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.