Ratio & Proportion – Quick & Easy Study Notes
(For Delhi Patwari / Lekhpal – Quantitative Aptitude)
1. What Is a Ratio?
A ratio compares two numbers.
Think of it as “how many times one thing contains another.”
| Everyday Example | Math Statement |
|---|---|
| 3 apples to 5 oranges | 3 : 5 |
| 2 cars in a parking lot for every 1 bus | 2 : 1 |
- Short sentence: A ratio tells you how many parts of one quantity exist for each part of another.
2. What Is a Proportion?
A proportion says two ratios are equal.
Proportion = ratio 1 = ratio 2
| Everyday Example | Math Statement |
|---|---|
| If 4 shirts cost the same as 6 pants, the ratio of shirts to pants = 4:6. | 4 : 6 = 2 : 3 (after simplifying) |
- Short sentence: A proportion is an equality between two ratios.
3. Why Do We Need Ratios & Proportions?
- Divide a total into parts (like splitting land, money or time).
- Compare sizes, speeds, densities, etc.
- Solve “if‑then” problems quickly (e.g., “If A:B = 2:3 and B = 15, find A.”)
4. Basic Rules & Tips
| Rule | Explanation | Quick Check |
|---|---|---|
| Cross‑Multiplication | To check if a:b = c:d, compute a × d and b × c. If equal, the proportion holds. | 2×9 = 18 and 3×6 = 18 → 2:3 = 6:9 |
| Simplify Ratios | Divide both numbers by their greatest common divisor (GCD). | 8:12 → divide by 4 → 2:3 |
| Scale a Ratio | Multiply both numbers by the same factor to get a new ratio. | 3:5 → ×2 → 6:10 |
| Find Missing Term | If a:b = c:d and one of a, b, c, d is missing, use cross‑multiplication: a × d = b × c. | 4:? = 8:12 → 4×12 = ?×8 → ? = 6 |
5. Step‑by‑Step Example
Problem:
A farmer has 90 kg of rice and 30 kg of wheat.
He wants to mix them in the same proportion as a recipe that uses 6 kg rice for every 2 kg wheat.
How much of each grain should he use?
Solution:
-
Find the ratio in the recipe
- Rice : Wheat = 6 : 2 → simplify → 3 : 1.
-
Set up the proportion
- Let (R) = amount of rice, (W) = amount of wheat.
- ( \frac{R}{W} = \frac{3}{1} ).
-
Use the available amounts
- He has 90 kg rice and 30 kg wheat.
- The ratio 90 : 30 = 3 : 1, exactly the same as the recipe.
-
Answer
- He can use all 90 kg rice and all 30 kg wheat.
- The mixture keeps the recipe’s ratio.
Tiny Example in Pure Numbers
| Item | Quantity | Ratio to Others |
|---|---|---|
| A | 6 | 6 : 4 = 3 : 2 |
| B | 4 | 4 : 6 = 2 : 3 |
- Cross‑multiply: 6 × 6 = 36, 4 × 4 = 16 → not equal → not a proportion.
6. Common Question Types
| Question Type | How to Tackle |
|---|---|
| Find a missing number | Use cross‑multiplication. |
| Compare two ratios | Reduce both to simplest form; compare numerators & denominators. |
| Scale a ratio | Multiply both numbers by the same factor. |
| Check if a proportion holds | Cross‑multiply and compare products. |
| Convert a proportion to a percentage | Divide each part by the total and multiply by 100. |
7. Quick Practice Problems
-
Missing Term
5 : 8 = 10 : (x) → Find (x). -
Simplify
Simplify 48 : 72. -
Check Proportion
Is 7 : 14 = 3 : 6? -
Scale Up
Scale 4 : 9 by a factor of 3. -
Real‑World
If 15 kg of flour is needed for 10 kg of sugar, how much flour is needed for 18 kg of sugar?
(Answers: 1. 20, 2. 2 : 3, 3. No, 4. 12 : 27, 5. 27 kg flour)
8. Quick Recap
- Ratio = “part1 : part2”
- Proportion = “ratio1 = ratio2”
- Cross‑multiply to check a proportion.
- Simplify by dividing by the GCD.
- Scale by multiplying both parts by the same number.
- Use these rules for quick mental math in exams.
Final Tip
Always write everything down step‑by‑step.
Even if you’re a quick solver, a clear written path prevents mistakes and helps when you review later.
Good luck with your studies and exams!