Why Ratio Problems Show Up Everywhere
Ever wondered why recipes, map distances, and even video game scores use ratios? That’s the same idea you’ll see in your ICSE maths exam.
💡 In Simple Words: A ratio tells you how many parts of one thing match parts of another. When two ratios are equal, that’s a proportion. Solving a word problem means turning a story into a ratio or proportion and then finding the missing number.
What Exactly Is a Ratio?
A ratio is just a way of comparing two quantities. Think of it as a fraction without the division sign – like 3:2 means "for every 3 of the first thing, there are 2 of the second." You can write it as 3/2, 3 to 2, or 3‑2.
Understanding Proportion
A proportion says two ratios are the same. For example, 4:6 = 2:3 because both simplify to the same fraction (2/3). When you set up a proportion, you can cross‑multiply (multiply opposite corners) to find an unknown.
How to Tackle Ratio and Proportion Word Problems
Turning a story into math looks like a small puzzle. Follow these four steps:
- Read carefully – pick out the numbers and what they’re comparing.
- Assign variables – use letters for the unknowns.
- Write the ratio or proportion – match the parts of the story to a mathematical expression.
- Solve – use cross‑multiplication or simple algebra, then check your answer.
Example 1: Simple Ratio Word Problem
Problem: A fruit seller has apples and bananas in the ratio 5:3. If there are 40 apples, how many bananas are there?
Solution:
- Identify the ratio: apples : bananas = 5 : 3.
- Let the number of bananas be b. Because 5 parts correspond to 40 apples, one part = 40 ÷ 5 = 8.
- Bananas = 3 parts × 8 = 24.
Check: 40 : 24 simplifies to 5 : 3, so we’re good.
Example 2: Proportion Word Problem (Cross‑Multiplication)
Problem: In a map, 1 cm represents 5 km. If two towns are 12 cm apart on the map, how many kilometres apart are they in real life?
Solution:
- Set up the proportion: 1 cm / 5 km = 12 cm / x km.
- Cross‑multiply: 1 × x = 12 × 5.
- Solve: x = 60 km.
So the towns are 60 kilometres apart.
Example 3: Mixed Ratio and Scale Factor
Problem: A recipe for 4 servings needs 200 g of flour. How much flour is needed for 10 servings?
Solution:
- Find the scale factor: 10 servings ÷ 4 servings = 2.5.
- Multiply the original amount by the scale factor: 200 g × 2.5 = 500 g.
That’s the amount for 10 servings.
Quick Reference Table
| Step | What to Do | Key Formula |
|---|---|---|
| 1. Identify numbers | Pick out all quantities mentioned. | — |
| 2. Write ratio | Place numbers in “a:b” form. | a:b |
| 3. Form proportion (if needed) | Set two ratios equal. | a/b = c/d |
| 4. Cross‑multiply | Multiply opposite corners. | a·d = b·c |
| 5. Solve for unknown | Isolate the variable. | Variable = (known × known) / known |
Common Mistakes to Avoid
- Mixing up the order of the ratio – always keep the same objects in the same positions.
- Forgetting to simplify ratios before setting up a proportion – a simplified ratio makes cross‑multiplication easier.
- Skipping the “check” step – plug your answer back into the story to see if it makes sense.
Tips for the ICSE Exam
- Underline the numbers in the question; label what each represents.
- Write a short sentence like “Let x be the number of …” before you start algebra.
- Remember that “ratio word problems” can appear in geometry (similar triangles) and in data interpretation tables.
📝 Likely Exam Questions
- Question: The ages of two brothers are in the ratio 4:5. If the older brother is 15 years old, how old is the younger brother?
Answer: Let the younger’s age be 4x, older’s be 5x = 15 → x = 3 → younger = 4 × 3 = 12 years. - Question: A school bus travels 120 km in 3 hours. If it maintains the same speed, how far will it travel in 7.5 hours?
Answer: Speed = 120 km / 3 h = 40 km/h. Distance = 40 km/h × 7.5 h = 300 km. - Question: In a class, the ratio of boys to girls is 7:5. If there are 56 boys, how many students are there in total?
Answer: One part = 56 ÷ 7 = 8. Girls = 5 × 8 = 40. Total = 56 + 40 = 96. - Question: A map scale is 1 cm : 8 km. Two cities are 9.5 cm apart on the map. Find the real distance.
Answer: 9.5 cm × 8 km/cm = 76 km. - Question: A recipe uses sugar and butter in the ratio 3:2. If you have 450 g of sugar, how much butter do you need?
Answer: One part = 450 g ÷ 3 = 150 g. Butter = 2 × 150 g = 300 g.