terms of the second fraction ļ, are multiplied by the denominator of the first, it becomes . The fractions and â have a common denominator, and are respectively equal to the given fractions, viz : =, and =1. (Art. 116.) Hence,

125. To reduce fractions to a common denominator.

Multiply each numerator into all the denominators except its own for a new numerator, and all the denominators together for a common denominator. 2. Reduce ì, , and to a common denominator.

Operation. 1 X4 X6=24 3x2x6=36 the three numerators. 5X2X4140) 2 X4 X6=48, the common denominator. The fractions required are 2 , to, and 48. Obs. It is manifest that the process of reducing fractions to a common denominator, does not change their value, for it is simply multiplying each numerator and denominator of the given fractions by the same number. (Art. 116.) 3. Reduce , \, and , to a common denominator.

Ans. 4. O, 128 4. Reduce , s, and to a common denominator.

Reduce the following fractions to a common denominator. 5. Reduce , , s, and s. 6. Reduce ģ,

6. Reduce, , $, and . 37. Reduce 3, 4, 16, and 15. 8. Reduce 10, 4, 13, and

QUEST.-125. How are fractions reduced to a common denominator ? Obs. Does the process of reducing fractions to a common denominator al. es the value of the fractions ? Why not?

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