Ratio & Proportion – Study Notes
Program: Goa Police Constable
Subject: Quantitative Aptitude
Topic: Ratio & Proportion
1. Simple Explanation
| Concept | What it means | Why it matters |
|---|---|---|
| Ratio | Comparison of two (or more) numbers using division. | Shows how much one quantity contains of another. |
| Proportion | An equation that states two ratios are equal. | Helps solve for unknowns when quantities are in a fixed relationship. |
Example:
If 3 apples cost ₹12 and 5 apples cost ₹20, the ratio of apples to price is (3:12 = 1:4). Since both sets of apples are sold at the same price per apple, the two ratios are in proportion.
2. Key Points
| # | Item | Details | Quick Tip |
|---|---|---|---|
| 1 | Ratio notation | (a:b) or (\frac{a}{b}) | Use colon for clarity in text; use fraction for calculations. |
| 2 | Simplifying a ratio | Divide both terms by their GCD. | Keeps numbers small and easier to work with. |
| 3 | Proportionality | (\frac{a}{b} = \frac{c}{d}) → (ad = bc) (Cross‑multiplication). | Most reliable test for proportion. |
| 4 | Finding a missing term | If (a:b = c:?) → (b \times c = a \times ?). | Cross‑multiply, then divide. |
| 5 | Mixed ratio | Combination of whole numbers and fractions (e.g., 3 ½ : 5 ¼). | Convert to whole numbers first (multiply by 2 or 4). |
| 6 | Compound ratio | (a:b:c) can be treated as two successive proportions: (a:b = b:c). | Useful in multi‑step problems. |
| 7 | Working with percentages | If “x% of a = b”, then (x = \frac{b}{a} \times 100). | Often appears in tax, discount, and interest problems. |
| 8 | Ratios of ratios | ((a:b) : (c:d) = \frac{a}{b} \div \frac{c}{d} = \frac{a \times d}{b \times c}). | Handy for comparing rates. |
| 9 | Checking consistency | In real problems, ensure the given ratios are consistent; inconsistent data → No solution. | Quick sanity check before calculation. |
| 10 | Use of algebra | Set unknowns as variables, apply cross‑multiplication, solve linear equations. | Keeps track of units and signs. |
3. Worked Example
Problem (typical for Goa Police Constable exam):
A factory produces 1200 units of Product A and 1800 units of Product B in one month. In the next month, 1400 units of Product A and 2100 units of Product B are produced.
(i) Find the ratio of production of A to B in the first month.
(ii) In the second month, if the ratio of production of A to B is the same as in the first month, how many units of Product A were actually produced (assume the second‑month B production is 2100)?
Step‑by‑Step Solution
-
First‑month ratio (A:B)
[ \text{Ratio} = 1200:1800 = \frac{1200}{1800} ] Simplify by dividing by GCD = 600:
[ 1200 ÷ 600 = 2,\quad 1800 ÷ 600 = 3 \Rightarrow \text{Ratio} = 2:3 ] -
Second‑month assumption
Let actual A production = (x).
Given B production = 2100.
Ratio must still be (2:3): [ \frac{x}{2100} = \frac{2}{3} ] -
Solve for (x)
Cross‑multiply: [ 3x = 2 \times 2100 \ 3x = 4200 \ x = \frac{4200}{3} = 1400 ] So, 1400 units of Product A were produced, matching the problem’s assertion. -
Verification
[ \frac{1400}{2100} = \frac{14}{21} = \frac{2}{3} ] Ratio holds.
Summary of the Worked Example
| Month | A Units | B Units | Ratio (A:B) |
|---|---|---|---|
| 1 | 1200 | 1800 | 2 : 3 |
| 2 (given) | 1400 | 2100 | 2 : 3 |
The second month’s data confirm the proportionality.
4. Quick Revision Summary
- Ratio = comparison of two numbers (use colon or fraction).
- Proportion = equality of two ratios; test via cross‑multiplication.
- Simplify ratios by dividing by GCD.
- Missing term: cross‑multiply → solve for unknown.
- Mixed/compound ratios → convert to whole numbers, treat as successive proportions.
- Percentages ↔ ratios: ( % = \frac{part}{whole} \times 100 ).
- Check consistency before solving; inconsistent data → “No solution.”
- Common exam use:
- Finding missing parts of a mixture.
- Comparing rates (e.g., speed, price per unit).
- Solving word problems involving “in the ratio of ... to ...”.
Keep the cross‑multiplication rule in mind:
[
\frac{a}{b} = \frac{c}{d} ;\Longleftrightarrow; ad = bc
]
Practice by converting real‑world