Ratio, Proportion, Ages & Partnership — IBPS PO Quant

This chapter is one idea wearing four hats. A ratio splits a quantity into parts; a proportion equates two ratios; ages are ratios that change over time; partnership is a profit-ratio from capital and time. All of them are solved fastest by introducing a single variable k for one "part" and working with parts instead of totals. IBPS asks 1–3 of these directly and hides many more inside DI and mixtures. Master the k-method and this becomes reliable one-line arithmetic.

1. What IBPS actually asks

• Ratio division: "Divide ₹6,300 among A, B, C in 2:3:4."
• Proportion: "If a:b = 3:4 and b:c = 6:5, find a:b:c."
• Ages: "The ratio of ages is 5:7; after 6 years it's 3:4 — find present ages."
• Partnership: "A invests ₹X for m months, B ₹Y for n months — split the profit."

2. The k-method (the core tool)

A ratio a:b:c means the quantities are ak, bk, ck for some common multiplier k. Introduce k and every ratio problem becomes one equation:

• Divide ₹6,300 in 2:3:4 ⇒ parts total 2k+3k+4k = 9k = 6300 ⇒ k = 700. Shares: ₹1,400, ₹2,100, ₹2,800.
• Work in parts: find the value of one part (k), then multiply out. Never juggle the total directly.

$a:b:c=ak:bk:ck,\text{one part }k=\frac{\text{total}}{\text{sum of ratio terms}}$

3. Combining ratios

To chain a:b = 3:4 and b:c = 6:5, make b common:

• b is 4 in the first, 6 in the second ⇒ LCM(4,6) = 12. Scale: a:b = 9:12, b:c = 12:10 ⇒ a:b:c = 9:12:10.
• Proportion property: if a:b = c:d, then ad = bc (product of extremes = product of means). Use it to find a missing term.
• Componendo–dividendo: if a/b = c/d, then (a+b)/(a−b) = (c+d)/(c−d) — a shortcut for "sum and difference" ratio questions.

4. Ages — ratios that shift in time

Age problems are ratios with a +t or −t on each part:

• Present ages in 5:7 ⇒ write 5k and 7k.
• "After 6 years, ratio is 3:4" ⇒ (5k+6)/(7k+6) = 3/4. Cross-multiply: 4(5k+6) = 3(7k+6) ⇒ 20k+24 = 21k+18 ⇒ k = 6. Present ages: 30 and 42.
• Every person's age changes by the same amount over a period — add/subtract the same t to each part.

5. Partnership — profit by capital × time

Profit shares in the ratio of each partner's capital × months invested:

$\text{share ratio}=C_1{T}_1:C_2{T}_2:\dots$

• A: ₹12,000 for 12 months = 144(k); B: ₹18,000 for 8 months = 144(k) ⇒ ratio 1:1.
• Simple partnership (same time) ⇒ ratio of capitals only.
• Adjustments: a "working partner" salary/commission is taken out first, then the remainder split by the capital×time ratio.

7. Common traps

• Adding ratios of different totals — you can't add 2:3 and 3:5 unless the wholes match; convert to values first.
• Forgetting the same-time simplification in partnership — equal periods reduce it to a capital ratio.
• In ages, adding t to the ratio number instead of the actual age — write 5k+t, not (5+t)k.
• Mean proportional vs third proportional: mean of a and b is √(ab); third proportional to a and b is b²/a.

8. The protocol

1. Introduce k for one part; solve for k, then multiply out.
2. Combine ratios by making the linking term common (LCM).
3. Ages: add/subtract the same t to each k-part, then cross-multiply.
4. Partnership: ratio = capital × time; remove any salary/commission first.
5. Watch bases — never add ratios with different wholes; convert to values.