Ratio & Proportion
Overview
Ratio & Proportion is a high-scoring topic in Maharashtra Police Bharti mathematics section. Questions typically appear in straightforward forms—comparing quantities, finding unknown values, or distributing amounts among partners. Mastering this topic also strengthens your foundation for related concepts like percentages, mixtures, and time-work problems.
The exam tests your ability to simplify ratios quickly, identify direct versus inverse relationships, and solve partnership problems where profits are shared based on investment and time. Most questions can be solved in 30-60 seconds if you know the core principles.
Students must focus on ratio simplification, the product rule for proportions, and the three types of partnership calculations. Speed and accuracy matter equally here.
Key Concepts
- Ratio compares two quantities of the same unit. Written as a:b or a/b. The ratio 4:6 simplifies to 2:3 by dividing both terms by their HCF.
- Proportion states that two ratios are equal. If a:b = c:d, then a×d = b×c (cross-multiplication rule). This is called the "product of means equals product of extremes."
- Direct Proportion: When one quantity increases, the other increases proportionally. Example: More workers → more work done (in same time).
- Inverse Proportion: When one quantity increases, the other decreases proportionally. Example: More workers → less time needed (for same work).
- Compounded Ratio: When ratios a:b and c:d are compounded, result is (a×c):(b×d).
- Duplicate Ratio of a:b is a²:b². Sub-duplicate Ratio is √a:√b. Triplicate Ratio is a³:b³.
- Partnership: When multiple people invest money, profit is divided in the ratio of (Capital × Time) for each partner.
- Componendo-Dividendo: If a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d). Useful for quick calculations in specific problems.
Formulas / Key Facts
| Concept | Formula/Rule |
|---|---|
| Ratio of a to b | a:b (always express in lowest terms) |
| Proportion check | If a:b :: c:d, then a×d = b×c |
| Mean Proportional | Between a and b is √(a×b) |
| Third Proportional | To a and b is b²/a |
| Fourth Proportional | To a, b, c is (b×c)/a |
| Direct Proportion | x₁/y₁ = x₂/y₂ |
| Inverse Proportion | x₁×y₁ = x₂×y₂ |
| Simple Partnership | Profit ratio = Capital ratio (when time is equal) |
| Compound Partnership | Profit ratio = (C₁×T₁) : (C₂×T₂) : (C₃×T₃) |
| Distribution formula | A's share = (A's ratio part / Total ratio parts) × Total amount |
Worked Examples
Example 1: Basic Ratio Problem
Q: Divide Rs 1750 among A, B, C in the ratio 3:5:7.
Solution:
- Total parts = 3 + 5 + 7 = 15
- Value of 1 part = 1750 ÷ 15 = Rs 116.67 (or work with fractions)
- A's share = (3/15) × 1750 = Rs 350
- B's share = (5/15) × 1750 = Rs 583.33
- C's share = (7/15) × 1750 = Rs 816.67
Answer: A = Rs 350, B = Rs 583.33, C = Rs 816.67
Example 2: Finding Unknown in Proportion
Q: If 4:x :: x:9, find x.
Solution:
- This is continued proportion: x is the mean proportional
- Using property: x² = 4 × 9 = 36
- x = √36 = 6
Answer: x = 6
Example 3: Partnership Problem
Q: A invests Rs 20,000 for 12 months. B invests Rs 30,000 for 8 months. If total profit is Rs 18,000, find each partner's share.
Solution:
- A's investment-time = 20,000 × 12 = 2,40,000
- B's investment-time = 30,000 × 8 = 2,40,000
- Profit ratio = 2,40,000 : 2,40,000 = 1:1
- Total parts = 2
- A's profit = (1/2) × 18,000 = Rs 9,000
- B's profit = (1/2) × 18,000 = Rs 9,000
Answer: Both get Rs 9,000 each
Example 4: Inverse Proportion
Q: 15 workers complete a task in 12 days. How many days will 20 workers take?
Solution:
- More workers = Less days (Inverse proportion)
- Workers × Days = Constant
- 15 × 12 = 20 × x
- 180 = 20x
- x = 9 days
Answer: 9 days
Example 5: Compound Ratio
Q: Find the compound ratio of 2:3, 4:5, and 6:7.
Solution:
- Multiply all first terms: 2 × 4 × 6 = 48
- Multiply all second terms: 3 × 5 × 7 = 105
- Compound ratio = 48:105
- Simplify by HCF (3): 16:35
Answer: 16:35
Common Mistakes
- Not simplifying ratios to lowest terms → Always divide both terms by HCF before proceeding. Ratio 12:18 must become 2:3.
- Confusing direct and inverse proportion → Ask: "Does increasing one quantity increase or decrease the other?" If opposite directions, it's inverse.
- Forgetting time factor in partnership → Profit sharing depends on BOTH capital AND time invested. Rs 10,000 for 6 months ≠ Rs 10,000 for 12 months.
- Adding instead of multiplying in compounded ratio → Compounded ratio of a:b and c:d is (a×c):(b×d), NOT (a+c):(b+d).
- Ratio order errors → If asked "ratio of A to B," answer must be A:B, not B:A. Read questions carefully.
- Mixing units before forming ratio → Convert to same unit first. Ratio of 2 kg to 500 gm is 2000:500 = 4:1, not 2:500.
Quick Reference
- Ratio a:b means a/b; always simplify using HCF
- In proportion a:b :: c:d, cross-multiply: a×d = b×c
- Mean proportional of a and b = √(a×b)
- Direct: x increases → y increases; use x₁/y₁ = x₂/y₂
- Inverse: x increases → y decreases; use x₁×y₁ = x₂×y₂
- Partnership profit ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂)
- To distribute total in ratio a:b:c → Each share = (part/sum of parts) × Total