Ratio & Proportion – Study Notes
Program: Tamil Nadu Forest Guard
Subject: Quantitative Aptitude
1. Simple Explanation
-
Ratio – A comparison of two numbers using “:”.
[ \text{Example: } 3:2 ;\text{means}; \frac{3}{2};\text{or};\frac{3}{2} ] -
Proportion – An equation stating that two ratios are equal.
[ \frac{a}{b} = \frac{c}{d};;\Longleftrightarrow;; a \times d = b \times c ] -
Direct Proportion – When one quantity increases, the other increases in the same way.
[ y = kx \quad (k\text{ is a constant}) ] -
Inverse Proportion – When one quantity increases, the other decreases in such a way that their product is constant.
[ y = \frac{k}{x} ]
2. Key Points (Cheat‑Sheet)
| Topic | Formula / Rule | How to Use |
|---|---|---|
| Basic Proportion | (\displaystyle \frac{a}{b} = \frac{c}{d}) | Cross‑multiply: (a \times d = b \times c) |
| Missing in Ratio | (\displaystyle \frac{a}{b} = \frac{c}{x}) | (x = \frac{b \times c}{a}) |
| Direct Proportion | (y = kx) | Find (k) using known pair ((x_1,y_1)): (k = \frac{y_1}{x_1}). |
| Inverse Proportion | (y = \frac{k}{x}) | Find (k) using known pair ((x_1,y_1)): (k = x_1y_1). |
| Ratio Simplification | (\frac{a:b}{c:d} \Longrightarrow \frac{a}{b} \times \frac{d}{c}) | Reduce to simplest integers. |
| Changing Ratios | If ratio changes by factor (m:n), multiply numerator & denominator by (m) and (n) respectively. | (\displaystyle \frac{a}{b} \times \frac{m}{n}) |
| Finding Unknowns | Set up proportion with known and unknown terms. | Use cross‑multiplication or algebraic manipulation. |
3. Worked Example
Problem
The ratio of boys to girls in a school is 5 : 3.
If there are 200 students in total, how many boys and how many girls are there?
Step‑by‑Step Solution
-
Express the ratio as a proportion.
[ \frac{\text{boys}}{\text{girls}} = \frac{5}{3} ] -
Let number of boys = (5x), number of girls = (3x) (common factor (x)).
Because the ratio 5:3 tells us that for every 5 boys there are 3 girls. -
Total students =
[ 5x + 3x = 200 ;;\Longrightarrow;; 8x = 200 ] -
Solve for (x).
[ x = \frac{200}{8} = 25 ] -
Find boys and girls.
[ \text{Boys} = 5x = 5 \times 25 = 125 ] [ \text{Girls} = 3x = 3 \times 25 = 75 ]
Answer
Boys = 125, Girls = 75
4. Quick Revision Summary
- Ratio = (a:b) → (a) compared with (b).
- Proportion = (\frac{a}{b} = \frac{c}{d}) → Cross‑multiply: (ad = bc).
- Direct Proportion: (y = kx ;(k = \frac{y}{x})).
- Inverse Proportion: (y = \frac{k}{x} ;(k = xy)).
- Finding a missing term:
- If (\frac{a}{b} = \frac{c}{x}), then (x = \frac{b,c}{a}).
- If (\frac{a}{b} = \frac{x}{d}), then (x = \frac{a,d}{b}).
- Check your work
- Simplify the ratio back to its simplest form.
- Verify by multiplying cross‑terms.
- Practice tip: Convert every ratio problem into a proportion; cross‑multiply; solve for the unknown.