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Cambridge IGCSE0607

Ratio and proportion

International Mathematics 0607 Chapter Notes

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Ratio and proportion
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Ratio and proportion notes

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In this chapter

  1. 1Ratios in their simplest form
  2. 2Dividing a quantity in a given ratio
  3. 3Working from one known share
  4. 4Proportional reasoning in context
  5. 5Map scales and best value

1. Ratios in their simplest form

A ratio compares quantities. To give it in its simplest form, divide every part by the same number until no number divides them all.

If the quantities are in different units, change them to the same unit first. The final ratio has no units.

For example, 20 : 30 : 40 in its simplest form is 2 : 3 : 4.

a : b = (a ÷ n) : (b ÷ n)

Examiner insight

“A few left their answer as 10 : 8 and some other candidates cancelled incorrectly.” “Candidates should be aware that ratios do not strictly include units.”

Worked example 12 marks

Write the ratio 30 g : 6 kg in its simplest form.

  1. 1

    change to the same unit: 6 kg = 6000 g, so 30 : 6000 [1]

  2. 2

    divide both parts by 30: 1 : 200 [1]

2. Dividing a quantity in a given ratio

Add the numbers in the ratio to find the total number of parts. Divide the quantity by this total to find the value of one part. Multiply one part by each number in the ratio.

Check your answer: the shares add up to the quantity.

one part = quantity/(total number of parts)

share = one part × its number in the ratio

480 shared in the ratio x : y gives 480x/(x + y) and 480y/(x + y)

Examiner insight

“Many candidates got as far as the value of one part and then either stopped or went on to give the amount of the smaller share.”

Worked example 12 marks

Share $165 in the ratio 3 : 8.

  1. 1

    total parts = 3 + 8 = 11, so one part = 165 ÷ 11 = 15 dollars [1]

  2. 2

    shares: 3 × 15 = 45 and 8 × 15 = 120, so 45 dollars and 120 dollars [1]

3. Working from one known share

Sometimes the question gives one share, not the whole quantity. Divide that share by its own number in the ratio to find one part. Then multiply to find the other share or the total.

Read the last line of the question. It may ask for one share, a difference, or the total.

Examiner insight

“Many candidates incorrectly divided 56 by 13 and then multiply their answer by 7 and 2.”

Worked example 13 marks

Dev and Esi share marbles in the ratio 7 : 5. Esi gets 35 marbles. Work out the total number of marbles they share.

  1. 1

    Esi's share is 5 parts, so one part = 35 ÷ 5 = 7 [1]

  2. 2

    total parts = 7 + 5 = 12 [1]

  3. 3

    total = 12 × 7 = 84 marbles [1]

4. Proportional reasoning in context

Two quantities in proportion are multiplied by the same number. Find the amount for one unit, then multiply by the new number. This adapts a recipe.

Finish the method. Dividing 100 g by 35 g gives the number of batches, not the number of biscuits: multiply by the 20 biscuits in a batch. For the greatest number that can be made, round down.

new amount = (old amount)/(old number) × new number

Examiner insight

“This part was less well done with the most common error was to divide 100 by 35 and not multiplying that by 20.”

Worked example 14 marks

These are the ingredients for making 20 ginger biscuits.

(a) Laila makes 48 biscuits. Work out the amount of sugar Laila needs.

(b) Girish has 100 g of ginger and large amounts of the other ingredients. Work out the greatest number of biscuits Girish can make.

ingredientamount (g)
butter100
sugar75
ginger35
syrup100
flour250
  1. 1

    (a) sugar = 75 ÷ 20 × 48 [1]

  2. 2

    = 180 g [1]

  3. 3

    (b) biscuits = 100 ÷ 35 × 20. On the calculator enter 100 ÷ 35 × 20 and read 57.142… [1]

  4. 4

    only whole biscuits can be made: round down to 57 [1]

5. Map scales and best value

A map scale such as 1 : 50 000 is a ratio. 1 cm on the map is 50 000 cm on the ground. Multiply to find a real distance. Divide to find a distance on the map. Then change to the unit asked for.

To find the best value, work out the cost of the same amount for each offer. The offer with the lower cost for the same amount is better value.

Worked example 13 marks

A shop sells 600 g of jam for 2.40 dollars and 1.5 kg of jam for 5.70 dollars. Which is better value?

  1. 1

    use the same unit: 600 g = 0.6 kg [1]

  2. 2

    cost per kg: 2.40 ÷ 0.6 = 4.00 and 5.70 ÷ 1.5 = 3.80, in dollars [1]

  3. 3

    the 1.5 kg jar is better value, because its cost per kg is lower [1]

Recap

  • Simplest form: change to the same unit, then divide every part by the same number.
  • A ratio has no units in the final answer.
  • Dividing a quantity: total parts, then one part, then each share.
  • Given one share: divide by its own number in the ratio to find one part.
  • Check whether the question asks for one share, a difference or the total.
  • Proportion: find one unit, then multiply. Round down for the greatest number that can be made.
  • Map scale 1 : n: multiply by n for the real distance, then change the unit.
  • Best value: compare the cost of the same amount.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write 12 : 18 in its simplest form.1 mark
  2. Do not use a calculator. Write 24 : 36 : 60 in its simplest form.1 mark
  3. Write 2.4 kg : 800 g in its simplest form.2 marks
  4. Do not use a calculator. Share $84 in the ratio 2 : 5.2 marks
  5. Do not use a calculator. Mia and Noor share money in the ratio 3 : 5. Mia receives $27. How much more than Mia does Noor receive?3 marks
  6. Do not use a calculator. The scale of a map is 1 : 50 000. Two towns are 6.4 cm apart on the map. Work out the real distance between the towns, in km.3 marks
  7. 12 muffins need 180 g of flour. Zara has 500 g of flour. Work out the greatest number of muffins she can make.3 marks
  8. A shop sells the same juice in three ways. A 330 ml can costs 0.66 dollars. A 1.5 litre bottle costs 2.85 dollars. A pack of 6 cans costs 3.60 dollars. Which is the best value? Show your working.4 marks
  9. A gardener makes plant feed. 12 litres of feed need 45 g of nitrate and 20 g of potash, mixed with water. She has 0.4 kg of nitrate, 150 g of potash and large amounts of water. She pours the feed into bottles that each hold 700 ml. Work out the greatest number of bottles she can fill.5 marks
  10. Do not use a calculator. A walking route has three stages. The lengths of the stages are in the ratio 3 : 5 : 4. On a map with scale 1 : 25 000, the second stage is 6 cm longer than the first stage. Work out the real length of the whole route, in km.5 marks

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