Solved Questions

How to Use These Questions

The 36 questions below come in mixed order. An election question sits next to a population question and a ratio, so before you pick a method, decide what the 100% is: the old value, the valid votes, the income or the whole mixture.

Work each one on paper with a timer running, then read the answer. Most should take 36 to 45 seconds. When one takes you over a minute, find the step that slowed you down, usually a percentage you didn't turn into a fraction or a base you worked out twice, and try the question again the next day.

About a fifth of the questions are basics, conversions and ratios. A quarter are on percentage change: successive changes, population and error questions. Another quarter use the product rule, including income, expenditure and savings. The rest are applications: elections, exams, spending chains and mixtures. Recent papers have a similar mix, and income and savings questions come up often in them.

A question marked [Tier 2] needs one step more than a usual Tier 1 question. Try those too: the setup is the same, only longer.

Solved Examples

Question 1

In an election between two candidates, 10% of the voters on the list did not vote and 6% of the votes cast were invalid. The winner got 55% of the valid votes and won by 1,692 votes. How many voters were on the list?

Answer. The loser got 45%, so the winner was ahead by 10% of the valid votes. 10% of the valid votes is 1,692, so there were 16,920 valid votes. These are 94% of the votes cast, so 16,920 ÷ 0.94 = 18,000 votes were cast. And 18,000 is 90% of the list, so the list had 18,000 ÷ 0.9 = 20,000 voters.

Question 2

36 paise is what percent of ₹4.80?

Answer. First I put both in paise: ₹4.80 = 480 paise. Then 36 ÷ 480 = 0.075, which is 7.5%. If you forget to change the units, you divide 36 by 4.8, get 7.5 and end up with 750%.

Question 3

The price of a book was raised by 18%, and it now costs ₹413. What was the old price?

Answer. The new price is 118% of the old one. So the old price = 413 ÷ 1.18 = ₹350. Check: 18% of 350 is 63, and 350 + 63 = 413.

Question 4

The price of tea goes up by 20%, and a family's spending on tea goes up by only 8%. By what percent did they cut the amount of tea they use?

Answer. Spending = price × amount. So the amount now = 1.08 ÷ 1.2 = 0.9 of what it was. They cut their tea by 10%. The quick wrong answer is 20 − 8 = 12%.

Question 5

Kunal spends 20% of his salary on rent and 15% of his salary on his children's school fees. He spends 40% of what is left on food and 25% of the rest on travel. He saves ₹5,850. What is his salary?

Answer. Rent and fees are both "of the salary", so I add them first: 35% gone, 65% left. Food then leaves 0.6 of that, and travel leaves 0.75 of what remains. So his savings are 0.65 × 0.6 × 0.75 = 0.2925 of the salary. Salary = 5,850 ÷ 0.2925 = ₹20,000. Check: ₹13,000 is left after rent and fees, ₹7,800 after food and ₹5,850 after travel.

Question 6

A student was supposed to multiply a number by $\frac{4}{7}$, but he multiplied it by $\frac{7}{4}$. His answer came out 99 more than the right answer. What was the number?

Answer. The gap between the two multipliers is $\frac{7}{4} - \frac{4}{7} = \frac{49 - 16}{28} = \frac{33}{28}$. So $\frac{33}{28}$ of the number is 99. That means $\frac{1}{28}$ of it is 3, and the number is 84. Check: $\frac{7}{4}$ × 84 = 147 and $\frac{4}{7}$ × 84 = 48, and 147 − 48 = 99.

Question 7

35% of a number is 153 more than 18% of the same number. Find the number.

Answer. The gap between 35% and 18% is 17% of the number, and that is 153. So 1% is 9, and the number is 900. Check: 315 − 162 = 153.

Question 8

A man saves 40% of his income. His income goes up by 15% and his savings go up by 30%. By what percent does his expenditure go up?

Answer. I take his income as 100, so he spends 60 and saves 40. New income 115, new savings 40 × 1.3 = 52. So the new expenditure is 115 − 52 = 63. It went from 60 to 63, which is up 5%.

Question 9

A student needs 36% marks to pass. He got 123 marks and failed by 39 marks. What are the maximum marks?

Answer. The pass mark is 123 + 39 = 162. That is 36% of the maximum, so the maximum = 162 ÷ 0.36 = 450. The trap is to do 123 ÷ 0.36, which doesn't even come out whole.

Question 10

A man's salary is raised by 10%, then raised by 10% again, and then cut by 20%. What is the net change?

Answer. Multipliers: 1.1 × 1.1 × 0.8 = 1.21 × 0.8 = 0.968. His salary is now 96.8% of the start, so it fell by 3.2%. Adding the rates (10 + 10 − 20) would say there is no change, but the cut is taken on a bigger salary.

Question 11

The length of a rectangle is increased by 30%. By what percent must the breadth be cut so that the area stays the same?

Answer. 30% is $\frac{3}{10}$. If one side goes up by $\frac{3}{10}$, the other must go down by $\frac{3}{3 + 10} = \frac{3}{13}$, which is $\mathbf{23\frac{1}{13}\%}$. Check: $\frac{13}{10} \times \frac{10}{13} = 1$. Cutting the breadth by 30% would make the area 1.3 × 0.7 = 0.91 of the old one, 9% less.

Question 12

In an election, A got 42% of the valid votes, B got 35% and C got the rest. A beat B by 2,394 votes, and 5% of the votes cast were invalid. How many votes did C get, and how many votes were cast?

Answer. A beat B by 42 − 35 = 7% of the valid votes. So 7% is 2,394, 1% is 342, and there were 34,200 valid votes. C got 100 − 42 − 35 = 23%, which is 23 × 342 = 7,866 votes. Valid votes are 95% of the votes cast, so 36,000 votes were cast (34,200 ÷ 0.95).

Question 13

If 40% of (A + B) = 70% of (A − B), find A : B.

Answer. Cross-swap: (A + B) : (A − B) = 70 : 40 = 7 : 4. Then componendo and dividendo: A : B = (7 + 4) : (7 − 4) = 11 : 3. Check with A = 11 and B = 3: 40% of 14 = 5.6 and 70% of 8 = 5.6.

Question 14

A town has 80,000 people. Every year, births add 6% and deaths take away 1% of the population. What will the population be after 2 years?

Answer. The net growth is 6 − 1 = 5% a year. Two years at 5% means × $1.05^2$, which is × 1.1025. So 80,000 × 1.1025 = 88,200.

Question 15

The price of grapes goes up by 25%, so ₹1,200 now buys 3 kg fewer. What is the new price per kg, and what was the old price?

Answer. Shortcut: 25% of the money ÷ the change in kg gives the new price. 25% of 1,200 = 300, and 300 ÷ 3 = ₹100 a kg now. The old price was 100 ÷ 1.25 = ₹80 a kg. Check: ₹1,200 bought 15 kg before and buys 12 kg now.

Question 16

In an exam, 35% of the students failed in Hindi, 28% failed in English and 15% failed in both. If 676 students passed in both subjects, how many students took the exam?

Answer. Failed in at least one subject = 35 + 28 − 15 = 48%. I take away the 15 because those students were counted in both the 35 and the 28. So 52% passed both, and 52% is 676. That makes 1% = 13, so 1,300 students took the exam.

Question 17

A shop's sales fell by 36%. By what percent must they now rise to get back to where they were?

Answer. 36% is $\frac{9}{25}$. To undo a fall of $\frac{9}{25}$, the rise is $\frac{9}{25 - 9} = \frac{9}{16}$, which is 56.25%. Check with sales of 25: they fall to 16, and 16 must grow by 9 to get back, which is $\frac{9}{16}$ of 16. It is not 36% again, because the rise starts from the lower figure.

Question 18 [Tier 2]

The radius of a cylinder is increased by 25% and its height is cut by 20%. Find the percent change in its volume and in its curved surface area.

Answer. Volume = $\pi r^2 h$, so the radius counts twice. The multiplier is $1.25^2 \times 0.8 = 1.5625 \times 0.8 = 1.25$, so the volume goes up 25%. Curved surface area = $2\pi rh$, so the radius counts only once: 1.25 × 0.8 = 1. The curved surface area does not change. In fractions it is even quicker: $\frac{5}{4}$ and $\frac{4}{5}$ cancel.

Question 19

The interest rate on a deposit falls from 8% to 6.5%. By how many percentage points did it fall, and by what percent?

Answer. Percentage points are a plain subtraction: 8 − 6.5 = 1.5 percentage points. As a percent of the old rate it is 1.5 ÷ 8 = 0.1875, so 18.75%. These are two different things, and SSC likes to put both in the options.

Question 20

50 litres of milk and water has 24% water. How much milk must be added to make the water 20% of the mixture?

Answer. This time it's the water that stays fixed: 24% of 50 = 12 litres. Those 12 litres must become 20% of the new mixture, so the new total = 12 ÷ 0.2 = 60 litres. So 10 litres of milk must be added.

Question 21 [Tier 2]

A machine loses 10% of its value every year. After 3 years it is worth ₹1,45,800. What did it cost new, and how much value did it lose in the second year?

Answer. Three years at a 10% loss means × $0.9^3$, which is × 0.729. So it cost 1,45,800 ÷ 0.729 = ₹2,00,000. After one year it was worth ₹1,80,000 and after two years ₹1,62,000. So in the second year it lost ₹18,000. The third year's loss was only ₹16,200, because each year's 10% is taken on a smaller value.

Question 22

A woman's income goes up by 16% and her expenditure goes up by 20%, but her savings in rupees stay the same. Her income was ₹45,000. How much does she save?

Answer. Her savings didn't change, so all the extra income went into extra spending. Extra income = 16% of 45,000 = ₹7,200. That is 20% of her old expenditure, so she used to spend 7,200 ÷ 0.2 = ₹36,000. Her savings = 45,000 − 36,000 = ₹9,000.

Question 23 [Tier 2]

The pass mark in a test is 40%. Asha scored 10% less than the pass mark and Bina scored 10% more than the pass mark. Bina got 40 marks more than Asha. Find the maximum marks.

Answer. Careful here: "10% less than the pass mark" means 10% of the pass mark, not 10 percentage points. So Asha got 0.9 × 40% = 36% of the maximum, and Bina got 1.1 × 40% = 44%. The gap of 8% is 40 marks, so the maximum is 500. Check: pass mark 200, Asha 180, Bina 220.

Question 24

20% of 35% of 60% of a number is 84. Find the number.

Answer. "Of" means multiply, and the order doesn't matter. 0.2 × 0.35 × 0.6 = 0.042. So 0.042 of the number is 84, and the number is 84 ÷ 0.042 = 2,000.

Question 25

A city's population grew by 8% in 2023 and fell by 5% in 2024. At the end of 2024 it was 1,02,600. What was it at the start of 2023?

Answer. The multiplier for the two years is 1.08 × 0.95 = 1.026. So the start was 1,02,600 ÷ 1.026 = 1,00,000.

Question 26

X's income is 25% more than Y's. X spends 85% of his income and Y spends 75% of hers. Y saves ₹1,500 more than X. Find both incomes.

Answer. 25% more means X : Y = 5 : 4, so I take Y's income as 4k and X's as 5k. X saves 15% of 5k = 0.75k. Y saves 25% of 4k = k. Y saves 0.25k more, and that is ₹1,500, so k = 6,000. Y earns ₹24,000 and X earns ₹30,000. Check: X saves ₹4,500 and Y saves ₹6,000.

Question 27

A man gave 35% of his money to his wife. He then gave 25% of the remaining money to each of his two sons. He donated half of what was left and kept the rest, which was ₹26,000. How much money did he have at first?

Answer. After the wife's share, 65% is left. The two sons together take 50% of that, which halves it, and the donation halves it again. So he kept 0.65 × 0.5 × 0.5 = 0.1625 of his money, and 26,000 ÷ 0.1625 = ₹1,60,000. Check: wife ₹56,000, each son ₹26,000, donation ₹26,000, and he kept ₹26,000.

Question 28

Two numbers are 30% and 44% less than a third number. What percent of the first number is the second?

Answer. I take the third number as 100. Then the first is 70 and the second is 56. So the second is $\frac{56}{70}$ of the first, which is 80%. Don't subtract 44 − 30 = 14 and say 86%: the base here is the first number, 70, not 100.

Question 29

A number is increased by 40% and the result is then decreased by 40%. The final number is 1,512. What was the number at the start?

Answer. The same rate up and down is always a loss of $\frac{40^2}{100}$ = 16%. So 1,512 is 84% of the start, and the start = 1,512 ÷ 0.84 = 1,800. Check: 1,800 becomes 2,520, which becomes 1,512.

Question 30

The numerator of a fraction is increased by 20% and its denominator is decreased by 10%. The new fraction is $\frac{16}{21}$. Find the original fraction.

Answer. The fraction gets multiplied by $\frac{1.2}{0.9} = \frac{4}{3}$. So the original × $\frac{4}{3}$ = $\frac{16}{21}$, and the original is $\frac{16}{21} \times \frac{3}{4} = \mathbf{\frac{4}{7}}$. Check: 4 × 1.2 = 4.8 and 7 × 0.9 = 6.3, and $\frac{4.8}{6.3} = \frac{16}{21}$.

Question 31 [Tier 2]

15 litres of a 30% acid solution are mixed with 25 litres of an 18% acid solution. What is the strength of the mixture? If 10 litres of water are then added, what is the new strength?

Answer. Acid in the first: 30% of 15 = 4.5 litres. Acid in the second: 18% of 25 = 4.5 litres. Together that's 9 litres of acid in 40 litres, which is 22.5%. Adding 10 litres of water keeps the 9 litres of acid but makes the total 50, so the strength becomes $\frac{9}{50}$ = 18%. The plain average of 30 and 18 is 24, which is wrong because the volumes are not equal.

Question 32 [Tier 2]

A price is increased by 30% and then decreased by x%. The net result is a 4% increase. Find x.

Answer. With multipliers: $1.3 \times \left(1 - \frac{x}{100}\right) = 1.04$. So $1 - \frac{x}{100}$ = 1.04 ÷ 1.3 = 0.8, and x = 20. The formula agrees: 30 − x − 0.3x = 4, so 1.3x = 26 and x = 20.

Question 33

A town has 25,000 people, and 60% of them are men. 70% of the men and 45% of the women can read and write. What percent of the whole town can read and write?

Answer. Men: 60% of 25,000 = 15,000, and 70% of them is 10,500. Women: 10,000, and 45% of them is 4,500. Together 15,000 people can read and write, and 15,000 ÷ 25,000 = 0.6, so 60%. Quicker: 0.6 × 70 + 0.4 × 45 = 42 + 18 = 60.

Question 34

A car increases its usual speed by 25% and reaches 12 minutes early. How long does the journey usually take?

Answer. Speed × time is fixed for the same distance. Speed up by $\frac{1}{4}$ means time down by $\frac{1}{5}$. So $\frac{1}{5}$ of the usual time is 12 minutes, and the usual time is 60 minutes. If you take 12 minutes as 25% of the time, you get 48 minutes, which is wrong.

Question 35 [Tier 2]

If (x − y) is 15% of (x + y), what percent of x is y?

Answer. (x − y) : (x + y) = 15 : 100 = 3 : 20. Componendo and dividendo: x : y = (20 + 3) : (20 − 3) = 23 : 17. So y is $\frac{17}{23}$ of x, which is $\mathbf{73\frac{21}{23}\%}$. Check with x = 23 and y = 17: x − y = 6 and x + y = 40, and 6 is 15% of 40.

Question 36 [Tier 2]

Fresh mangoes are 85% water, and dried mango slices are 25% water. How many kg of dried slices can be made from 60 kg of fresh mangoes?

Answer. Only water dries off, so the pulp stays the same. Pulp in the fresh mangoes = 15% of 60 = 9 kg. In the dried slices this 9 kg is 75% of the weight, so the slices weigh 9 ÷ 0.75 = 12 kg. Dividing by 0.25 instead would give 36 kg, which mixes up the water share with the pulp share.