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VCE Foundation Mathematics (Year 12) · Number and Arithmetic · 10m read

Ratios and Proportions

Ever wondered how chefs know exactly how much salt to add when they triple a cake recipe? That's ratios at work. A ratio is just a way of comparing amounts—like saying a smoothie needs two parts fruit to one part yogurt. You'll use ratios to split things fairly, scale recipes up or down, read maps, and solve real problems where proportions matter. Since VCAA tests these skills constantly in practical contexts, mastering ratios now makes both the exam and everyday cooking, building, and budgeting way easier.

Learning objective: Use ratios to share quantities, scale measurements and interpret maps and recipes.

By the end of this lesson you can:

  • I always find total parts before solving a sharing problem.
  • I simplify ratios by dividing both sides by the highest common factor.
  • I convert between different units before writing a ratio.

What are ratios and proportions

A ratio compares two or more quantities showing how much of one thing there is relative to another. "2:3" means for every 2 of the first you have 3 of the second. A proportion states that two ratios are equal.

Ratios appear in VCAA Foundation Maths in three recurring contexts: mixing ingredients or paint, sharing money or prizes, and reading maps and scale diagrams. The first move on almost every ratio question is the same — find the total number of parts.

Smoothie logic. Mix banana to mango in a ratio of 2:3. For every 200 mL of banana you need 300 mL of mango. Double the recipe: 400 mL and 600 mL — the ratio is still 2:3. That's proportional thinking, in one glass.

Ratio toolkit

  • Total parts: Add all ratio numbers. 2:3 has 5 total parts.
  • Each share: Total ÷ total parts × your share.
  • Scale up/down: Multiply both sides by the same number. 2:3 = 4:6.
  • Unit rate: Divide until one side is 1. 150 km in 3 h = 50 km/h.

Quick reference

| Idea | Example |
|---|---|
| Ratio | 2 : 5 |
| Simplify | 10 : 25 = 2 : 5 |
| Share $70 in 2 : 5 | $20 and $50 |
| Proportion | 3 for $6 → 7 for $14 |

Worked example (Easy)

Mia and Jake share a $480 prize in the ratio 3:5. How much does each person receive?

  1. Total parts. Add the ratio numbers. 3 + 5 = 8
  2. Value of one part. Divide the total by total parts. $480 ÷ 8 = $60
  3. Mia's share. Multiply her ratio value by the value of one part. 3 × $60 = $180
  4. Jake's share. Same logic for Jake. 5 × $60 = $300
  5. Check. Shares must sum to the original total. $180 + $300 = $480 ✓

Answer: Mia receives $180 and Jake receives $300.

Practice questions

  1. A concrete mix uses cement, sand and gravel in ratio 1:2:4. A builder needs 28 kg total. How much sand is needed?

    1. 8 kg
    2. 4 kg
    3. 14 kg
    4. 2 kg

    (2 marks)

  2. A map scale is 1:50,000. Two towns are 6 cm apart on the map. What is the real distance in km?

    1. 3 km
    2. 30 km
    3. 300 km
    4. 0.3 km

    (2 marks)

  3. Priya and Sam share profits in the ratio 5:3. Last month's profit was $1,240. This month's profit increased 20%. How much does Priya receive this month?

    1. $930
    2. $825
    3. $775
    4. $744

    (3 marks)

  4. A tuck shop makes punch by mixing apple juice and orange juice in the ratio 3:2. They make 5 L total. Apple juice costs $3.50/L and orange juice costs $2.80/L. What is the total cost?

    (3 marks)

  5. Simplify the ratio 12 : 18 : 30 to its lowest terms.

    1. 2 : 3 : 5
    2. 4 : 6 : 10
    3. 6 : 9 : 15
    4. 1 : 2 : 3

    (2 marks)

  6. A car travels 240 km on 16 L of fuel. What is the fuel-use rate in L/100 km?

    1. 6.67 L/100 km
    2. 6 L/100 km
    3. 15 L/100 km
    4. 24 L/100 km

    (2 marks)

Worked example (Medium)

A paint mix uses white, blue and yellow in the ratio 5:2:1. A painter needs 16 L of mix in total. How much of each colour is needed, and what is the new ratio if they double the yellow?

  1. Total parts. Add the ratio numbers. 5 + 2 + 1 = 8
  2. One part. Divide total by total parts. 16 ÷ 8 = 2 L
  3. Each colour. Multiply each ratio number by one part. White 10 L, Blue 4 L, Yellow 2 L
  4. Doubling yellow. Yellow goes from 1 to 2 in the ratio. 5 : 2 : 2
  5. Simplify (if possible). No common factor. Final ratio 5 : 2 : 2

Answer: White 10 L, Blue 4 L, Yellow 2 L. After doubling yellow the new ratio is 5 : 2 : 2.

Common mistakes

  • Forgetting to add the total parts first A ratio of 3:5 is 8 parts, not 3 or 5. Skipping this turns the answer into the wrong fraction of the total. Fix: Always write "Total parts = 3 + 5 = 8" as your first line.
  • Mixing units inside one ratio A ratio of "30 cm : 1 m" is meaningless until both sides use the same unit. Fix: Convert to a common unit FIRST: 30 cm : 100 cm = 3 : 10.
  • Treating a 2:3 share as 2 out of 3 2:3 means 2 parts out of 5, not 2 out of 3. Fix: Build a denominator from the *sum*: 2/(2+3) = 2/5.

Going deeper — direct vs inverse proportion

If two quantities are in DIRECT proportion, doubling one doubles the other (e.g. cost vs number of items). If they're in INVERSE proportion, doubling one HALVES the other (e.g. workers vs time to finish a job).

Distance, speed and time use both: at a fixed speed, distance is in direct proportion to time. At a fixed distance, time is in inverse proportion to speed. Recognising which type of proportion appears in a question saves you from setting up the wrong equation.

Summary

  • A ratio compares parts of a whole; total parts = sum of the ratio numbers.
  • Each share = total ÷ total parts × that share.
  • Convert to a common unit before writing or simplifying a ratio.
  • Maps and scale models use ratio: real distance = map distance × scale factor.

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