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Simplifying algebraic fractions

Cancel common factors to simplify algebraic fractions; add and subtract algebraic fractions with numeric denominators; multiply and divide algebraic fractions.

Worked examples

Simplifying a single algebraic fraction

Straightforward

Problem

Simplify

\dfrac{10x^2}{4x}

.

1

Write out the factors in the numerator and denominator.

\dfrac{10x^2}{4x} = \dfrac{10 \times x \times x}{4 \times x}

2

Identify the highest common factor (HCF) of the numerator and denominator.

The HCF of 10 and 4 is 2, and

x

appears in both. So the HCF is

2x

.

3

Divide both numerator and denominator by the HCF.

\dfrac{10x^2 \div 2x}{4x \div 2x} = \dfrac{5x}{2}

Answer

\dfrac{5x}{2}

Adding algebraic fractions with different denominators

Moderate

Problem

Simplify

\dfrac{x}{4} + \dfrac{x}{6}

.

1

Find the lowest common denominator (LCD) of 4 and 6.

Multiples of 4: 4, 8, 12, 16,

\ldots

Multiples of 6: 6, 12, 18,

\ldots

The LCD is 12.

2

Convert each fraction to an equivalent fraction with denominator 12.

\dfrac{x}{4} = \dfrac{x \times 3}{4 \times 3} = \dfrac{3x}{12}, \qquad \dfrac{x}{6} = \dfrac{x \times 2}{6 \times 2} = \dfrac{2x}{12}

3

Add the numerators, keeping the denominator the same.

\dfrac{3x}{12} + \dfrac{2x}{12} = \dfrac{3x + 2x}{12} = \dfrac{5x}{12}

Answer

\dfrac{5x}{12}

Dividing algebraic fractions

Challenging

Problem

Simplify

\dfrac{9x^2}{4} \div \dfrac{3x}{8}

.

1

Rewrite the division as multiplication by the reciprocal of the second fraction.

\dfrac{9x^2}{4} \div \dfrac{3x}{8} = \dfrac{9x^2}{4} \times \dfrac{8}{3x}

2

Multiply numerators together and denominators together.

\dfrac{9x^2 \times 8}{4 \times 3x} = \dfrac{72x^2}{12x}

3

Simplify by cancelling common numeric and variable factors.

\dfrac{72x^2}{12x} = \dfrac{72}{12} \times \dfrac{x^2}{x} = 6 \times x = 6x

Answer

6x

Practise

Q1·Straightforward

Which of the following is the simplified form of

\dfrac{6x}{9}

?

$\dfrac{x}{3}$$\dfrac{2x}{3}$$\dfrac{6}{9}$$\dfrac{2}{3}$

Explanation

The HCF of 6 and 9 is 3. Dividing:

\dfrac{6x}{9} = \dfrac{6x \div 3}{9 \div 3} = \dfrac{2x}{3}

.

Q2·Straightforward

Which of the following is the simplified form of

\dfrac{4x^2}{8x}

?

$\dfrac{x^2}{2}$$\dfrac{x}{2}$$\dfrac{x}{8}$$\dfrac{4x}{8}$

Explanation

\dfrac{4x^2}{8x} = \dfrac{4 \times x \times x}{8 \times x}

. The HCF is

4x

. Cancelling:

\dfrac{4x^2 \div 4x}{8x \div 4x} = \dfrac{x}{2}

.

Q3·Straightforward

Which of the following is the simplified form of

\dfrac{15m}{10}

?

$\dfrac{5m}{4}$$\dfrac{3m}{2}$$\dfrac{m}{2}$$\dfrac{3m}{5}$

Explanation

The HCF of 15 and 10 is 5. Dividing:

\dfrac{15m}{10} = \dfrac{15m \div 5}{10 \div 5} = \dfrac{3m}{2}

.

Q4·Straightforward

Simplify

\dfrac{12y}{4y}

. Enter the numeric value.

Explanation

\dfrac{12y}{4y} = \dfrac{12}{4} = 3

. The

y

terms cancel completely.

Q5·Moderate

Which of the following is the simplified form of

\dfrac{x}{3} + \dfrac{x}{4}

?

$\dfrac{7x}{12}$$\dfrac{2x}{7}$$\dfrac{x}{7}$$\dfrac{2x}{12}$

Explanation

The LCD of 3 and 4 is 12. Converting:

\dfrac{x}{3} = \dfrac{4x}{12}

and

\dfrac{x}{4} = \dfrac{3x}{12}

. Adding:

\dfrac{4x}{12} + \dfrac{3x}{12} = \dfrac{7x}{12}

.

Q6·Moderate

Which of the following is the simplified form of

\dfrac{x}{2} + \dfrac{x}{5}

?

$\dfrac{7x}{10}$$\dfrac{2x}{7}$$\dfrac{x}{10}$$\dfrac{3x}{7}$

Explanation

The LCD of 2 and 5 is 10. Converting:

\dfrac{x}{2} = \dfrac{5x}{10}

and

\dfrac{x}{5} = \dfrac{2x}{10}

. Adding:

\dfrac{5x}{10} + \dfrac{2x}{10} = \dfrac{7x}{10}

.

Q7·Moderate

Which of the following is the simplified form of

\dfrac{3x}{4} - \dfrac{x}{8}

?

$\dfrac{5x}{8}$$\dfrac{x}{4}$$\dfrac{x}{2}$$\dfrac{2x}{4}$

Explanation

The LCD of 4 and 8 is 8. Converting:

\dfrac{3x}{4} = \dfrac{6x}{8}

. Subtracting:

\dfrac{6x}{8} - \dfrac{x}{8} = \dfrac{5x}{8}

.

Q8·Moderate

Which of the following is the simplified form of

\dfrac{2x}{3} - \dfrac{x}{6}

?

$\dfrac{x}{2}$$\dfrac{x}{3}$$\dfrac{3x}{9}$$\dfrac{x}{6}$

Explanation

The LCD of 3 and 6 is 6. Converting:

\dfrac{2x}{3} = \dfrac{4x}{6}

. Subtracting:

\dfrac{4x}{6} - \dfrac{x}{6} = \dfrac{3x}{6} = \dfrac{x}{2}

.

Q9·Challenging

Simplify

\dfrac{3x}{4} \times \dfrac{8}{x}

. Enter the numeric value.

Explanation

\dfrac{3x}{4} \times \dfrac{8}{x} = \dfrac{3x \times 8}{4 \times x} = \dfrac{24x}{4x} = 6

. Both the

x

terms and the factor of 4 cancel.

Q10·Challenging

Simplify

\dfrac{2x}{5} \div \dfrac{x}{10}

. Enter the numeric value.

Explanation

\dfrac{2x}{5} \div \dfrac{x}{10} = \dfrac{2x}{5} \times \dfrac{10}{x} = \dfrac{2x \times 10}{5 \times x} = \dfrac{20x}{5x} = 4

.

Q11·Challenging

Which of the following is the simplified form of

\dfrac{6x^2}{5} \times \dfrac{10}{3x}

?

$2x$$4x$$\dfrac{4x^2}{3}$$\dfrac{x}{4}$

Explanation

\dfrac{6x^2}{5} \times \dfrac{10}{3x} = \dfrac{6x^2 \times 10}{5 \times 3x} = \dfrac{60x^2}{15x} = \dfrac{60}{15} \times \dfrac{x^2}{x} = 4x

.

Q12·Challenging

Which of the following is the simplified form of

\dfrac{4x}{9} \div \dfrac{2x^2}{3}

?

$\dfrac{2}{3x}$$\dfrac{8x^3}{27}$$\dfrac{2x}{3}$$\dfrac{3x}{2}$

Explanation

\dfrac{4x}{9} \div \dfrac{2x^2}{3} = \dfrac{4x}{9} \times \dfrac{3}{2x^2} = \dfrac{4x \times 3}{9 \times 2x^2} = \dfrac{12x}{18x^2} = \dfrac{2}{3x}

.

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