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4. Reduce 3c+5 to a fraction whose denominator is 16c2.

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5. Reduce a-b to a fraction, whose denominator is a2-2ab+b2. a3—3a2b+3ab2—b3___(a—b)3 a2-2ab+b2 (a-b)

Ans.

ART. 136.-To convert a fraction to an equivalent one, having a denominator equal to some multiple of the denominator of the given fraction.

1. Reduce to a fraction, whose denominator is bc.

It is evident, that the terms must be multiplied by the same quantity, so as not to change the value of the fraction. It is then required to find, what the denominator, b, must be multiplied by, that the product shall become bc; but, it is evident, this multiple will be found, by dividing be by b, which gives the quotient, c. Then, multiplying both terms of the fraction by c, the result is b

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which is equal to the given fraction, and has, for its denominator bc. Hence, the

RULE,

FOR CONVERTING A FRACTION TO AN EQUIVALENT ONE, HAVING A GIVEN DENOMINATOR.

Divide the given denominator by the denominator of the given fraction, and multiply both terms by the quotient.

REMARK. This rule is perfectly general, but it is never applied, except where the required denominator is a multiple of the given one. In other cases, it would produce a complex fraction. Thus, if it is required to convert into an equivalent fraction, whose denominator is 5, the numerator of the new fraction would be 24.

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3. Convert to an equivalent fraction, having the denomina

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Ans.

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REVIEW.

134. If each fraction is not in its lowest terms, before commencing the operation, what is to be done? 135. How do you reduce an entire quantity to the form of a fraction having a given denominator?

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6. Convert to an equivalent fraction, having the denomi

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nator a2(b+c)2.

CASE V.

a3 (b+c)

Ans.

a2 (b+c) 2°

ADDITION AND SUBTRACTION OF FRACTIONS.

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ART. 137.-1. Let it be required to find the sum of Here, both parts being of the same kind, that is, fifths, we may add them together, and the sum is 6 fifths, (§).

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2. Let it be required to find the sum of and

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Here, the parts being of the same kind, that is, mths, we may, as in the first case, add the numerators, and write the result over the common denominator.

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Here, the parts not being of the same kind, that is, the denominators being different, we can not add the numerators together, and call them by the same name. We may, however, reduce them to a common denominator, and then add them together.

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

ART. 138. It is obvious, that the same principles would apply, if it were required to find the difference between two fractions; that is, if their denominators were the same, the numerators might be subtracted; but, if their denominators were different, it would be necessary to reduce them to the same denominator, before performing the subtraction. Hence, the RULE,

FOR THE SUBTRACTION OF FRACTIONS.

Reduce the fractions, if necessary, to a common denominator; then subtract the numerator of the fraction to be subtracted from the numerator of the other, and place the remainder over the common denominator.

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When entire quantities and fractions are to be added together, they may be connected by the sign of addition, or the entire quan tities and the fractions may be reduced to a common denominator, and the addition then performed.

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REVIEW.-136. How do you convert a fraction to an equivalent one, having a given denominator? Explain the operation by an example. 137. When fractions have the same denominator, how do you add them together? When fractions have different denominators, how do you add them together?

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REVIEW. 138. If two fractions have the same denominator, how do you find their difference? When two fractions have different denominators, how do you find their difference?

CASE VI.

TO MULTIPLY ONE FRACTIONAL QUANTITY BY ANOTHER.

ART. 139. To multiply a fraction by an entire quantity, or an entire quantity by a fraction.

It is evident, from Prop. I., Art. 122, that in multiplying the numerator of a fraction by an entire quantity, the fraction is increased as many times as there are units in the multiplier.

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Thus, taken twice, is ; and taken m times, is

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b Again, when two quantities are to be multiplied together, either may be made the multiplier (Art. 67); to multiply 4 by is the same as to multiply by 4. Or, to multiply m by is the b'

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FOR THE MULTIPLICATION OF A FRACTION BY AN ENTIRE QUANTITY,
OR OF AN ENTIRE QUANTITY BY A FRACTION.

Multiply the numerator by the entire quantity, and write the product over the denominator.

Since (See Art. 125,) dividing the denominator of a fraction increases the value of the fraction, as many times as there are units in the divisor, it is evident, that any fraction will be multiplied by an entire quantity, if the denominator of the fraction be divided by the entire quantity. Thus, in multiplying by 2, we may divide the denominator by 2, and the result will be, which is the same as to multiply by 2, and reduce the resulting fraction to its lowest terms. Hence, in multiplying a fraction and an entire quantity together, we should always divide the denominator of the fraction by the entire quantity, when it can be done without a remainder.

REMARK.-The expression, "What is two thirds of 6?" has the same meaning, as "What is the product of 6 multiplied by ?" The reason of the rule for the multiplication of an entire quantity by a fraction, may be shown otherwise, thus: one third of a is ; two thirds is twice as much as

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REVIEW.-139. How do you multiply a fraction by an entire quantity, or an entire quantity by a fraction? When the denominator of the fraction is a multiple of the entire quantity, what is the shortest method of finding their product?

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