The arithmetical process at p. 48 is only a convenient method of putting down the work, where the divisors or common measures that will suit are found by inspection or by trials.

ALGEBRAIC FRACTIONS.

ALGEBRAIC FRACTIONS have the same names and rules of operation, as numerical fractions in common arithmetic; as appears in the following Rules and Cases.

CASE I.

To reduce a mixed quantity to an improper fraction.

MULTIPLY the integer by the denominator of the fraction, and to the product add the numerator, or connect it with its proper sign, + or ; then the denominator being set under this sum, will give the improper fraction required.

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

DIVIDE the numerator by the denominator, for the integral part; and set the remainder, if any, over the denominator, for the fractional part; the two joined together will be the mixed quantity required.

To reduce fractions to a common denominator.

MULTIPLY each numerator by all the denominators except its own for a new

* In such examples as these the multiplication may be advantageously performed by detached coefficients.

numerator of the equivalent fraction; and all the denominators together for a common denominator to all the fractions equivalent to the given ones *.

It will be convenient in putting down the work to write all the numerators in succession in a vertical column, and commence the factors which follow by the denominators of those fractions which succeed them.

For this is only multiplying the numerator and denominator of each fraction by equal quantities, which does not alter its value.

CASE IV.

To reduce fractions to their least common denominator.

FIND the least common multiple M of their denominators for a new denominator. Divide M by each of the denominators D, D1, D1⁄2 ............, and multiply the corresponding numerators N, N,, N2, by these quotients, for the corresponding new numerators.

Here abc is the least common multiple of the denominators.

Exx. 2, 3, 4. Reduce Ex. 6, 8, 9, of the last case to their least common denominators.

CASE V.

To reduce a fraction to its lowest terms.

FIND the greatest common measure of its numerator and denominator. Then divide both the terms of the fraction by the common measure thus found, and it will reduce it to its lowest terms at once, as was required. Or, divide the terms by any quantity which it may appear will divide them both, as explained in art. (6), p. 134.

b

c2

Here a + b is the greatest common measure, by which, dividing the nume

rator and denominator, we have

for the fraction in its lowest terms.

Ir the fractions have a common denominator, add all the numerators together; then under their sum set the common denominator, and it is done.

If they have not a common denominator, reduce them to one (the least), and then add them as before.

* In the addition of mixed quantities, it is best to bring the fractional parts only to a common denominator, and to annex their sum to the sum of the integers, with the proper sign. And the same rule may be observed for mixed quantities in subtraction also. If, however, in the sum of the fractions thus obtained there should happen to appear integer quantities, these will be better brought out and united with the integers of the given quantities.

Though, it must be further remarked, it is often the more convenient method to thus reduce the quantities to a mixed state, especially in the arithmetical part of a process; yet it also frequently happens (and this more particularly where the formula itself is the object of consideration) that the more elegant result is obtained by reducing the whole to the form of one fraction.

See also, the note to addition of fractions in the arithmetic, p. 50.