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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.

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51. Reduce of 19 of

Operation.

2

10 3 2

of

Ans.

7-7

to a simple fraction.

First, we cancel the 4 and 3 in the numerator, then the 12 in the denominator, which is equal to the factors 4 and 3. Finally, we cancel the 5 in the de

nominator and the factor 5 in the numerator 10, placing the other factor 2 above. We have 2 left in the numerator, and 7 in the denominator.

52. Reduce

53. Reduce

Ans. 4.

of of 19 to a simple fraction.
ofofofto a simple fraction.

54. Reduce of of of to a simple fraction.

55. Reduce

10
6

of 12 of 39 to a simple fraction.

56. Reduce of 15 of of to a simple fraction.

57. Reduce of 19 of 13

20

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58. Reduce of of of to a simple fraction.

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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 ?

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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, 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, are multiplied by the denominator of the first, it becomes. The fractions and have a common denominator, and are respectively equal to the given fractions, viz, and . (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 1, 2, and to a common denominator.

Operation. 1x4×6=24)

3×2×6=36 the three numerators.

5×2×4 40

2×4×6=48, the common denominator.
The fractions required are 24, 38, and 48.

48' 489

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, 1, and

to a common denominator. 35 Ans. 84 140, 140, 140. 4. Reduce,, and to a common denominator. Reduce the following fractions to a common denomi

nator.

5. Reduce,,, and . 6. Reduce 3, 4, 6, and 2.
7. Reduce,,, and 7. 8. Reduce,, 13, and .
612, 3.

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

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Operation. 2)4" 6" 8

2)2

"3" 4

1 3

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First find the least common multiple of all the given denominators, (Art. 102,) and it will be the least common denominator required. The next step is to re

2×2×3×2=24, the duce the given fractions to twentyleast com. denom. fourths without altering their value. This may evidently be done by multiplying both terms of each fraction by the number of times its denominator is contained in 24. Thus 4 the denominator of the first fraction, is contained in 24, 6 times; hence, multiplying both terms of the fraction by 6, it becomes 1. The denominator 6 is contained in 24, 4 times; hence, multiplying the second fraction by 4, it becomes. The denominator 8 is contained in 24, 3 times; and multiplying the third fraction by 3, it becomes 1. Therefore 14, and are the fractions required. Hence,

18 8

126. To reduce fractions to their least common denominator.

I. Find the least common multiple of all the denominators of the given fractions, and it will be the least common denominator. (Art. 102.)

II. Multiply each given numerator by the number of times its denominator is contained in the least common denominator, and place the respective products over the least common denominator.

QUEST.-126. How are fractions reduced to the least common denomi

nator?

OBS. Multiplying each numerator by the number of times its denominator is contained in the least common denominator, is, in effect, multiplying both terms of the given fractions by the same number. For if we multiply each denominator by the number of times it is contained in the least common denominator, the product will be equal to the least common denominator. Hence, the new fractions thus obtained must be of the same value as the given fractions. (Art. 116.)

14. Reduce,, and to the least com. denominator.

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2×3×2=12, the least com. denominator. Now 12-3x2=8, numerator of 1st.

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Ex. 1. What is the sum of 1, 2, 3, and ?

Suggestion. Since all these fractions have the same denominator, it is plain their numerators may be added as well as so many pounds or bushels, and their sum placed over the common denominator, will be the answer required. Thus 1 eighth and 2 eighths are 3 eighths, and 3 are 6 eighths, and 5 are 11 eighths. Ans. 1, or 18.

QUEST.-Obs. Does this process alter the value of the given fractions? Why?

2. What is the sum of 1, 4, 1, and 2?

3. What is the sum of 4. What is the sum of 5. What is the sum of 6. What is the sum of 7. What is the sum of 8. What is the sum of 9. What is the sum of

10. What is the sum of

19, 1, and 1?

6 2

3, 3, 3, and 8?

13,

10 5

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1, 1, 1, and 1?

5,

15,

4,

8 1

5, 2, 3, and 2?

25' 25'

7 4

25

15, 15, 19, and 10?
4, 64' 64'
20 30 3 5 and 84

16, 10

8 ?

1 8 and 1

45' 45' 45' 45'

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?

10, and 180?

100, 100, 100,

EXERCISES FOR THE SLATE.

11. What is the sum of and?

Suggestion. A difficulty presents itself here; for it is manifest that 1 half added to 1 third are neither 2 halves nor 2 thirds. (Art. 22.) This difficulty may be removed by reducing the given fractions to a common denominator. (Art. 125.) Thus,

1x3=3

1X22 the new numerators.

2x3=6, the common denominator.

The fractions reduced are 3 and 2, and may now be added. Thus 3+2=5.

12. What is the sum of 3, 4, and ?

Ans..

Ans. 12, or 12.

127. From these illustrations we deduce the following

RULE FOR ADDITION OF FRACTIONS.

Reduce the fractions to a common denominator; add their numerators, and place the sum over the common denominator.

OBS. 1. Compound fractions must, of course, be reduced to simple

QUEST.-127 How are fractions added? Obs. What must be done with compound fractions?

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