← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 1 Number · Chapter 6 · pp. 50–61

Ratio and proportion

Recipes, maps, shared prizes and train timetables all run on proportion. Two questions decide every problem in this chapter: do the two quantities go up together, or does one go down as the other goes up? Once you know that, the unitary method (“find one first”) solves almost everything.

Learning objectives

  1. Solve direct proportion problems by the ratio method and the unitary method.
  2. Divide a quantity in a given ratio.
  3. Solve inverse proportion problems.
  4. Increase and decrease a quantity by a given ratio.
  5. Use the compound measures speed, density and population density.

Book exercises: 6.1 to 6.7 and Student assessments 1 and 2 (pp. 50–61).

Direct or inverse?

hours worked → Direct wage ↑ as hours ↑ people working → Inverse time ↓ as people ↑
Direct proportion is a straight line through the origin: double one, double the other. Inverse proportion curves down: double one, halve the other.

🔑 Two methods for direct proportion

  • Ratio method: set up equal fractions, = , and solve.
  • Unitary method: find the value for one unit, then multiply up.
  • Put both quantities in the same units first (1½ hours = 90 minutes).

📘 Example 1 · A machine fills 500 bottles in 15 minutes. How many in 1½ hours? (unitary method)

  1. Same units: 1½ hours = 90 minutes.
  2. Find one: in 1 minute it fills 500 ÷ 15 bottles.
  3. Multiply up: in 90 minutes it fills 90 × 500 ÷ 15 = 3000 bottles.

📘 Example 2 · Tin and copper are mixed 8 : 3. How much tin goes with 36 g of copper?

  1. 3 g of copper mixes with 8 g of tin.
  2. So 1 g of copper mixes with g of tin.
  3. 36 g of copper mixes with 36 × = 96 g of tin.

✏️ Try it

  1. A machine prints four books in 10 minutes. How many in 2 hours?
  2. Sand and gravel are mixed 5 : 3. How much gravel goes with 750 kg of sand?
Answers

1. 2 hours = 120 min, 120 ÷ 10 × 4 = 48 books · 2. 750 ÷ 5 × 3 = 450 kg

Next: dividing in a ratio →

Dividing a quantity in a given ratio

🔑 Key ideas

  • Add the ratio numbers to get the total number of parts.
  • Divide the quantity by the total to find one part.
  • Multiply one part by each number in the ratio. The shares must add back to the whole.
4 m4 m4 m 4 m4 m 3 parts = 12 m 2 parts = 8 m 20 m shared 3 : 2 → 5 parts of 4 m
Draw the ratio as equal boxes. 3 : 2 is five boxes, so each box is 20 ÷ 5 = 4 m.

📘 Example 3 · Divide 20 m in the ratio 3 : 2

  1. Total parts: 3 + 2 = 5.
  2. One part: 20 ÷ 5 = 4 m.
  3. Shares: 3 × 4 = 12 m and 2 × 4 = 8 m.
  4. Check: 12 + 8 = 20. ✔

📘 Example 4 · Red, blue, white and green cars are made 7 : 5 : 3 : 1. How many of 48 000 are white?

  1. Total parts: 7 + 5 + 3 + 1 = 16.
  2. One part: 48 000 ÷ 16 = 3000 cars.
  3. White is 3 parts: 3 × 3000 = 9000 white cars.

🧪 Try it: ratio sharer

    ✏️ Try it

    1. Divide 72 in the ratio 2 : 3 : 4.
    2. The angles of a triangle are in the ratio 3 : 5 : 4. Find each angle.
    3. 55% of students are boys. What is the ratio of boys to girls?
    Answers

    1. 9 parts of 8: 16, 24, 32 · 2. 12 parts of 15°: 45°, 75°, 60° · 3. 55 : 45 = 11 : 9

    Next: inverse proportion →

    Inverse proportion

    🔑 Key ideas

    • More of one means less of the other: more pickers, fewer days.
    • The product stays the same: people × days is the total work.
    • Unitary method: find how long one would take (multiply), then share it out (divide).
    8 people × 6 days= 48 person-days 8 wide6 12 people × 4 days= 48 person-days 12 wide4
    The job is the same size, so the rectangle keeps its area. Make it wider (more people) and it must get shorter (fewer days).

    📘 Example 5 · 8 people pick the apples in 6 days. How long will 12 people take?

    1. Decide: more people means fewer days, so it is inverse.
    2. One person would take 6 × 8 = 48 days.
    3. 12 people take 48 ÷ 12 = 4 days.

    📘 Example 6 · A cyclist averages 27 km/h for 4 hours. What speed covers the same distance in 3 hours?

    1. Less time needs more speed, so it is inverse.
    2. Doing it in 1 hour would need 27 × 4 = 108 km/h.
    3. In 3 hours: 108 ÷ 3 = 36 km/h.

    ✏️ Try it

    1. Six people dig a trench in 8 hours. How long would 4 people take?
    2. Chairs are in 35 rows of 18. How many rows with 21 chairs per row?
    Answers

    1. 48 ÷ 4 = 12 hours · 2. 630 ÷ 21 = 30 rows

    Next: increase and decrease by a ratio →

    Increase and decrease by a given ratio

    🔑 Key ideas

    • Increase by a ratio 3 : 2 → multiply by . Decrease by 2 : 3 → multiply by .
    • The new size goes on top of the fraction, the old size underneath.
    • If lengths are multiplied by k, areas are multiplied by k² and volumes by k³.
    • A map scale 1 : 25 000 means 1 cm on the map is 25 000 cm (250 m) in real life.
    12 × 8 × 3/2 18 × 12 width 12 × 3/2 = 18 · height 8 × 3/2 = 12
    Enlarging a 12 cm × 8 cm photo in the ratio 3 : 2 multiplies every length by 3/2.

    📘 Example 7 · A slide 5 cm wide by 3 cm tall is projected to an image 1.5 m wide

    1. Same units: 1.5 m = 150 cm.
    2. Scale factor: 150 ÷ 5 = 30, so the ratio of enlargement is 30 : 1.
    3. Height of the image: 3 × 30 = 90 cm.

    📘 Example 8 · Map scale 1 : 25 000. Two villages are 8 cm apart on the map

    1. Real distance = 8 × 25 000 = 200 000 cm.
    2. 200 000 cm ÷ 100 = 2000 m, and 2000 m ÷ 1000 = 2 km.

    ✏️ Try it

    1. Increase 40 by a ratio of 5 : 4.
    2. Decrease 210 by a ratio of 3 : 7.
    3. A 7 cm by 4 cm rectangle is enlarged 2 : 1. By what ratio does its area grow?
    Answers

    1. 40 × 5/4 = 50 · 2. 210 × 3/7 = 90 · 3. 28 cm² → 112 cm², so 4 : 1 (that is 2²)

    Next: compound measures →

    Compound measures

    🔑 Key ideas

    • A compound measure is built from two others: speed (km/h), density (g/cm³), population density (people/km²).
    • Cover the one you want in the triangle: what's left tells you to multiply (side by side) or divide (one over the other).
    • Units must match: a time in minutes must become hours before you use km/h.
    • Decimal hours → minutes: multiply the decimal part by 60.
    Distance Speed Time Mass Density Volume Population Pop. density Area
    Formula triangles. Cover Speed and you see Distance over Time, so speed = distance ÷ time. Cover Distance and you see Speed × Time.
    MeasureFormulaRearrangedUnits
    Speeddistance ÷ timeD = S × T, T = D ÷ Skm/h, m/s
    Densitymass ÷ volumeM = D × V, V = M ÷ Dkg/m³, g/cm³
    Population densitypopulation ÷ areaP = PD × A, A = P ÷ PDpeople/km²

    📘 Example 9 · A train covers 140 km in 1½ hours, including 15 minutes stopped at stations

    1. Average speed overall = 140 ÷ 1.5 = 93⅓ km/h.
    2. Same units for the stop: 15 minutes = ¼ hour, so moving time = 1½ − ¼ = 1¼ hours.
    3. Average speed while moving = 140 ÷ 1.25 = 112 km/h.

    📘 Example 10 · At 120 km/h, how long does the 140 km journey take?

    1. Time = distance ÷ speed = 140 ÷ 120 = 1.1666… hours.
    2. Convert the decimal part to minutes: 0.1666… × 60 = 10.
    3. Time = 1 hour 10 minutes.

    📘 Example 11 · A village of 540 people covers 8 km². The density must not exceed 110 people/km²

    1. Population density now = 540 ÷ 8 = 67.5 people/km².
    2. Most people allowed = 110 × 8 = 880.
    3. Extra people the village can take = 880 − 540 = 340.

    ✏️ Try it

    1. Aluminium has a density of 2900 kg/m³. What is the mass of 4 m³?
    2. A 5 cm cube of metal has a mass of 1.1 kg. Find its density in g/cm³.
    Answers

    1. 2900 × 4 = 11 600 kg · 2. 1100 g ÷ 125 cm³ = 8.8 g/cm³ (brass)

    Next: summary and quiz →

    Chapter summary

    • Direct proportion: both go up together. Find one unit, then multiply up.
    • Dividing in a ratio: add the parts, find one part, multiply by each number. Check the shares add up.
    • Inverse proportion: one goes up as the other goes down. The product stays the same.
    • Increase or decrease by a ratio a : b: multiply by a/b. Areas scale by k², volumes by k³.
    • Speed = distance ÷ time, density = mass ÷ volume, population density = population ÷ area. Match the units first.

    📄 Practice questions

    The book's Exercises 6.1 to 6.7 and Student assessments 1 and 2 are in the printable assignment for this chapter.

    Mini-quiz

    1. A machine fills 500 bottles in 15 minutes. How many does it fill in 90 minutes?
    2. Tin and copper are mixed 8 : 3. How much tin goes with 36 g of copper?
    3. Divide 20 m in the ratio 3 : 2.
    4. Angles of a triangle are in the ratio 1 : 2 : 3. What is the largest angle?
    5. 8 people pick the apples in 6 days. How long do 12 people take?
    6. A 12 cm by 8 cm photo is enlarged in the ratio 3 : 2. What is the new size?
    7. A train travels 140 km at 120 km/h. How long does the journey take?
    8. A village of 540 people has an area of 8 km². What is its population density?

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 6 (Ratio and proportion). Explanations, the Try it answers, the tool and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.