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86. To reduce fractions to equivalent fractions having a common denominator.

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OPERATION.

a

by

с

b x

=

=

a b x

b2 x y

bcy

b2 x y

We multiply the numerator and denominator of each fraction by the denominator of the other (Art. 83, Cor.). This must reduce them to equivalent fractions having a common denominator, as the new denominator

of each fraction is the product of the same factors.

a

by

C

OR,

a x

b x y

cy

b x

bxy

In the second operation we find the least common multiple, bxy, of the denominators by and bx; as each denominator is contained in this multiple, each fraction can be reduced to a fraction with this multiple as a de

nominator, by multiplying its numerator and denominator by the quotient arising from dividing this multiple by its denominator. Hence,

RULE.

Multiply all the denominators together for a common denominator, and multiply each numerator into the continued product of all the denominators, except its own, for new numerators. Or,

Find the least common multiple of the denominators for the least common denominator. For new numerators, multiply each numerator by the quotient arising from dividing this multiple by its denominator.

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OPERATION.

b

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b + c

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If anything is divided into equal parts, a number of these parts represented by b, added to a number represented by c, gives b+c of

these parts. In the example given, a unit is divided into x equal parts, and it is required to find the sum of b and c of these parts; i. e. с b + c

b

2

+

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It is evident, therefore, that fractions that have a common denominator can be added by adding their numerators. But fractions that do not have a common denominator can be reduced to equivalent fractions having a common denominator. Hence,

RULE.

Reduce the fractions, if necessary, to equivalent fractions having a common denominator; then write the sum of the numerators over the common denominator.

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Reduce the fractions, if necessary, to equivalent fractions having a common denominator; then subtract the numerator of the subtrahend from that of the minuend, and write the result over the common denominator.

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89. To reduce a mixed quantity to an improper fraction.

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8x + a

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As eight eighths make a unit, there will be in x units eight times x

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Hence,

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