A ratio is simply a way of comparing two quantities of the same kind. When you write a : b, you are saying that for every a units of one thing, there are b units of the other. Nothing more mysterious than that.
A proportion is the statement that two ratios are equal: a : b = c : d, or in fraction form, a/b = c/d. This is the backbone of almost every "find the unknown" problem you see in CSAT.
Here is the analogy that makes it stick: think of a ratio as a recipe. If a dal recipe calls for 2 cups of lentils and 1 cup of water, the ratio is 2:1. Proportion asks: if you scale the recipe to 6 cups of lentils, how much water do you need? You are holding the ratio constant — that is proportion.
The critical vocabulary you need to lock down:
a : b, a is the antecedent, b is the consequent.a : b compounded with c : d gives ac : bd. You multiply across.a² : b².√a : √b — the square-root version.b : a.a : x = x : b, then x = √(ab). This is the geometric mean, and it appears in CSAT in disguise.Direct vs. Inverse Proportion:
Direct proportion means two quantities rise and fall together: y = kx. If you double one, you double the other.
Inverse proportion means one rises as the other falls: xy = k. If you double one, the other halves.
The trap CSAT sets: presenting a scenario that looks like direct proportion but is actually inverse (or vice versa). Always ask yourself — "if I increase this variable, does the other increase or decrease?" Answer that first, then set up your equation.
Every ratio problem involving multiple groups comes down to one operation: equalize the common quantity using LCM.
Look — when you are told "ratio of A to B is 3:4" and "ratio of B to C is 5:6", you cannot directly compare A and C because the value of B is different in each statement. The fix is to make B the same number in both:
A : B = 3 : 4 = 15 : 20
B : C = 5 : 6 = 20 : 24
Now B = 20 in both, so A : B : C = 15 : 20 : 24.
The LCM of 4 and 5 is 20. Multiply the first ratio by 5 (to make B = 20) and the second by 4 (same reason). This is the standard chain-ratio technique and it solves 80% of "two-company" or "three-group" ratio problems on UPSC CSAT.
Given a/b = c/d:
ad = bc (product of means = product of extremes)b/a = d/ca/c = b/d(a+b)/b = (c+d)/d(a-b)/b = (c-d)/d(a+b)/(a-b) = (c+d)/(c-d)Componendo-Dividendo is the most useful shortcut in the list. It eliminates fractions in one step. When a problem gives you (x+y)/(x-y) = 5/3, you can immediately read off x/y = (5+3)/(5-3) = 8/2 = 4. Standard solving with substitution: 6 steps. Componendo-Dividendo: 1 step.
If y ∝ x, then y₁/x₁ = y₂/x₂.
Classic application: "If 6 workers complete a wall in 10 days, how many days for 15 workers?" — but wait, this is inverse proportion (more workers, fewer days). The test-taker who sets this up as direct proportion gets the wrong answer and often cannot figure out why.
Rule: More workers → fewer days → inverse proportion. Set up as 6 × 10 = 15 × d, giving d = 4.
If y ∝ 1/x, then x₁y₁ = x₂y₂.
The real danger zone is combined proportion problems: "If 8 pipes fill a tank in 6 hours, how long for 12 pipes if each pipe now works at 3/4 efficiency?" You have two variables changing simultaneously. Handle them one at a time:
8 × 6 / 12 = 4 hours.4 ÷ (3/4) = 16/3 hours.Never try to combine both changes in one ratio fraction. Sequential adjustment prevents errors.
If X : Y = Y : Z, then Y² = XZ. This means Y is the geometric mean of X and Z.
This is exactly the structure of the T20 cricket PYQ below. The trap: the problem does not tell you whether the sequence X, Y, Z is increasing or decreasing. Both are valid, and they give different orderings. Recognizing this means you pick "cannot be determined" with confidence — rather than wasting two minutes trying to find a unique answer.
Partnership is ratio applied to time and money. If A invests ₹P₁ for t₁ months and B invests ₹P₂ for t₂ months, their profit ratio is:
A's share : B's share = P₁t₁ : P₂t₂
This is just the weighted ratio. Nothing new — same LCM-equalization instinct applies when three partners are involved.
When two ratios share a common term (like female workers in both companies), immediately find the LCM of the two values of that term and scale both ratios up. This avoids fractions entirely.
Example: A has male:female = 13:10, B has male:female = 7:5. Female appears as 10 and 5. LCM = 10. Scale B's ratio by 2: B becomes 14:10. Now both have 10 females. Total A = 23, Total B = 24. Done.
Standard approach (setting variable f and solving two simultaneous expressions): ~50 seconds. LCM approach: ~15 seconds.
When you see an equation of the form (a+b)/(a-b) = p/q, apply Componendo-Dividendo immediately: a/b = (p+q)/(p-q).
Example: (x+3)/(x-3) = 7/2. By C-D: x/3 = (7+2)/(7-2) = 9/5. So x = 27/5.
Standard cross-multiply + expand + collect terms: 6 algebraic steps. C-D: 2 steps.
Whenever a problem says "ratio of A to B equals ratio of B to C" (or any middle term appears on both sides of an equality of ratios), immediately write B² = AC. Then ask: can multiple integer solutions satisfy the total sum? If yes, the answer is "cannot be determined."
Example: X + Y + Z = 37, Y² = XZ. Try (16,12,9): sum = 37, 144 = 144. Try (9,12,16): sum = 37, 144 = 144. Two valid orderings → "cannot be determined."
Without this pattern recognition, you might spend 3 minutes trying to find a unique solution that does not exist.
Before writing any equation, answer two questions: (1) If variable A increases, does B increase or decrease? (2) Is there any third variable complicating the direction?
If B increases with A → direct proportion, use A₁/B₁ = A₂/B₂.
If B decreases with A → inverse proportion, use A₁B₁ = A₂B₂.
This takes 5 seconds but eliminates the most common CSAT arithmetic error (setting up the wrong proportion type). Students who skip this test make the mistake roughly 1 in 3 times on inverse proportion questions.
To compound three ratios a:b, c:d, e:f, write them as fractions and multiply: (a/b) × (c/d) × (e/f) = ace/bdf.
Example: Compound 3:4, 5:6, 7:8 = (3×5×7):(4×6×8) = 105:192 = 35:64.
Writing this as three separate multiplication steps: 9 individual multiplications. The one-line product approach groups them in 2 multiplications per numerator/denominator. Saves ~20 seconds on multi-ratio compounding.
Read the problem and route it through this decision tree in the exam hall:
Step 1 — Identify the structure.
middle² = product of extremes. Check for multiple solutions.Step 2 — Choose the method.
(sum)/(difference) form.Step 3 — Assign the simplest variable. Let the LCM value be 1 unit (or 10k, or whatever keeps arithmetic clean). Never assign x = some fractional starting point.
Step 4 — Verify with the original constraint. Plug back in. On a 2-minute problem, this takes 15 seconds and prevents careless errors from reaching the final answer.
If you reach Step 4 and the answer does not verify cleanly, go back to Step 1 and re-check whether you used direct or inverse proportion.
Why this question (6a205577ccfcbabc57a3a37b): This is a classic "equalize the common element" problem. The common element is female workers. UPSC tests whether you reach for LCM naturally or waste time setting up two simultaneous equations.
Solving path: Female workers are equal in both companies. Company A has male:female = 13:10, so the female group is a multiple of 10. Company B has male:female = 7:5, so the female group is a multiple of 5. LCM(10, 5) = 10. Set females = 10 in both. In A: males = 13, total = 23. In B: males = 14, total = 24. Ratio = 23:24. Answer: option (b).
Why this question (6a205577ccfcbabc57a3a382): This is the same structural problem as above, confirming UPSC has tested this pattern in the same exam cycle. Treat it as a reinforcement drill — if your method is correct, you should reach 23:24 in under 30 seconds the second time.
Solving path: Identical method. Female LCM = 10. A totals 23, B totals 24. Ratio = 23:24. The fact that this appeared twice tells you LCM-equalization is a core expected skill, not an edge case.
Why this question (6a2059edccfcbabc57a3a53b): This is a trap question that rewards pattern recognition over computation. The structure is "A:B = B:C with a fixed total" — which immediately signals geometric mean. The question tests whether you check for uniqueness before committing to an ordering.
Solving path: X + Y + Z = 37, and X/Y = Y/Z → Y² = XZ. You need integer (or at minimum positive real) solutions. Try X = 16, Y = 12, Z = 9: sum = 37, Y² = 144, XZ = 144. Valid. Ordering: X > Y > Z, so Value-I > Value-II > Value-III (option b). Now try X = 9, Y = 12, Z = 16: sum = 37, Y² = 144, XZ = 144. Also valid. Ordering: X < Y < Z, so Value-I < Value-II < Value-III (option a). Two valid solutions with different orderings → "Cannot be determined." Answer: option (d).
The mistake to avoid: finding the first valid solution (16,12,9) and immediately selecting option (b) without checking whether other solutions exist. UPSC specifically designs these to punish premature commitment.
Setting up direct proportion for an inverse relationship. The "more workers, more days" error is extremely common. Always do the More-Less test before writing any equation.
Not checking for multiple solutions in geometric mean problems. When you see X:Y = Y:Z, both (a,b,c) and (c,b,a) satisfy Y² = XZ if the sum constraint allows both. Never assume a unique ordering without testing the reverse sequence.
Forgetting to scale both parts of a ratio when equalizing. If you multiply the female term of company B by 2 to reach LCM = 10, you must also multiply the male term by 2. Students often equalize one term and forget the other, computing an incorrect total.
Treating "ratio of A to B" and "ratio of B to A" as equivalent. The problem says 13:10 (male:female). If you invert this to 10:13 by accident, your total flips from 23 to 23 but your male count is wrong, and the final ratio will be incorrect.
Using the wrong base in partnership problems when investment periods differ. Profit share depends on principal × time, not just principal. A common error is to divide profit in the ratio of principals alone when different partners invest for different durations.
Confusing compounded ratio with sum of ratios. Compounding 2:3 and 4:5 gives 8:15, not 6:8. Compounding means multiply the corresponding terms; it is not addition of numerators and denominators.