Ratio and Proportion
1. What is Ratio & Proportion?
Ratio and Proportion are basic arithmetic concepts used to compare quantities and maintain equality between two ratios. These topics are very common in Delhi Police and SSC exams to test your numerical reasoning and analytical skills.
Ratio: Comparison of Two Quantities
Proportion: Equality of Two Ratios
Tests Numerical Reasoning
Common in Competitive Exams
Simple Definition:
Ratio
A ratio is a comparison of two quantities of the same kind.
Example: If A = 10 and B = 5, then the ratio of A to B is 10:5 or 2:1.
Proportion
When two ratios are equal, they are said to be in proportion.
Example: 2:4 = 3:6 → both simplify to 1:2.
Examples:
Example 1: Ratio of 8 to 12 = 8:12 = 2:3
Example 2: If 2:3 = 6:9, then these are in proportion
Example 3: If 3 pens cost ₹15, 1 pen costs ₹5 (using ratio concept)
2. Important Concepts & Rules
| Concept | Description | Example |
|---|---|---|
| Ratio Formula | Ratio = First Quantity / Second Quantity | 4:5 → 4/5 |
| Proportion Formula | a:b = c:d → a/b = c/d | 2:3 = 4:6 |
| Mean Proportion | Between a & b = √(a×b) | Between 4 & 9 → √36 = 6 |
| Continued Proportion | a:b = b:c → then b² = a×c | If a=4, c=9 → b=6 |
| Inverse Proportion | When one increases, other decreases | Speed ↑ ⇒ Time ↓ |
| Compound Ratio | Multiply ratios directly | (a:b) × (c:d) = ac:bd |
3. Step-by-Step Method for Solving
| Step | Description |
|---|---|
| Step 1 | Convert all quantities into the same units (e.g., kg → g) |
| Step 2 | Simplify the ratio to its lowest form |
| Step 3 | Set up proportions correctly (cross-multiply if needed) |
| Step 4 | Apply formulas for mean/inverse/compound ratios |
| Step 5 | Check final ratio equality to ensure accuracy |
4. Real-Life Analogy
Think of a Ratio as a recipe
Ratio
→ Comparison of ingredients
(2 cups flour : 1 cup sugar)
Proportion
→ Maintaining equality of ratios
(4 cups flour : 2 cups sugar)
If a cake requires flour and sugar in the ratio 2:1, doubling both keeps the same taste (proportion). So, Ratio = comparison, and Proportion = maintaining equality of ratios.
5. Common Delhi Police / SSC Exam Facts
Ratio & Proportion questions are frequent in Arithmetic section
Often combined with Partnership, Time & Work, and Mixture problems
Units must always be the same before forming a ratio
Remember: If a:b = c:d ⇒ ad = bc (cross-multiplication rule)
Inverse and compound ratio-based questions appear in higher-level exams
Always simplify ratios to their lowest terms for accuracy
6. Practice Questions (Exam-Type)
| No. | Question | Solution | Answer |
|---|---|---|---|
| 1 | Find the ratio of 20 and 45 | 20:45 = 4:9 | 4:9 |
| 2 | If 3:5 = 9:x, find x | Cross multiply → 3x = 45 → x = 15 | 15 |
| 3 | Find mean proportion between 9 and 16 | √(9×16) = √144 = 12 | 12 |
| 4 | If a:b = 2:3 and b:c = 4:5, find a:c | a:b × b:c = 2:3 × 4:5 = 8:15 | 8:15 |
| 5 | If x:y = 4:5, find inverse ratio | Inverse → 1/x : 1/y = 5:4 | 5:4 |
7. Quick Recap
| Concept | Summary |
|---|---|
| Ratio | Comparison of two quantities of same kind |
| Proportion | Equality of two ratios |
| Key Rule | If a:b = c:d ⇒ ad = bc |
| Mean Proportion | √(a×b) |
| Exam Tip | Always simplify ratios to lowest form |
You've completed Ratio and Proportion!
Courage Tip: Practice ratio and proportion problems regularly, focusing on unit conversion and simplification. These concepts form the foundation for many advanced topics in quantitative aptitude.
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