Ratio & Proportion – Himachal Pradesh PSC (Quantitative Aptitude)
Quick fact:
Ratio is a comparison of two or more quantities.
Proportion is an equation that states two ratios are equal.
1. Simple Explanation
| Concept | Meaning | Example |
|---|---|---|
| Ratio | A comparison of two numbers, written as a : b or a/b. |
The ratio of boys to girls in a class is 3:2. |
| Proportion | An equality of two ratios. | If a : b = c : d, then a/b = c/d. |
| Direct Proportion | When one quantity increases, the other increases proportionally. | x : y = 2 : 3 ⇒ x/y = 2/3. |
| Inverse Proportion | When one quantity increases, the other decreases proportionally. | x : y = 3 : 2 ⇒ x·y = constant. |
2. Key Points & Formulas
| Topic | Formula | Notes |
|---|---|---|
| Basic Ratio | a : b = a/b |
Use when you need the numeric value of a ratio. |
| Proportion Test | a : b = c : d ⇔ a·d = b·c |
Cross‑multiplication check. |
| Direct Proportion | a : b = c : d and b = k·a |
Scale factor k = b/a. |
| Inverse Proportion | x·y = k (constant) |
If x doubles, y halves. |
| Mixed Proportion | x : y = a : b and y : z = c : d |
Find x : z by multiplying: x : z = (a·c) : (b·d). |
| Ratio Involving Sum/Difference | If a : b = c : d, then a+b : a = c+d : c |
Useful for “parts of a whole” problems. |
| Percentage from Ratio | ratio part / total part × 100% |
e.g., 3:7 → 30% for part 3. |
| Least Common Multiple (LCM) | For equalizing ratios | Use LCM to avoid fractions when adding/subtracting ratios. |
| Unit Rate | total / quantity |
e.g., 5 km in 10 min → 0.5 km/min. |
3. Worked Example
Problem
A factory produces 120 units of Product A and 180 units of Product B in 5 days.
How many units of Product A and Product B will it produce in 12 days if the production rate remains unchanged?
Step‑by‑Step Solution
-
Find daily production ratio.
Total units in 5 days = 120 + 180 = 300.
Daily production = 300 / 5 = 60 units/day. -
Determine daily production of each product.
RatioA : B = 120 : 180 = 2 : 3.
Let daily A =2x, daily B =3x.
2x + 3x = 60⇒5x = 60⇒x = 12.So daily production:
- Product A =
2x = 24units/day - Product B =
3x = 36units/day
- Product A =
-
Scale to 12 days.
- Product A:
24 × 12 = 288units - Product B:
36 × 12 = 432units
- Product A:
Answer
Product A: 288 units
Product B: 432 units
4. Quick Revision Summary
| ✔ | Item | How to Remember |
|---|---|---|
| ✔ | Direct Proportion | x/y = constant → x = k·y |
| ✔ | Inverse Proportion | x·y = constant → if one doubles, other halves |
| ✔ | Cross‑Multiplication | a:b = c:d → a·d = b·c |
| ✔ | Mixed Proportion | x:y = a:b & y:z = c:d → x:z = (a·c):(b·d) |
| ✔ | Adding/Subtracting Ratios | Use LCM or common denominator |
| ✔ | Percentage from Ratio | (part/whole)×100 |
| ✔ | Unit Rate | total/quantity |
Quick Tips for PSC Exam
- Always check the units – make sure you’re comparing like quantities.
- Simplify ratios before multiplying – reduces error.
- Use cross‑multiplication for any proportion check – fastest method.
- For mixed ratios, remember the “chain” rule – multiply numerators and denominators separately.
- Keep a one‑liner cheat sheet in your notes:
a:b = c:d ⇔ a·d = b·c x:y = k (direct) → x = k·y x·y = k (inverse) → y = k/x
Use these notes to practice a variety of ratio & proportion problems, and you’ll be ready for the PSC quantitative aptitude section!