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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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Teacher resources

The teacher copies: slides with the questions built in, the answers, and anything else attached to this lesson for whoever is teaching it.

Student handouts

The same files the students see, to print or hand out.

Last Lesson and Before

Answer each one, then check.

  1. 1

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

    Show answerHide answer

    0.25

  2. 2

    Write 30% as a decimal.

    Show answerHide answer

    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?

    Show answerHide answer

    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%, \(0.\dot{3}\)

Show the solutionHide the solution
  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.

Questions and answers

8 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Non-calculator 1 mark Easier

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

Mark scheme — 1 mark available

  • 35% — B1

Model answer

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

2. Exam question Non-calculator 2 marks Easier

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

Mark scheme — 2 marks available

  • At least three correctly converted to decimals or percentages — M1
  • 0.6, \(\frac{5}{8}\), 65%, \(\frac{2}{3}\) — A1

Model answer

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

3. Exam question Non-calculator 1 mark Easier

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

Mark scheme — 1 mark available

  • A correct explanation referring to the prime factors of 40 being 2 and 5 — C1

Model answer

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

4. Exam question Non-calculator · Higher 3 marks Easier

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

Mark scheme — 3 marks available

  • \(100x = 45.45\ldots\) written with \(x = 0.45\ldots\) — M1
  • \(99x = 45\) or \(\frac{45}{99}\) — M1
  • Simplified correctly to \(\frac{5}{11}\) — A1

Model answer

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

5. Exam question Non-calculator · Higher 3 marks Easier

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

Mark scheme — 3 marks available

  • 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

Model answer

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

6. Multiple choice 1 mark Easier

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

  1. A 38%
  2. B 3.8%
  3. C 0.375%
  4. D 37.5% Correct

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

7. Multiple choice 1 mark Core

Which of these fractions gives a recurring decimal?

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

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

8. Multiple choice 1 mark Stretch

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

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

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