A ratio is a way of comparing two quantities of the same kind. When you say the ratio of red candies to blue candies is 3 : 5, you are not saying there are exactly 3 and 5 candies — you are saying for every 3 red, there are 5 blue. The actual numbers could be 6 and 10, or 30 and 50. The ratio captures the relationship, not the absolute count.
Think of it like a recipe. If a recipe says use flour and sugar in a ratio of 2 : 1, you can make a small batch (200g : 100g) or a large batch (1kg : 500g). The ratio stays fixed regardless of the batch size.
Proportion is the statement that two ratios are equal. When you write a : b = c : d, you are saying these two ratios are equivalent. This is called a proportion. The four quantities a, b, c, d are called the terms. a and d are the extremes; b and c are the means. The golden rule of proportion: product of extremes = product of means, i.e., a × d = b × c.
In Indian classrooms, you will often hear these two variants:
The analogy that sticks: a ratio is like a zoom factor on a map. The map does not show real distances — it shows a scaled version. A proportion says two different maps have the same zoom factor.
For UP Police Constable, these questions are direct and arithmetic-heavy. You will almost never need algebra. The bottleneck is speed — recognizing the type, applying the right operation, and moving on in under 30 seconds.
A ratio a : b is written in its simplest form by dividing both terms by their HCF.
Example: 18 : 100. HCF of 18 and 100 is 2. Simplified ratio = 9 : 50.
This is exactly what the PYQ on John and Sara's investments tests. Never leave a ratio un-simplified in your final answer — all options in the paper will be in simplest form.
If a : b = 3 : 5 and a = 24, then:
b = (5/3) × 24 = 40
The logic: a corresponds to 3 parts, so one part = 24 ÷ 3 = 8. Then b = 5 parts = 5 × 8 = 40.
This "unit part" method is faster in the exam hall than setting up an equation.
When you have a : b and b : c and need a : b : c, the trick is to make b the same number in both ratios.
Example: a : b = 3 : 2 and b : c = 8 : 7.
b = 2. In the second, b = 8. LCM of 2 and 8 is 8.3 : 2 = 12 : 8.b = 8: 8 : 7.a : b : c = 12 : 8 : 7.When b is already the same in both ratios, you can read the answer directly without any scaling.
For a chain of three ratios — P : Q = 5 : 6 and Q : R = 14 : 15:
Q is 6 in the first and 14 in the second. LCM(6, 14) = 42.P : Q = 5 × 7 : 6 × 7 = 35 : 42.Q : R = 14 × 3 : 15 × 3 = 42 : 45.P : R = 35 : 45 = 7 : 9.If a total of 1280 people is split in the ratio 15 : 1 (students : teachers):
1280 × (1/16) = 80.This is the most common format. Total × (required part / total parts).
Two quantities x and y are in direct proportion if x / y = constant, i.e., x₁ / y₁ = x₂ / y₂.
Meena earns ₹9,000 in 25 days. Days needed for ₹10,800:
9000 / 25 = 360.10800 / 360 = 30.Alternatively, using direct proportion: 25 / 9000 = d / 10800 → d = 25 × 10800 / 9000 = 30.
This is a classic question type. "Find the number to subtract from both terms of 15 : 19 to get 3 : 4."
Let the number be x:
(15 - x) / (19 - x) = 3 / 4
Cross-multiply: 4(15 - x) = 3(19 - x)
60 - 4x = 57 - 3x
3 = x
So x = 3. Check: (15-3) : (19-3) = 12 : 16 = 3 : 4. Correct.
This is an advanced application. If income ratio is 10 : 6 and expenditure ratio is 18 : 10, and savings are ₹5,200 and ₹3,600:
10x and 6x; expenditures be 18y and 10y.10x - 18y = 5200 and 6x - 10y = 3600.3x - 5y = 1800.5x - 9y = 2600.x = 1000, so incomes = ₹10,000 and ₹6,000.This type needs two equations. Don't try to shortcut it — set up the algebra cleanly and it solves in under 90 seconds.
When ratio is given and one actual value is known, find "one part" first, then scale.
If red : blue = 3 : 5 and red = 24: one part = 24 ÷ 3 = 8. Blue = 5 × 8 = 40.
This avoids setting up fractions. Standard fraction method: 3 steps. Unit part method: 2 steps. Saves roughly 10-12 seconds per question.
To combine a : b and b : c, find LCM of the two values of b and scale both ratios.
Example: a : b = 3 : 2, b : c = 8 : 7. LCM(2, 8) = 8. Scale first ratio by 4: 12 : 8. Second stays 8 : 7. Answer: 12 : 8 : 7. No algebra needed. Standard equation method: 5 steps. LCM bridge: 2 steps.
For "ratio to find one group from total" questions: answer = Total × (that group's ratio number) ÷ (sum of all ratio numbers). Teachers in 1280, ratio 15 : 1 → Teachers = 1280 × 1 ÷ 16 = 80. This is a one-line calculation. Avoids setting up let-statements entirely.
For "what to subtract from p : q to get r : s" — always set up (p - x)/(q - x) = r/s and cross-multiply immediately. Do not try to guess or use options first; elimination is slower here.
(15-x)/(19-x) = 3/4 → 60 - 4x = 57 - 3x → x = 3. Time: 20 seconds. Checking all 4 options: 40-60 seconds.
When asked for a simplified ratio, quickly check if options share an obvious common factor with each other. The correct answer is always fully reduced. If you see options like 9:50, 5:30, 5:20, 2:70 — check which one cannot be reduced further. 9:50 — HCF is 1, fully reduced. 5:30 — HCF is 5, reduces to 1:6. Eliminate non-reduced options immediately. This cuts checking time by half.
When you see a ratio-proportion question in the exam hall, run this decision tree:
Step 1 — What is given?
Step 2 — Simplify at the start, not the end. Reduce all ratios to lowest terms before calculating. This keeps numbers small and prevents arithmetic errors.
Step 3 — Check your answer is fully simplified.
All answer options will be in reduced form. If your answer is 35 : 45, reduce to 7 : 9 before looking at options.
Step 4 — Direct vs. Inverse proportion check. If both quantities move in the same direction → Direct. If opposite → Inverse. Multiply or divide accordingly.
Target: standard ratio questions in under 30 seconds, combined ratio in under 45 seconds, income-expenditure type in under 90 seconds.
Why this question: Tests the most basic application — ratio given, one value known, find the other. This is the floor-level question every aspirant must get right.
Solving path: Red : Blue = 3 : 5. Red = 24. One part = 24 ÷ 3 = 8. Blue = 5 × 8 = 40. Check option C.
Why this question: Direct proportion disguised as a word problem. The trigger phrase is "at the same rate" — that locks in direct proportion.
Solving path: Daily earning = 9000 ÷ 25 = 360. Days = 10800 ÷ 360 = 30. Option D.
Why this question: Combined ratio — the most frequently tested ratio variant. Makes b common using LCM.
Solving path: a : b = 3 : 2 = 12 : 8. b : c = 8 : 7. b is now 8 in both. So a : b : c = 12 : 8 : 7. Option B.
Why this question: Ratio from absolute values requires simplification. Tests whether you can reduce 18000 : 100000 correctly under pressure.
Solving path: 18000 : 100000. Divide both by 2000: 9 : 50. Option A. Do not divide by 1000 first and stop at 18 : 100 — check if it reduces further.
Why this question: Chain ratio with three terms and two steps of LCM scaling. Tests whether you know to find LCM of the middle term.
Solving path: P : Q = 5 : 6. Q : R = 14 : 15. LCM(6, 14) = 42. Scale first by 7: 35 : 42. Scale second by 3: 42 : 45. So P : R = 35 : 45 = 7 : 9. Option A.
Why this question: Total split by ratio. One of the most templated question types — learn the formula once and apply forever.
Solving path: Students : Teachers = 15 : 1. Total parts = 16. Teachers = 1280 × (1/16) = 80. Option D.
Why this question: "Subtract from ratio" questions appear regularly. Tests cross-multiplication under time pressure.
Solving path: (15 - x)/(19 - x) = 3/4. Cross-multiply: 60 - 4x = 57 - 3x. So x = 3. Option D.
Not simplifying the initial ratio. Working with 18 : 100 instead of 9 : 50 makes cross-multiplication messy and leads to calculation errors. Always simplify before computing.
Adding ratio parts instead of multiplying in the "split" method. The answer is Total × (part / total-parts), not Total × part. For teachers in 1280 with ratio 15:1, total parts = 16, not 15 and 1 computed separately.
Confusing direct and inverse proportion. If more days → more money, that is direct. If more workers → fewer days, that is inverse. The phrase "at the same rate" always signals direct proportion. Do not multiply when you should divide.
Stopping at 18 : 100 instead of 9 : 50. When reducing a ratio, check if the result can be reduced again. Divide by HCF, not just by any common factor.
Wrong LCM when combining ratios. In P : Q = 5 : 6 and Q : R = 14 : 15, some candidates find LCM(5, 14) = 70 (the P terms) instead of LCM(6, 14) = 42 (the Q terms). Always find LCM of the common middle term.
Cross-multiplying incorrectly. In (15-x)/(19-x) = 3/4, the cross-multiplication is 4 × (15-x) = 3 × (19-x). A common error is 4 × 15 - x = 3 × 19 - x, forgetting to distribute the 4 and 3 over the bracket. Always expand fully.