How to Solve: Ratio and Proportion Word Problems

How to Solve: Ratio and Proportion Word Problems

GCSE & A-Level Maths

Introduction

"Mastering ratio and proportion word problems unlocks 10–15% of your GCSE Maths exam marks—and real-life skills like splitting bills, scaling recipes, or even mixing paint for DIY projects. One wrong step here could cost you 4–6 marks in a single question. Let’s make sure you never lose those marks again."

What You Need To Know First

Before diving in, ensure you understand: 1. Simplifying ratios – Dividing both parts by the same number (e.g., 6:9 simplifies to 2:3). 2. Unitary method – Finding the value of one unit first (e.g., if 5 apples cost £2, one apple costs £0.40). 3. Basic algebra – Solving equations like 3x = 12 to find x.

If any of these feel shaky, pause and review them first.

Key Vocabulary

Formulas To Know

1. Ratio to Fraction
2. Formula: Part A : Part B = A / (A + B)
3. Example: In a ratio 3:5, Part A is 3/(3+5) = 3/8 of the whole.

4. MEMORISE THIS – Not given on exam sheets.

5. Direct Proportion

6. Formula: y = kx (where k is the constant of proportionality).
7. Example: If y is directly proportional to x, and y = 10 when x = 2, then k = 5 (since 10 = 5 × 2).

8. MEMORISE THIS – Not given on exam sheets.

9. Inverse Proportion

10. Formula: y = k/x (where k is the constant).
11. Example: If y is inversely proportional to x, and y = 4 when x = 3, then k = 12 (since 4 = 12/3).

12. MEMORISE THIS – Not given on exam sheets.

13. Scaling Ratios

14. Formula: New value = Original value × (New ratio part / Original ratio part)
15. Example: If a recipe uses 200g flour for a ratio 2:3, to make the ratio 4:6, multiply by 4/2 = 2 → 200g × 2 = 400g.
16. Given on exam sheet (but you must know how to use it).

Step-by-Step Method

Step 1: Read the Question Twice

• Underline the numbers, units, and key words (e.g., "total," "share," "proportional").
• Circle what the question is asking for (e.g., "Find the number of boys").

Step 2: Write Down the Given Ratio

• If the ratio is not simplified, simplify it first.
• Example: 15:25 → Divide both by 5 → 3:5.
• If the ratio is part-to-part, decide if you need part-to-whole.

Step 3: Assign a Variable to One Part

• Let one part of the ratio be x.
• Example: If the ratio is 3:5, let the first part = 3x and the second part = 5x.
• This turns the ratio into an equation you can solve.

Step 4: Use the Total (If Given)

• If the question gives a total quantity, add the parts and set equal to the total.
• Example: "The ratio of boys to girls is 3:5. There are 40 students in total."
  • 3x + 5x = 40 → 8x = 40 → x = 5.
  • Boys = 3x = 15, Girls = 5x = 25.

Step 5: Solve for the Unknown

• Use algebra to find x, then multiply to find the required part.
• If the question asks for a new ratio, adjust the parts accordingly.

Step 6: Check Units and Answer the Question

• Does your answer make sense? (e.g., Can you have 1.5 people? No—round if needed.)
• Does it match the units asked for? (e.g., £, kg, cm).

Step 7: Verify with a Quick Calculation

• Plug your answer back into the original problem to check.
• Example: If boys = 15 and girls = 25, total = 40 (matches the question).

WORKED EXAMPLE (Using the Steps Above)

Question: "The ratio of red to blue marbles in a bag is 4:7. There are 55 marbles in total. How many blue marbles are there?"

Step 1: Read the Question Twice

• Underline: ratio 4:7, 55 marbles, how many blue marbles.
• What’s asked? Number of blue marbles.

Step 2: Write Down the Given Ratio

• Ratio = 4:7 (already simplified).

Step 3: Assign a Variable to One Part

• Let red marbles = 4x, blue marbles = 7x.

Step 4: Use the Total

• Total marbles = 4x + 7x = 11x.
• Given total = 55 → 11x = 55.

Step 5: Solve for x

• 11x = 55 → x = 55 ÷ 11 = 5.
• Blue marbles = 7x = 7 × 5 = 35.

Step 6: Check Units and Answer

• 35 is a whole number (no decimals).
• Units: marbles (matches the question).
• Answer: 35 blue marbles.

Step 7: Verify

• Red marbles = 4x = 20.
• Total = 20 + 35 = 55 (correct).

Worked Examples

Example 1 – Basic (Direct Proportion)

Question: "A recipe uses 300g of flour for 4 people. How much flour is needed for 10 people?"

Solution:

1. Understand: Flour is directly proportional to the number of people.
2. Find the constant (k):
3. 300g = k × 4 → k = 300 ÷ 4 = 75g per person.
4. Scale up:
5. For 10 people: 75 × 10 = 750g.
6. Answer: 750g of flour.

What we did and why: - Used y = kx because flour increases with people. - Found k first, then multiplied for the new quantity.

Example 2 – Medium (Part-to-Whole)

Question: "In a class, the ratio of boys to girls is 5:8. What fraction of the class are girls?"

Solution:

1. Simplify ratio: 5:8 (already simplified).
2. Total parts: 5 + 8 = 13.
3. Fraction of girls: 8/13.
4. Answer: 8/13 of the class are girls.

What we did and why: - Added the ratio parts to find the whole (13). - Girls are 8 parts out of 13, so the fraction is 8/13.

Example 3 – Exam-Style (Inverse Proportion)

Question: "It takes 6 workers 8 hours to build a wall. How long would it take 4 workers?"

Solution:

1. Understand: More workers = less time (inverse proportion).
2. Use formula: Workers × Time = Constant.
3. 6 × 8 = 48 (constant).
4. Find new time:
5. 4 × Time = 48 → Time = 48 ÷ 4 = 12 hours.
6. Answer: 12 hours.

What we did and why: - Recognised inverse proportion (more workers = less time). - Used Workers × Time = Constant to solve.

Common Mistakes

Exam Traps

1-Minute Recap (Night Before the Exam)

"Okay, listen up—this is your 60-second cheat sheet for ratio and proportion word problems. First, read the question twice. Underline the ratio, the total (if given), and what you’re asked to find. Simplify the ratio straight away—no excuses. Next, assign a variable: if the ratio is 3:5, write 3x and 5x. If there’s a total, add them up and solve for x. For direct proportion, use y = kx; for inverse, use y = k/x. Always check units—if the answer is 3.5 people, you’ve messed up. And watch out for exam traps: hidden totals, inverse proportion, and ratio changes. Finally, verify—plug your answer back in to make sure it works. You’ve got this!