Study Notes: Ratio and Proportion

Overview

Ratio and Proportion is a foundational topic in RRB NTPC Mathematics, appearing in 4–6 questions every year. This topic tests your ability to compare quantities, share resources, and solve partnership/mixture problems. Mastery here directly improves your speed in profit-loss, time-work, and mixture-alligation questions since those topics build on ratio concepts.

The exam focuses on three core areas: basic ratio operations and simplification, partnership problems (how profits split based on investments), and mean/third proportional calculations. You'll also face word problems involving ages, money division, ingredient mixing, and business scenarios. The key to scoring well is recognizing the ratio pattern hidden in the problem statement and setting up the equation correctly. Most errors happen in the setup phase, not the calculation phase.

Practice converting word problems into ratio notation within 10–15 seconds. Once you can write "A:B = 3:5" from "A's share is 3 parts when B's is 5 parts," the rest becomes mechanical arithmetic. This topic rewards systematic approach over shortcuts.

Key Concepts

• **Ratio** expresses how many times one quantity contains another. If A:B = 3:5, then A = 3x and B = 5x for some common multiplier x. The ratio does not tell absolute values, only relative proportions.

• **Proportion** is an equation of two ratios: a:b = c:d means a/b = c/d, which gives ad = bc (cross-multiplication property). Use this to find any one unknown when three terms are known.

• **Compound ratio** of two ratios a:b and c:d is ac:bd. For three ratios a:b, c:d, e:f, compound ratio is ace:bdf. This appears when combining multiple conditions (e.g., age ratios at different times).

• **Duplicate ratio** of a:b is a²:b². Triplicate ratio is a³:b³. Sub-duplicate ratio is √a:√b. These rarely appear directly but may feature in geometry mensuration problems.

• **Mean proportional** between a and c is b where a:b = b:c, giving b² = ac, so b = √(ac). Third proportional to a and b is c where a:b = b:c. Fourth proportional to a, b, c is d where a:b = c:d.

• **Partnership** problems split profit/loss in the ratio of (capital × time). If A invests ₹3000 for 4 months and B invests ₹5000 for 6 months, profit ratio = (3000×4):(5000×6) = 12000:30000 = 2:5.

• In **componendo-dividendo**, if a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d). Useful for quickly solving certain proportion equations, though not essential if you're comfortable with cross-multiplication.

• **Direct proportion**: when A increases, B increases proportionally (workers and output). **Inverse proportion**: when A increases, B decreases (speed and time for fixed distance). Identify the type before setting up the equation.

Formulas / Key Facts

• **a:b = c:d ⇔ a/b = c/d ⇔ ad = bc** — Cross-multiplication is the workhorse identity.
• **Compound ratio of a:b and c:d = ac:bd** — Multiply numerators and denominators separately.
• **If A:B = m:n, then A = m/(m+n) × Total and B = n/(m+n) × Total** — Direct conversion from ratio to actual amount.
• **Mean proportional between a and c: b = √(ac)** — Used when three quantities form continuous proportion.
• **Third proportional to a, b is c where b² = ac ⇒ c = b²/a**.
• **Fourth proportional to a, b, c is d where a:b = c:d ⇒ d = bc/a**.
• **Partnership profit ratio = (Capital₁ × Time₁):(Capital₂ × Time₂):...** — Both investment amount and duration matter.
• **Duplicate ratio of a:b = a²:b²; Sub-duplicate = √a:√b**.
• **Componendo-dividendo: a/b = c/d ⇒ (a+b)/(a−b) = (c+d)/(c−d)** — Helps simplify nested proportion equations.

Worked Examples

**Example 1: Basic Ratio Division** Divide ₹7200 among A, B, C in ratio 2:3:4. *Solution:* Total parts = 2 + 3 + 4 = 9. A's share = (2/9) × 7200 = ₹1600. B's share = (3/9) × 7200 = ₹2400. C's share = (4/9) × 7200 = ₹3200. **Check:** 1600 + 2400 + 3200 = 7200 ✓

**Example 2: Partnership Problem** A invests ₹40,000 for 6 months, B invests ₹50,000 for 8 months. If total profit is ₹13,600, find each person's share. *Solution:* Profit ratio = (40000 × 6):(50000 × 8) = 240000:400000 = 3:5 (divide both by 80000). Total parts = 3 + 5 = 8. A's profit = (3/8) × 13600 = ₹5100. B's profit = (5/8) × 13600 = ₹8500.

**Example 3: Mean Proportional** Find the mean proportional between 9 and 25. *Solution:* Mean proportional b = √(9 × 25) = √225 = 15. **Verification:** 9:15 = 15:25 ⇒ 9/15 = 15/25 ⇒ 3/5 = 3/5 ✓

**Example 4: Compound Ratio Application** The ratio of boys to girls is 5:3, and the ratio of girls to teachers is 12:1. Find the ratio of boys to teachers. *Solution:* Make the common term (girls) equal: 5:3 and 12:1. Multiply first ratio by 4: (5×4):(3×4) = 20:12. Now girls = 12 in both ratios. Boys:Girls = 20:12 and Girls:Teachers = 12:1. So Boys:Teachers = 20:1.

Common Mistakes

• **Mistake:** Adding ratios directly — thinking 2:3 plus 4:5 equals 6:8. **Fix:** You cannot add ratios like fractions. Convert each to actual quantities with a common total, or find a common term if linking two ratios.

• **Mistake:** Forgetting to include time in partnership problems — using only capital invested. **Fix:** Partnership profit always depends on (Capital × Time). If time isn't mentioned, assume equal duration for all partners.

• **Mistake:** Confusing mean proportional with average. If asked for mean proportional of 4 and 16, writing (4+16)/2 = 10. **Fix:** Mean proportional is √(4×16) = √64 = 8, not the arithmetic mean. The term "mean" here refers to geometric mean.

• **Mistake:** Incorrect simplification — reducing 12:18 to 2:3 by subtracting 10 from both terms. **Fix:** Always divide both terms by their HCF. 12:18 → divide by 6 → 2:3. Never add/subtract the same number to simplify.

• **Mistake:** In compound ratio, adding instead of multiplying. For ratios 2:3 and 4:5, writing compound as (2+4):(3+5) = 6:8. **Fix:** Compound ratio is (2×4):(3×5) = 8:15. Multiply corresponding terms.

Quick Reference

• **a:b = ma:mb** — Ratios remain unchanged when both terms are multiplied/divided by the same non-zero number.
• **To divide amount M in ratio p:q → First part = p/(p+q) × M, Second part = q/(p+q) × M**.
• **Partnership: Profit ∝ (Capital × Time)** — Both factors multiply, not add.
• **Mean proportional between a and c is √(ac)** — Not (a+c)/2.
• **Cross-multiply to solve proportions: a:b = c:d ⇒ ad = bc** — Then isolate the unknown.
• **Compound ratio of multiple ratios: multiply all numerators, multiply all denominators** — ace:bdf for a:b, c:d, e:f.