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Cambridge IGCSE0580

Ratio and proportion

Mathematics 0580 Chapter Notes

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Ratio and proportion
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In this chapter

  1. 1Ratios in their simplest form
  2. 2Dividing a quantity in a given ratio
  3. 3Using a difference between two shares
  4. 4A ratio that applies to part of a quantity
  5. 5Changing a ratio
  6. 6Recipes and map scales
  7. 7Best value

1. Ratios in their simplest form

A ratio is written as a : b, or a : b : c for three quantities. If the quantities are in different units, change them to the same unit first.

To give a ratio in its simplest form, divide every term by the same common factor. Stop when no whole number divides every term. A ratio in simplest form does not contain decimals.

20 : 30 : 40 = 2 : 3 : 4

Examiner insight

“Division by 3 and by 2 was often more successful than a single division by 6.” “Some attempted to make one of the ratios equal to 1 not appreciated that ratios in simplest form do not contain decimals.”

Worked example 12 marks

Do not use a calculator. The table gives the cost of a ticket for each type of seating. Find the ratio of the cost of the tickets stalls : circle : private box. Give your answer in its simplest form.

type of seatingcost of a ticket
stallschar"24 30
circlechar"24 24
private boxchar"24 72
  1. 1

    write the ratio 30 : 24 : 72 and divide each term by the common factor 6 [1]

  2. 2

    30 : 24 : 72 = 5 : 4 : 12 [1]

2. Dividing a quantity in a given ratio

To share a quantity in a ratio, add the ratio terms to find the total number of parts.

Divide the quantity by the number of parts to find one part. Multiply one part by each term of the ratio. Check that the shares add up to the quantity.

one part = quantity/(total number of parts)

Examiner insight

“Another common error was to divide 4620 by the ratio parts 2 and by 3 as given in the question.”

Worked example 12 marks

Do not use a calculator. The total number of apples and oranges in a bowl is 20. The ratio apples : oranges = 7 : 3. Work out the number of apples.

  1. 1

    total parts = 7 + 3 = 10, so one part = 20 ÷ 10 = 2 [1]

  2. 2

    apples = 7 × 2 = 14 [1]

3. Using a difference between two shares

Sometimes a question gives how much more one share is than another, not the total.

The difference between the ratio terms is a number of parts. That number of parts equals the given difference. Divide to find one part, then find each share.

one part = (difference in amount)/(difference in parts)

Examiner insight

“Most candidates were familiar with ratio questions in which they are given the value of one of the items but many were unsure how to deal with this question, in which they were given the difference in value of two terms.”

Worked example 13 marks

Ruta and Zoli share some money in the ratio 7 : 3. Ruta receives char"24 420 more than Zoli. Work out how much money Ruta and Zoli each receive.

  1. 1

    difference in parts = 7 - 3 = 4, so 4 parts = char"24 420 [1]

  2. 2

    one part = 420 ÷ 4 = char"24 105, so Ruta = 7 × 105 = char"24 735 [1]

  3. 3

    Zoli = 3 × 105 = char"24 315 [1]

4. A ratio that applies to part of a quantity

Read which quantity the ratio is about. It may apply only to part of the whole.

First find the part the ratio describes, such as the tickets sold. Use that amount, not the full total.

Examiner insight

“Some correctly found the number of seats sold in the stalls as 1000 and made no further progress.”

Worked example 13 marks

Do not use a calculator. There are 1200 seats in the stalls. One evening, 5/6 of the tickets for seats in the stalls are sold. On this evening, the number of tickets sold for seats is in the ratio stalls : circle = 10 : 7. Find the number of tickets sold for seats in the circle.

  1. 1

    stalls tickets sold = 5/6 × 1200 = 1000 [1]

  2. 2

    10 parts = 1000, so one part = 100 [1]

  3. 3

    circle tickets sold = 7 × 100 = 700 [1]

5. Changing a ratio

When more items are added, the ratio can change. Work in steps.

Find the original amounts, add the new amounts, then use the new ratio to find the unknown.

Worked example 14 marks

Do not use a calculator. A bag contains 45 balls. The balls are red or green. The balls are in the ratio red balls : green balls = 2 : 3. Carlos puts 4 more red balls and x more green balls into the bag. The balls are now in the ratio red balls : green balls = 1 : 2. Find the value of x.

  1. 1

    one part = 45 ÷ 5 = 9, so red = 18 and green = 27 [1]

  2. 2

    new red = 18 + 4 = 22 [1]

  3. 3

    new ratio 1 : 2, so new green = 2 × 22 = 44 [1]

  4. 4

    x = 44 - 27 = 17 [1]

6. Recipes and map scales

To adapt a recipe, find the amount for one person or one unit, then multiply. Or multiply every amount by the same scale factor.

A map scale such as 1 : 50 000 means 1 cm on the map is 50 000 cm in real life. Multiply a map length by the second term. Then change cm to m or km.

real length = map length × 50 000 (scale 1 : 50 000)

100 000 cm = 1 km

Worked example 12 marks

Do not use a calculator. A map has a scale of 1 : 50 000. Two towns are 7 cm apart on the map. Find the real distance between the towns, in km.

  1. 1

    real distance = 7 × 50 000 = 350 000 cm [1]

  2. 2

    350 000 ÷ 100 000 = 3.5 km [1]

7. Best value

To find the best value, compare the cost of the same amount of each item. Work out the cost per unit mass, or the mass per unit cost.

Use the same unit for every item. Write the units beside each answer, so you pick the right one.

cost per 100 g = cost/(mass in g) × 100

Examiner insight

“Common errors included selecting the incorrect bag after correct comparisons seen, calculations not evaluated to enough accuracy for comparison, and incorrect methods involving the differences between costs and masses.” “However a significant number of candidates who did the correct divisions then chose the incorrect figure for the best value possibly because they rarely wrote down the units they were working with.”

Worked example 13 marks

A 300 g jar of jam costs char"24 2.10. A 500 g jar of the same jam costs char"24 3.25. Which jar is better value? Show how you decide.

  1. 1

    300 g jar: 2.10 ÷ 3 = char"24 0.70 per 100 g [1]

  2. 2

    500 g jar: 3.25 ÷ 5 = char"24 0.65 per 100 g [1]

  3. 3

    the 500 g jar costs less per 100 g, so it is better value [1]

Recap

  • Simplest form: divide every term by the same common factor. Use whole numbers only.
  • Change to the same unit before you simplify a ratio.
  • Sharing: total parts, then one part, then each share. Check the shares add to the total.
  • A difference between shares equals the difference in parts times one part.
  • If the ratio applies to part of a quantity, find that part first.
  • Map length times the scale gives the real length. Convert cm to km by dividing by 100 000.
  • Best value: compare the cost of the same amount of each item and write the units.

End-of-chapter exercise

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

  1. Do not use a calculator. Write the ratio 18 : 24 in its simplest form.1 mark
  2. Do not use a calculator. Write the ratio 0.5 m : 80 cm in its simplest form.2 marks
  3. Do not use a calculator. Share char"24 72 in the ratio 5 : 3.2 marks
  4. Blue and yellow paint are mixed in the ratio 3 : 5. Sofia uses 2.4 litres of blue paint. Work out the amount of yellow paint she uses.2 marks
  5. Do not use a calculator. A recipe for 6 people uses 240 g of flour. Work out the amount of flour needed for 9 people.2 marks
  6. Do not use a calculator. A map has a scale of 1 : 25 000. Two villages are 6 km apart. Find the distance between the villages on the map, in cm.3 marks
  7. A 250 g tub of butter costs char"24 1.80. A 400 g tub costs char"24 2.80. Which tub is better value? Show how you decide.3 marks
  8. Ana and Ben share some money in the ratio 4 : 7. Ben receives char"24 45 more than Ana. Work out how much each receives.3 marks
  9. Do not use a calculator. The students at a school walk, take the bus or cycle to school. The numbers are in the ratio walk : bus : cycle = 5 : 4 : 3. There are 240 more students who walk than cycle. Some students change from the bus to cycling. The ratio is now walk : bus : cycle = 10 : 5 : 9. Work out how many students changed from the bus to cycling.5 marks
  10. Concrete is made from cement, sand and gravel with masses in the ratio 1 : 2 : 4. A builder needs 1540 kg of concrete. Shop A sells cement in 25 kg bags at char"24 6.80 each. Shop B sells cement in 40 kg bags at char"24 10.70 each. The builder buys whole bags from one shop only. Work out which shop is cheaper for the builder, and by how much.6 marks

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