117. The denominator of a fraction shows into how many parts a unit is to be divided, and the numerator shows how many of those parts are to be taken. Fractions can only be added when they are like parts of unity; that is, when they have a common denominator. In that case, the numerator of each fraction will indicate how many times the common fractional unit is repeated in that fraction, and the sum of the numerators will indicate how many times this result is repeated in the sum of the fractions. Hence we have the following

RULE.

Reduce the fractions to a common denominator; then add the numerators together, and write their sum over the common denomi

nator.

If there are mixed quantities, we may add the entire and fractional parts separately.

Бас

6

Adding the numerators, we obtain

It is plain that three sixths of x and two sixths of a make five sixths of x.

5. What is the sum of 2a, 3a+5' and a+8x?

6. What is the sum of a+x,

9

Ans. 6a+

and a-x?

2a

a+2x

4x

58x

45

a

Ans. a+x+2+

a2-ax

Ans. a.

2

2

118. Fractions can only be subtracted when they are like parts of unity; that is, when they have a common denominator. In that case, the difference of the numerators will indicate how many times the common fractional unit is repeated in the difference of the fractions. Hence we have the following

RULE.

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

and it is plain that ten fifteenths of x, diminished by nine fifteenths of x, equals one fifteenth of x.

It must be remembered that a minus sign before the dividing line of a fraction affects the quotient (Art. 111); and since a quantity is subtracted by changing its sign, the result of the subtraction in this case is

which fractions may be reduced to a common denominator, and the like terms united as in addition.

a

119. Let it be required to multiply by &

First let us multiply by c. According to the first principle of Art. 109, the product must be

ac

But the proposed multiplier was ; that is, we have used a multiplier d times too great. We must therefore divide the result by d; and, according to the second principle of Art. 109, we obtain

ac

bd

a с ас

; that is, &=
bd bd

Hence we have the following

RULE.

Multiply the numerators together for a new numerator, and the denominators for a new denominator.

Entire and mixed quantities should first be reduced to frac tional forms. Also, if there are any factors common to the numerator and denominator of the product, they should be canceled.