Percentage is one of those topics where students lose marks not because the concept is hard — it is not — but because they slow down on the "reverse" questions (what percent less, what percent more) and misread which number is the base.
Here is the core idea: percent means per hundred. When you say 40%, you mean 40 out of every 100. That is it. Everything else — percentage increase, percentage decrease, comparison between two quantities — is a direct application of this one idea with the base changing depending on context.
Think of it like this: imagine two friends, Ravi and Ram, standing on different floors of a building. Their incomes are the heights of their floors. When you ask "how much taller is Ravi's floor than Ram's floor?", you are comparing the gap to Ram's floor (Ram is the base). When you ask "how much shorter is Ram's floor than Ravi's floor?", the same gap is now compared to Ravi's floor (Ravi is the base). Same gap, different bases — different percentage answers. This is the single most tested trap in Bihar Police Constable percentage problems.
The core formula:
Percentage change:
X is what percent of Y:
If you internalize these three and learn which number goes in the denominator in each situation, you have already handled 80% of the percentage questions you will see in this exam.
Look at this question type: "A is 20% more than B. By what percent is B less than A?"
Students instinctively write 20%. That is wrong. The base has shifted.
% less = (A - B)/A × 100 = 0.2B / 1.2B × 100 = 100/6 = 16.67%The formula for this conversion is critical enough to memorize:
If A is r% more than B, then B is less than A by:
If A is r% less than B, then B is more than A by:
Knowing these cold — not calculating them — is how you save 20-30 seconds per question:
| Fraction | Percentage | |----------|------------| | 1/2 | 50% | | 1/3 | 33.33% | | 1/4 | 25% | | 1/5 | 20% | | 1/6 | 16.67% | | 1/7 | 14.28% | | 1/8 | 12.5% | | 1/9 | 11.11% | | 1/10 | 10% | | 1/12 | 8.33% | | 2/3 | 66.67% | | 3/4 | 75% | | 3/5 | 60% |
For almost every percentage comparison problem, your first move should be: assume the reference quantity = 100. Do not work with variables unless forced.
Example: "80% of A = 50% of B. Find B as % of A."
Let A = 100. Then 80% of 100 = 80. So 50% of B = 80 → B = 160. Therefore B = 160% of A. Done. No algebra needed.
If a value changes by a% and then by b%, the net change is:
(Use negative values for decreases.)
Example: Price increases 20% then decreases 10%.
Net = 20 + (−10) + (20 × −10)/100 = 10 − 2 = 8% net increase.
This distinction rarely appears in Bihar Police Constable but is worth knowing for data interpretation questions. If a rate goes from 20% to 25%, it increased by 5 percentage points but by 25% relative to the original. The question wording tells you which one is asked.
"A number when increased by 20% gives 240. Find the original number."
Do not subtract 20% from 240 (that gives 192, which is wrong). Instead:
Or using fractions: 120% of x = 240 → x = 240 × (100/120) = 200.
When A is r% more than B, and you need 'B is what % less than A', use: r/(100+r) × 100. No setup needed. If A is 24% more than B: 24/124 × 100 = 600/31 ≈ 19.35%. Standard method (assign variables, compute difference, divide by new base): 6 steps. This formula: 2 steps, under 15 seconds.
Any time you see "x% of A = y% of B, find A as % of B" — immediately let A = 100. Compute the other side numerically. No equations, no cross-multiplication. Example: 80% of A = 50% of B, A = 100 → 80 = 50% of B → B = 160. Standard method (algebraic equation): 4 steps. Substitution: 2 steps, under 10 seconds.
Whenever you see 33.33%, 66.67%, 16.67%, 83.33% in options, immediately convert the answer to a fraction: 1/3, 2/3, 1/6, 5/6. Match to the fraction equivalent of your computed answer rather than struggling with the decimal. This eliminates rounding errors entirely. Standard method (decimal matching): prone to 0.01 errors. Fraction method: exact, 0 errors.
For back-to-back percentage changes a% then b%: net% = a + b + ab/100. Use negatives for decrease. Example: +30% then −20%: 30 + (−20) + (30 × −20)/100 = 10 − 6 = 4% net increase. Standard method (compute intermediate value, then second change): 5 steps. Formula: 3 numbers plugged in, 1 step, under 12 seconds.
"After a 15% increase, value is 345. Find original." Never subtract 15% from 345. Use: original = 345/1.15 = 345 × (20/23) = 300. The trick: convert 1.15 to a fraction (23/20), flip it to 20/23, multiply. Avoids the classic error of subtracting % from the increased value. Saves 1 step and eliminates the most common wrong answer.
When you see a percentage question in the exam hall, run through this decision tree in order:
Step 1 — Identify the base. Who is being compared to whom? The question says "percent of" — that word tells you the denominator.
Step 2 — Is it a "more than / less than" comparison? If yes, check whether the base has shifted. Use the flip formula r/(100±r) × 100 immediately.
Step 3 — Is there an unknown quantity? Assume it equals 100. Compute everything numerically.
Step 4 — Is it a successive change problem? Use a + b + ab/100. Do not compute step by step.
Step 5 — Does the answer have a recurring decimal? Convert to a fraction and match with options as fractions.
Step 6 — Stuck? Pick the two most different-looking options and plug them back into the original condition. Eliminate. Never leave a percentage question unattempted — the structure almost always allows elimination of at least two options.
Time target: 60-90 seconds per percentage question.
Why this question: This is the classic "A is more than B, B is less than A by how much?" trap. The exact base-flip pattern appears repeatedly.
Solving path: Let Ram = 100. Ravi = 124. Difference = 24. Ram is less than Ravi by (24/124) × 100 = 2400/124 = 600/31 = 19 and 11/31 percent. Look at the option — 600/31% is the answer. The trap: students write 24% directly. That is wrong because 24 compared to 124 (Ravi's income) is the correct denominator, not 100.
Why this question: This tests whether you can set up percentage-of-percentage relationships without getting tangled in variables. The assume-100 trick resolves this in two lines.
Solving path: Let A = 100. Then 80% of A = 80. Given 80% of A = 50% of B, so 50% of B = 80, meaning B = 160. So B = 160% of A, hence x = 160. The trap here is misreading the question — B = x% of A means x = 160, not that B itself is 160.
Why this question: This is a straightforward "what percent of total" question, but the numbers do not divide cleanly. Tests whether you can handle approximate answers and match to the nearest option.
Solving path: Percentage spent = (300/2080) × 100. Simplify: 300/2080 = 15/104. Now 15/104 × 100 = 1500/104 ≈ 14.42%. The nearest option is 15. The trap: students compute 300/2000 = 15% exactly and pick 15 without noticing the denominator is 2080, not 2000. Here the approximation still gives 15, but you should be aware of the rounding.
Why this question: This tests the "p is 6 times q — by what percent is q less than p?" structure, which is a direct application of the base-flip concept with a multiplier.
Solving path: p = 6q. So q is less than p by (p − q)/p × 100 = (6q − q)/(6q) × 100 = 5/6 × 100 = 83.33% = 83 and 1/3 percent. Note that the explanation in the answer key references 88 1/3 as per the answer key, so go with the official answer key for this one. The calculation method (5/6 × 100) is the standard approach regardless of which option is marked correct in the key.
Using the wrong base in "more than / less than" problems. If A is 30% more than B, B is NOT 30% less than A. The denominator changes. Always identify which quantity is the base before computing.
Subtracting a percentage directly from the increased value. If a price increases 20% to reach 600, the original is not 600 − 20% of 600 = 480. It is 600/1.2 = 500. The percentage was applied to the original, not the new value.
Misidentifying the "whole" in fraction problems. "300 is what percent of 2080?" — the whole is 2080, it goes in the denominator. Not 300. Students sometimes flip this when the part is mentioned first in the sentence.
Treating percentage points as percentages. If a quantity goes from 40% to 50%, it increased by 10 percentage points, but the percentage increase is (10/40) × 100 = 25%. In straightforward Bihar Police Constable questions this distinction is less common, but watch for it in data-heavy questions.
Rounding mid-calculation and getting wrong answers. Do not round 600/31 to 19.4 and then try to match it with options. Keep the fraction form (600/31) and see if any option says 600/31 directly. It usually does.
Forgetting the negative sign in successive decrease problems. If the price decreases 10% and then increases 5%, the net is: −10 + 5 + (−10 × 5)/100 = −5 − 0.5 = −5.5%, meaning a 5.5% net decrease. Students often compute −10 + 5 = −5% and stop, missing the cross-product term.