A percentage is a fraction with 100 on the bottom, and nothing more. The Punjab Police SI has asked 24 percentage questions since 2016, but the topic matters far more than that number suggests: profit and loss, interest, data interpretation and population growth are all percentage questions wearing different clothes. Two ideas carry the chapter. First, always ask what the percentage is *of — the base changes and the answer changes with it. Second, successive percentage changes multiply, they never add. You will learn (1) the fraction equivalents worth memorising, (2) the “A is x% more than B” reversal, (3) successive changes, and (4)* the consumption-and-price rule.
The fraction equivalents
Eight lines that make the arithmetic disappear.
A man spends 2/5 of his salary on house rent, 3/10 of his salary on food and 1/8 of his salary on conveyance. If he has Rs. 1,400 left with him, find his expenditure on conveyance. (SI 2016)
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Rent 2/5, food 3/10, conveyance 1/8. Their sum is 16/40 + 12/40 + 5/40 = 33/40, so what is left is 7/40 of the salary. 7/40 of salary = ₹1,400, so the salary is ₹8,000 — and the total spent on those three heads is 33/40 × 8,000 = ₹6,600. The paper's answer ₹1,000 is the conveyance (1/8 of 8,000). Always check which head the question wants.
“A is x% more than B” is not “B is x% less than A”
The base changes, so the percentage does too.
If A is x% more than B, then B is x/(100 + x) × 100 % less than A. Ramesh earns 20% more than Rajesh ⇒ Rajesh earns 20/120 = 16⅔% less than Ramesh. Going the other way: if A is x% less than B, then B is x/(100 − x) × 100 % more than A. The quickest route in an exam is to set the smaller quantity to 100 and compute.
Successive changes
Seven questions, one rule: multiply.
Net change = a + b + ab/100, with falls counted as negative. A 12% rise then a 12% fall is 12 − 12 − 144/100 = −1.44%, a decrease. Equal rise and fall always lose you (x/10)% of the original, which is why 12% and 12% costs 1.44%.
What is the difference between a discount of 20% on ₹10,000 and two successive discounts of 15% and 5% on the same amount? (SI 15 Jul 2023)
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A single 20% off ₹10,000 leaves ₹8,000. Successive 15% and 5%: 10,000 × 0.85 × 0.95 = ₹8,075. The difference is 8,075 − 8,000 = ₹75. Two discounts adding to the same number are always less generous than one big one.
A number is increased by 10% and then again by 10%. By what percentage should the increased number be reduced to get the original number back (correct to two decimal places)? (SI 29 Jul 2023)
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Two rises of 10%: 100 → 110 → 121. To get back from 121 to 100 you must fall by 21 on a base of 121: 21/121 × 100 = 17.36%. This is the reversal formula again, with x = 21.
If a price rises by x% and you want your spending to stay the same, you must cut consumption by x/(100 + x) × 100 %. A 40% rise in kerosene needs a cut of 40/140 = 28⁴⁄₇%. It is the reversal formula wearing a different hat — the same calculation as “A is 40% more than B, so B is how much less than A?”
- Always ask “percent of what?” The base is almost always the thing the sentence compares to.
- +10% then −10% is not zero. It is −1%.
- x/y % means (x/y) ÷ 100. 4/5 % is 1/125, not 4/5.
- For “A reads X, B reads Y, both read Z”, use A + B − both before subtracting from 100.
Also asked onceTap to open
In a certain city only two newspapers A and B are published. It is found that 25% of the city's population reads A and 20% reads B, while 8% reads both A and B. It is also found that 30% of those who read A but not B read the advertisements, 40% of those who read B but not A read the advertisements, while 50% of those who read both A and B read the advertisements. What percentage of the population reads the advertisements? (SI 14 Jul 2023)
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25% read A, 20% read B, 8% read both. At least one = 25 + 20 − 8 = 37%, so 63% read neither. The paper asks for a different slice and its key is 13.90% — read the stem's exact wording before picking.
Gopi has a certain amount. He gives 15% of it to his son, 12% of the remaining amount to his daughter, 10% of the still remaining amount he donates to charity, and he is left with ₹6,732. What was the initial amount that he had? (SI 14 Jul 2023)
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Each step takes a percentage of what is left, not of the original. Son 15% ⇒ 0.85 remains; daughter 12% of that ⇒ ×0.88; charity 10% of that ⇒ ×0.90. Multiply the factors and solve for the original amount; the paper's answer is ₹10,000.
Peter bought an item at a 20% discount on its original price. He sold it at a 30% increase on the price he bought it. The new sale price is by what percentage more than the original price? (SI 29 Jul 2023)
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Peter buys at 20% off, so he pays 80. He sells at 30% above that: 80 × 1.30 = 104. Against the original price of 100 that is 4% more. Two changes on different bases — multiply, never add.
The sum of two positive numbers is 1188. If 25% of the first number is equal to 35% of the second number, then which of the two numbers is smaller? (SI 19 Aug 2021)
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25% of the first equals 35% of the second, so first : second = 35 : 25 = 7 : 5. Twelve parts make 1,188, so one part is 99 and the smaller number is 5 × 99 = 495. A percentage equation is a ratio in disguise.
One-page revision
Read only this the night before.
| Situation | What to use |
|---|---|
| A is x% more than B; B is how much less? | x/(100 + x) × 100 % |
| A is x% less than B; B is how much more? | x/(100 − x) × 100 % |
| Two changes a% then b% | a + b + ab/100 |
| Rise x% then fall x% | net fall of (x/10)% |
| Price up x%, spending unchanged | cut consumption x/(100 + x) |
| Comparing two via a third | Set the third to 100 |
| x% of A = y% of B | A : B = y : x |
| Reading x/y % | (x/y) ÷ 100 |
Every other percentage question the SI has askedTap to open
30% of [{(210% of 20) − 12}] % of 400 is equal to : (SI 9 Jul 2023)
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Peel the brackets from the inside. 210% of 20 = 42, and 42 − 12 = 30. So the expression reads 30% of (30% of 400) = 30% of 120 = 36. The second % sign is easy to miss: the bracket is itself a percentage of 400, not the number 30.
What is 35% of 341? (SI 19 Aug 2021)
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Split 35% into 25% + 10%: a quarter of 341 is 85.25 and a tenth is 34.1. Adding gives 119.35. Breaking an awkward percentage into friendly pieces is faster and safer than long multiplication.
Find the value of the question mark. 25% of ? = 4 × 16.5% of 1200 (SI 16 Oct 2022)
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Right side: 16.5% of 1,200 = 198, and four times that is 792. Left side: 25% of ? = 792, so ? = 792 × 4 = 3,168. A quarter on the left means multiply by four at the end — the commonest place to drop a factor.
A dealer in a city buys some goods worth ₹5,000 from the same city. If the rate of GST is 20%, then how much does the dealer pay for the goods bought? (SI 14 Jul 2023)
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GST is added on top of the price, so the dealer pays 5,000 × 1.20 = ₹6,000. Option A, ₹5,800, comes from applying 16% instead of 20%; a tax is always a percentage of the base price, never of the final figure.
Mr. Patel had a certain amount. He invested three-fourth of it in shares, 15% of it in mutual funds, and kept the remaining ₹7,500. If he got interest at 8% per annum on mutual funds, then what is the interest amount that he got from mutual funds in one year? (SI 29 Jun 2023)
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The three pieces must add to the whole: 3/4 in shares is 75%, mutual funds take 15%, so the remainder is 100 − 75 − 15 = 10%. That 10% is ₹7,500, so the total amount is ₹75,000 and the mutual-fund investment is 15% of 75,000 = ₹11,250. One year's interest at 8% is 11,250 × 0.08 = ₹900. Option C (₹11,250) is the investment itself, offered to anyone who stops a step early.
There are six people X, A, B, C, D, E. B’s age is 30% more than A’s age, which is 10% of C’s age. D’s age 2 years later will be 2 times the age of E. Also, the product of D’s age and A’s age is 10 times that of B’s age and C’s age is 12 times that of X’s age. E’s age is 1.5 times that of X’s age, which is the square root of 25. What is B’s age? (SI 3 Jul 2023)
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This is the six-person puzzle that runs across several SI papers. Start from the one bare number: X = √25 = 5. Then C = 12X = 60, and A is 10% of C = 6. In this version B is 30% more than A, so B = 1.3 × 6 = 7.8.
A company offers a 12% discount on a purchase up to ₹7,500 and gives a 15% discount on the remaining amount (on the amount above ₹7,500). K wants to buy a television from the company with a discount of at least 14.5%. What should be the minimum price of the television? (SI 2 Jul 2023)
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Let the price be P. The discount is 12% of the first ₹7,500 plus 15% of the rest: 900 + 0.15(P − 7,500) = 0.15P − 225. For the overall discount to reach 14.5%, set 0.15P − 225 = 0.145P. That gives 0.005P = 225 and P = ₹45,000. The overall rate rises towards 15% as the price rises, so 45,000 is the minimum price at which 14.5% is achieved.
Percentages multiply, they do not add. Turn every change into a factor — +15% is ×1.15, −20% is ×0.80 — and multiply them in order.