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Dividing the denominator by 5 it becomes, or 3.

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7 m3 c + 42 m2 a c3

by

5.

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Dividing the denominator by bit becomes, or a.

b

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In fact

as much as

multiplied by 6 is — = 1, and being a times

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b

must give a product a times as large, or a

times 1, which is a.

Hence, if a fraction be multiplied by its denominator, the product will be the numerator.

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Two ways have been shown to multiply fractions, and two

In

by

17 a2.

17 a2

by

10 ab-2 c.

by a x2-3 am +b.

ways to divide them.

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Reducing Fractions to Lower Terns.

XVII. If both numerator and denominator be multiplied by the same number, the value of the fraction will not be altered.

Arith. Art. XIX.

For multiplying the numerator multiplies the fraction, and multiplying the denominator divides it; hence it will be multiplied and the product divided by the multiplier, which reproduces the multiplicand.

α

In other words, signifies that a contains 5 a certain num

b

ber of times, if a is as large or larger than b; or a part of one time, if b is larger than a. Now it is evident that 2 a will contain 2 b just as often, since both numbers are twice as large as before.

So dividing both numerator and denominator, both divides and multiplies by the same number.

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Hence, if a fraction contain the same factor both in the numerator and denominator, it may be rejected in both, that is, both may be divided by it. This is called reducing fractions to lower terms.

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7. Reduce

to its lowest terms

108 a x2+81 x 90 m2 x3

8. Divide 35 a b m3 x3 by 7 a3n m3 x.

Write the divisor under the dividend in the form of a fraction, and reduce it to its lowest terms.

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21. Divide 18 a2 m2-54 a m2 + 42 a3 m*

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22. Divide (a+b) (13 ac+bc) by (m2—c) (a+b).

23. Divide 3 c2 (a-2c)3 by 2 b c3 (a — 2 c)3. 24. Divide 36 b3 c2 (2 a + d)2 (7 b — d)3

by 1263 (2a + d)2 (7 b — d)3 (a —d).

Addition and Subtraction of Fractions.

α

XVIII. Add together and and.

This addition may be expressed by writing the fractions one after the other with the sign of addition between them; thus

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N. B. When fractions are connected by the signs and —, the sign should stand directly in a line with the line of the fraction.

It is frequently necessary to add the numerators together, in which case, the fractions, if they are not of the same denomination, must first be reduced to a common denominator, as in Arithmetic, Art. XIX.

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These must be reduced to a common denominator. It has been shown above that if both numerator and denominator be multiplied by the same number, the value of the fraction will not be altered. If both the numerator and denominator of the first fraction be multiplied by 7, and those of the second by 5, the fractions become and. They are now both of the same denomination, and their numerators may be added. The answer is 3.

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Multiply both terms of the first by d, and of the second by b, they become and

a d b c

The denominators are now alike

b d b ď

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In all cases the denominators will be alike if both terms of each fraction be multiplied by the denominators of all the others. For then they will all consist of the same factors.

Applying this rule to the above example, the fractions bead fh befh bdeh bdfg bdfh' bdfh' bdfh bdfh

come

and

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