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115. It is evident from the preceding articles, that, multiplying the numerator by any number, has the same effect on the value of the fraction, as dividing the denominator by that number. (Arts. 111, 114.)

Dividing the numerator has the same effect, as multiplying the denominator. (Arts. 112, 113.)

116. If the numerator and denominator are both multiplied or both divided by the same number, the value of the fraction will not be altered. (Art. 88.)

Thus, 12=3. Now if the numerator and denominator are both multiplied by 2, the fraction becomes 24; whose value is 3. If both terms are divided by 2, the fraction becomes 9, whose value is 3; that is, 12-24 = =3.

=

117. Since the value of a fraction is the quotient of the numerator divided by the denominator, it follows, that If the numerator and denominator are equal, the value is a unit or one.

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If the numerator is greater than the denominator, the value is greater than one. Thus, 2, 13, &c.

If the numerator is less than the denominator, the value is less than one. Thus, =1 third of one, =4 fifths of one, &c.

118. It will be seen from the preceding exercises, that fractions may be added, subtracted, multiplied, and divided, as well as whole numbers.

OBS. 1. In order to perform these operations, it is often necessary to make certain changes in the terms of the fractions.

QUEST.-115. What may be done to the denominator to produce the same effect on the value, as multiplying the numerator by a given number? What, to produce the same effect as dividing the numerator by a given number? 116. What is the effect if the numerator and denominator are both multiplied or both divided by the same number? 117. When the numerator and denominator are equal, what is the value of the fraction? When the numerator is the larger, what? When smaller, what?

2. It is evident that any changes may be made in the terms of a fraction, which do not alter the quotient; for, if the quotient is not altered, the value remains the same. Thus the terms of the fraction

may be changed into 2, 8, 16, &c., without altering its value; for in each case the quotient of the numerator divided by the denominator is 2. Hence, for any given fraction, we may substitute any other fraction, which will give the same quotient.

REDUCTION OF FRACTIONS.

119. The process of changing the terms of a fraction into others, without altering its value, is called reduction of fractions.

2)

CASE I.

EXERCISES FOR THE SLATE.

6

Ex. 1. Reduce to its lowest terms.

First Operation.
again, 3). Ans.

Dividing both terms of the fraction by 2, it becomes again, dividing terms are the lowest to

both by 3, we obtain, whose
which the given fraction can be reduced.

Second Operation. 6). Ans.

If we divide both terms by 6, their greatest common divisor, (Art. 96.) the given fraction will be reduced to

its lowest terms by a single division. Hence,

120. To reduce a fraction to its lowest terms.

Divide the numerator and denominator by any number which will divide them both without a remainder; and thus continue the operation, till there is no number greater than 1 that will divide them exactly.

Or, divide both the numerator and denominator by their greatest common divisor: the two quotients thus arising will be the least terms to which the given fraction can be reduced. (Art. 96.)

OBS. 1. A fraction is reduced to its lowest terms, when its numerator and denominator are expressed in the smallest numbers possible.

QUEST-Obs. What changes may be made in the terms of a fraction? 119. What is meant by reduction of fractions? 120. How is a fraction reduced to its lowest terms? Obs. What is meant by lowest terms? Is the value of a fraction altered by reducing it to its lowest terms?

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.

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Suggestion. The object is to find a whole or mixed number, whose value is equal to the given fraction. But the value of a fraction is the quotient of the numerator divided by the denominator. (Art. 110.) Hence,

Operation. 5)17

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 29 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 15 to an improper fraction.

Operation.

153

4

63 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.

:

The

Multiply the whole number by the denominator of the fraction to the product add the given numerator. 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.

60

Thus 25 may be expressed by 25, 100, or 400, &c., for 25-25-100-400, &c. So 12=1=2=36 for the quotient of each of these numerators divided by its denominator, is 12.

Ans. 25.

32. Reduce 83 to an improper fraction. Reduce the following numbers to improper fractions:

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. 35. Reduce 237. 37. Reduce 64.

34. Reduce 165.
36. Reduce 4512.
38. Reduce 5623.

39. Reduce 304.

40. Reduce 725.

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 12, or 4. (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 all the denominators together for a new denomi

nator.

46. Reduce of of to a simple fraction.

48

Ans. 4, or 21. 47. Reduce of 19 of 2 to a simple fraction. 48. Reduce of of to a simple fraction. 49. Reduce of of 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 de

nominator.

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

QUEST.-What is of ? How does it appear that multiplying the 123. How then are denominator by 3 gives 1 third of the fraction? 124. How may the operacompound fractions reduced to simple ones? tion be shortened? Obs. How does it appear that this method will give the true answer?

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