Ratio & Proportion – Study Notes
(Odisha Forest Guard – Quantitative Aptitude)
1. Simple Explanation
- Ratio is a comparison of two or more numbers using division (e.g., 2 : 3).
- Proportion is an equality of two ratios (e.g., 2 : 3 = 4 : 6).
- If two ratios are equal, the cross‑multiplication of their terms gives the same product.
Key Idea:
When you’re given a ratio and need to find a missing part, set up a proportion and solve for the unknown using cross‑multiplication or the property of equality of ratios.
2. Key Points
| Concept | Formula / Rule | Example |
|---|---|---|
| Basic Ratio | a : b | 5 : 7 |
| Proportion | a : b = c : d | 5 : 7 = 10 : ? |
| Cross‑Multiplication | a·d = b·c | 5·? = 7·10 → ? = 14 |
| Direct Proportion | If (y) varies directly as (x): (y = kx) | If 4 apples cost ₹8 → 1 apple costs ₹2 |
| Inverse Proportion | If (y) varies inversely as (x): (y = \frac{k}{x}) | If 3 workers finish in 12 days → 6 workers finish in 6 days |
| Part of a Whole | If ratio is (m:n) and total = (T): part1 = ( \frac{m}{m+n} \times T) | 2:3 of 100 → part1 = ( \frac{2}{5} \times 100 = 40) |
| Unit Ratio | Convert to a common unit when items differ in size | 3 kg of rice vs 5 kg of wheat |
| Percentage from Ratio | ( \frac{part}{whole} \times 100% ) | 2:3 of 10 → 40% |
3. Worked Example
Problem (Typical Forest Guard Question)
The forest area of Mayurbhanj district is divided into two zones: Zone‑A and Zone‑B. The ratio of their areas is 3 : 5. If the total area is (4{,}200,\text{km}^2), what is the area of Zone‑B?
Step‑by‑Step Solution
-
Identify the ratio
- Zone‑A : Zone‑B = 3 : 5
- Let the common multiplier be (x).
- Zone‑A = (3x)
- Zone‑B = (5x)
-
Set up the proportion with the total area
[ 3x + 5x = 4{,}200 ] [ 8x = 4{,}200 ] -
Solve for (x)
[ x = \frac{4{,}200}{8} = 525 ] -
Find Zone‑B’s area
[ \text{Zone‑B} = 5x = 5 \times 525 = 2{,}625,\text{km}^2 ]
Answer: Zone‑B area = 2,625 km².
Quick Check
- Zone‑A = (3 \times 525 = 1{,}575,\text{km}^2)
- Sum = (1{,}575 + 2{,}625 = 4{,}200,\text{km}^2) ✔️
4. Quick Revision Summary
| Topic | What to Remember | Quick Tip |
|---|---|---|
| Ratio | Comparison of two numbers | Write as (a:b) |
| Proportion | Two ratios equal | Use cross‑multiplication |
| Cross‑Multiplication | (a\cdot d = b\cdot c) | Solve for unknown easily |
| Direct Proportion | (y = kx) | Same ratio, same scale factor |
| Inverse Proportion | (y = \frac{k}{x}) | Increase in one → decrease in other |
| Part of Whole | (\frac{m}{m+n} \times T) | Convert ratio to fraction of total |
| Common Unit | Convert items to same unit first | Avoid mistakes in mixed units |
| Percentage | (\frac{part}{whole}\times100%) | Helpful for “percentage of a total” problems |
Final Tips for the Exam
- Read the question carefully. Identify whether it’s a direct or inverse proportion.
- Set up the proportion before doing any calculation.
- Check units – if the items differ in size, convert them.
- Verify your answer by plugging it back into the original ratio.
- Practice with problems on area, population, resources, and time‑work situations – all common in Forest Guard tests.
Good luck!