Start With the Meaning
A percentage is a quantity expressed out of one hundred. If 18 out of 60 questions are correct, the percentage is 18/60 × 100 = 30%. Always identify the base before calculating: 20% of 50 is different from 20% of 80.
To convert a percentage to a fraction, divide by 100 and simplify. To convert a fraction to a percentage, multiply by 100. These simple conversions make mental estimates faster during timed practice.
Increase and Decrease
If a quantity increases by r percent, multiply it by 1 + r/100. A 15% increase changes 200 to 200 × 1.15 = 230. If it decreases by r percent, multiply by 1 − r/100. A 15% decrease changes 200 to 170.
A common error is to assume that a 20% increase followed by a 20% decrease returns to the original value. Starting at 100, the increase gives 120 and the decrease gives 96. The combined change is a 4% decrease.
For successive changes a and b, the net percentage change is a + b + ab/100 when both are expressed with their signs. Use a negative value for a decrease, then estimate the result with an actual number to catch sign errors.
Compare Parts With the Correct Base
If A is 25% more than B, then A = 1.25B and B is not 25% less than A. The decrease from A back to B is 25/125 × 100 = 20%. This difference matters in ratio and data-interpretation questions.
When a question says a score rises from 40 to 50, the increase is 10 points but the percentage increase is 10/40 × 100 = 25%. Percentage change uses the original value as the denominator unless the question specifies another base.
Reverse Percentage Problems
A reverse percentage question gives you the result after a change and asks for the original value — the direction most students get backwards under time pressure. If a price after a 20% discount is ₹400, the temptation is to add back 20% of 400, giving ₹480. That is wrong, because the 20% discount was taken off the original price, not off 400. The correct approach: 400 represents 80% of the original, so the original is 400 ÷ 0.8 = ₹500. Check it: 20% of 500 is 100, and 500 − 100 = 400, which matches.
The same logic applies to increases. If a quantity after a 25% increase is 150, that 150 represents 125% of the original, so the original is 150 ÷ 1.25 = 120. The general rule: to reverse a percentage change of r%, divide by (1 ± r/100) — do not apply the percentage to the final value as if it were the original.
Mental-Math Shortcuts
Common percentages map to simple fractions, and converting is usually faster than long multiplication: 50% is half, 25% is a quarter, 10% is one-tenth, found by moving the decimal point, 5% is half of 10%, and 1% is one-hundredth. Most percentages you will actually need can be built from these: 15% of a number is 10% plus half of that 10%, and 35% is 25% plus 10%. Practising this kind of decomposition is faster to learn than memorising a conversion table, because it works for any starting percentage.
For percentages of round numbers, it is often faster to work the calculation in the easier direction: finding 40% of 25 is awkward directly, but 25% of 40 gives the identical result, 10, and is easier to see immediately. This works because a% of b always equals b% of a — useful to recognise in a timed test rather than derive from scratch each time.
Common Exam Traps
Beyond the successive-change trap and the reverse-percentage direction error already covered, two more patterns cause a disproportionate number of avoidable mistakes. The first is applying a percentage to the wrong base in a multi-step question — recompute the base at each step rather than assuming it stays the same as the first line of the question. The second is rounding too early in a multi-step calculation, which compounds a small early error into a wrong final answer; keep intermediate values exact, as fractions or unrounded decimals, and only round the final result.
Worked Practice
If a learner answers 36 of 45 questions correctly, the accuracy is 36/45 × 100 = 80%. If the same learner improves accuracy by 10 percentage points, the new accuracy is 90%; if accuracy improves by 10 percent, it is 88%. Percentage points and percent change are not interchangeable.
For every problem, write the base, estimate the direction of the answer, calculate, and check whether the result is plausible. This four-step habit prevents many avoidable errors.
One habit worth keeping: practice questions should be checked against the stated assumptions. In data interpretation, read the table title, units, and time period before applying a percentage formula.
The ThinkZene Quant Desk writes worked examples for aptitude learners and emphasises estimation, clear units, and checking each result.