Percentage Aptitude Questions: Shortcuts and 15 Solved Problems for TNPSC, Railway and Bank Exams

Every percentage calculation is one of three basic forms:

• What per cent is A of B? (A ÷ B) × 100. Example: 18 minutes out of 2 hours is 18 ÷ 120 × 100 = 15%. Put both in the same unit first.

• What is x% of a number? The number × x ÷ 100.

• You know x% of a number; what is the number? The known value × 100 ÷ x. Example: if 12% of a number is 54, the number is 54 × 100 ÷ 12 = 450.

The six shortcuts below are faster ways of doing these same three steps.

Per cent means "out of 100", so every percentage is a fraction. Examiners choose numbers that turn into neat fractions, and once you know this table, 37.5% of a number is simply three-eighths of it.

Learn it in both directions. Two examples:

• 62.5% of 320 = 5/8 × 320 = 200.

• 16 2/3% of 540 = 1/6 × 540 = 90.

For other numbers, build up from 10%. 10% of 7,450 is 745, 5% is half of that (372.5), and 15% is the two added together (1,117.5).

x% of y = (x × y) ÷ 100 = y% of x. Swap the numbers whenever one way round is easier.

Example: 18% of 50 is not obvious, but 50% of 18 is 9, so 18% of 50 is also 9.

Percentage change = (change ÷ original value) × 100. The original value goes at the bottom, never the new one. Getting that right eliminates most trap options.

Example: a price rises from ₹250 to ₹300. The change is ₹50, so the rise is 50 ÷ 250 × 100 = 20%. If it falls back from ₹300 to ₹250, the fall is 50 ÷ 300 × 100 = 16 2/3%. The same ₹50, but a different base.

For "what per cent more or less" questions:

• If A is r% more than B, then B is less than A by r ÷ (100 + r) × 100 per cent.

• If A is r% less than B, then B is more than A by r ÷ (100 − r) × 100 per cent.

Example: if A's salary is 25% more than B's, B's salary is 25 ÷ 125 × 100 = 20% less than A's. The same rule answers the price and consumption question: if the price of sugar rises by 25%, a family must cut its consumption by 20% to keep its spending the same.

When a value changes by a% and then by b%, the net change is:

a + b + (a × b) ÷ 100 per cent

Use a minus sign for a decrease. A positive answer is a net increase, a negative one a net decrease.

• A rise of 10%, then 20%: 10 + 20 + (10 × 20) ÷ 100 = 32% increase.

• A rise of 20%, then a fall of 20%: 20 − 20 − (20 × 20) ÷ 100 = −4%, so a 4% decrease. A rise and an equal fall never cancel out.

The same formula gives the change in the area of a rectangle when its length and breadth change, and two years of growth at the same rate (shortcut 6).

If a value is V after an r% increase, the original was V × 100 ÷ (100 + r). After an r% decrease, it was V × 100 ÷ (100 − r).

Example: a phone costs ₹13,200 after a 12% price cut. The original price was 13,200 × 100 ÷ 88 = ₹15,000.

Don't take 12% of ₹13,200 and add it back. That gives ₹14,784, a typical wrong option.

When something grows by r% every year on the previous year's figure, the growth compounds:

• Value after n years = P × (1 + r/100) × (1 + r/100) × …, with one bracket for each year.

• For a yearly fall in value (depreciation), use (1 − r/100) in each bracket instead.

Example: a town of 50,000 people grows by 10% a year. After 2 years it has 50,000 × 1.1 × 1.1 = 60,500 people. With shortcut 4, two years at 10% is 10 + 10 + 1 = 21%, and 21% of 50,000 is 10,500, which gives the same 60,500.

Example: a scooter worth ₹60,000 loses 10% of its value every year. After 2 years it is worth 60,000 × 0.9 × 0.9 = ₹48,600. By shortcut 4, that is −10 − 10 + 1 = −19%.

To go back in time ("what was the population 2 years ago?"), divide by the same factors instead of multiplying.

Most word problems become one line once you decide what 100% stands for.

Marks. The pass mark is 35%. A student scores 210 and fails by 35 marks. The pass mark is 210 + 35 = 245, which is 35% of the maximum, so the maximum is 245 × 100 ÷ 35 = 700.

Votes. In a two-candidate election, the winner gets 58% of the valid votes and wins by 2,400. The loser has 42%, so the margin is 16% of the votes. If 16% is 2,400, then 100% is 15,000 valid votes.

Profit and loss questions are percentage questions with a fixed base: profit or loss per cent is worked out on the cost price unless the question says otherwise.

Example: bought for ₹400 and sold for ₹500, the profit is ₹100, which is 100 ÷ 400 × 100 = 25%. Bought for ₹500 and sold for ₹400, the loss is the same ₹100 but only 20%, because the base is now ₹500. Marked price, discount and dishonest-dealer questions all build on this same rule.

Cover the answer under each question, allow yourself about 40 seconds for each one, and write down your choice before you look.

1. What is 37.5% of 480?

(A) 160 · (B) 175 · (C) 180 · (D) 192

Answer: (C) 180. From the table, 37.5% = 3/8, so 3/8 × 480 = 3 × 60 = 180.

2. What is the value of 28% of 25 + 25% of 28?

(A) 14 · (B) 7 · (C) 12.5 · (D) 28

Answer: (A) 14. By shortcut 2, 28% of 25 is the same as 25% of 28, which is a quarter of 28 = 7. The two terms are equal, so the total is 7 + 7 = 14.

3. If 15% of a number is 63, what is 40% of the same number?

(A) 152 · (B) 160 · (C) 172 · (D) 168

Answer: (D) 168. You don't need the number itself: 40% is 40/15 = 8/3 times 15%, so the answer is 63 × 8 ÷ 3 = 168. (The number is 420, and 40% of 420 is 168.)

4. The fare for a bus journey goes up from ₹40 to ₹50. By what percentage has it increased?

(A) 20% · (B) 25% · (C) 10% · (D) 30%

Answer: (B) 25%. The rise is ₹10 on an original fare of ₹40: 10 ÷ 40 × 100 = 25%. Option (A), 20%, comes from wrongly dividing by the new fare of ₹50.

5. Ravi's salary is 60% more than Kumar's. By what percentage is Kumar's salary less than Ravi's?

(A) 60% · (B) 40% · (C) 37.5% · (D) 62.5%

Answer: (C) 37.5%. By shortcut 3, the answer is 60 ÷ (100 + 60) × 100 = 60 ÷ 160 × 100 = 37.5%. Check: if Kumar earns 100, Ravi earns 160, and Kumar's is 60 less than 160, which is 37.5% of 160.

6. A student needs 40% of the maximum marks to pass. He scores 232 marks and fails by 28 marks. What are the maximum marks?

(A) 600 · (B) 580 · (C) 700 · (D) 650

Answer: (D) 650. The pass mark is 232 + 28 = 260, and 260 is 40% of the maximum, so the maximum is 260 × 100 ÷ 40 = 650.

7. After a 20% rise, the price of a gas cylinder is ₹1,080. What was the price before the rise?

(A) ₹900 · (B) ₹864 · (C) ₹880 · (D) ₹960

Answer: (A) ₹900. By shortcut 5, the old price is 1,080 × 100 ÷ 120 = ₹900. Option (B) is the trap: taking 20% off ₹1,080 gives ₹864, but the 20% was worked out on the old price, not the new one.

8. A shopkeeper raises the price of a shirt by 25% and then gives a 20% discount on the new price. Compared with the original price, the final price is:

(A) 5% more · (B) 5% less · (C) the same · (D) 1% less

Answer: (C) the same. By shortcut 4: 25 − 20 − (25 × 20) ÷ 100 = 25 − 20 − 5 = 0. Check with ₹100: it becomes ₹125, and 20% off ₹125 is ₹25, which brings it back to ₹100.

These take two or three steps. Write each step down; most mistakes in medium questions come from doing the second step in your head.

9. In an election between two candidates, 10% of the voters on the list did not vote. The winner got 55% of the votes cast and won by 1,800 votes. If every vote cast was valid, how many voters were on the list?

(A) 18,000 · (B) 20,000 · (C) 19,800 · (D) 22,000

Answer: (B) 20,000. The winner had 55% and the loser 45% of the votes cast, so the margin of 1,800 is 10% of the votes cast, and the votes cast were 18,000. Those 18,000 are 90% of the list, so the list had 18,000 × 100 ÷ 90 = 20,000 voters. Option (A) stops one step short.

10. The population of a town is 64,000. It grows by 5% every year. What will the population be after 2 years?

(A) 70,400 · (B) 71,040 · (C) 70,560 · (D) 70,000

Answer: (C) 70,560. After one year: 64,000 × 1.05 = 67,200. After two: 67,200 × 1.05 = 70,560. By shortcut 4, two years at 5% is 5 + 5 + 0.25 = 10.25%, and 10.25% of 64,000 is 6,560. Option (A) wrongly adds a flat 10%.

11. A machine bought for ₹1,25,000 loses 20% of its value every year. What will it be worth after 3 years?

(A) ₹50,000 · (B) ₹80,000 · (C) ₹62,500 · (D) ₹64,000

Answer: (D) ₹64,000. Each year the value is multiplied by 0.8: ₹1,25,000 becomes ₹1,00,000, then ₹80,000, then ₹64,000. Option (A) wrongly takes 60% off the original price in one go, and option (B) is the value after only 2 years.

12. Because the price of rice fell by 20%, a man can buy 5 kg more rice for ₹800. What is the reduced price of rice per kg?

(A) ₹32 · (B) ₹40 · (C) ₹30 · (D) ₹36

Answer: (A) ₹32. A 20% fall saves 20% of ₹800 = ₹160 on the old quantity, and that ₹160 buys the extra 5 kg at the new price, so the new price is 160 ÷ 5 = ₹32 per kg. Check: the old price was ₹40, so ₹800 bought 20 kg; at ₹32 it buys 25 kg, which is 5 kg more. Option (B) is the old price, not the reduced one.

13. The length of a rectangle is increased by 30% and its breadth is decreased by 20%. What is the change in its area?

(A) 10% increase · (B) 4% increase · (C) 6% decrease · (D) 4% decrease

Answer: (B) 4% increase. Area is length × breadth, so shortcut 4 applies: 30 − 20 − (30 × 20) ÷ 100 = 30 − 20 − 6 = 4, a 4% increase. Check: 1.3 × 0.8 = 1.04.

14. Meena spends 30% of her monthly salary on rent and 25% of what remains on food. She saves the rest, which is ₹15,750. What is her monthly salary?

(A) ₹28,000 · (B) ₹35,000 · (C) ₹32,500 · (D) ₹30,000

Answer: (D) ₹30,000. After rent, 70% of the salary is left. Food takes a quarter of that, so three-quarters of it is saved: 70% × 3/4 = 52.5% of the salary. If 52.5% is ₹15,750, the salary is 15,750 × 100 ÷ 52.5 = ₹30,000. Check: rent ₹9,000 leaves ₹21,000; food ₹5,250 leaves ₹15,750.

15. In an examination, 35% of the students failed in Mathematics, 25% failed in English and 10% failed in both. If 450 students passed in both subjects, how many students wrote the examination?

(A) 1,125 · (B) 900 · (C) 1,500 · (D) 750

Answer: (B) 900. Students who failed in at least one subject = 35 + 25 − 10 = 50%, because the 10% who failed in both were counted twice. So 50% passed in both, and if 50% is 450, the total is 900. Option (A) forgets to subtract the 10% (450 ÷ 40% = 1,125).

Count a guessed answer as wrong when you score yourself. A lucky guess is still a gap.

Percentage rewards a little daily practice more than long weekend sessions. A routine that works for most aspirants:

• Write the fraction table from memory every morning for a week, until you no longer need to.

• Do 10 mixed percentage questions a day against the clock, aiming for 40 seconds each.

• Once a week, take a full mock test, so percentage questions come mixed in with everything else, as they will in the exam hall.

On Thervu, the question banks for RRB NTPC, RRB Group D, TN Police SI and SBI Clerk and PO each have a separate Percentage topic. Choosing a single topic is a Pro feature; free accounts get daily mock tests across the full syllabus and Quick Practice on a few topics in each subject. Tests run in Tamil or English, are marked the moment you submit, and the questions you get wrong come back for revision. See plans for what the free plan includes.

Preparing for Group 4? Our 90-day Group 4 study plan puts simplification and percentage in week 1, next to polity. For the General Studies side of the paper, try our 30 GK questions with answers. For the railway exams, our RRB Group D cut-off post shows how close the RRB Chennai cut-offs run, which is why a few quick maths marks matter.