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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.
Last Lesson and Before
Answer each one, then check.
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1
What is the HCF of 12 and 18?
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6
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2
Last lesson: work out \(\frac{3}{5}\) of 40.
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24
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3
Simplify \(\frac{15}{25}\).
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\(\frac{3}{5}\)
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4
How many grams are in 1.2 kg?
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1200 g
Learning Objectives
What a Ratio Is
A ratio compares amounts: 3 red counters to 5 blue is written \(3 : 5\).
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The order matters
\(3 : 5\) (red to blue) is not the same as \(5 : 3\).
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Simplify
Divide every part by the HCF: \(12 : 18 = 2 : 3\).
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Same units first
45 cm : 1.2 m becomes 45 : 120 (both in cm), then \(3 : 8\).
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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.
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- 1 Put both in the same units 1.2 m = 120 cm
- 2 Write the ratio in cm \(45 : 120\)
- 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.
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Total parts
In the ratio \(3 : 5\) there are \(3 + 5 = 8\) parts.
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As fractions
The first amount is \(\frac{3}{8}\) of the total and the second is \(\frac{5}{8}\).
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Not the same thing
\(3 : 5\) does NOT mean \(\frac{3}{5}\) of the total - it means \(\frac{3}{8}\).
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Unit ratios
Divide to make one part 1: \(4 : 10 = 1 : 2.5\), and \(10 : 4 = 2.5 : 1\).
Sharing with a Bar Model
A bar model makes a ratio visible. The ratio \(3 : 4\) has 7 equal parts. £84 shared into 7 parts is £12 a part, so the shares are \(3 \times 12 = £36\) and \(4 \times 12 = £48\).
7 parts share £84, so each part is £12.
Sharing in a Ratio
Share £84 between Amy and Ben in the ratio \(3 : 4\).
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- 1 Add the parts \(3 + 4 = 7\) parts
- 2 Find one part \(84 \div 7 = 12\)
- 3 Multiply for each person Amy \(3 \times 12 = 36\), Ben \(4 \times 12 = 48\)
- 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 The difference in parts \(5 - 2 = 3\) parts
- 2 3 parts are 18 years, so one part is \(18 \div 3 = 6\)
- 3 Multiply Father \(5 \times 6 = 30\), son \(2 \times 6 = 12\)
- 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...?
- 1Simplify a ratio.
- 2Simplify a ratio with different units.
- 3Write a ratio as \(1 : n\) or \(n : 1\).
- 4Write a ratio as fractions of the total.
- 5Share an amount in a ratio.
- 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 \(1 : n\).
Practice questions
Have a go at each one before you open its answer.
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Question 1 Non-calculator 1 mark
Write \(24 : 36 : 60\) in its simplest form.
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Model answer
\(2 : 3 : 5\)
Mark scheme
- \(2 : 3 : 5\) — B1
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Question 2 Non-calculator 3 marks
Share £450 between Asha, Ben and Cal in the ratio \(2 : 3 : 4\).
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Model answer
\(2 + 3 + 4 = 9\) parts; \(450 \div 9 = 50\). Asha £100, Ben £150, Cal £200.
Mark scheme
- 9 parts, or \(450 \div 9\) — M1
- 50 — M1
- £100, £150, £200 — A1
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Question 3 Non-calculator 3 marks
The ratio of boys to girls in a club is \(4 : 5\). There are 45 girls. How many members does the club have?
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Model answer
5 parts = 45, so 1 part = 9. Boys: \(4 \times 9 = 36\). Total \(36 + 45 = 81\).
Mark scheme
- \(45 \div 5 = 9\) — P1
- 36 boys — P1
- 81 — A1
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Question 4 Non-calculator 3 marks
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?
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Model answer
Difference in parts: \(7 - 3 = 4\). 4 parts = 28, so 1 part = 7. Total \(10 \times 7 = 70\) beads.
Mark scheme
- 4 parts identified as the difference — P1
- 1 part = 7 — P1
- 70 — A1
Quick check
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What is \(18 : 12\) in its simplest form?
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B: \(3 : 2\)
Divide both by the HCF, 6.
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Juice and water are mixed in the ratio \(1 : 4\). What fraction of the drink is juice?
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C: \(\frac{1}{5}\)
There are \(1 + 4 = 5\) parts, and 1 of them is juice.
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Write \(5 : 8\) in the form \(1 : n\).
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A: \(1 : 1.6\)
Divide both parts by 5: \(1 : 1.6\).
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