parta, then the fraction, mat hate toasted of tvile the atm

bey of these part, and and have been

manner it might be expressed by

In the same

Hence, the value of a fraction is not altered by multiplying or dividing both its terms by the same quantity.

REDUCTION.

PROBLEM L

To reduce an integer to the form of a fraction.

If the denominator be given, multiply the integer by it for

the numerator, and under the product place the denominator. If no denominator is given, place unit for it.

Hence, a mixed quantity may be reduced to the form of a fraction by multiplying the integer by the denominator of the fraction, and adding the numerator to the product for the numerator, below which place the denominator.

1. Reduce 3a to a fraction, of which the denominator is 26.

To reduce an improper fraction to an integer or a mixed quantity.

Divide the numerator by the denominator, the quotient is the integer, the remainder, with the divisor below it, constitutes the fraction.

* When a fraction has the sign

x2 + xy + y2.

before it, all the signs of the nu

merator are to be changed. Here ab-a2 becomes -ab +

2

To reduce fractions of different denominators to others of the same value which have a common denominator.

Multiply each of the numerators into all the denominators, except its own, for the new numerators, and all the denominators together for the common denominator:

To reduce a fraction to lower terms.

Divide its numerator and denominator by any quantity which measures both.

The greatest divisor of the coefficients is found as in arithmetic, and the greatest simple divisor of the letters is discovered by inspection.

To find the greatest compound divisor.

Divide the greater by the less and the divisor by the remain

der continually, till nothing remain: the last divisor is the greatest common measure.

NOTE. The several divisors must be first divided by the greatest simple quantity which measures all their terms before they are used. Also the dividend must be sometimes multiplied by a simple quantity to make the division succeed. And any compound quantity in a remainder which does not measure the divisor from which it proceeds, may be taken out of it.

What is the greatest common measure of

a + — b✩

1.

2.

Divide by a+x.

5a5+10a4x+5a3 x2

a3x+2a2x2 + 2ax3 + x2

by a+x.

by a+b2.

ADDITION AND SUBTRACTION.

REDUCE the fractions to a common denominator, if they have different ones; then add or subtract their numerators, and

* In fractions like this, where a letter is but of one dimension in either the numerator or the denominator, divide it into two parts, one of which has that letter in every term; then find the common measure of these two parts, and try whether it will divide the other quantity. Here the parts of the denominator are 4adq+24ad and — 7fgq-42fg, and the common measure of these is q +6, which succeeds.

under the sum or the remainder write the common denominator, for the sum or the difference of the fractions.

MULTIPLICATION AND DIVISION.

MULTIPLY the numerators together for the numerator of the product, and the denominators together for its denominator. In division, invert the divisor and multiply as before.