126. To reduce a mixed quantity to the form of a fraction.

This case is the converse of the last, and may be explained by it. Hence the following

RULE. Multiply the entire part by the denominator of the fraction; add the numerator if the sign of the fraction be plus, but subtract it if the sign be minus, and write the result over the denomi

127. To transfer a factor from the denominator to the numerator, or the reverse.

ахт

Let us take any fraction, as and multiply both numerator

byn

and denominator by y, observing that any factor having zero for its exponent is equal to unity, (88, 1), and may therefore be omitted. We shall have

If we multiply both numerator and denominator of the same frac tion by a-", we shall have

In like manner we may transfer any factor having a negative ex

ах

ponent. For example, let us take the fraction, and multiply " b

both numerator and denominator by x; we shall have

ах-8

ах-8+8 ахо α

b

bxs bxbx

By the same principle also, any fraction may be reduced to the form of an entire quantity; thus,

In all operations of this kind, the intermediate steps may be omitted, and the results obtained by the following

RULE. I. To transfer any factor from the denominator to the nu merator, or the reverse:— ·Change the sign of its exponent.

II. To reduce any fraction to the form of an entire quantity :-Transfer all the factors of the denominator to the numerator, observing to change the signs of the exponents of the factors transferred.

EXAMPLES FOR PRACTICE.

In each of the following fractions, transfer the unknown factors, or factors containing unknown quantities, to the numerator.

In each of the following fractions, transfer the known factors to the depominator, and the unknown factors to the numerator.

In the following fractions, transfer the factors having negative

Reduce each of the following fractions to the form of an entire

128. To reduce one or more fractions to a common denominator.

We have seen (124) that a fraction may be reduced to lower terms by division. Conversely, a fraction must be reduced to higher terms by multiplication, and each of the higher denominators it may have, must be some multiple of its lowest denominator. Hence,

1.-A common denominator to which two or more fractions may be reduced, must be a common multiple of their lowest denominators; and

2. The least common denominator of two or more fractions, must be the least common multiple of their denominators.

We find by inspection that the least common multiple of the given denominators is ab3. And

If, therefore, we multiply both numerator and denominator of the first fraction by b2, and of the second by a, we shall reduce the two fractions to their least common denominator, al3. Thus,

c>b2=¿1c, new numerator of first fraction;

daad, new numerator of second fraction.
с d b'c ad
a2b ab3 a2 2

Hence

=

Ans.

From these principles and illustrations we deduce the following RULE. I. Find the least common multiple of all the denominators, for the least common denominator.

II. Divide this common denominator by each of the given denominators, and multiply each numerator by the corresponding quotient. The products will be the new numerators.

NOTE.-Mixed numbers should first be reduced to fractions, and all fractions to their lowest terms.

EXAMPLES FOR PRACTICE.

In each of the following examples, reduce the fractions and mixed quantities to their least common denominator :

ay-1

ay1-y

ays - y