Math · Aptitude · Chapter 02

⚖️ Ratio & Proportion

Compare quantities, unitary method, partnership basics.

💡 Comparing Two Quantities

A ratio compares two amounts. 3 : 5 means "3 parts to 5 parts". A ratio is just a fraction written differently — 3:5 = 3/5.

A proportion is when two ratios are equal: 2 : 3 = 4 : 6. Multiply or divide both sides by the same number — the ratio stays equivalent.

Simplify a ratio

12 : 18 → divide by 6 → 2 : 3

Divide N in ratio a:b

a's share = (a/(a+b)) × N

Compounded ratio

(a:b) × (c:d) = ac : bd

Mean proportion

x² = a × b → x = √(ab)

Dividing money in a ratio — most common SSC question type:

Example: Divide ₹6,000 between A and B in ratio 2:3.
Total parts = 2+3 = 5. A gets (2/5)×6000 = ₹2,400. B gets (3/5)×6000 = ₹3,600.

⚡ Unitary method — solves "rule of 3" questions instantly

If N units cost ₹X, find cost of M units:

1 unit = X/N. So M units = (X/N) × M = (X × M) / N.

Example: 5 pens cost ₹40, what do 8 pens cost? (40 × 8)/5 = ₹64.

⚡ Direct vs Inverse proportion

Direct: as one ↑, other ↑. More workers → more output. More speed → more distance covered.
Inverse: as one ↑, other ↓. More workers → less time. More speed → less travel time.

Inverse formula: N₁ × T₁ = N₂ × T₂ (work-rate principle).
If 4 men do work in 12 days, 6 men do it in: 4 × 12 = 6 × T → T = 8 days.

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Dividing in a Ratio

Animation

Bar widths visualize how the total splits. The longer bar belongs to the bigger ratio.

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Ratio Calculators

Calculator

💵 Divide amount in ratio a:b

Total ₹ in ratio :

A = ₹2,400 · B = ₹3,600

📦 Unitary method

If units cost ₹, then units cost ₹64

1 unit = ₹40 ÷ 5 = ₹8. So 8 units = 8 × ₹8 = ₹64.

🔄 Inverse proportion (work-time)

If men finish in days, then men finish in 8 days

Practice (SSC): If A:B = 2:3 and B:C = 4:5, find A:B:C.

Answer: 8 : 12 : 15. Make B equal in both ratios. A:B = 2:3 = 8:12 (×4). B:C = 4:5 = 12:15 (×3). Now combine: A : B : C = 8 : 12 : 15.

Practice (Banking): ₹4900 is divided among A, B, C in ratio 3 : 4 : 7. What does B get?

Answer: ₹1,400. Total parts = 3+4+7 = 14. B's share = (4/14) × 4900 = (2/7) × 4900 = ₹1,400.