📚 Delhi Police Constable – Quantitative Aptitude
Topic: Ratio & Proportion
Ratio : A comparison of two quantities by division (e.g., 3 : 4).
Proportion : The equality of two ratios (e.g., 3 : 4 = 6 : 8).
1. Simple Explanation
Ratio tells us how many times one quantity contains another.
Proportion tells us that two ratios are equal, meaning one ratio is a scaled‑up or scaled‑down version of the other.
Example:
If the number of men and women in a group is 3 : 4 and the total is 70, we can find the individual counts by setting up a proportion.
2. Key Points
| Concept | Definition | Common Uses |
|---|---|---|
| Ratio | (a:b) or (a/a') | Comparing parts, speed, density, etc. |
| Proportion | (a:b = c:d) | Solving for unknowns, mixing problems, scale drawings |
| Direct Proportion | (y = kx) | (a:b = c:d) where (k) is constant |
| Inverse Proportion | (y = k/x) | (a:b = d:c) (note the order reversal) |
| Mixed Proportion | (a : b = c : d + e : f) | Complex word problems |
| Unit Rate | (a/b) | Find per‑unit value |
| Cross‑Multiplication | (a \times d = b \times c) | Quick checking of proportion |
| Conversion of Ratios | Reduce to lowest terms | Easier calculations |
Useful Formulae
| Situation | Formula | Explanation |
|---|---|---|
| Direct proportion | ( \frac{a}{b} = \frac{c}{d} \Rightarrow a \times d = b \times c ) | Cross‑multiply to solve for unknown |
| Inverse proportion | ( \frac{a}{b} = \frac{d}{c} ) | Same cross‑multiplication rule |
| Finding part from whole | ( \frac{a}{a+b} = \frac{c}{t} ) | (c = \frac{t \times a}{a+b}) |
| Percentage change | ( \text{New} = \text{Old} \times (1 \pm \frac{\text{change %}}{100}) ) | Often appears with ratios |
3. Problem‑Solving Techniques
| Step | What to Do | Why It Works |
|---|---|---|
| 1. Identify the type (direct, inverse, mixed) | Read the wording carefully | Determines the proportion form |
| 2. Write the known ratio | Convert to a fraction or set up (a:b) | Establishes the relationship |
| 3. Set up the proportion | ( \frac{a}{b} = \frac{c}{d} ) | Creates an equation |
| 4. Use cross‑multiplication | (a \times d = b \times c) | Eliminates fractions |
| 5. Solve for the unknown | Isolate the variable | Gives the answer |
| 6. Check units | Ensure consistency (e.g., km/h, kg) | Prevents errors |
Tip: For word problems, always sketch a quick diagram or write the ratio in terms of variables first.
4. Worked Example
Problem (from previous DPC exam):
The number of boys to girls in a school class is 5 : 6. If there are 77 students in total, how many boys and how many girls are there?
Step‑by‑Step Solution
-
Let
- (b =) number of boys
- (g =) number of girls
-
Ratio given
[ \frac{b}{g} = \frac{5}{6} ] -
Set up proportion with total
[ b + g = 77 ] -
Express one variable in terms of the other
From the ratio:
[ b = \frac{5}{6} g \quad \text{or} \quad g = \frac{6}{5} b ] -
Substitute in the total equation
[ \frac{5}{6} g + g = 77 ] Combine terms:
[ \frac{11}{6} g = 77 ] -
Solve for (g)
[ g = 77 \times \frac{6}{11} = 77 \times \frac{6}{11} = 77 \times \frac{6}{11} = 7 \times 6 = 42 ] -
Find (b)
[ b = 77 - g = 77 - 42 = 35 ] -
Result
- Boys: 35
- Girls: 42
Quick Check
Ratio ( \frac{35}{42} = \frac{5}{6}) ✓
Total (35 + 42 = 77) ✓
5. Quick Revision Summary
- Ratio = comparison of two numbers.
- Proportion = equality of two ratios.
- Direct Proportion: both variables increase together.
- Inverse Proportion: one