Operations with Fractions (AQA GCSE Maths)

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Mark

Last updated

16 March 2023

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Adding & Subtracting Fractions

Dealing with mixed numbers

Always turn mixed numbers into top heavy fractions before adding or subtracting

Adding & subtracting

Adding and subtracting are treated in exactly the same way:
- Find the lowest common denominator (the smallest whole number that each denominator divides)
- Write each fraction as an equivalent fraction over this denominator (by multiplying top-and-bottom by the same amount)
- Add (or subtract) the numerators and write this over a single lowest common denominator
  - do not add the denominators
- Check for any cancellation (or if asked to turn top heavy fractions back into mixed numbers)

Worked example

(a) Find $\frac{2}{3} + \frac{1}{5}$

Find the lowest common denominator of 3 and 5

15 is the smallest number that divides both 3 and 5

the lowest common denominator is 15

Write both fractions as equivalent fractions over 15 (by multiplying top and bottom by the same amount)

$\frac{2 \times 5}{3 \times 5} + \frac{1 \times 3}{5 \times 3} = \frac{10}{15} + \frac{3}{15}$

Add the numerators and write over a single denominator

$\frac{10 + 3}{15} = \frac{13}{15}$

There is no cancellation

$\frac{13}{15}$

(b) Find $3 \frac{3}{4} - \frac{5}{8}$ giving your answer as a mixed number

Change the mixed number into a top heavy fraction (by multiplying the denominator, 4, by the whole number, 3, then adding the numerator, 3)

$\frac{4 \times 3 + 3}{4} = \frac{15}{4}$

To find $\frac{15}{4} - \frac{5}{8}$ first find the lowest common denominator of 4 and 8

8 is the smallest number that divides both 4 and 8

the lowest common denominator is 8

Write both fractions as equivalent fractions over 8 (by multiplying top and bottom by the same amount)

$\frac{15 \times 2}{4 \times 2} - \frac{5 \times 1}{8 \times 1} = \frac{30}{8} - \frac{5}{8}$

Subtract the numerators and write over a single denominator

$\frac{30 - 5}{8} = \frac{25}{8}$

Change into a mixed number (by dividing 25 by 8 to get 3 remainder 1)

$= 3 \frac{1}{8}$

There is no more cancellation

$3 \frac{1}{8}$

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Multiplying Fractions

Dealing with mixed numbers

Always turn mixed numbers into top heavy fractions before multiplying

Multiplying fractions

Cancel any numbers on the tops of the fractions with numbers on the bottoms of the fractions (either fraction)
Multiply the tops
Multiply the bottoms
Cancel again if possible
Turn top heavy fractions back into mixed numbers (if necessary / asked for)

Worked example

Find $\frac{4}{15} \times \frac{25}{11}$

The 15 and 25 can be cancelled before multiplying (to make the next step easier)

$\frac{4}{5 \times 3} \times \frac{5 \times 5}{11} = \frac{4}{5 \times 3} \times \frac{5 \times 5}{11} = \frac{4}{3} \times \frac{5}{11}$

Multiply the numerators together and the denominators together

$\frac{4 \times 5}{3 \times 11}$

There is no further cancelling that can be done

$\frac{20}{33}$

Dividing Fractions

Dealing with mixed numbers

Always turn mixed numbers into top heavy fractions before dividing

Dividing fractions

Never try to divide fractions
Instead “flip’n’times” (flip the second fraction and change ÷ into ×)
So $\div \frac{a}{b}$ becomes $\times \frac{b}{a}$
Then multiply the fractions (multiply tops and multiply bottoms)
Cancel the final answer (if possible)

Worked example

Divide $3 \frac{1}{4}$ by $\frac{3}{8}$ , giving your answer as a mixed number

Rewrite $3 \frac{1}{4}$ as an improper fraction

$3 \frac{1}{4} = \frac{12}{4} + \frac{1}{4} = \frac{13}{4}$

Turn the division into a multiplication, using the fact that dividing by a fraction is the same as multiplying by its reciprocal

$\frac{13}{4} \div \frac{3}{8} = \frac{13}{4} \times \frac{8}{3}$

Multiply the fractions

$\frac{13}{4} \times \frac{8}{3} = \frac{13 \times 8}{4 \times 3} = \frac{104}{12}$

Simplify the fraction, by dividing the numerator and denominator by 4

$\frac{104}{12} = \frac{26}{3}$

Rewrite as a mixed number

$\frac{26}{3} = \frac{24}{3} + \frac{2}{3} = 8 \frac{2}{3}$

$8 \frac{2}{3}$

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Operations with Fractions (AQA GCSE Maths)

Revision Note

Dealing with mixed numbers

Adding & subtracting

Dealing with mixed numbers

Multiplying fractions

Dealing with mixed numbers

Dividing fractions

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Author: Mark