Ratio & Proportion – Study Notes
Program: Forest Officer
Subject: Quantitative Aptitude
1. Simple Explanation
- Ratio is a comparison of two numbers by division.
- Example: If a forest has 20 saplings and 10 old trees, the ratio saplings : old trees = 20 : 10 = 2 : 1.
- Proportion is an equation stating that two ratios are equal.
- Example:
[ \frac{3}{5} = \frac{9}{x} ]
Here, 3:5 is proportional to 9:x.
- Example:
- In real‑world questions, ratios often represent “per unit” relationships (e.g., 10 ha of forest to 50 hectares of water → ratio 10 : 50 = 1 : 5).
2. Key Points (Quick Reference)
| Concept | Symbol | What It Means | Example |
|---|---|---|---|
| Ratio | a : b | Comparative measure | 8:2 = 4:1 |
| Proportion | a:b = c:d | Two ratios equal | 4:1 = 12:3 |
| Finding a Missing Term | a:b = c:? | Cross‑multiply: (a \times ? = b \times c) | 5:10 = 2:? → (5×? = 10×2 → ? = 4) |
| Direct Proportion | (y = kx) | y increases as x increases | If 5 km² forest yields 10 ha timber, 10 km² yields 20 ha |
| Inverse Proportion | (xy = k) | If one increases, the other decreases | 3 workers finish 9 days → 9 workers finish ? days → (3×9 = 9×? → ? = 3) |
| Percentage Change | % = (\frac{new-old}{old}\times100) | Used to compare growth/decline | Forest area 120 ha → 150 ha → (\frac{30}{120}×100 = 25%) |
| Unit Conversion | Convert to same base before ratio | 2 km : 200 m → 2000 m : 200 m = 10 : 1 |
3. Worked Example
Problem (Typical for Forest Officer Exams)
The forest department plans to plant saplings in a protected area.
The ratio of saplings to mature trees planted in the first year is 7 : 3.
In the second year, the saplings planted are 21 000 and mature trees planted are 9 000.
Find:
- The ratio of saplings to mature trees in the second year.
- If the same ratio is maintained for the next year, how many saplings should be planted if 5 000 mature trees are planted?
Solution
1. Ratio in the second year
- Saplings = 21 000
- Mature trees = 9 000
[ \text{Ratio} = \frac{21,000}{9,000} = \frac{21}{9} = \frac{7}{3} ]
Answer: The ratio remains 7 : 3 (same as first year).
2. Saplings for next year
Given the ratio stays 7 : 3 and mature trees = 5 000.
Let saplings = (x).
[ \frac{x}{5,000} = \frac{7}{3} \quad\Rightarrow\quad x = 5,000 \times \frac{7}{3} = \frac{35,000}{3} \approx 11,666.7 ]
Since saplings must be whole numbers, round to 11 667 saplings.
Answer: Plant ≈ 11 667 saplings.
4. Quick Revision Summary
- Ratio → a:b (compare two numbers).
- Proportion → a:b = c:d (two ratios equal).
- Missing term → cross‑multiply: (a × ? = b × c).
- Direct proportion → (y = kx).
- Inverse proportion → (xy = k).
- Percentage change → (\frac{new-old}{old}×100).
- Always convert units to the same base before comparing ratios.
- Check consistency: If a ratio is given for one year, verify it in subsequent years.
Quick Practice Question
In a forest reserve, the ratio of male to female deer is 3 : 2.
If 90 male deer are counted, how many female deer are there?
Answer:
[
\frac{3}{2} = \frac{90}{x} \Rightarrow 3x = 180 \Rightarrow x = 60
]
So, there are 60 female deer.
Tip for Exam:
- Write down the ratio first.
- Translate the problem into a proportion if a missing quantity is requested.
- Always double‑check units and round only when necessary (e.g., whole animals, trees, etc.).
Good luck with your studies!