ICSE Class 10 Maths: Ratio & Proportion Word Problems Explained

Why Ratio Problems Matter in Real Life

Ever wondered how a recipe tells you to use 2 cups of flour for every 1 cup of sugar? That’s a ratio in action, and the same idea shows up in shopping discounts, maps, and even video games.

💡 In Simple Words: A ratio compares two quantities, like “2 to 1”. Proportion says two ratios are equal, like “2 : 1 = 4 : 2”. When you see a word problem, you just turn the story into those two‑number comparisons and solve.

What Are Ratio Word Problems?

A ratio word problem gives a situation and asks you to find missing numbers using a comparison. The keyword “ratio” often appears, but you might also see “as … as”, “to”, or “per”.

Key Vocabulary

• Ratio: a way to show how big one thing is compared to another (e.g., 3 to 5).
• Proportion: an equation that says two ratios are the same (e.g., 3 : 5 = 6 : 10).
• Unitary method: solving by first finding the value of one single unit, then scaling up.

Step‑by‑Step Method for Solving Ratio Problems

Here’s a simple checklist you can follow for most ICSE‑style questions:

1. Read the problem carefully and underline the quantities being compared.
2. Write the given ratio in “a : b” form.
3. Set up a proportion if a second ratio is mentioned, or use the unitary method if you need a single unit.
4. Cross‑multiply (multiply opposite corners) to solve for the unknown.
5. Check the answer by plugging it back into the story.

Worked Example 1 – Direct Proportion

Problem: A school bus can seat 40 students. If 5 buses are needed for a field trip, how many students can the whole fleet carry?

Solution:

• Ratio of buses to seats = 1 : 40.
• We have 5 buses, so set up the proportion 1 : 40 = 5 : x.
• Cross‑multiply: 1 × x = 5 × 40 ⇒ x = 200.
• Answer: The 5 buses can carry 200 students.

Worked Example 2 – Inverse Proportion

Problem: Three workers can paint a fence in 6 hours. How long will it take 6 workers to finish the same fence, assuming they work at the same speed?

Solution:

• When more workers are added, the time taken decreases – that’s an inverse proportion.
• Set up the product of workers and time as a constant: 3 × 6 = 18.
• Let the unknown time be y. Then 6 × y = 18 ⇒ y = 3 hours.
• Answer: Six workers need 3 hours.

When to Use Direct vs. Inverse Proportion

Common Mistakes to Avoid

• Mixing up “per” and “to”. “2 per 5” means 2 for every 5, not 2 : 5.
• Skipping the cross‑multiply step and trying to divide directly.
• Forgetting to check units (hours vs. minutes, meters vs. centimeters).

📝 Likely Exam Questions

1. Question: The ratio of boys to girls in a class is 3 : 4. If there are 21 boys, how many girls are there?
   Answer: Set up 3 : 4 = 21 : x → 3x = 84 → x = 28 girls.
2. Question: A recipe uses 250 g flour for every 150 g sugar. How much flour is needed for 375 g sugar?
   Answer: Ratio flour : sugar = 250 : 150 = 5 : 3. So 5 parts flour = 375 g sugar → 1 part = 75 g → flour = 5 × 75 = 375 g.
3. Question: If 8 machines produce 240 widgets in 4 hours, how many widgets will 12 machines produce in the same time?
   Answer: Direct proportion: 8 : 12 = 240 : x → 8x = 2880 → x = 360 widgets.
4. Question: Six cyclists travel a distance in 2 hours. How long will it take four cyclists to travel the same distance at the same speed?
   Answer: Inverse proportion: 6 × 2 = 4 × y → y = 3 hours.