# Multiplying And Dividing Algebraic Fractions

Here we will learn about multiplying and dividing algebraic fractions, including algebraic fractions with monomial and binomial numerators and denominators.

There are also multiplying and dividing algebraic fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is multiplying and dividing algebraic fractions?

Multiplying and dividing algebraic fractions is the skill of multiplying and dividing two or more fractions that contain algebraic terms.

For example, \cfrac{2a}{5}, \cfrac{3-gh}{4f}, or \cfrac{1}{x-x^{2}} \, .

To do this we must combine our knowledge of multiplying and dividing fractions with our understanding of algebra.

Step-by-step guide: Multiplying and dividing fractions

Step-by-step guide: Algebraic terms

### Multiplication with algebraic fractions

To multiply with fractions, we multiply the numerators together, and multiply the denominators together. This is the same for algebraic fractions, but we need to take extra care when multiplying algebraic terms or expressions.

For example, \cfrac{3x^3}{a} \times \cfrac{5x}{2b}=\cfrac{3x^3\times 5x}{a\times 2b}=\cfrac{15x^4}{2ab} \, .

### Division with algebraic fractions

To divide with fractions, we first write the reciprocal of the dividing fraction and then multiply the numerators together, and multiply the denominators together. This is the same for algebraic fractions, but we need to take extra care when multiplying algebraic terms or expressions.

For example, \cfrac{4b}{3} \div \cfrac{7a}{b}=\cfrac{4b}{3} \times \cfrac{b}{7a}=\cfrac{4b\times b}{3\times 7a}=\cfrac{4b^2}{21a} \, .

## How to multiply algebraic fractions

In order to multiply algebraic fractions:

1. Multiply the numerators together and multiply the denominators together.
2. Simplify the fraction if possible.

## Multiplying algebraic fractions examples

### Example 1: multiplying algebraic fractions with monomial expressions.

Write as a single fraction in its simplest form, \cfrac{2}{a}\times\cfrac{5}{b} \, .

1. Multiply the numerators together and multiply the denominators together.

\cfrac{2\times5}{a\times b}=\cfrac{10}{ab}

2Simplify the fraction if possible.

The fraction \cfrac{10}{ab} cannot be simplified as the numerator and the denominator do not have any common factors.

### Example 2: multiplying algebraic fractions involving a binomial expression

Write as a single fraction in its simplest form, \cfrac{4x}{5}\times\cfrac{x+7}{8} \, .

Multiply the numerators together and multiply the denominators together.

Simplify the fraction if possible.

### Example 3: multiplying algebraic fractions involving binomial expressions

Write as a single fraction in its simplest form, \cfrac{3x+9}{4x}\times\cfrac{5}{2x+6} \, .

Multiply the numerators together and multiply the denominators together.

Simplify the fraction if possible.

## How to divide algebraic fractions

In order to divide algebraic fractions:

1. Find the reciprocal of the dividing fraction and rewrite the question with multiplication instead of division.
2. Multiply the numerators together and multiply the denominators together.
3. Simplify the fraction if possible.

## Dividing algebraic fractions examples

### Example 4: dividing with algebraic fractions with monomial expressions

Write as a single fraction in its simplest form, \cfrac{10}{3c}\div\cfrac{5}{3d} \, .

Find the reciprocal of the dividing fraction and rewrite the question with multiplication instead of division.

Multiply the numerators together and multiply the denominators together.

Simplify the fraction if possible.

### Example 5: dividing with algebraic fractions involving a binomial expression

Write as a single fraction in its simplest form, \cfrac{3x}{4}\div\cfrac{x+1}{8} \, .

Find the reciprocal of the dividing fraction and rewrite the question with multiplication instead of division.

Multiply the numerators together and multiply the denominators together.

Simplify the fraction if possible.

### Example 6: dividing with algebraic fractions with binomial expressions

Write as a single fraction in its simplest form, \cfrac{x^{2}-9}{x+1}\div\cfrac{x+3}{2} \, .

Find the reciprocal of the dividing fraction and rewrite the question with multiplication instead of division.

Multiply the numerators together and multiply the denominators together.

Simplify the fraction if possible.

Step-by-step guide: Difference of two squares

### Common misconceptions

• Cross multiplying vs multiplying fractions
When multiplying fractions, we multiply the numerators together, and multiply the denominators together. Some students can confuse this with a method referred to as ‘cross multiplying’ which some people teach as a method for dividing by a fraction.

For example,
Incorrectly choosing to cross multiply,

The correct solution is

• Common denominators are not needed
When we add or subtract fractions we must ensure there is a common denominator. A mistake often made by students is to believe this rule also applies to the multiplication of fractions. When we are multiplying fractions, we can choose to write the fractions with a common denominator but this simply makes the process lengthier and it is not necessary.

For example,
Without writing the fractions with a common denominator,

Writing the fractions with a common denominator first,

### Practice multiplying and dividing algebraic fractions questions

1) Write as a single fraction in the simplest form, \cfrac{3}{2e}\times\cfrac{8}{f}.

\cfrac{3f}{16e}

\cfrac{16e}{3f}

\cfrac{24}{2ef}

\cfrac{12}{ef}

\cfrac{3}{2e}\times\cfrac{8}{f}=\cfrac{3\times{8}}{2e\times{f}}=\cfrac{24}{2ef}=\cfrac{12}{ef}

2) Write as a single fraction in the simplest form, \cfrac{2x+3}{2}\times\cfrac{2(2x+1)}{3}.

\cfrac{6x+9}{4(2x+1)}

\cfrac{4x^{2}+8x+3}{3}

\cfrac{14x+13}{6}

\cfrac{4x^{2}+8x+3}{6}

\cfrac{2x+3}{2}\times\cfrac{2(2x+1)}{3}=\cfrac{(2x+3)\times{(2(2x+1))}}{6}=\cfrac{8x^{2}+16x+6}{6}=\cfrac{4x^{2}+8x+3}{3}

3) Write as a single fraction in the simplest form, \frac{4x+4}{7}\times\frac{3x}{2x+2}.

\cfrac{8(x+1)^{2}}{21x}

\cfrac{12x^{2}+12}{14x+4}

\cfrac{6x}{7}

\cfrac{12x^{2}}{14}

\cfrac{4x+4}{7}\times\cfrac{3x}{2x+2}=\cfrac{(4x+4)\times{3x}}{7\times(2x+2)}=\cfrac{3x\times{4(x+1)}}{7\times2(x+1)}=\cfrac{12x(x+1)}{14(x+1)}=\cfrac{6x}{7}

4) Write as a single fraction in the simplest form, \cfrac{15}{4x^{2}}\div\cfrac{3x}{2}.

\cfrac{5}{2x^{3}}

\cfrac{13}{x}

\cfrac{5}{2x}

\cfrac{5}{2x^{2}}

\cfrac{15}{4x^{2}}\div\cfrac{3x}{2}=\cfrac{15}{4x^{2}}\times\cfrac{2}{3x}=\cfrac{30}{12x^{3}}=\cfrac{5}{2x^{3}}

5) Write as a single fraction in the simplest form, \cfrac{5+x}{12}\div\cfrac{x^{2}}{4}.

\cfrac{1+x}{12-x^{2}}

\cfrac{20+x}{12x^{2}}

\cfrac{5+x}{3x^{2}}

\cfrac{5}{3x}

\cfrac{5+x}{12}\div\cfrac{x^{2}}{4}=\cfrac{5+x}{12}\times\cfrac{4}{x^{2}}=\cfrac{4(5+x)}{12x^{2}}=\cfrac{5+x}{3x^{2}}

6) Write as a single fraction in the simplest form, \cfrac{x^{2}-36}{4x-8}\div\cfrac{x-6}{2}.

\cfrac{x^{2}-72}{4(x-2)(x-6)}

\cfrac{x-18}{2}

\cfrac{x+6}{2(x-2)}

\cfrac{6-x}{2x}

\cfrac{x^{2}-36}{4x-8}\div\cfrac{x-6}{2}=\cfrac{x^{2}-36}{4x-8}\times\cfrac{2}{x-6}=\cfrac{(x^{2}-36)\times{2}}{(4x-8)\times(x-6)}=\cfrac{2(x+6)(x-6)}{4(x-2)(x-6)}

=\cfrac{2(x+6)}{4(x-2)}=\cfrac{x+6}{2(x-2)}

### Multiplying and dividing algebraic fractions GCSE questions

1. Write as a single fraction, \cfrac{2}{x}\times\cfrac{3}{y}\div\cfrac{5}{z}.

(3 marks)

\cfrac{6}{xy}\div\cfrac{5}{z}

(1)

\cfrac{6}{xy}\times\cfrac{z}{5}

(1)

\cfrac{6z}{5xy}

(1)

2. Triangle ABC is a right angled triangle where BC is perpendicular to AC and x > 0.

(a) Calculate the area of the triangle.

(b) As the value of x increases, what happens to the area of the triangle?

(4 marks)

(a) (\cfrac{1}{x}\times\cfrac{x}{2})\div{2}

(1)

\cfrac{x}{2x}(=\cfrac{1}{2})\div{2}

\cfrac{1}{2}\times\cfrac{1}{2}

(1)

\cfrac{1}{4}

(1)

(b) Stays the same.

The area is independent of the value of x.

(1)

3. Use factorisation to fully simplify \cfrac{4x^{2}-16}{x+1}\div\cfrac{x-2}{3x+3} into the form a(x+b) where a and b are integers.

(5 marks)

4x^{2}-16=4(x^{2}-4)

(1)

4(x^{2}-4)=4(x+2)(x-2)

(1)

3x+3=3(x+1)

(1)

\cfrac{4(x+2)(x-2)}{x+1}\div\cfrac{x-2}{3(x+1)}=\cfrac{4(x+2)(x-2)}{x+1}\times\cfrac{3(x+1)}{x-2}

(1)

\cfrac{4(x+2)(x-2)\times{3}(x+1)}{(x+1)(x-2)}=12(x+2)

(1)

## Learning checklist

You have now learned how to:

• Multiply and divide with algebraic fractions

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