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Maths · Fractions, ratio and percentages

Ratio and proportion

When two quantities are in proportion, knowing one tells you the other. Recipes, best buys and currency are direct proportion; more workers finishing a job faster is inverse proportion.

  • 5 key terms
  • All boards
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Last Lesson and Before

Answer each one, then check.

  1. 1

    Last lesson: share 20 in the ratio \(1 : 4\).

    Show answerHide answer

    4 and 16

  2. 2

    Work out \(12 \times 1.5\).

    Show answerHide answer

    18

  3. 3

    Work out \(450 \div 6\).

    Show answerHide answer

    75

  4. 4

    Last lesson: write \(4 : 10\) in the form \(1 : n\).

    Show answerHide answer

    \(1 : 2.5\)

Learning Objectives

  1. 1Solve direct proportion problems with the unitary method.
  2. 2Compare best buys.
  3. 3Convert between currencies.
  4. 4Solve inverse proportion problems.
  5. 5Recognise direct and inverse proportion from a graph.

Direct Proportion

Two quantities are in direct proportion when they increase at the same rate: double one and the other doubles.

  • The unitary method

    Find the value of ONE, then multiply: if 6 pens cost £4.20, one costs 70p, so 10 cost £7.

  • Scaling

    Multiply both by the same number: 6 people to 9 people is \(\times 1.5\).

  • The graph

    A straight line through the origin.

  • Examples

    Cost and number of items, recipe amounts and servings, pounds and euros.

A Recipe

A recipe for 6 people uses 450 g of flour. How much flour is needed for 10 people?

Show the solutionHide the solution
  1. 1 Find the amount for 1 person \(450 \div 6 = 75\) g
  2. 2 Multiply by 10 \(75 \times 10 = 750\) g

Answer750 g of flour

Best Buy

A 500 g bag of rice costs £1.85. A 750 g bag costs £2.70. Which is better value?

Show the solutionHide the solution
  1. 1 Find the cost of the same amount of each, say 100 g \(185 \div 5 = 37\)p and \(270 \div 7.5 = 36\)p
  2. 2 Compare 36p is less than 37p

AnswerThe 750 g bag is better value (36p per 100 g).

Currency Conversion

Exchange rates are direct proportion.

  • Multiply to convert from pounds

    If £1 = €1.15, then £240 is \(240 \times 1.15 = 276\) euros, €276.

  • Divide to convert back

    €92 is \(92 \div 1.15 = 80\) pounds, £80.

  • Check the size

    Each pound buys more than one euro, so the euro amount should be bigger.

Inverse Proportion

Two quantities are in inverse proportion when one goes up as the other goes down, and their product stays the same.

  • The idea

    Double the number of workers and the job takes half the time.

  • The method

    Find the total "work": 6 painters \(\times\) 8 days = 48 painter-days. Then divide.

  • The graph

    A curve that gets closer and closer to both axes.

  • Examples

    Workers and time, speed and journey time, people sharing a prize.

An Inverse Proportion Problem

6 painters take 8 days to paint a building. How long would 4 painters take, working at the same rate?

Show the solutionHide the solution
  1. 1 Find the total work \(6 \times 8 = 48\) painter-days
  2. 2 Divide by the new number of painters \(48 \div 4 = 12\)

Answer12 days

Shopping Detective

Which is the best buy? (a) Cereal: 500 g for £2.40 or 750 g for £3.45. (b) Juice: 1 litre for £1.10 or 1.5 litres for £1.62. (c) Eggs: 6 for £1.35 or 15 for £3.30. Then: 3 friends share a prize and get £40 each. How much would each get if 5 friends shared it?

1. Compare the same amount of each.

2. Show the unit price.

3. Decide, and say why.

A good answer shows: (a) 48p vs 46p per 100 g: 750 g box. (b) 11p vs 10.8p per 100 ml: 1.5 litres. (c) 22.5p vs 22p per egg: 15 eggs. Prize: \(3 \times 40 = 120\); \(120 \div 5 = £24\) each.

Can I...?

  1. 1Use the unitary method.
  2. 2Scale a recipe.
  3. 3Compare best buys.
  4. 4Convert between currencies.
  5. 5Solve inverse proportion problems.
  6. 6Recognise direct and inverse proportion graphs.

Summary & Exam Focus

  • Direct proportion: find one, then multiply.
  • Best buy: compare the price of the same amount.
  • Currency: multiply one way, divide the other.
  • Inverse proportion: the product stays the same.

Exam focus

It takes 5 machines 12 hours to make an order. How long would 3 machines take? (2 marks) (2 marks)

Ask first: if one quantity doubles, does the other double (direct) or halve (inverse)? Getting that right is most of the marks.

Key terms

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

Direct proportion
Two quantities that increase at the same rate; their ratio stays the same.
Inverse proportion
As one quantity increases, the other decreases; their product stays the same.
Unitary method
Finding the value of one item first, then scaling up.
Exchange rate
How much of one currency you get for one unit of another.
Best buy
The option that gives the most for your money.

Practice questions

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

  1. Question 1 Calculator 3 marks

    A recipe for 8 biscuits uses 120 g of butter and 200 g of flour. Jo wants to make 20 biscuits. She has 250 g of butter. Does she have enough butter? Show how you decide.

    Show answerHide answer

    Model answer

    One biscuit uses \(120 \div 8 = 15\) g of butter. 20 biscuits need \(20 \times 15 = 300\) g. She has 250 g, so she does not have enough.

    Mark scheme

    • \(120 \div 8\) or \(20 \div 8 = 2.5\) — P1
    • 300 g — A1
    • No, with 300 g compared to 250 g — C1
  2. Question 2 Calculator 3 marks

    Toothpaste is sold in two sizes: 75 ml for £1.80 and 125 ml for £2.85. Which is the better value? You must show your working.

    Show answerHide answer

    Model answer

    75 ml: \(180 \div 75 = 2.4\)p per ml. 125 ml: \(285 \div 125 = 2.28\)p per ml. The 125 ml tube is better value.

    Mark scheme

    • A correct method to compare, e.g. price per ml for one size — M1
    • Both correct: 2.4p and 2.28p per ml — A1
    • 125 ml, from correct figures — C1
  3. Question 3 Calculator 2 marks

    It takes 5 machines 12 hours to make an order. How long would 3 machines take, working at the same rate?

    Show answerHide answer

    Model answer

    \(5 \times 12 = 60\) machine-hours. \(60 \div 3 = 20\) hours.

    Mark scheme

    • \(5 \times 12\) or 60 — M1
    • 20 hours — A1
  4. Question 4 Calculator 2 marks

    The exchange rate is £1 = $1.26. Lara changes £350 into dollars. How many dollars does she get?

    Show answerHide answer

    Model answer

    \(350 \times 1.26 = 441\) dollars

    Mark scheme

    • \(350 \times 1.26\) — M1
    • $441 — A1

Quick check

  1. 4 pens cost £2.40. How much do 7 pens cost?

    1. A£3.60
    2. B£4.20
    3. C£4.80
    4. D£2.80
    Show answerHide answer

    B: £4.20

    One pen costs \(2.40 \div 4 = 60\)p, so 7 cost \(7 \times 60 = 420\)p = £4.20.

  2. 4 people take 6 hours to paint a fence. How long would 8 people take?

    1. A3 hours
    2. B12 hours
    3. C6 hours
    4. D2 hours
    Show answerHide answer

    A: 3 hours

    Inverse proportion: \(4 \times 6 = 24\) person-hours, and \(24 \div 8 = 3\) hours.

  3. Which graph shows direct proportion?

    1. AA curve getting closer to both axes
    2. BA horizontal straight line
    3. CA straight line crossing the y-axis at 5
    4. DA straight line through the origin
    Show answerHide answer

    D: A straight line through the origin

    Direct proportion is always a straight line through the origin.

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