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Maths · Fractions, ratio and percentages

Fractions, decimals and percentages

The same amount can be written three ways. Moving between fractions, decimals and percentages lets you compare and order anything - and at Higher, turn a recurring decimal into an exact fraction.

  • 4 key terms
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Last Lesson and Before

Answer each one, then check.

  1. 1

    Write \(\frac{1}{4}\) as a decimal.

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    0.25

  2. 2

    Write 30% as a decimal.

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    0.3

  3. 3

    Last lesson: what multiplier decreases by 20%?

    Show answerHide answer

    0.8

  4. 4

    Which is bigger: 0.45 or 0.405?

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    0.45

Learning Objectives

  1. 1Convert between fractions, decimals and percentages.
  2. 2Order a mixture of fractions, decimals and percentages.
  3. 3Know which fractions give terminating decimals and which recur.
  4. 4Convert a recurring decimal to a fraction. (Higher)

Equivalents Worth Knowing

  • \(\frac{1}{2}\)

    Decimal: 0.5. Percentage: 50%

  • \(\frac{1}{4}\)

    Decimal: 0.25. Percentage: 25%

  • \(\frac{3}{4}\)

    Decimal: 0.75. Percentage: 75%

  • \(\frac{1}{5}\)

    Decimal: 0.2. Percentage: 20%

  • \(\frac{1}{8}\)

    Decimal: 0.125. Percentage: 12.5%

  • \(\frac{1}{10}\)

    Decimal: 0.1. Percentage: 10%

  • \(\frac{1}{3}\)

    Decimal: \(0.\dot{3}\). Percentage: \(33\frac{1}{3}\%\)

  • \(\frac{2}{3}\)

    Decimal: \(0.\dot{6}\). Percentage: \(66\frac{2}{3}\%\)

Converting Between the Three

Each conversion is one step.

  1. 1 Fraction to decimal

    Divide the top by the bottom: \(\frac{3}{8} = 3 \div 8 = 0.375\).
  2. 2 Decimal to percentage

    Multiply by 100: \(0.375 = 37.5\%\).
  3. 3 Percentage to fraction

    Write over 100 and simplify: \(35\% = \frac{35}{100} = \frac{7}{20}\).
  4. 4 Decimal to fraction

    Use place value: \(0.36 = \frac{36}{100} = \frac{9}{25}\).

Terminating and Recurring Decimals

Some fractions end; others repeat for ever.

  • Terminating

    The decimal stops: \(\frac{3}{8} = 0.375\).

  • Recurring

    A digit or group of digits repeats for ever: \(\frac{1}{3} = 0.333\ldots = 0.\dot{3}\) and \(\frac{1}{7} = 0.\dot{1}4285\dot{7}\).

  • Dot notation

    A dot over one digit repeats it; dots over the first and last digit repeat the whole group.

  • The rule

    A fraction in its simplest form terminates only if the prime factors of its denominator are just 2s and 5s: \(\frac{7}{40}\) terminates because \(40 = 2^3 \times 5\).

Ordering a Mixture

Write in order, smallest first: \(\frac{3}{8}\), 0.38, 37%, \(0.\dot{3}\)

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  1. 1 Change everything to decimals \(\frac{3}{8} = 0.375\), \(0.38\), \(37\% = 0.37\), \(0.\dot{3} = 0.333\ldots\)
  2. 2 Compare the decimals \(0.333\ldots < 0.37 < 0.375 < 0.38\)
  3. 3 Write the original forms in order \(0.\dot{3}\), 37%, \(\frac{3}{8}\), 0.38

Answer\(0.\dot{3}\), 37%, \(\frac{3}{8}\), 0.38

A Recurring Decimal to a Fraction

Show that \(0.\dot{4}\dot{5} = \dfrac{5}{11}\)

Show the solutionHide the solution
  1. 1 Let \(x\) be the decimal \(x = 0.454545\ldots\)
  2. 2 Two digits repeat, so multiply by 100 \(100x = 45.454545\ldots\)
  3. 3 Subtract to remove the repeating part \(100x - x = 45\), so \(99x = 45\)
  4. 4 Divide and simplify \(x = \dfrac{45}{99} = \dfrac{5}{11}\)

Answer\(0.\dot{4}\dot{5} = \dfrac{45}{99} = \dfrac{5}{11}\)

When the Repeat Starts Later

Write \(0.2\dot{3}\) as a fraction in its simplest form.

Show the solutionHide the solution
  1. 1 Let \(x\) be the decimal \(x = 0.2333\ldots\)
  2. 2 Multiply by 10 and by 100, so both have the same repeating tail \(10x = 2.333\ldots\) and \(100x = 23.333\ldots\)
  3. 3 Subtract \(100x - 10x = 21\), so \(90x = 21\)
  4. 4 Divide and simplify \(x = \dfrac{21}{90} = \dfrac{7}{30}\)

Answer\(\dfrac{7}{30}\)

Terminate or Recur?

Without dividing, predict whether each fraction terminates or recurs, then check with a calculator: \(\frac{3}{20}\), \(\frac{5}{12}\), \(\frac{7}{16}\), \(\frac{2}{15}\), \(\frac{9}{25}\), \(\frac{4}{9}\). Higher: write \(0.\dot{7}\) and \(0.1\dot{6}\) as fractions.

1. Write each denominator in prime factors.

2. Predict, then check.

3. Higher: use the algebra method.

A good answer shows: Terminate: \(\frac{3}{20}\) (0.15), \(\frac{7}{16}\) (0.4375), \(\frac{9}{25}\) (0.36). Recur: \(\frac{5}{12}\) (\(0.41\dot{6}\)), \(\frac{2}{15}\) (\(0.1\dot{3}\)), \(\frac{4}{9}\) (\(0.\dot{4}\)). Higher: \(0.\dot{7} = \frac{7}{9}\), \(0.1\dot{6} = \frac{15}{90} = \frac{1}{6}\).

Can I...?

  1. 1Convert fractions to decimals and percentages.
  2. 2Convert percentages to fractions.
  3. 3Order fractions, decimals and percentages.
  4. 4Use dot notation for recurring decimals.
  5. 5Say whether a fraction terminates or recurs.
  6. 6Convert a recurring decimal to a fraction. (Higher)

Summary & Exam Focus

  • Fraction to decimal: divide. Decimal to percentage: multiply by 100.
  • To order a mixture, change everything to decimals.
  • A fraction terminates only if its denominator's prime factors are 2 and 5.
  • (Higher) Recurring to fraction: multiply to line up the repeats, subtract, divide.

Exam focus

Show that \(0.\dot{4}\dot{5} = \dfrac{5}{11}\) (3 marks) (3 marks)

For "show that" with recurring decimals, write out \(x = \ldots\) and \(100x = \ldots\) with at least two repeats of the digits, and show the subtraction. The algebra is the method mark.

Key terms

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

Terminating decimal
A decimal that stops, e.g. 0.375.
Recurring decimal
A decimal in which a digit or group of digits repeats for ever.
Dot notation
Dots above digits to show which digits recur.
Equivalent
Equal in value, though written differently.

Practice questions

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

  1. Question 1 Non-calculator 1 mark

    Write \(\frac{7}{20}\) as a percentage.

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    Model answer

    \(\frac{7}{20} = \frac{35}{100} = 35\%\)

    Mark scheme

    • 35% — B1
  2. Question 2 Non-calculator 2 marks

    Write these numbers in order of size, starting with the smallest: \(\frac{2}{3}\), 0.6, 65%, \(\frac{5}{8}\)

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    Model answer

    As decimals: 0.666..., 0.6, 0.65, 0.625. Order: 0.6, \(\frac{5}{8}\), 65%, \(\frac{2}{3}\).

    Mark scheme

    • At least three correctly converted to decimals or percentages — M1
    • 0.6, \(\frac{5}{8}\), 65%, \(\frac{2}{3}\) — A1
  3. Question 3 Non-calculator 1 mark

    Explain why \(\frac{7}{40}\) can be written as a terminating decimal.

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    Model answer

    \(40 = 2^3 \times 5\): its only prime factors are 2 and 5.

    Mark scheme

    • A correct explanation referring to the prime factors of 40 being 2 and 5 — C1
  4. Question 4 Non-calculator · Higher 3 marks

    Show that \(0.\dot{4}\dot{5} = \dfrac{5}{11}\)

    Show answerHide answer

    Model answer

    \(x = 0.4545\ldots\), \(100x = 45.4545\ldots\), so \(99x = 45\) and \(x = \dfrac{45}{99} = \dfrac{5}{11}\).

    Mark scheme

    • \(100x = 45.45\ldots\) written with \(x = 0.45\ldots\) — M1
    • \(99x = 45\) or \(\frac{45}{99}\) — M1
    • Simplified correctly to \(\frac{5}{11}\) — A1
  5. Question 5 Non-calculator · Higher 3 marks

    Write \(0.1\dot{3}\dot{6}\) as a fraction in its simplest form.

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    Model answer

    \(x = 0.13636\ldots\). \(1000x = 136.3636\ldots\) and \(10x = 1.3636\ldots\). Subtracting: \(990x = 135\), so \(x = \dfrac{135}{990} = \dfrac{3}{22}\).

    Mark scheme

    • Two multiples of \(x\) with the same recurring tail, e.g. \(1000x\) and \(10x\) — M1
    • \(990x = 135\) or \(\frac{135}{990}\) — M1
    • \(\frac{3}{22}\) — A1

Quick check

  1. What is \(\frac{3}{8}\) as a percentage?

    1. A38%
    2. B3.8%
    3. C0.375%
    4. D37.5%
    Show answerHide answer

    D: 37.5%

    \(3 \div 8 = 0.375\), and \(0.375 \times 100 = 37.5\%\).

  2. Which of these fractions gives a recurring decimal?

    1. A\(\frac{3}{20}\)
    2. B\(\frac{5}{12}\)
    3. C\(\frac{7}{16}\)
    4. D\(\frac{9}{25}\)
    Show answerHide answer

    B: \(\frac{5}{12}\)

    \(12 = 2^2 \times 3\): it has a prime factor other than 2 and 5, so \(\frac{5}{12}\) recurs.

  3. (Higher) What is \(0.\dot{7}\) as a fraction?

    1. A\(\frac{7}{10}\)
    2. B\(\frac{7}{100}\)
    3. C\(\frac{7}{9}\)
    4. D\(\frac{77}{100}\)
    Show answerHide answer

    C: \(\frac{7}{9}\)

    \(10x - x = 7\), so \(9x = 7\) and \(x = \frac{7}{9}\).

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