Ratio and Proportion (RRB)

A ratio compares two quantities of the same kind: 4:6 = 2:3. It has no units. A proportion is the equality of two ratios: a:b :: c:d ⟹ a/b = c/d ⟹ a × d = b × c (product of extremes = product of means).

Key facts:

• If a:b = c:d, then (a+b):(c+d) = a:c and (a−b):(c−d) = a:c — useful in age and partnership problems.
• The "duplicate ratio" of a:b is a²:b², the "sub-duplicate" is √a:√b, "triplicate" is a³:b³.
• Compounded ratio: (a:b) × (c:d) = ac:bd.

Dividing N in a given ratio a:b:c:
Share = (a / (a+b+c)) × N, etc. Example: divide ₹720 in 2:3:4. Shares = 720×2/9, 720×3/9, 720×4/9 = 160, 240, 320.

Partnership — profit is shared in the ratio of capital × time:

• Equal time, capitals A:B → profit in A:B.
• Different times t₁ and t₂ → profit in A·t₁ : B·t₂.
• A sleeping partner contributes capital only — same formula still applies.

Example: P invests ₹6,000 for 12 months; Q invests ₹9,000 for 8 months. Ratio of profit = 6000×12 : 9000×8 = 72000 : 72000 = 1:1.

Continued proportion: a, b, c are in continued proportion if b² = ac. b is the mean proportional between a and c, and is √(ac).

Inverse proportion: if x increases as y decreases (xy = k), they are inversely proportional. Speed × time = distance is a classic case — at a fixed distance, speed and time are inversely proportional.

Trick 1 — Ratio adjustment: if A:B = 5:7 and we add 4 to both, the ratio becomes 9:11 — not the same as 5:7. So adding the same number changes the ratio. This is the basis for "X added to both terms" style problems.

Example: What must be added to each term of 5:9 to make it 7:11?
Let x be added. (5+x)/(9+x) = 7/11 ⟹ 11(5+x) = 7(9+x) ⟹ 55+11x = 63+7x ⟹ 4x = 8 ⟹ x = 2.

Trick 2 — Three-way ratio combination: when A:B = 2:3 and B:C = 4:5, you can't just write A:B:C = 2:3:5. You must make B equal in both ratios:

• A:B = 2:3 → multiply by 4 → 8:12.
• B:C = 4:5 → multiply by 3 → 12:15.
• So A:B:C = 8:12:15.

Trick 3 — Partnership with capital change mid-year: split each partner's contribution into time-windows.
Example: A puts ₹10,000 for 6 months, then adds ₹2,000 (so ₹12,000 for the remaining 6 months). A's effective capital = 10000×6 + 12000×6 = 60,000 + 72,000 = 132,000 (units of rupee-month).

Trick 4 — Coin problems: if coins of ₹1, ₹2, ₹5 are in ratio 3:2:1 and total = ₹120, treat the ratio in terms of value: (1×3) : (2×2) : (5×1) = 3:4:5. So value of ₹1-coins is 3/12 of 120 = ₹30, meaning 30 coins of ₹1.

Common RRB exam trap: "Ratio of incomes is 4:5 and expenditures 3:4, savings are equal." Set incomes = 4x, 5x; expenditures = 3y, 4y; equate savings (4x−3y = 5x−4y) and solve.