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Maths · Fractions, ratio and percentages

Fractions

Adding, subtracting, multiplying and dividing fractions and mixed numbers, finding fractions of amounts, and working back to the whole - the fraction skills every other topic leans on.

  • 6 key terms
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Before We Start

Answer each one, then check.

  1. 1

    Simplify \(\frac{12}{18}\).

    Show answerHide answer

    \(\frac{2}{3}\)

  2. 2

    What is the lowest common multiple of 4 and 6?

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    12

  3. 3

    Work out \(\frac{3}{4}\) of 20.

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    15

  4. 4

    Write \(2\frac{1}{3}\) as an improper fraction.

    Show answerHide answer

    \(\frac{7}{3}\)

Learning Objectives

  1. 1Add and subtract fractions and mixed numbers.
  2. 2Multiply and divide fractions and mixed numbers.
  3. 3Find a fraction of an amount, and write one quantity as a fraction of another.
  4. 4Find the whole amount when you know a fraction of it.

Equivalent Fractions and Mixed Numbers

Multiplying or dividing the top and bottom by the same number gives an equivalent fraction.

  • Simplify

    Divide the top and bottom by their HCF: \(\frac{12}{18} = \frac{2}{3}\) (dividing by 6).

  • Mixed to improper

    \(2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}\).

  • Improper to mixed

    \(\frac{17}{5} = 3\frac{2}{5}\), because \(17 \div 5 = 3\) remainder 2.

  • Always finish simply

    Give answers in their simplest form, as a mixed number if the question uses them.

Adding and Subtracting

Fractions can only be added or subtracted when their denominators are the same.

  • Common denominator

    Use the lowest common multiple of the denominators: for \(\frac{1}{4} + \frac{1}{6}\), use 12.

  • Change each fraction

    \(\frac{1}{4} = \frac{3}{12}\) and \(\frac{1}{6} = \frac{2}{12}\), so the answer is \(\frac{5}{12}\).

  • Mixed numbers

    Change them to improper fractions first - it avoids borrowing mistakes.

  • Never add the bottoms

    \(\frac{1}{4} + \frac{1}{6}\) is NOT \(\frac{2}{10}\).

Adding Mixed Numbers

Work out \(2\frac{3}{4} + 1\frac{5}{6}\)

Show the solutionHide the solution
  1. 1 Change to improper fractions \(\dfrac{11}{4} + \dfrac{11}{6}\)
  2. 2 Common denominator: the LCM of 4 and 6 is 12 \(\dfrac{33}{12} + \dfrac{22}{12}\)
  3. 3 Add the numerators \(\dfrac{55}{12}\)
  4. 4 Change back to a mixed number \(55 \div 12 = 4\) remainder 7

Answer\(4\dfrac{7}{12}\)

Subtracting Mixed Numbers

Work out \(3\frac{1}{5} - 1\frac{2}{3}\)

Show the solutionHide the solution
  1. 1 Change to improper fractions \(\dfrac{16}{5} - \dfrac{5}{3}\)
  2. 2 Common denominator 15 \(\dfrac{48}{15} - \dfrac{25}{15}\)
  3. 3 Subtract the numerators \(\dfrac{23}{15}\)
  4. 4 Change back \(1\dfrac{8}{15}\)

Answer\(1\dfrac{8}{15}\)

Multiplying and Dividing

No common denominator is needed - but mixed numbers must be made improper first.

  • Multiply

    Multiply the tops and multiply the bottoms: \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\).

  • Cancel first

    Divide a top and a bottom by a common factor before multiplying to keep numbers small.

  • Reciprocal

    Flip a fraction to get its reciprocal: the reciprocal of \(\frac{4}{5}\) is \(\frac{5}{4}\).

  • Divide

    Dividing by a fraction is the same as multiplying by its reciprocal: keep, flip, change.

Multiplying Mixed Numbers

Work out \(2\frac{1}{4} \times 1\frac{1}{3}\)

Show the solutionHide the solution
  1. 1 Change to improper fractions \(\dfrac{9}{4} \times \dfrac{4}{3}\)
  2. 2 Cancel: the 4s cancel, and 9 and 3 share a factor of 3 \(\dfrac{3}{1} \times \dfrac{1}{1}\)
  3. 3 Multiply \(3\)

Answer\(3\)

Dividing Mixed Numbers

Work out \(3\frac{1}{3} \div 1\frac{1}{4}\)

Show the solutionHide the solution
  1. 1 Change to improper fractions \(\dfrac{10}{3} \div \dfrac{5}{4}\)
  2. 2 Keep, flip, change \(\dfrac{10}{3} \times \dfrac{4}{5}\)
  3. 3 Multiply \(\dfrac{40}{15} = \dfrac{8}{3}\)
  4. 4 Change back \(2\dfrac{2}{3}\)

Answer\(2\dfrac{2}{3}\)

Fractions of Amounts

Divide by the bottom, multiply by the top.

  • A fraction of an amount

    \(\frac{3}{5}\) of 40: \(40 \div 5 = 8\), \(8 \times 3 = 24\).

  • One amount as a fraction of another

    15 out of 40 is \(\frac{15}{40} = \frac{3}{8}\). Make sure both are in the same units first.

  • Finding the whole

    If \(\frac{3}{5}\) of a number is 24, then \(\frac{1}{5}\) is 8, so the whole is \(5 \times 8 = 40\).

  • The rest

    If \(\frac{2}{5}\) is spent, \(1 - \frac{2}{5} = \frac{3}{5}\) is left.

Fraction Target

Using each of the digits 1, 2, 3 and 4 once, make two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) whose sum is as close to 1 as possible. Then find the pair whose product is as small as possible.

1. Try several combinations.

2. Work out each sum exactly.

3. Explain why your best answer is best.

A good answer shows: Closest sum to 1: \(\frac{1}{4} + \frac{2}{3} = \frac{11}{12}\) (or \(\frac{1}{3} + \frac{2}{4} = \frac{5}{6}\), further away). Smallest product: \(\frac{1}{3} \times \frac{2}{4} = \frac{1}{6}\) or \(\frac{1}{4} \times \frac{2}{3} = \frac{1}{6}\).

Can I...?

  1. 1Simplify fractions.
  2. 2Convert between mixed numbers and improper fractions.
  3. 3Add and subtract fractions.
  4. 4Add and subtract mixed numbers.
  5. 5Multiply fractions and mixed numbers.
  6. 6Divide fractions and mixed numbers.
  7. 7Find a fraction of an amount.
  8. 8Find the whole from a fraction of it.

Summary & Exam Focus

  • Add and subtract: common denominator first; mixed numbers become improper.
  • Multiply: tops times tops, bottoms times bottoms; cancel first.
  • Divide: keep, flip, change.
  • Fraction of an amount: divide by the bottom, multiply by the top.

Exam focus

Work out \(1\frac{3}{4} \div 2\frac{1}{3}\). Give your answer in its simplest form. (3 marks) (3 marks)

Show every step - the improper fractions, the flipped fraction, the multiplication. On the non-calculator paper each step can earn a method mark.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Numerator
The top number of a fraction.
Denominator
The bottom number of a fraction.
Improper fraction
A fraction whose numerator is bigger than its denominator, e.g. \(\frac{7}{3}\).
Mixed number
A whole number and a fraction, e.g. \(2\frac{1}{3}\).
Reciprocal
1 divided by a number; for a fraction, the fraction flipped.
Common denominator
A denominator shared by two fractions, so they can be added or subtracted.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 2 marks

    Work out \(\dfrac{2}{3} + \dfrac{1}{4}\)

    Show answerHide answer

    Model answer

    \(\dfrac{8}{12} + \dfrac{3}{12} = \dfrac{11}{12}\)

    Mark scheme

    • Both fractions written with a common denominator — M1
    • \(\frac{11}{12}\) — A1
  2. Question 2 Non-calculator 2 marks

    Work out \(2\frac{1}{2} \times 1\frac{3}{5}\)

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    Model answer

    \(\dfrac{5}{2} \times \dfrac{8}{5} = \dfrac{40}{10} = 4\)

    Mark scheme

    • Both written as improper fractions: \(\frac{5}{2}\) and \(\frac{8}{5}\) — M1
    • \(4\) — A1
  3. Question 3 Non-calculator 3 marks

    Work out \(1\frac{3}{4} \div 2\frac{1}{3}\). Give your answer in its simplest form.

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    Model answer

    \(\dfrac{7}{4} \div \dfrac{7}{3} = \dfrac{7}{4} \times \dfrac{3}{7} = \dfrac{21}{28} = \dfrac{3}{4}\)

    Mark scheme

    • Both written as improper fractions — M1
    • Multiplying by the reciprocal: \(\frac{7}{4} \times \frac{3}{7}\) — M1
    • \(\frac{3}{4}\) — A1
  4. Question 4 Non-calculator 3 marks

    Sam spends \(\frac{2}{5}\) of his wages on rent and \(\frac{1}{4}\) on food. He saves the rest. His wages are £1200. How much does he save?

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    Model answer

    Fraction spent: \(\frac{2}{5} + \frac{1}{4} = \frac{8}{20} + \frac{5}{20} = \frac{13}{20}\). Fraction saved: \(\frac{7}{20}\). \(\frac{7}{20} \times 1200 = 420\). He saves £420.

    Mark scheme

    • \(\frac{13}{20}\) or \(\frac{7}{20}\), or £480 and £300 found — P1
    • \(\frac{7}{20} \times 1200\) or \(1200 - 780\) — P1
    • £420 — A1

Quick check

  1. What is \(\frac{1}{3} + \frac{1}{4}\)?

    1. A\(\frac{2}{7}\)
    2. B\(\frac{1}{7}\)
    3. C\(\frac{7}{12}\)
    4. D\(\frac{1}{12}\)
    Show answerHide answer

    C: \(\frac{7}{12}\)

    Common denominator 12: \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\).

  2. What is \(\frac{3}{4} \div \frac{1}{2}\)?

    1. A\(1\frac{1}{2}\)
    2. B\(\frac{3}{8}\)
    3. C\(\frac{2}{3}\)
    4. D\(\frac{1}{4}\)
    Show answerHide answer

    A: \(1\frac{1}{2}\)

    Keep, flip, change: \(\frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = 1\frac{1}{2}\).

  3. \(\frac{3}{5}\) of a number is 36. What is the number?

    1. A21.6
    2. B12
    3. C108
    4. D60
    Show answerHide answer

    D: 60

    \(\frac{1}{5}\) is \(36 \div 3 = 12\), so the number is \(5 \times 12 = 60\).

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