Fractions

1. Find equivalent fractions and write a fraction in its simplest form

   Multiplying or dividing the top and bottom of a fraction by the same number keeps its value the same - so $\frac{2}{3}$ and $\frac{8}{12}$ are the same amount. To simplify, divide top and bottom by their highest common factor: $\frac{18}{24}$ simplifies to $\frac{3}{4}$ by dividing both by 6.

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2. Convert between improper fractions and mixed numbers

   An improper fraction has a top bigger than its bottom, like $\frac{9}{2}$. Divide to convert: 9 ÷ 2 is 4 remainder 1, so $\frac{9}{2}$ = 4 $\frac{1}{2}$. Going the other way, multiply the whole number by the denominator and add the numerator: 3 $\frac{2}{5}$ = $\frac{17}{5}$.

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3. Order fractions by writing them with a common denominator

   You can only compare fractions fairly when the bottoms match. To order $\frac{2}{3}$ and $\frac{3}{5}$, rewrite both over 15: $\frac{10}{15}$ and $\frac{9}{15}$ - now it is clear that $\frac{2}{3}$ is bigger. The lowest common multiple of the denominators is the quickest common bottom to use.

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4. Add and subtract fractions, including mixed numbers

   Make the denominators match first, then add or subtract the tops only: $\frac{1}{2}$ + $\frac{1}{6}$ becomes $\frac{3}{6}$ + $\frac{1}{6}$ = $\frac{4}{6}$ = $\frac{2}{3}$. With mixed numbers, deal with the whole parts and the fraction parts separately - and if the fractions add past 1, carry into the wholes.

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5. Multiply and divide fractions, including mixed numbers

   To multiply, go straight across the tops and bottoms: $\frac{2}{3}$ × $\frac{4}{5}$ = $\frac{8}{15}$ - no common denominator needed. To divide, flip the second fraction and multiply instead: $\frac{3}{4}$ ÷ $\frac{2}{5}$ = $\frac{3}{4}$ × $\frac{5}{2}$ = $\frac{15}{8}$. Convert mixed numbers to improper fractions before you start.

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6. Find a fraction of an amount, and write one quantity as a fraction of another

   For a fraction of an amount, divide by the bottom then multiply by the top: $\frac{3}{5}$ of £40 is 40 ÷ 5 × 3 = £24. To write one quantity as a fraction of another, put the first over the second and simplify: 15 minutes out of an hour is $\frac{15}{60}$ = $\frac{1}{4}$.

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• Adding the denominators as well as the numerators, so $\frac{1}{2}$ + $\frac{1}{3}$ becomes $\frac{2}{5}$.

  Make the denominators the same first, then add only the numerators: $\frac{1}{2}$ + $\frac{1}{3}$ = $\frac{3}{6}$ + $\frac{2}{6}$ = $\frac{5}{6}$.

• Flipping the first fraction instead of the second when dividing, so $\frac{3}{4}$ ÷ $\frac{2}{5}$ turns into $\frac{4}{3}$ × $\frac{2}{5}$.

  Keep the first fraction, change ÷ to ×, then flip only the second: $\frac{3}{4}$ ÷ $\frac{2}{5}$ = $\frac{3}{4}$ × $\frac{5}{2}$ = $\frac{15}{8}$.

• Multiplying mixed numbers part by part, so 2 $\frac{1}{2}$ × 1 $\frac{1}{2}$ becomes 2 $\frac{1}{4}$.

  Turn mixed numbers into improper fractions before multiplying: 2 $\frac{1}{2}$ × 1 $\frac{1}{2}$ = $\frac{5}{2}$ × $\frac{3}{2}$ = $\frac{15}{4}$ = 3 $\frac{3}{4}$.

• Taking the smaller fraction from the bigger one when subtracting mixed numbers, so 3 $\frac{1}{4}$ − 1 $\frac{1}{2}$ becomes 2 $\frac{1}{4}$.

  Change both to improper fractions over the same denominator: 3 $\frac{1}{4}$ − 1 $\frac{1}{2}$ = $\frac{13}{4}$ − $\frac{6}{4}$ = $\frac{7}{4}$ = 1 $\frac{3}{4}$.

• Leaving an answer such as $\frac{6}{8}$ when the question asks for the simplest form.

  Divide the top and bottom by their highest common factor: $\frac{6}{8}$ = $\frac{3}{4}$. Then check that no number bigger than 1 still divides both.

More Number topics

• Percentages
• Standard Form
• Surds
• Indices
• Bounds & Rounding
• Factors & Multiples

Keep reading

• How do I revise for GCSE maths?
• What's on the GCSE maths syllabus?
• What are the GCSE Maths grade boundaries?
• The GCSE Maths syllabus, explained