GCSE Maths Number - How to Solve: Ratio and Proportion Word Problems

How to Solve: Ratio and Proportion Word Problems

GCSE / A-Level (Physics, Chemistry, Biology) – Complete Guide

Introduction

"Mastering ratio and proportion word problems unlocks 5–10% of your GCSE/A-Level Physics, Chemistry, and Biology exams—questions on dilution, reaction rates, genetics, and even medical dosing all rely on this. One wrong step here could cost you 4–6 marks in a single question. Today, you’ll learn the exact method to solve them every time."

WHAT YOU NEED TO KNOW FIRST

Before starting, you must already understand: 1. Basic fractions and decimals – How to simplify, multiply, and divide them. 2. Unit conversion – How to switch between grams, moles, cm³, etc. 3. Direct and inverse proportion – If A doubles, does B double (direct) or halve (inverse)?

KEY TERMS & FORMULAS

Key Terms

Formulas

1. Ratio to Fraction
2. A ratio a:b can be written as the fraction a/b or b/a.

3. MEMORISE THIS: Always match the order of the question.

4. Proportion Equation

5. If a/b = c/d, then a × d = b × c (cross-multiplication).

6. MEMORISE THIS: Used to solve for an unknown.

7. Direct Proportion Formula

8. y = kx (where k is the constant of proportionality).

9. MEMORISE THIS: If y doubles when x doubles, use this.

10. Inverse Proportion Formula

11. y = k/x (where k is the constant).

12. MEMORISE THIS: If y halves when x doubles, use this.

13. Dilution Formula (Chemistry)

14. C₁V₁ = C₂V₂ (where C = concentration, V = volume).
15. GIVEN ON EXAM SHEET (but you must know how to apply it).

STEP-BY-STEP METHOD

Follow these exact steps for every ratio/proportion word problem.

Step 1: Read the Question Twice

• Underline key numbers and units.
• Circle what you’re solving for (e.g., "Find the mass of X").

Step 2: Identify the Type of Proportion

• Direct? (More A → More B)
• Inverse? (More A → Less B)
• Ratio only? (No change in total, just parts)

Step 3: Write Down the Given Ratio or Proportion

• If a ratio is given (e.g., 3:5), write it as a fraction: 3/5 or 5/3.
• If a proportion is given (e.g., "4 g of X reacts with 6 g of Y"), write 4/6 = x/y.

Step 4: Assign Variables to Unknowns

• Let x = the unknown quantity.
• If two unknowns, use x and y (e.g., "Find the masses of A and B in a 2:3 ratio").

Step 5: Set Up the Equation

• Direct proportion? Use y = kx or a/b = c/d.
• Inverse proportion? Use y = k/x or a × b = c × d.
• Ratio problem? Use part/whole = x/total.

Step 6: Solve for the Unknown

• Cross-multiply if needed.
• Isolate x using algebra.
• Check units—do they match the question?

Step 7: Verify the Answer

• Does it make sense? (e.g., If you diluted a solution, the concentration should decrease.)
• Plug x back into the original ratio to check.

WORKED EXAMPLES

Example 1 – Basic (Direct Proportion)

Question: "A car travels 60 km in 1.5 hours. How far will it travel in 4 hours at the same speed?"

Step-by-Step Solution: 1. Read twice: Underline "60 km", "1.5 hours", "4 hours". Circle "how far". 2. Identify proportion: More time → more distance → direct proportion. 3. Write ratio: 60 km / 1.5 h = x km / 4 h. 4. Assign variable: Let x = distance in 4 hours. 5. Set up equation: 60/1.5 = x/4. 6. Solve:
- Cross-multiply: 60 × 4 = 1.5 × x
- 240 = 1.5x
- x = 240 / 1.5 = 160 km 7. Verify: 1.5 h → 60 km, so 4 h → 160 km (makes sense).

What we did and why: We used direct proportion because distance increases with time at constant speed. Cross-multiplication gives the unknown distance.

Example 2 – Medium (Inverse Proportion)

Question: "It takes 6 workers 8 days to build a wall. How many days will it take 4 workers?"

Step-by-Step Solution: 1. Read twice: Underline "6 workers", "8 days", "4 workers". Circle "how many days". 2. Identify proportion: Fewer workers → more time → inverse proportion. 3. Write ratio: 6 workers × 8 days = 4 workers × x days. 4. Assign variable: Let x = days for 4 workers. 5. Set up equation: 6 × 8 = 4 × x. 6. Solve:
- 48 = 4x
- x = 12 days 7. Verify: Fewer workers → longer time (makes sense).

What we did and why: We used inverse proportion because more workers mean less time. The product of workers and days stays constant.

Example 3 – Exam-Style (Chemistry Dilution)

Question: "A student has 250 cm³ of 0.8 mol/dm³ HCl. What volume of water must be added to dilute it to 0.2 mol/dm³?"

Step-by-Step Solution: 1. Read twice: Underline "250 cm³", "0.8 mol/dm³", "0.2 mol/dm³". Circle "volume of water". 2. Identify proportion: Dilution → direct proportion (C₁V₁ = C₂V₂). 3. Write given: C₁ = 0.8, V₁ = 250, C₂ = 0.2, V₂ = ? 4. Assign variable: Let V₂ = final volume. 5. Set up equation: 0.8 × 250 = 0.2 × V₂. 6. Solve:
- 200 = 0.2V₂
- V₂ = 1000 cm³ 7. Find water added: 1000 cm³ (final) – 250 cm³ (initial) = 750 cm³. 8. Verify: Lower concentration → larger volume (makes sense).

What we did and why: We used the dilution formula (C₁V₁ = C₂V₂) because moles of solute stay the same. The difference in volumes gives the water added.

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP (Night Before Exam)

"Listen up—this is your 60-second cheat sheet for ratio and proportion word problems. First, read the question twice and underline numbers. Next, decide: direct or inverse? If more A means more B, it’s direct (y = kx). If more A means less B, it’s inverse (y = k/x). Write the ratio as a fraction, assign x to the unknown, and cross-multiply. For dilution, use C₁V₁ = C₂V₂—just plug in the numbers. Always check units and verify—does your answer make sense? If you’re stuck, ask: ‘What happens if I double this?’ That’ll tell you if it’s direct or inverse. Now go crush those questions!"