Elementary Fractions.

94. DEFINITION. If a quantity x be divided into b equal parts, and a of these parts be taken, the result is called the fraction of x. If x be the unit, the fraction of x is called

α

a

a

b

simply "the fraction"; so that the fraction represents a Ъ equal parts, b of which make up the unit.

95. In this chapter we propose to deal only with the easier kinds of fractions, where the numerator and denominator are simple expressions.

Their reduction and simplification will be performed by the usual arithmetical rules. For the proofs of these rules the reader is referred to the Elementary Algebra for Schools, Chapter XV.

Rule. To reduce a fraction to its lowest terms: divide numerator and denominator by every factor which is common to them both, that is by their highest common factor.

Dividing numerator and denominator of a fraction by a common factor is called cancelling that factor.

Multiplication and Division of Fractions.

96. Rule. To multiply algebraical fractions: as in Arithmetic, multiply together all the numerators for a new numerator, and all the denominators for a new denominator.

2a 5x2 362 2a × 5x2 × 3b2 5x

Example 1. X

X

3b 2a2b 2x 3b × 2a2b × 2x 2α

by cancelling like factors in numerator and denominator.

all the factors cancelling each other.

97. Rule. To divide one fraction by another: invert the divisor and proceed as in multiplication.

Reduction to a Common Denominator.

98. In order to find the sum or difference of any fractions, we must, as in Arithmetic, first reduce them to a common denominator; and it is most convenient to take the lowest common multiple of the denominators of the given fractions.

Example. Express with lowest common denominator the fractions a b с

3xy' 6xyz' 2yz

The lowest common multiple of the denominators is 6xyz. Multiplying the numerator of each fraction by the factor which is required to make its denominator 6xyz, we have the equivalent fractions

2az b 3cx

6xyz' 6xyz' 6xyz

Note. The same result would clearly be obtained by dividing the lowest common denominator by each of the denominators in turn, and multiplying the corresponding numerators by the respective quotients.

Addition and Subtraction of Fractions.

99. Rule. To add or subtract fractions: express all the fractions with their lowest common denominator; form the algebraical sum of the numerators, and retain the common denominator.

5x 3 7x Example 1. Simplify +x3 4

The least common denominator is 12.

20x+9x14x15x5x

The expression

=

Example 2. Simplify

10x

Gab-5ab- ab 0

5x 2x

The expression

10x

= = 0.

10x

The expression = 6ax-cy, and admits of no further simplification.

" 3a3c2

Note. The beginner must be careful to distinguish between erasing equal terms with different signs, as in Example 2, and cancelling equal factors in the course of multiplication, or in reducing fractions to lowest terms. Moreover, in simplifying fractions he must remember that a factor can only be removed from numerator and denominator when it divides each taken as a whole.

Thus in 6ax-cy c cannot be cancelled because it only divides cy

3a3c2

and not the whole numerator. Similarly a cannot be cancelled because it only divides 6ax and not the whole numerator. The fraction is therefore in its simplest form.

When no denominator is expressed the denominator 1 may be understood.

If a fraction is not in its lowest terms it should be simplified before combining it with other fractions.

CHAPTER XIII.

SIMULTANEOUS EQUATIONS.

100. CONSIDER the equation 2x+5y=23, which contains two unknown quantities.

From this it appears that for every value we choose to give to x there will be one corresponding value of y. Thus we shall be able to find as many pairs of values as we please which satisfy the given equation.

For instance, if x=1, then from (1) we obtain

21

y

But if also we have a second equation containing the same unknown quantities, such as 3x+4y=24,

we have from this

y=

24-3x
4

.(2).

If now we seek values of x and y which satisfy both equations, the values of y in (1) and (2) must be identical.

Substituting this value in the first equation, we have

Thus, if both equations are to be satisfied by the same ́alues of x and y, there is only one solution possible.