A ratio compares two quantities without saying how big either one is. “A : B = 3 : 4” tells you nothing about A except that for every 3 parts of A there are 4 of B — and that single idea, thinking in parts, solves almost all 41 ratio and partnership questions the Punjab Police SI has asked since 2016. Give the parts a name. If A : B = 3 : 4, write A = 3k and B = 4k. Now any sentence in the question becomes an equation in one unknown. You will learn (1) the parts method, (2) how to chain A : B with B : C, (3) mean and third proportionals, (4) direct and inverse variation, and (5) partnership, where profit follows capital × time.
Think in parts
One letter turns a ratio into algebra.
The ratio does not fix the numbers — it fixes their shape. Writing A = 3k and B = 4k keeps that shape and gives you one unknown to solve for. A sum of parts is especially easy: if A : B : C = 3 : 4 : 5 and the total is ₹960, there are 12 parts, each part is ₹80, and the shares are 240, 320 and 400.
₹20,250 is divided among A, B and C in such a way that the share of A is 7/8 of the combined share of B and C. Find A's share. (SI 31 Jul 2023)
Show answer
A's share is 7/8 of (B + C), so A : (B + C) = 7 : 8 and the total is 15 parts. ₹20,250 ÷ 15 = ₹1,350 per part, so A gets 7 × 1,350 = ₹9,450. “x is p/q of the rest” always becomes the ratio p : q.
A sum of ₹970 is to be divided among A, B and C such that if they get ₹2, ₹5 and ₹7 more respectively, their shares will be in the ratio 3 : 4 : 5. What will be B's share (in ₹)? (SI 15 Jul 2023)
Show answer
After adding ₹2, ₹5 and ₹7 the shares are in 3 : 4 : 5, and the total becomes 970 + 14 = ₹984. Twelve parts, so each part is ₹82. C's new share is 5 × 82 = ₹410, and his original share was 410 − 7 = ₹403. The paper's answer, ₹323, is B's original share: 4 × 82 − 5 = 323.
Chaining two ratios
Twelve of the 30.
If A : B = 2 : 4, B : C = 3 : 5 and C : D = 4 : 6, then A : D is equal to ______. (SI 20 Jun 2023)
Show answer
A : B = 2 : 4, B : C = 3 : 5, C : D = 4 : 6. Multiply along the chain: A/D = (A/B)(B/C)(C/D) = (2/4)(3/5)(4/6) = 24/120 = 1/5. So A : D = 1 : 5, which the paper writes as 6 : 30. Multiplying the fractions along the chain is always quicker than scaling each ratio.
Two numbers are in the ratio 37 : 40. If 7 is subtracted from each of them, the new ratio becomes 67 : 73. Find the smaller number. (SI 16 Oct 2022)
Show answer
Numbers are 37k and 40k. After subtracting 7 from each the ratio is 67 : 73: (37k − 7)/(40k − 7) = 67/73. Cross-multiply: 73(37k − 7) = 67(40k − 7) ⇒ 2701k − 511 = 2680k − 469 ⇒ 21k = 42 ⇒ k = 2. The smaller number is 37 × 2 = 74.
Proportion: mean and third
Two definitions, nine questions.
Direct: y = kx. Double one, double the other. Cost of petrol, weight of paper, train fare against the price of diesel — all direct. Inverse: y = k/x. Double one, halve the other. When the stem says “varies inversely as the cube”, it means x = k/y³, so a change in y by a factor f changes x by a factor 1/f³.
’x’ varies inversely as cube of ‘y’. It is given that y = 6 for x = 1. Find the value of ‘x’ for y = 8. (SI 29 Jun 2023)
Show answer
x varies inversely as y³, so x = k/y³. With y = 6 and x = 1: k = 1 × 216 = 216. Then at y = 8: x = 216/512 = 27/64.
- A ratio has no units. 15 weeks against 150 days must both become days first: 105 : 150 = 7 : 10.
- Mean proportional is √(ac); third proportional is b²/a. Two different things.
- Adding a constant to both sides of a ratio changes it. 3 : 4 is not 4 : 5.
Also asked onceTap to open
A precious stone weighing 48 g costs ₹1,94,560. It accidentally breaks into two pieces having weights in the ratio of 3 : 5. If the price varies directly as the cube of the weights, then what is the loss (in ₹) incurred due to the breaking of the stone? (SI 29 Jul 2023)
Show answer
The price varies as the square of the weight. 48 g in the ratio 3 : 5 gives pieces of 18 g and 30 g. The original price is proportional to 48² = 2304; the two pieces to 18² + 30² = 324 + 900 = 1224. New total = 1,94,560 × 1224/2304 = ₹1,03,360, so the loss is 1,94,560 − 1,03,360 = ₹1,36,800 by the paper's reading of what is asked.
Which number should be subtracted from each of the numbers 10, 12, 19 and 24 so that the resulting numbers are in proportion? (SI 29 Jun 2023)
Show answer
Let x be subtracted. Then (10 − x) : (12 − x) = (19 − x) : (24 − x). Cross-multiply: (10 − x)(24 − x) = (12 − x)(19 − x) ⇒ 240 − 34x + x² = 228 − 31x + x² ⇒ −3x = −12 ⇒ x = 4. The x² always cancels — these are linear, not quadratic.
For any two numbers a, b; (a + b) : (a − b) = 4 : 3. Find the value of 1/a : 1/b. (SI 14 Jul 2023)
Show answer
(a + b) : (a − b) = 4 : 3 ⇒ 3a + 3b = 4a − 4b ⇒ a = 7b. Then 1/a : 1/b = b : a = 1 : 7. Reciprocals always reverse a ratio.
A vertical pole of height 6 m 30 cm casts a shadow 3 m 15 cm long. Find : (i) the length of the shadow of another pole 8 m 5 cm high (ii) the height of a pole whose shadow is 5 m long (SI 2 Jul 2023)
Show answer
Shadows are directly proportional to heights, and 6.30 m casts 3.15 m — a ratio of exactly 2 : 1. So a pole 8.05 m high casts 8.05/2 = 4.025 m, and a shadow of 5 m belongs to a pole 2 × 5 = 10 m high.
Every other ratio question the SI has askedTap to open
If A : B = 1.75 : 1.50 and B : C = 3.2 : 5.6, then find A : B : C. (SI 2 Jul 2023)
Show answer
Clear the decimals first. A : B = 1.75 : 1.50 = 175 : 150 = 7 : 6, and B : C = 3.2 : 5.6 = 32 : 56 = 4 : 7. B is 6 in one and 4 in the other, so scale to 12: double the first to 14 : 12 and triple the second to 12 : 21. Hence A : B : C = 14 : 12 : 21.
If x : y = 11 : 4, then find the following ratio. (4x + 5y) ∶ (4x − 5y) = ? : ? (SI 20 Jun 2023)
Show answer
Write x = 11k and y = 4k. Then 4x + 5y = 44k + 20k = 64k and 4x − 5y = 44k − 20k = 24k. The ratio is 64 : 24 = 8 : 3. The k cancels, which is why you never need the actual values.
There are two teams in a game. The first team consists of three players : A, B and C. The second team consists of each of their parents, D, E and F. Each member of one team is of the same age as his/her teammates. The first team to second team members’ ages are in the ratio 2 : 5. Four years later the ratio of their ages becomes 5 : 11, respectively. How old were the players of second team five years ago? (SI 9 Jul 2023)
Show answer
Take the ages as 2k and 5k. Four years later they are 2k + 4 and 5k + 4, and the new ratio is 5 : 11, so 11(2k + 4) = 5(5k + 4). That gives 22k + 44 = 25k + 20, so 3k = 24 and k = 8. The second team is 5k = 40 now, and five years ago they were 35. Read the last line: it asks for five years ago, not now — 40 is on offer as option D.
Find a third proportional to the numbers 4, 3. (SI 9 Jul 2023)
Show answer
The third proportional to a and b is b²/a, so here it is 3² ÷ 4 = 9/4 = 2.25.
What is the third proportional to 17 and 136? (SI 29 Jun 2023)
Show answer
Third proportional = b²/a = 136² ÷ 17. Since 136 = 8 × 17, this is 136 × 136/17 = 136 × 8 = 1088.
If 0.25 : x : : 5 : 9, then x = ? (SI 31 Jul 2023)
Show answer
0.25 : x :: 5 : 9 means 0.25/x = 5/9, so x = 0.25 × 9/5 = 0.45. In a proportion the product of the means equals the product of the extremes: 5x = 0.25 × 9.
Find the unknown value of 'x' in the equal proportion (4x + 1) : 2 = (x + 3) : 5. (SI 24 Aug 2021)
Show answer
Cross-multiply: 5(4x + 1) = 2(x + 3), so 20x + 5 = 2x + 6 and 18x = 1, giving x = 1/18. The options are written as ratios, so the answer is 1 : 18.
The price of diesel has been increased by 10 percent. A train fare is directly proportional to the price of diesel. Earlier a passenger used to travel 80 km by train for ₹250. How much money will he now spend (approximately) to travel 750 km by the same train? (SI 29 Jun 2023)
Show answer
Fare rises with the diesel price, so the ₹250 for 80 km becomes 250 × 1.1 = ₹275 for 80 km. Fare also rises with distance, so 750 km costs 275 × 750/80 = ₹2,578.125, which is approximately ₹2,578. Two proportionalities in one question: apply the price rise first, then scale the distance.
One third of Gowri’s marks in History equals her marks in Geography. If she obtained 160 marks in the two subjects together, how many marks did she get in Geography ? (SI 2016)
Show answer
A third of History equals Geography, so History : Geography = 3 : 1 — four parts in all. 160 ÷ 4 = 40 per part, so Geography is 40 and History is 120.
A certain amount is distributed among X, Y and Z. X gets 4/5 and Y gets 1/6 of the whole amount. If Z gets ₹15, then Y gets ______. (SI 16 Oct 2022)
Show answer
X takes 4/5 and Y takes 1/6, so Z takes the rest: 1 − 4/5 − 1/6 = (30 − 24 − 5)/30 = 1/30. Z's 1/30 is ₹15, so the whole amount is ₹450 and Y gets 1/6 of it = ₹75.
Partnership
Eleven questions, and one product decides them all.
This single product, often called the capital-month, settles every partnership question. When all partners stay in for the same period the time cancels and the profit follows the capital ratio alone — that is the simple case. When they join or leave at different times, or add money part-way, you must build the product month by month — the compound case. For a partner who changes his investment mid-year, add the segments: ₹60,000 for 4 months and ₹55,000 for 8 months is 240,000 + 440,000 = 680,000 capital-months.
- Splitting the profit by capital when the partners were in for different lengths of time. Always build capital × months first.
- Paying the working partner out of his capital share instead of before it. The salary or management fee comes off the top, and he still takes his capital share afterwards.
- Forgetting the tax. If the profit is quoted before tax, deduct it before dividing.
- Treating a mid-year top-up as if it had been there all year. Split that partner's contribution into two segments and add the products.
One-page revision
Read only this the night before.
| Situation | What to do |
|---|---|
| Ratio a : b given | Write ak and bk |
| Total shared in a ratio | Total ÷ sum of parts = one part |
| A : B and B : C | Make the shared term match |
| Long chain A:B:C:D | Multiply the fractions |
| Mean proportional of a, c | √(ac) |
| Third proportional to a, b | b²/a |
| Direct variation | y = kx |
| Inverse variation | y = k/x |
| Reciprocals 1/a : 1/b | = b : a |
| Partnership profit | Ratio of capital × months |
| Working partner | Salary off the top, then the split |
Never work with the ratio itself — work with k. The moment you write A = 3k, every sentence in the question turns into an equation you can solve.