Ever wondered how to split pizza slices fairly while still getting a perfect score in math?
💡 In Simple Words: A ratio tells you how many parts of one thing compare to another, like 3 red balls for every 2 blue balls. A proportion says two ratios are equal, so you can swap one pair for the other and keep the balance.
What is a Ratio?
A ratio is just a way of saying "this many of A for every this many of B". Think of it as a recipe: if a cake needs 2 cups of flour for every 1 cup of sugar, the ratio of flour to sugar is 2:1.
What is a Proportion?
A proportion is an equation that says two ratios are the same. It’s like saying the recipe works whether you make a small cake or a big one – the relationship stays equal.
How to Solve Ratio and Proportion Word Problems (step‑by‑step)
When a question gives you a story, you first pick out the numbers that are being compared, turn them into a ratio, then write a proportion to find the missing piece.
Quick Reference Table
| Step | What to Do | Why |
|---|---|---|
| 1 | Read the story carefully | Find the two groups being compared |
| 2 | Write the given ratio | Shows the relationship in numbers |
| 3 | Set up a proportion | Connect known ratio to unknown one |
| 4 | Cross‑multiply | Turns the proportion into an easy equation |
| 5 | Solve for the unknown | Gets the answer you need |
| 6 | Check the answer | Make sure it fits the story |
Worked Example 1
Problem: A classroom has boys and girls in the ratio 3:4. If there are 21 boys, how many girls are there?
Solution: The given ratio is 3 boys : 4 girls. Write it as 3/4 = 21/x, where x is the number of girls. Cross‑multiply: 3·x = 4·21 → 3x = 84 → x = 28. So there are 28 girls.
Worked Example 2
Problem: In a recipe, 5 L of water mixes with 2 L of syrup to make a drink. If you want to make 21 L of the drink, how much syrup do you need?
Solution: The water‑to‑syrup ratio is 5:2, or 5/2 = water/syrup. Let syrup be s litres. Then water = (5/2)·s. Total drink = water + syrup = (5/2)s + s = (7/2)s. Set (7/2)s = 21 → s = 21·2/7 = 6 L. You need 6 L of syrup.
📝 Likely Exam Questions
- The ratio of red to blue marbles in a bag is 7:5. If there are 56 red marbles, how many blue marbles are there?
Answer: 7/5 = 56/x → 7x = 280 → x = 40 blue marbles. - A map uses a scale of 1 cm : 8 km. If two cities are 3.5 cm apart on the map, what is the real distance?
Answer: Set proportion 1 cm/8 km = 3.5 cm / d km → d = 3.5×8 = 28 km. - In a garden, the length to width ratio is 4:3. If the perimeter is 56 m, find the length.
Answer: Let length = 4k, width = 3k. Perimeter 2(4k+3k)=56 → 14k=56 → k=4 → length = 4k = 16 m. - Six pens cost the same as nine pencils. If a pencil costs ₹12, what is the cost of one pen?
Answer: Ratio pens:pencils = 9:6 = 3:2. So 3 pens cost the same as 2 pencils → 3p = 2·12 → p = 8 ₹ per pen.