Ratio & Proportion

Study Materials › Arithmetic › Ratio & Proportion

Ratio is the backbone of the arithmetic section — partnership, ages, mixtures and most data interpretation questions are ratio problems underneath. The habit that pays off most is converting a ratio into parts immediately, then finding the value of one part.

Core Concepts

Dividing an amount in the ratio a : b : c → shares are a/(a+b+c), b/(a+b+c) and c/(a+b+c) of the total.

Proportion: if a : b = c : d then ad = bc (product of extremes = product of means).

Direct proportion: one goes up, the other goes up — a/b is constant.

Inverse proportion: one goes up, the other goes down — ab is constant.

Compounded ratio of a : b and c : d is ac : bd.

Work in parts. If a sum is split 4 : 7 : 5, do not carry fractions — say "16 parts", find one part, and multiply. Nearly every ratio question in these papers becomes a one-line calculation once you do this.

Partnership

Profit is shared in the ratio of capital × time.

Simple partnership — equal time: profit ratio = capital ratio.

Compound partnership — unequal time: profit ratio = C₁T₁ : C₂T₂.

A working partner's salary or commission is taken out first; only the remainder is divided.

Worked Examples from Real Papers

TS Police SI 2022 — Q6

An amount x is distributed to three persons A, B and C. A gets 25% of x. Twice the difference of B and C equals the share of A. Find the ratio A : B : C.

A = x/4. Given 2(B − C) = x/4, so B − C = x/8.
B + C = x − x/4 = 3x/4.
Adding: 2B = 3x/4 + x/8 = 7x/8, so B = 7x/16. Then C = 3x/4 − 7x/16 = 5x/16.
A : B : C = 4x/16 : 7x/16 : 5x/16 = 4 : 7 : 5.

TS Police SI 2022 — Q10

₹5,600 is distributed among 12 people consisting of men, women and children. The ratio of the total amounts given to all men, all women and all children is 9 : 4 : 1, while the ratio of the amounts received by each man, woman and child is 3 : 2 : 1. Find the amount received by each woman.

Totals: 14 parts on ₹5,600 → men ₹3,600, women ₹1,600, children ₹400.
Let each man get 3k, each woman 2k, each child k, with counts m, w and c.
3km = 3600 → m = 1200/k · 2kw = 1600 → w = 800/k · kc = 400 → c = 400/k.
Total people: (1200 + 800 + 400)/k = 2400/k = 12, so k = 200.
Each woman receives 2k = ₹400.

TS Police SI 2022 — Q23

P started a business with ₹5,00,000. After 3 months Q joined with ₹4,00,000. At the end of the year Q received a total of ₹50,000, including 20% of the profit for managing the business. Find the amount P received.

Capital × time → P: 5,00,000 × 12 = 60 lakh-months; Q: 4,00,000 × 9 = 36 lakh-months.
Ratio P : Q = 60 : 36 = 5 : 3.
Let total profit be T. Q's management share is 0.2T; the remaining 0.8T is split 5 : 3, so Q gets (3/8)(0.8T) = 0.3T.
Q's total = 0.2T + 0.3T = 0.5T = 50,000 → T = 1,00,000.
P receives (5/8)(0.8 × 1,00,000) = ₹50,000.

TS Police SI 2022 — Q40

The ratio of the ages of Ram and Rahim five years from now is 11 : 10. Four years ago the ratio was 8 : 7. Find the ratio of their present ages.

Let present ages be R and H.
(R + 5)/(H + 5) = 11/10 → 10R − 11H = 5
(R − 4)/(H − 4) = 8/7 → 7R − 8H = −4
Solving the pair gives H = 25 and R = 28.
Present ratio = 28 : 25.
Note: age questions almost always reduce to two linear equations. Set them up carefully and the algebra is routine — the marks are lost in translating the sentence, not in solving.

Ratios are not amounts. A common wrong option is built by treating the ratio terms themselves as rupees or years. Always finish by converting parts back into the unit the question asked for.

Also see: Averages · Time, Speed & Distance · Percentages