Adding and subtracting rational expressions - Higher

Adding and subtracting algebraic fractions is a similar process to adding and subtracting normal fractions.

Fractions can only be added or subtracted when there is a common denominator and algebraic fractions are the same in this method.

Example

Write \frac{2}{y} + \frac{1}{y} as a single fraction.

In this example, the denominators of the two fractions are the same, so the numerators can simply be added.

This gives \frac{2}{y} + \frac{1}{y} = \frac{2 + 1}{y} =  \frac{3}{y}.

This fraction cannot be simplified so \frac{3}{y} is the final answer.

Question

Write \frac{5}{3t} - \frac{2}{7t} as a single fraction.

The denominators of each fraction are different, 3t and 7t, so a common denominator must be created. This is found by working out the lowest common multiple of 3t and 7t which is 21t.

Remember, if it is difficult to work out the lowest common multiple of two expressions, a common denominator can always be found by simply multiplying the denominators together. Remember this may mean the fractions need simplifying at the end.

\frac{5}{3t} - \frac{2}{7t}

To create a common denominator of 21t, the first fraction's numerator and denominator must be multiplied by 7 and the second fraction's numerator and denominator must be multiplied by 3:

\frac{35}{21t} - \frac{6}{21t}

Now the denominators are the same, the numerators can be subtracted:

\frac{35 - 6}{21t} = \frac{29}{21t}

This fraction cannot be simplified any further, so \frac{29}{21t} is the final answer.

Question

Simplify \frac{2}{y + 4} + \frac{3}{y - 2}.

These fractions do not have a common denominator. There are also no common factors between the denominators, so the only way to create a common denominator is to multiply the two expressions together.

(y + 4) \times (y - 2) can be written as (y + 4)(y - 2).

\frac{2}{y + 4} (multiply numerator and denominator by y - 2) + \frac{3}{y - 2} (multiply numerator and denominator by y + 4) = \frac{2(y - 2)}{(y + 4)(y - 2)} + \frac{3(y + 4)}{(y - 2)(y + 4)}

Expanding brackets means everything inside the bracket has to be multiplied by the term outside the bracket.

2(y - 2) = 2 \times y + 2 \times -2 = 2y + -4 = 2y - 4

3(y + 4) = 3 \times y + 3 \times 4 = 3y + 12

This gives \frac{2y - 4}{(y + 4)(y - 2)} + \frac{3y + 12}{(y - 2)(y + 4)}.

Now the denominators are the same, the numerators can be simplified:

= \frac{2y - 4 + 3y + 12}{(y - 2)(y + 4)}

Collect the like terms: 2y - 4 + 3y + 12. 2y + 3y = 5y. -4 + 12 = 8.

= \frac{5y + 8}{(y - 2)(y + 4)}

This fraction cannot be simplified, so this is the final answer.