Fractions in Disguise
Why This Matters
Walk past any shop and you see them: “Mega Sale — up to 50% off!” Open your report card and you read “Hiya scored 83%”. Look at a food packet and it says “60% cocoa”.
That little sign — % — is everywhere. In sales, in marks, in news, in cricket, in your phone’s battery.
But what does it really mean? And why is it so useful that the whole world agrees to use it?
Here is the secret. A percentage is not some new, scary thing. It is just a fraction wearing a disguise. Once you see the fraction hiding behind the % sign, percentages become easy — and even fun.
This chapter teaches you to see through the disguise. You will learn what “per cent” means, how to turn any fraction, ratio, or decimal into a percentage (and back again), how to find a percentage of an amount, and how to handle real money problems like discounts, profit, and loss. By the end, a “30% off” sticker will hold no mystery for you.
The Big Idea
The word per cent comes from the Latin “per centum”, which means out of a hundred. So a percentage is simply a fraction whose denominator is fixed at 100. 25% means 25 out of every 100 — written as 25/100. Because the bottom is always 100, percentages are easy to compare, easy to picture, and easy to turn into decimals. To change any fraction into a percentage, you just multiply it by 100. To find a percentage of a quantity, you write the percentage as a fraction over 100 and multiply. That single idea — “out of a hundred” — runs through this entire chapter.
Let’s Break It Down
What “per cent” really means: out of 100
Let us slow right down on the most important idea.
Per cent means “out of 100”. When we say 25%, we mean 25 out of every 100. It could be 25 marks out of 100 marks, 25 rupees out of 100 rupees, or 25 people out of 100 people. The thing being counted changes, but the meaning of the % sign does not: for every 100, take this many.
So a percentage is really a fraction with a fixed denominator of 100:
25% = 25/100
50% = 50/100 = 1/2
20% = 20/100 = 1/5
That is the whole secret. A percentage is a “fraction in disguise” — a fraction that has been forced to have 100 on the bottom.
The picture below makes this concrete. A big square is cut into 100 tiny squares, so the whole square is “100”. When we shade some of them, the number shaded is the percentage.
So whenever you see a % sign, quietly say to yourself: “out of 100”. That habit alone will carry you through most percentage problems.
Changing a fraction into a percentage — and why it works
Suppose Surya mixes paint for a sunset. The red paint makes up 3/4 of the mixture. What percentage of the colour is red?
We want to write 3/4 as “something out of 100”. So we find an equal fraction that has 100 on the bottom.
Multiply top and bottom by the same number (this does not change the value, it only renames it):
3/4 = (3 × 25)/(4 × 25) = 75/100 = 75%
So the red paint is 75% of the mixture. Nice and clean.
Before we go further, let us refresh the idea of equal fractions, because every percentage conversion leans on it.
But there is an even quicker rule. A fraction is “out of 1 whole”, while a percentage is “out of 100”. So to find the percentage, you just ask: if the whole is 100, how many parts is this? That is found by multiplying the fraction by 100.
To write any fraction as a percentage: multiply the fraction by 100, then add the % sign.
3/4 × 100 = 75, so 3/4 = 75%
This is faster, and it always works — even when 100 does not divide neatly. Let us watch it on a fraction that is not so tidy.
Surya wants to save 2/5 of his prize money for a new canvas. What percentage of his money is that?
- To turn a fraction into a percentage, multiply it by 100. So we work out 2/5 × 100.
- 2/5 × 100 = (2 × 100)/5 = 200/5.
- Now divide: 200 ÷ 5 = 40.
- So 2/5 = 40%. Surya is saving 40% of his prize money. (Check: 40/100 = 4/10 = 2/5. Correct.)
Going the other way is just as easy. A percentage is already a fraction over 100, so to write a percentage as a fraction, put it over 100 and simplify.
24% = 24/100 = 12/50 = 6/25
The FDP trio: fractions, decimals and percentages
A fraction, a decimal, and a percentage can all describe the same amount. They are three names for one thing. We call them the FDP trio — Fractions, Decimals, Percentages.
Why is the decimal version handy? Because our number system is base 10, and 100 fits perfectly with decimals. 31% = 31/100 = 0.31. So a percentage turns into a decimal just by dividing by 100 (move the point two places left), and back into a percentage by multiplying by 100.
The picture below shows the three names for the value 3/4 and how to hop between them.
Here is a small table of the trio so you can see the pattern. Read across each row — the three boxes always mean the same amount.
| Per cent | Fraction | Decimal |
|---|---|---|
| 50% | 50/100 = 1/2 | 0.5 |
| 25% | 25/100 = 1/4 | 0.25 |
| 75% | 75/100 = 3/4 | 0.75 |
| 10% | 10/100 = 1/10 | 0.1 |
| 1% | 1/100 | 0.01 |
| 5% | 5/100 = 1/20 | 0.05 |
Turning a ratio into a percentage
Sometimes a quantity is split as a ratio. To prepare a millet porridge, the ratio of millet to water is 2 : 7. What percentage of the mixture is millet?
First, find the whole. The mixture has 2 parts millet and 7 parts water, so the whole is 2 + 7 = 9 parts. Millet is 2 out of 9 parts, which is the fraction 2/9.
Now turn that fraction into a percentage by multiplying by 100:
2/9 × 100 = 22.22% (millet)
Water is the rest: 100 − 22.22 = 77.78%
The picture below shows the 9 parts split into millet and water.
So the rule is short: a ratio becomes a fraction of the whole, and then a fraction becomes a percentage.
Finding a percentage of a quantity
Now the most useful skill of all: finding a percentage of some amount. “25% of 120 grams”, “80% of 75 marks”, “18% GST on ₹8250” — all the same kind of question.
The method follows straight from the meaning. y% of an amount = (y/100) × amount. You write the percentage as a fraction over 100 and multiply.
So 25% of 120 = (25/100) × 120 = 30. The picture below shows why this is exactly one quarter of the bar.
There is a friendly shortcut here. 25% is the same as 1/4 (a quarter), because 25/100 = 1/4. So “25% of 40” is just “a quarter of 40”, which is 10 — no pen needed. Some percentages have easy fraction-twins like this: 50% = 1/2, 10% = 1/10, 20% = 1/5, 75% = 3/4.
Two mental tricks make harder ones easy too:
- 20% of a number is double 10% of it. (20 parts is twice 10 parts.) So if 10% of 80 is 8, then 20% of 80 is 16.
- Percentages add up. 25% = 20% + 5%, so (25% of y) = (20% of y) + (5% of y). You can build awkward percentages from easy ones.
Let us put the main method to work on a marks problem.
The maximum marks in a test are 75. A student needs 80% to get an A grade. How many marks is that?
- We need 80% of 75. Write the percentage as a fraction over 100 and multiply: 80% of 75 = (80/100) × 75.
- Simplify the fraction first: 80/100 = 4/5. So we need 4/5 × 75.
- 4/5 × 75 = (4 × 75)/5 = 300/5 = 60.
- So the student needs at least 60 marks out of 75 to get an A grade. (Check with decimals: 0.8 × 75 = 60. Same answer.)
Percentages can be more than 100
So far every percentage was 100 or less. But a percentage can also be more than 100. That just means more than the whole.
Kishanlal’s daily sales target is ₹5000 (that is his “100%”). One day he sells ₹6000. What percentage of his target is that?
6000/5000 × 100 = 120%
He achieved 120% of his target — that is, 20% more than the target. A percentage over 100 is normal whenever the amount is bigger than the base.
Another example: a farmer harvested 260 kg last year and 650 kg this year. This year as a percentage of last year is 650/260 × 100 = 250%. That means this year’s harvest is 2.5 times last year’s.
So remember: 100% is the whole. Below 100% is part of it. Above 100% is more than it.
Percentage increase and decrease
Percentages are perfect for describing change. Prices go up, crowds go down, populations grow. We measure these changes in percentages.
Here is the one rule you must hold tight:
Percentage change = (amount of change ÷ ORIGINAL amount) × 100
The word that trips everyone is ORIGINAL. You always divide by the amount you started with, not the amount you ended with. Let us see both directions.
Three years ago, 1 kg of tomatoes cost ₹30. Now it costs ₹42. The increase is ₹42 − ₹30 = ₹12. The original price was ₹30, so:
Percentage increase = (12 / 30) × 100 = 40%
Now a decrease. A theatre’s footfall was 160 before COVID and is 100 now. The decrease is 160 − 100 = 60. The original was 160, so:
Percentage decrease = (60 / 160) × 100 = 37.5%
The picture below shows both side by side, so you can see that “change over original” is the same idea whether things go up or down.
Why divide by the original and not the new value? Because percentage change answers the question “how big is this change compared with where we started?”. The starting amount is the fair yardstick — it is the thing that changed. If you divided by the new value instead, you would be measuring against a different whole and get a different, misleading answer.
One neat link: saying “the population is 165% of the old population” means the same as saying it “increased by 65%”. Because 100% is the old amount you already had, and the extra 65% is the growth: 100% + 65% = 165%.
Using percentages: comparing, discount, profit and loss
The biggest reason percentages exist is to compare fairly. Two fractions with different bottoms are hard to compare at a glance. Turn both into percentages — same base of 100 — and the winner is obvious.
Eesha scored 42 out of 50 in English and 70 out of 80 in Science. Which is better? The totals are different, so we cannot just compare 42 and 70. Turn both into percentages:
English: 42/50 × 100 = 84%
Science: 70/80 × 100 = 87.5%
Now it is clear — her Science score is better. That is the power of a common base of 100.
Percentages also run the world of buying and selling. Three prices matter here, so let us define them clearly:
- Cost price (CP) — what the shopkeeper paid to buy the item.
- Marked price (MP) — the price on the tag (the MRP or quoted price).
- Selling price (SP) — what the customer actually pays, after any discount or bargaining.
If SP is more than CP, the shopkeeper makes a profit. If SP is less than CP, it is a loss. The picture below follows a sweater through all three prices.
So the two key money formulas are:
Profit = SP − CP and Profit % = (Profit / CP) × 100
Loss = CP − SP and Loss % = (Loss / CP) × 100
A discount is a percentage cut from the marked price. “35% off” means the price drops by 35% of the marked price. Let us solve a full discount-and-profit problem.
A cooker has a marked price (MRP) of ₹1800. A store offers 35% discount on it. The cost price was ₹900. Find the selling price, and the profit percentage made on the sale.
- First the discount. 35% of ₹1800 = (35/100) × 1800 = 35 × 18 = ₹630. This is how much the price is cut.
- Selling price = marked price − discount = 1800 − 630 = ₹1170.
- Now the profit. Profit = SP − CP = 1170 − 900 = ₹270. (SP is more than CP, so it really is a profit.)
- Profit % is always on the cost price: (270 / 900) × 100 = 30%. So the cooker sells for ₹1170, at a profit of 30%.
Loss works the same way, just with SP below CP. Shyamala bought a vase for ₹2650 and sells it at an 18% loss. A quick way: a loss of 18% means she keeps 100% − 18% = 82% of the cost. So SP = 82% of 2650 = 0.82 × 2650 = ₹2173.
Common Mistakes
These three slips catch students again and again. Spot them now and they will not catch you.
To find the percentage increase, divide the change by the new (final) amount.
The new amount is the number sitting right in front of you at the end, so it feels like the natural thing to divide by — students reach for the most recent figure.
Percentage change is always divided by the ORIGINAL amount, not the new one. For a price rising from 30 to 42, the increase 12 is divided by the starting price 30, giving 40%. Dividing by the new value would answer a different question and give the wrong figure.
A 30% + 20% discount is the same as a 50% discount.
In ordinary arithmetic 30 + 20 really does make 50, so it feels obvious that the two discounts should add up to one big 50% discount.
The second discount is taken on the already-reduced price, not the original — this is called compounding. On a ₹200 cake: 30% off gives ₹140, then 20% off ₹140 gives ₹112. A straight 50% off ₹200 gives ₹100. So 30% + 20% leaves you paying MORE than a single 50% off. The percentages do not simply add.
A 50% profit margin can be cancelled by giving a 50% discount, so the shop breaks even.
The two numbers are both 50%, so it looks like one undoes the other exactly, leaving no gain and no loss.
The 50% profit is added to the cost price, but the 50% discount is taken off the bigger selling price — so they are percentages of different amounts. If cost is x, the selling price is 1.5x, and 50% off that is 0.75x. Selling at 0.75x means a 25% LOSS, not break-even. Percentages of different bases never cancel like this.
Quick Check
Try these four. Each checks one idea from the chapter.
What does 25% mean?
What is 3/5 written as a percentage?
A price rises from ₹40 to ₹50. What is the percentage increase?
A shopkeeper buys an item for ₹200 and sells it for ₹250. What is the profit percentage?
Practice Problems
Try each one yourself first. Only then tap to see the full solution.
Easy
Express 9/20 as a percentage.
To turn a fraction into a percentage, multiply by 100. 9/20 × 100 = (9 × 100)/20 = 900/20 = 45. So 9/20 = 45%. (Check: 45/100 = 9/20. Correct.)
Nandini has 25 marbles, of which 15 are white. What percentage of her marbles are white?
White marbles are 15 out of 25, which is the fraction 15/25. Multiply by 100: 15/25 × 100 = 1500/25 = 60. So 60% of her marbles are white. (You can also simplify first: 15/25 = 3/5, and 3/5 = 60%.)
Find 16% of 250.
A percentage of an amount means write the percentage over 100 and multiply. 16% of 250 = (16/100) × 250. = 16 × 250 / 100 = 4000/100 = 40. So 16% of 250 = 40.
Medium
A clothing shop offers a 25% discount on a shirt with an original price of ₹300. How much will Anwar pay?
First find the discount. 25% of 300 = (25/100) × 300 = 75. So the price is cut by ₹75. Selling price = original − discount = 300 − 75 = ₹225. A quicker way: paying after a 25% discount means paying 100% − 25% = 75% of the price. 75% of 300 = 0.75 × 300 = ₹225. Same answer. So Anwar pays ₹225.
The price of petrol was ₹60 in 2015 and ₹100 in 2025. What is the percentage increase?
Find the change first: 100 − 60 = ₹40 increase. Percentage change uses the ORIGINAL amount, which is the 2015 price of ₹60. Percentage increase = (40 / 60) × 100. = 4000/60 = 66.66%. So the price of petrol increased by about 66.66%.
A number increased by 20% becomes 90. What is the number?
Let the original number be the whole, 100%. After a 20% increase, it becomes 100% + 20% = 120% of the original. So 120% of the number = 90. That means (120/100) × number = 90, so 1.2 × number = 90. Number = 90 ÷ 1.2 = 75. Check: 20% of 75 = 15, and 75 + 15 = 90. Correct. The number is 75.
Challenge
Samson bought a car for ₹4,40,000 after getting a 15% discount from the dealer. What was the original (marked) price of the car?
A 15% discount means Samson paid 100% − 15% = 85% of the original price. So 85% of the original = ₹4,40,000. That is (85/100) × original = 4,40,000. Original = 4,40,000 ÷ (85/100) = 4,40,000 × 100 / 85. = 4,40,00,000 / 85 = 5,17,647 (approximately ₹5,17,647). So the original price was about ₹5,17,647. (Common trap: do NOT just add 15% of 4,40,000 — that 15% would be on the wrong amount. The discount was on the original, not on the price paid.)
A bakery offers '30% + 20% off' on a cake worth ₹200. A rival offers a flat '50% off'. Which is cheaper, and by how much?
‘30% + 20%’ is applied one after the other (compounding), not added. First discount: 30% of 200 = 60, so the price becomes 200 − 60 = ₹140. Second discount: 20% of 140 = 28, so the price becomes 140 − 28 = ₹112. So ‘30% + 20%’ gives a final price of ₹112. The flat 50% off: 50% of 200 = 100, so the price is 200 − 100 = ₹100. The flat 50% off (₹100) is cheaper than 30% + 20% (₹112), by ₹12. Lesson: two discounts in a row are NOT the same as adding the percentages.
Surbhi keeps a 50% profit margin, then offers a 50% discount to clear stock, thinking she breaks even. If she sold goods (originally bought for some amount) for ₹12,000 after the discount, how much did she lose, and what is the loss percentage?
Let the cost price (what she bought the goods for) be x. With a 50% profit margin, her selling price is x + 50% of x = 1.5x. A 50% discount on that is half of 1.5x, which is 0.75x. So after the discount she receives 0.75x. We are told this is ₹12,000. So 0.75x = 12,000, giving x = 12,000 ÷ 0.75 = ₹16,000 (her cost). Loss = cost − received = 16,000 − 12,000 = ₹4,000. Loss % = (4000 / 16000) × 100 = 25%. So she lost ₹4,000, a 25% loss — not break-even. The 50% discount is taken on the bigger marked price, so it outweighs the 50% margin.
Summary
- Per cent means “out of 100”. A percentage is simply a fraction with denominator 100: x% = x/100. The symbol % is read “per cent”.
- Fraction → percentage: multiply the fraction by 100 and add the % sign. So 3/4 × 100 = 75%. Percentage → fraction: put it over 100 and simplify, so 24% = 24/100 = 6/25.
- A fraction, a decimal, and a percentage are three names for one amount (the FDP trio): 3/4 = 0.75 = 75%. Divide top by bottom to get the decimal; × 100 to get the percentage.
- A ratio becomes a percentage by writing each part as a fraction of the whole (the sum of all parts), then multiplying by 100.
- Percentage of a quantity: y% of an amount = (y/100) × amount. Easy twins help: 25% = 1/4, 50% = 1/2, 10% = 1/10.
- A percentage can be more than 100 — it simply means more than the whole (120% = 1.2 times; 250% = 2.5 times).
- Percentage change = (change ÷ ORIGINAL amount) × 100 — always divide by the amount you started with, for both increase and decrease.
- In trade, Profit = SP − CP and Loss = CP − SP, with the percentage taken on the cost price. A discount is a percentage off the marked price. Two discounts in a row do NOT add up.
What’s Next
You can now see through the disguise: every percentage is a friendly fraction with 100 on the bottom, and that one idea handles marks, sales, discounts, profit and loss. That is a skill you will use for the rest of your life — in shops, banks, news, and exams.
Next we go back to geometry and meet one of the most famous results in all of mathematics. The next chapter is Pythagoras Theorem. There you will learn the beautiful rule connecting the three sides of a right-angled triangle — and why it is true, not just that it is. See you there!
Frequently Asked Questions
What does per cent actually mean?
Per cent comes from the Latin 'per centum', meaning 'out of a hundred'. So 25 per cent (25%) means 25 out of every 100 — like 25 marks out of 100, or 25 rupees out of every 100 rupees. A percentage is simply a fraction whose denominator is 100.
How do I change a fraction into a percentage?
Multiply the fraction by 100 and write the per cent sign. For example, 3/4 × 100 = 75, so 3/4 = 75%. This works because a percentage is per 100, so multiplying by 100 tells you how many parts there would be out of 100.
How do I find a percentage of a quantity?
Write the percentage as a fraction over 100 and multiply by the quantity. For example, 25% of 120 = (25/100) × 120 = 30. You can also turn the percentage into a decimal: 25% = 0.25, and 0.25 × 120 = 30.
What is the difference between percentage increase and percentage decrease?
Both are found the same way: change ÷ original amount × 100. If the new value is larger it is a percentage increase; if smaller it is a percentage decrease. The key is to always divide by the ORIGINAL amount, not the new one.
Can a percentage be more than 100?
Yes. A percentage over 100 just means more than the whole. If sales of 6000 are compared with a target of 5000, that is 6000/5000 × 100 = 120% — twenty per cent more than the target. 200% means twice the original value.