Ratio and Proportion — Study Notes

Overview

Ratio and Proportion forms a fundamental pillar of the Numerical & Mental Ability section in the UP Police Constable exam. This topic typically accounts for 3–5 questions and serves as the foundation for solving problems in profit-loss, partnership, mixture-alligation, and time-work. Mastery of this topic is non-negotiable because ratio problems appear both directly and embedded within other question types.

Students must develop fluency in three core areas: understanding and simplifying ratios, solving proportion problems using unitary method or cross-multiplication, and applying these concepts to real-world scenarios like age problems, mixture problems, and business partnerships. The exam tests not just mechanical calculation but also the ability to translate word problems into mathematical ratios quickly. A strong grip on direct and inverse proportions, compound ratios, and partnership division ensures you can tackle 15–20% of the numerical section confidently within strict time limits.

The key to success lies in recognizing ratio patterns instantly, avoiding calculation errors with fractions, and practicing enough to internalize the standard problem templates that appear year after year.

Key Concepts

• **Ratio** is a comparison of two quantities of the same kind, written as a:b or a/b. Ratios have no units and can be simplified like fractions (12:18 = 2:3).

• **Proportion** states that two ratios are equal: a:b = c:d, written as a:b::c:d. In proportion, the product of extremes equals the product of means (a×d = b×c).

• **Direct Proportion**: When one quantity increases, the other increases proportionally (more workers → more work). If a∝b, then a₁/a₂ = b₁/b₂.

• **Inverse Proportion**: When one quantity increases, the other decreases proportionally (more speed → less time). If a∝1/b, then a₁/a₂ = b₂/b₁.

• **Compound Ratio**: The ratio compounded from two or more ratios by multiplying corresponding terms. Compound ratio of a:b and c:d is ac:bd.

• **Partnership** applies ratio principles to profit/loss sharing based on capital invested and time period. Profit share ratio equals capital × time ratio.

• **Fourth Proportional**: If a:b = c:d, then d is the fourth proportional. To find d: d = (b×c)/a.

• **Third Proportional**: If a:b = b:c, then c is the third proportional to a and b. Formula: c = b²/a.

Formulas / Key Facts

• **Simplifying ratios**: Divide both terms by their HCF. Example: 48:64 = 3:4 (÷16).
• **Combining ratios**: If a:b = 2:3 and b:c = 4:5, make 'b' equal: a:b:c = 8:12:15.
• **Direct Proportion formula**: a₁/a₂ = b₁/b₂, cross-multiply to solve for unknown.
• **Inverse Proportion formula**: a₁/a₂ = b₂/b₁ (note the inversion on right side).
• **Compound ratio**: (a:b) compounded with (c:d) = ac:bd.
• **Mean Proportional**: If a:b = b:c, then b = √(a×c). 'b' is the mean proportional.
• **Partnership (equal time)**: Profit ratio = Capital ratio. If A invests 3000, B invests 5000, profit splits 3:5.
• **Partnership (different time)**: Profit ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂).
• **Dividend distribution**: If total profit = P and ratio = a:b:c, A's share = [a/(a+b+c)] × P.
• **Continued proportion**: a, b, c are in continued proportion if a:b = b:c, meaning b² = ac.

Worked Examples

**Example 1: Direct Proportion** If 15 workers can build a wall in 12 days, how many days will 20 workers take?

*Solution*: More workers → Less days (inverse relationship) Workers ratio = 15:20 = 3:4 Days ratio = 4:3 (inverted) If 15 workers → 12 days Then 20 workers → (15×12)/20 = 9 days

**Example 2: Compound Ratio** Find compound ratio of 2:3, 4:5 and 3:7.

*Solution*: Compound ratio = (2×4×3) : (3×5×7) = 24 : 105 = 8:35 (÷3)

**Example 3: Partnership** A starts business with ₹5000. After 4 months, B joins with ₹8000. After 8 months from start, C joins with ₹10000. If total profit after 12 months is ₹19,200, find each person's share.

*Solution*: A's contribution = 5000 × 12 = 60,000 B's contribution = 8000 × 8 = 64,000 (joined after 4 months, so 12-4 = 8 months) C's contribution = 10000 × 4 = 40,000 (joined after 8 months, so 12-8 = 4 months) Ratio = 60,000 : 64,000 : 40,000 = 15:16:10 (÷4000) Sum of ratio = 15+16+10 = 41 A's share = (15/41) × 19,200 = ₹7,024.39 ≈ ₹7,024 B's share = (16/41) × 19,200 = ₹7,492.68 ≈ ₹7,493 C's share = (10/41) × 19,200 = ₹4,682.93 ≈ ₹4,683

**Example 4: Fourth Proportional** Find the fourth proportional to 3, 7, and 9.

*Solution*: Let fourth proportional = x 3:7 = 9:x 3×x = 7×9 x = 63/3 = 21

Common Mistakes

**Mistake 1**: Confusing direct and inverse proportion → **Fix**: Ask yourself: if one increases, does the other increase (direct) or decrease (inverse)? More speed = less time is inverse.

**Mistake 2**: Not making middle terms equal when combining ratios like a:b = 2:3 and b:c = 5:7 → **Fix**: Find LCM of middle terms (b = 3 and 5, LCM = 15). Convert: a:b = 10:15 and b:c = 15:21, so a:b:c = 10:15:21.

**Mistake 3**: In partnership, forgetting to multiply capital by time when partners invest for different periods → **Fix**: Always use (Capital × Time) as the contribution measure, not just capital.

**Mistake 4**: Adding ratios directly (thinking 2:3 + 3:4 = 5:7) → **Fix**: Ratios cannot be added. Convert to fractions with common denominator or convert to actual values first.

**Mistake 5**: Calculation errors when distributing total amount in given ratio → **Fix**: Always verify that sum of distributed parts equals the total. Use formula: Each share = (part/sum of parts) × total.

Quick Reference

• Ratio simplification: Divide by HCF. Always express in lowest terms.
• Proportion test: Product of extremes = Product of means (a×d = b×c).
• Direct proportion: Same direction change. Use a₁/a₂ = b₁/b₂.
• Inverse proportion: Opposite direction change. Use a₁/a₂ = b₂/b₁.
• Partnership profit ratio: (Capital₁ × Time₁) : (Capital₂ × Time₂).
• To find share: (Individual ratio part / Sum of all parts) × Total amount.