# Adding and subtracting algebraic fractions

When adding and subtracting fractions, we must ensure that we have the same denominator.

### Step 1

Multiply the two terms on the bottom to get the same denominator.

### Step 2

Multiply the top number on the first fraction by the bottom number of the second fraction to get the new top number of the first fraction.

### Step 3

Multiply the top number on the second fraction by the bottom number of the first fraction to get the new top number of the second fraction.

### Step 4

Now add/subtract the top numbers and keep the bottom number so that there is now one fraction.

### Step 5

Simplify the fraction if required.

## Example

Calculate $$\frac{2}{5} + \frac{3}{7}$$

$=\frac{2\times 7}{35}+\frac{3\times 5}{35}$

$= \frac{{14}}{{35}} + \frac{{15}}{{35}} = \frac{{29}}{{35}}$

Now try the example questions that follow.

Question

Calculate $$\frac{2}{3} - \frac{y}{{18}}$$

$=\frac{2\times 18}{54}-\frac{3y}{54}$

$= \frac{{36}}{{54}} - \frac{{3y}}{{54}}$

$= \frac{{36 - 3y}}{{54}}$

$= \frac{{3(12 - y)}}{{54}}$

Take out a common factor of 3 on the numerator, then you notice that you can simplify by dividing top and bottom by 3.

$= \frac{{12 - y}}{{18}}$

Question

$\frac{x}{y} + \frac{y}{x}$

$= \frac{{{x^2}}}{{xy}} + \frac{{{y^2}}}{{xy}}$

$= \frac{{{x^2} + {y^2}}}{{xy}}$

Question

$\frac{2}{x} - \frac{5}{{x + 2}}$

$= \frac{{2(x + 2)}}{{x(x + 2)}} - \frac{{5 \times x}}{{x(x + 2)}}$

Multiply the brackets out on the numerator, but not the denominator as we are going to be subtracting the two numerators together so need to collect like terms then factorise if required.

$= \frac{{2x + 4}}{{x(x + 2)}} - \frac{{5x}}{{x(x + 2)}}$

$= \frac{{2x + 4 - 5x}}{{x(x + 2)}}$

$= \frac{{4 - 3x}}{{x(x + 2)}}$