Addition and Subtraction of Fractions

Two or more fractions with the same denominator are said to have a common denominator. The following rules for adding and subtracting fractions with a common denominator can be derived from the basic algebraic properties of real numbers and the definition of a quotient.

 

Rules for Addition Subtraction of Fractions with Common Denominator

If $$p$$, $$q$$ and $$r$$ are real numbers, $$q \ne 0$$, then $$\frac{p}{q} + \frac{r}{q} = \frac{{p + r}}{q}$$   and   $$\frac{p}{q} – \frac{r}{q} = \frac{{p – r}}{q}$$

For instance, $$\frac{3}{7} + \frac{2}{7} = \frac{{3 + 2}}{7} = \frac{5}{7}$$   and   $$\frac{3}{7} – \frac{2}{7} = \frac{{3 – 2}}{7} = \frac{1}{7}$$

Again, the same rules apply to adding and subtracting fractional expressions.

 

Example:

Perform each operation

(a) $$\frac{{5x}}{{2x – 1}} + \frac{{3x}}{{2x – 1}}$$
(b) $$\frac{{5x}}{{{{\left( {3x – 2} \right)}^2}}} – \frac{{3x}}{{{{\left( {3x – 2} \right)}^2}}}$$

Solution:

(a) $$\frac{{5x}}{{2x – 1}} + \frac{{3x}}{{2x – 1}} = \frac{{5x + 3x}}{{2x – 1}} = \frac{{8x}}{{2x – 1}}$$
(b) $$\frac{{5x}}{{{{\left( {3x – 2} \right)}^2}}} – \frac{{3x}}{{{{\left( {3x – 2} \right)}^2}}} = \frac{{5x – 3x}}{{{{\left( {3x – 2} \right)}^2}}} = \frac{{2x}}{{{{\left( {3x – 2} \right)}^2}}}$$

To add or subtract fractions that do not have a common denominator, you must rewrite the fractions so they do have denominators. To do this, multiply the numerator and denominator of each fraction by an appropriate number. For instance,

\[\frac{2}{3} + \frac{4}{5} = \frac{{2 \cdot 5}}{{3 \cdot 5}} + \frac{{3 \cdot 4}}{{3 \cdot 5}} = \frac{{10}}{{15}} + \frac{{12}}{{15}} = \frac{{10 + 12}}{{15}} = \frac{{22}}{{15}}\]

More generally, if $$q \ne s$$, you can always add $$p/q$$ and $$r/s$$ as follows:
\[\frac{p}{q} + \frac{r}{s} = \frac{{p \cdot s}}{{q \cdot s}} + \frac{{q \cdot r}}{{q \cdot s}} = \frac{{ps + qr}}{{qs}}\]

Example:

Add the expression $$\frac{{3x}}{{4x – 1}}$$ and $$\frac{{2x}}{{3x – 5}}$$

Solution:

$$\frac{{3x}}{{4x – 1}} + \frac{{2x}}{{3x – 5}} = \frac{{3x\left( {3x – 5} \right)}}{{\left( {4x – 1} \right)\left( {3x – 5} \right)}} + \frac{{\left( {4x – 1} \right)2x}}{{\left( {4x – 1} \right)\left( {3x – 5} \right)}}$$
$$ = \frac{{3x\left( {3x – 5} \right) + \left( {4x – 1} \right)2x}}{{\left( {4x – 1} \right)\left( {3x – 5} \right)}} = \frac{{9{x^2} – 15x + 8{x^2} – 2x}}{{\left( {4x – 1} \right)\left( {3x – 5} \right)}}$$
$$ = \frac{{17{x^2} – 17x}}{{\left( {4x – 1} \right)\left( {3x – 5} \right)}} = \frac{{17x\left( {x – 1} \right)}}{{\left( {4x – 1} \right)\left( {3x – 5} \right)}}$$