OpenRevise GCSE

Viewing as

Student Teacher

Teacher view: planning notes, the answers to every question, and the teacher copies of the files.

Maths · Fractions, ratio and percentages

Ratios

A ratio compares the sizes of two or more amounts. Simplifying ratios, writing them as fractions and sharing amounts in a given ratio all come down to one idea - finding the value of one part.

  • 4 key terms
  • All boards
Download the teacher pack · 5 files

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

    What is the HCF of 12 and 18?

    Show answerHide answer

    6

  2. 2

    Last lesson: work out \(\frac{3}{5}\) of 40.

    Show answerHide answer

    24

  3. 3

    Simplify \(\frac{15}{25}\).

    Show answerHide answer

    \(\frac{3}{5}\)

  4. 4

    How many grams are in 1.2 kg?

    Show answerHide answer

    1200 g

Learning Objectives

  1. 1Simplify ratios, including ratios with different units.
  2. 2Write a ratio in the form \(1 : n\) or \(n : 1\).
  3. 3Connect ratios and fractions.
  4. 4Share an amount in a given ratio.
  5. 5Solve ratio problems when you know the difference or one part.

What a Ratio Is

A ratio compares amounts: 3 red counters to 5 blue is written \(3 : 5\).

  • The order matters

    \(3 : 5\) (red to blue) is not the same as \(5 : 3\).

  • Simplify

    Divide every part by the HCF: \(12 : 18 = 2 : 3\).

  • Same units first

    45 cm : 1.2 m becomes 45 : 120 (both in cm), then \(3 : 8\).

  • Decimals and fractions

    Multiply every part to make whole numbers: \(1.5 : 2 = 3 : 4\).

Simplifying with Different Units

Write 45 cm : 1.2 m in its simplest form.

Show the solutionHide the solution
  1. 1 Put both in the same units 1.2 m = 120 cm
  2. 2 Write the ratio in cm \(45 : 120\)
  3. 3 The HCF of 45 and 120 is 15 \(45 \div 15 = 3\), \(120 \div 15 = 8\)

Answer\(3 : 8\)

Ratios and Fractions

A ratio tells you how many parts there are altogether.

  • Total parts

    In the ratio \(3 : 5\) there are \(3 + 5 = 8\) parts.

  • As fractions

    The first amount is \(\frac{3}{8}\) of the total and the second is \(\frac{5}{8}\).

  • Not the same thing

    \(3 : 5\) does NOT mean \(\frac{3}{5}\) of the total - it means \(\frac{3}{8}\).

  • Unit ratios

    Divide to make one part 1: \(4 : 10 = 1 : 2.5\), and \(10 : 4 = 2.5 : 1\).

Sharing in a Ratio

Share £84 between Amy and Ben in the ratio \(3 : 4\).

Show the solutionHide the solution
  1. 1 Add the parts \(3 + 4 = 7\) parts
  2. 2 Find one part \(84 \div 7 = 12\)
  3. 3 Multiply for each person Amy \(3 \times 12 = 36\), Ben \(4 \times 12 = 48\)
  4. 4 Check \(36 + 48 = 84\)

AnswerAmy £36, Ben £48

When You Know the Difference

The ages of a father and son are in the ratio \(5 : 2\). The father is 18 years older than the son. How old is each?

Show the solutionHide the solution
  1. 1 The difference in parts \(5 - 2 = 3\) parts
  2. 2 3 parts are 18 years, so one part is \(18 \div 3 = 6\)
  3. 3 Multiply Father \(5 \times 6 = 30\), son \(2 \times 6 = 12\)
  4. 4 Check the difference \(30 - 12 = 18\)

AnswerFather 30, son 12

Ratio Problems Relay

(a) Simplify \(24 : 36 : 60\). (b) Share 450 g of sweets in the ratio \(2 : 3 : 4\). (c) Paint is mixed blue to white in the ratio \(2 : 7\). How much white is mixed with 5 litres of blue? (d) Write \(8 : 5\) in the form \(n : 1\).

1. Decide what you know: the total, one part, or a difference.

2. Find the value of one part.

3. Answer the question asked.

A good answer shows: (a) \(2 : 3 : 5\) (b) 100 g, 150 g, 200 g (c) \(5 \div 2 = 2.5\) litres per part, so \(7 \times 2.5 = 17.5\) litres (d) \(1.6 : 1\)

Can I...?

  1. 1Simplify a ratio.
  2. 2Simplify a ratio with different units.
  3. 3Write a ratio as \(1 : n\) or \(n : 1\).
  4. 4Write a ratio as fractions of the total.
  5. 5Share an amount in a ratio.
  6. 6Solve problems when I know one part or the difference.

Summary & Exam Focus

  • Simplify by dividing every part by the HCF; use the same units first.
  • Total parts = sum of the ratio; each share is a fraction of the total.
  • Find the value of one part, then multiply.
  • A difference in amounts matches the difference in parts.

Exam focus

Red and blue beads are in the ratio \(3 : 7\). There are 28 more blue beads than red beads. How many beads are there altogether? (3 marks) (3 marks)

Decide what the number in the question stands for: the total, one person's share, or the difference. Each needs a different number of parts.

Key terms

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

Ratio
A comparison of the sizes of two or more quantities.
Simplest form
A ratio whose parts have no common factor other than 1.
Part
One share of a ratio; the total number of parts is the sum of the ratio.
Unit ratio
A ratio written with one part equal to 1, such as \(1 : n\).

Questions and answers

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

1. Exam question Non-calculator 1 mark Easier

Write \(24 : 36 : 60\) in its simplest form.

Mark scheme — 1 mark available

  • \(2 : 3 : 5\) — B1

Model answer

\(2 : 3 : 5\)

2. Exam question Non-calculator 3 marks Easier

Share £450 between Asha, Ben and Cal in the ratio \(2 : 3 : 4\).

Mark scheme — 3 marks available

  • 9 parts, or \(450 \div 9\) — M1
  • 50 — M1
  • £100, £150, £200 — A1

Model answer

\(2 + 3 + 4 = 9\) parts; \(450 \div 9 = 50\). Asha £100, Ben £150, Cal £200.

3. Exam question Non-calculator 3 marks Easier

The ratio of boys to girls in a club is \(4 : 5\). There are 45 girls. How many members does the club have?

Mark scheme — 3 marks available

  • \(45 \div 5 = 9\) — P1
  • 36 boys — P1
  • 81 — A1

Model answer

5 parts = 45, so 1 part = 9. Boys: \(4 \times 9 = 36\). Total \(36 + 45 = 81\).

4. Exam question Non-calculator 3 marks Easier

Red and blue beads are in the ratio \(3 : 7\). There are 28 more blue beads than red beads. How many beads are there altogether?

Mark scheme — 3 marks available

  • 4 parts identified as the difference — P1
  • 1 part = 7 — P1
  • 70 — A1

Model answer

Difference in parts: \(7 - 3 = 4\). 4 parts = 28, so 1 part = 7. Total \(10 \times 7 = 70\) beads.

5. Multiple choice 1 mark Easier

What is \(18 : 12\) in its simplest form?

  1. A \(6 : 4\)
  2. B \(3 : 2\) Correct
  3. C \(2 : 3\)
  4. D \(9 : 6\)

Why: Divide both by the HCF, 6.

6. Multiple choice 1 mark Core

Juice and water are mixed in the ratio \(1 : 4\). What fraction of the drink is juice?

  1. A \(\frac{1}{4}\)
  2. B \(\frac{4}{5}\)
  3. C \(\frac{1}{5}\) Correct
  4. D \(\frac{1}{3}\)

Why: There are \(1 + 4 = 5\) parts, and 1 of them is juice.

7. Multiple choice 1 mark Stretch

Write \(5 : 8\) in the form \(1 : n\).

  1. A \(1 : 1.6\) Correct
  2. B \(1 : 3\)
  3. C \(1 : 0.625\)
  4. D \(1 : 40\)

Why: Divide both parts by 5: \(1 : 1.6\).