Percentage is one of the most frequently tested topics in UP Police Constable quant — and it shows up disguised in profit-loss, interest, and data interpretation too. So getting this right is not optional.
Here is the core idea: "percent" means "per hundred" (Latin: per centum). When you say 40%, you mean 40 out of every 100 parts. That is it. Everything else — percentage change, reverse percentage, comparing two quantities — is just this one idea applied in different directions.
Think of it this way: imagine you have a chapati divided into 100 equal pieces. If someone takes 40 pieces, they took 40% of the chapati. If they took 32 pieces, that is 32%. The total (100 pieces) is your base — always called the denominator in the formula.
The general formula:
Now, where students go wrong — they mix up which quantity is the "whole." In the question "what percentage of V is H?", H is the part and V is the whole. In "what percentage of H is V?", V is the part and H is the whole. These give different answers. Identifying the base correctly is 80% of the battle in percentage problems.
The three types of questions UP Police Constable papers repeat:
Everything in the PYQs you will see below falls into one of these three buckets.
From Percentage = (Part / Whole) × 100, you can derive:
Commit these three rearrangements to memory — not by rote, but by understanding that you are always solving for the missing variable in a three-variable relationship.
Look — this is where most candidates drop marks. If 70% of a number is 56, what is the number?
Using the formula: Whole = (Part × 100) / Percentage = (56 × 100) / 70 = 80.
The shortcut: divide by the percentage and multiply by 100. Or better — use the fraction form: 70% = 7/10, so (7/10) × number = 56 → number = 56 × (10/7) = 80. Fraction form is faster than decimal division for clean percentages.
The key rule: the denominator is always the original (old) value, never the new one. This is the single most common error in percentage change problems.
Example: Revenue goes from 35 to 80. Increase = 45. Percentage increase = (45/35) × 100 = 128.57%. Note — the base is 35, not 80.
Instead of calculating increase/decrease separately, use multipliers:
This is faster than writing out the fraction and especially useful when changes are applied successively.
Example: A number is increased by 25%. New value = Original × 1.25. If original = 80, new = 80 × 1.25 = 100.
When a number is first increased by a% and then decreased by b%, the net change is NOT simply (a - b)%. The formula for net change is:
When a = b (same percentage up then down):
The negative sign means it is always a decrease. A 20% increase followed by 20% decrease = -(20²/100) = -4%. This is a direct PYQ pattern.
This is a useful symmetry property: 8% of 25 = 25% of 8 = 2. This matters when one calculation is easier than the other. 25% is just dividing by 4 — that is much faster than calculating 8% of 25 the long way.
Memorise these — they save 20-30 seconds per question:
| Percentage | Fraction | |-----------|---------| | 10% | 1/10 | | 12.5% | 1/8 | | 16.67% | 1/6 | | 20% | 1/5 | | 25% | 1/4 | | 33.33% | 1/3 | | 37.5% | 3/8 | | 50% | 1/2 | | 62.5% | 5/8 | | 75% | 3/4 |
When you see 16.67% in a question, immediately think 1/6. When you see 312.5%, think 25/8.
When the question asks "V is 32% of H, so H is what % of V?", you are being asked to flip the relationship.
If V = (32/100) × H, then H = (100/32) × V = (25/8) × V = 312.5% of V.
The rule: if A is P% of B, then B is (100/P) × 100 % of A, which simplifies to (10000/P)%.
So for 32%: 10000/32 = 312.5%. One division step.
Standard method: set up equation, solve for H, divide — 4 steps, ~40 seconds. This shortcut: one division, ~10 seconds.
When a value goes up by r% then down by r% (or any two successive changes), skip the two-step calculation entirely.
Formula: Net % = a - b - (ab/100) where increase is positive, decrease is negative.
For "20% increase then 20% decrease": 20 - 20 - (20×20)/100 = 0 - 4 = -4%.
Standard method: take 100, increase to 120, decrease 20% of 120 = 24, so 120 - 24 = 96. Change = -4%. That is 3 steps, ~35 seconds. Formula: 1 step, ~8 seconds. Use whenever both percentage values are given together.
When asked for 16% of 25 or 32% of 25, swap the percentages.
16% of 25 = 25% of 16 = 16/4 = 4. Done. 32% of 25 = 25% of 32 = 32/4 = 8. Done.
25% of anything is just dividing by 4 — you can do that instantly. This works whenever one side of the swap gives you a "clean" percentage (25%, 50%, 20%, 10%, etc.).
Standard method: 16 × 25 / 100 = 400/100 = 4 (~20 seconds). Swap method: 25/4 = 4 (~5 seconds). Saves one multiplication step.
When two candidates get a% and b% of total votes, the extra votes for the winner = (b - a)% of total.
You do not need to calculate each candidate's votes separately and subtract.
Example: P gets 42%, Q gets 58%, total = 4500. Difference = 16% of 4500 = 720. That is one multiplication instead of two multiplications and a subtraction.
Standard method: Q's votes = 58% of 4500 = 2610; P's votes = 42% of 4500 = 1890; difference = 720. Three steps, ~40 seconds. Shortcut: one step, ~12 seconds.
In table-based questions with a given total income or total marks, calculate only the specific percentage asked — do not calculate all rows unless asked.
Example: Total income = 57,000. House rent = 20%. Remaining = 80% of 57,000 = 45,600.
Instead of: computing 20% of 57,000 = 11,400 and subtracting from 57,000. Directly compute 80% = 0.8 × 57,000. One step vs two steps. When the table has 5-6 rows, this focus saves you from reading and computing unnecessary data — saves 30-45 seconds per DI set.
Read the question and classify it in the first 5 seconds:
Type 1 — Find the number (reverse percentage): "X% of a number is Y, find the number." Formula: Number = Y × 100 / X. Or use fraction: if X% = m/n, then number = Y × (n/m).
Type 2 — Find the percentage: "What % of A is B?" Formula: (B/A) × 100. Identify A as the base — A is the "of" quantity.
Type 3 — Percentage change: "Change from old to new." Formula: (New - Old) / Old × 100. Base is always OLD value.
Type 4 — Successive changes: Two percentage changes one after another. Use net change formula: a - b - ab/100. Saves you from two-step calculation.
Type 5 — Table/DI percentage: Only compute the row or column asked. Use multiplier directly on total.
Decision rule: if the question has two successive percentage changes, always reach for the formula. If it has a table, ignore rows not asked. If it says "what % of X is Y", the base is X, not Y.
Why this question: Tests the two-step chain — first find the original number using reverse percentage, then apply a new percentage change using the multiplier method.
Solving path: 70% of number = 56. So number = 56 × 100/70 = 80. Now increase by 25%: 80 × 1.25 = 100. Both steps use the core formulas — this is a Type 1 + multiplier chain.
Why this question: Classic reverse percentage with relationship flip. The most common trap: candidates calculate V% of H instead of H% of V.
Solving path: V = 32% of H means H = (100/32) × V = 3.125V. So H is 312.5% of V. Use the shortcut: 10000/32 = 312.5. The base in the final answer is V, not H.
Why this question: Standard table/DI application. Tests whether you can extract a single percentage from a multi-row table without getting distracted by other rows.
Solving path: House rent = 20% of 57,000 = 11,400. Remaining = 57,000 - 11,400 = 45,600. Or directly: (100 - 20)% = 80% of 57,000 = 45,600. Two-second identification of the relevant row, one multiplication.
Why this question: Percentage change from a data table — tests whether you correctly identify the old value (2020 = 35) as the base, not the new value (2022 = 80).
Solving path: Increase = 80 - 35 = 45. Percentage increase = (45/35) × 100. Now 45/35 = 9/7. 9/7 × 100 = 900/7 = 128.57%. Key: base is 35 (the older year), not 80.
Why this question: Tests basic percentage calculation with addition — a high-frequency format in school-mark-based questions.
Solving path: Add all marks: 72 + 88 + 65 + 79 + 81 = 385. Percentage = (385/500) × 100 = 77%. The addition step is where errors happen — add carefully, left to right: 72+88=160, 160+65=225, 225+79=304, 304+81=385.
Why this question: Election-vote type — tests the difference shortcut. Many candidates compute both candidates' votes separately and subtract. There is a faster path.
Solving path: Difference in % = 58 - 42 = 16%. Extra votes = 16% of 4500 = 720. One multiplication, not three operations.
Why this question: Successive percentage change — the trap is thinking that +20% then -20% cancels out to 0. It does not, and this is a direct formula application question.
Solving path: Net change = -(r²/100) = -(400/100) = -4%. It is a 4% decrease. Using numbers to verify: start with 100, increase by 20% → 120, decrease 20% of 120 = 24 → 96. Change = -4%. The formula gives the answer in one step.
Why this question: Percentage of a subset — here you count how many items fail a condition and express that count as a percentage of the total count.
Solving path: Check each subject against 70: Physics 78 ✓, Chemistry 82 ✓, Mathematics 90 ✓, Biology 74 ✓, English 68 ✗, Computer 86 ✓. Failed = 1. Percentage = (1/6) × 100 = 16.67%. Use the fraction-to-percentage table: 1/6 = 16.67%.
Flipping the base in percentage change. "From 35 to 80" means base = 35, not 80. The formula is (change/old) × 100. If you use the new value as base, you get a different (wrong) answer. Always ask: "what was it before?"
Assuming +r% then -r% = 0 net change. It is always a net decrease of r²/100 percent. When r = 20, that is a 4% net decrease. Many candidates write "0% change" and lose the mark.
Confusing "A is what % of B" with "B is what % of A". These are reciprocal questions. V is 32% of H is not the same as H is 32% of V. Read the sentence direction carefully — the "of" quantity is always the base (denominator).
Not converting 16.67% or 12.5% to fractions quickly. Candidates who try to compute (1/6) × 100 using long division in the exam hall lose 30-40 seconds. 1/6 = 16.67% is a standard fraction-percent pair that must be memorised.
Adding up all table rows when only one is asked. In DI/table percentage questions, the question asks for one specific row. Computing all rows wastes time and introduces addition errors. Anchor only on the asked row.
Percentage vs percentage points. If something rises from 42% to 58%, the rise is 16 percentage points, not 38% (which is 16/42 × 100). These are different — though UP Police papers mostly deal with absolute percentage point differences, not "percentage change in percentage."