In short

A percentage is a fraction whose denominator has been agreed in advance to be 100 — so 30\% means \dfrac{30}{100}, no more and no less. A ratio is the same idea written as a comparison: a : b means "for every a of the first there are b of the second." Both notations describe proportional relationships, and you can move freely between fractions, decimals, percentages, and ratios. One technique — the unitary method, "first find what one unit is worth, then scale up" — solves nearly every shop discount, GST, profit-and-loss, recipe-scaling, and ratio-sharing problem you will ever meet.

A shop sign in Sarojini Nagar says "30% OFF". The kurta is marked at ₹800. What do you actually pay?

You probably already know the answer — ₹560 — but think about what just happened in your head. You took a number (₹800), found a piece of it (30% of it = ₹240), and subtracted that piece from the whole. The whole calculation rests on a single idea: the symbol "30\%" is shorthand for the fraction \tfrac{30}{100}, and "30% of ₹800" is the multiplication \tfrac{30}{100} \times 800 = 240. Once you see the hidden denominator, every percentage problem becomes a fraction problem — and every fraction problem becomes one of the patterns from Fractions and Decimals.

This article is the operating manual for the three notations that describe the same kind of comparison — fractions, percentages, ratios — and the one universal technique (the unitary method) that handles every proportional problem in school arithmetic.

What a percentage actually is

The phrase per cent comes from a Latin word meaning per hundred. A percentage is a fraction whose denominator has been agreed in advance to be 100. So:

30\% = \frac{30}{100} \qquad 7\% = \frac{7}{100} \qquad 145\% = \frac{145}{100}

That is the entire definition. Everything else about percentages — converting them, comparing them, calculating with them — follows from this one fact.

The reason the notation exists at all is that it makes comparison cheap. If one bank pays 6.5\% interest and another pays 7\%, you can compare them at a glance because the denominator is already the same. Without percentages, you would have to find a common denominator first — the work that the convention has done in advance, once and for all.

There are three conversions worth memorising, because they appear in every real-world calculation:

Three notations for the number one quarter: fraction, decimal, and percentageA central rounded rectangle holds the number one quarter. Three arrows fan out to three labelled boxes around it. The first box says one quarter as a fraction. The second box says zero point two five as a decimal. The third box says twenty-five per cent as a percentage. The arrows are labelled multiply by one, divide by four, and multiply by one hundred respectively to show how the conversions work.one number1 / 4fraction0.25decimal (÷ by 4)25 %percentage (× 100)
One number in three notations. The fraction $\tfrac{1}{4}$, the decimal $0.25$, and the percentage $25\%$ are three different ways of writing exactly the same point on the number line. Choosing which notation to use is a matter of convenience — percentages are convenient for comparison, decimals are convenient for arithmetic, fractions are convenient for exact answers.

Computing a percentage of a number

The basic question is: what is X\% of Y? The answer is

\frac{X}{100} \times Y

For "30\% of 800":

\frac{30}{100} \times 800 = \frac{30 \times 800}{100} = \frac{24000}{100} = 240

So 30\% of 800 is 240, and the kurta you pay for is 800 - 240 = 560 rupees.

There is also a useful mental shortcut: 10\% of any number is just that number with the decimal point shifted one place to the left. 10\% of 800 is 80. So 30\% is three of those, or 240. 5\% is half of 10\%, or 40. Almost every quick percentage calculation in your head is a combination of "10\% of …" and "1\% of …", added or subtracted.

You can feel the relationship between the percentage and the answer directly. The figure below shows a bar of length \textsf{₹}1000. Drag the red marker. As you move it, the shaded portion shows the corresponding percentage of the bar, and the readouts show both the percentage and the rupee amount. This is the picture every "percentage of a number" question is secretly asking.

Interactive bar showing X percent of one thousand rupeesA horizontal bar from zero to one hundred percent on a number line. A draggable red point determines the cut-off — the portion of the bar to the left of the point is shaded in red, representing X percent of the total. Two live readouts above the bar show the current percentage and the corresponding rupee value when the total is one thousand rupees.total = ₹10000%25%50%75%100%↔ drag the red point
Drag the red point to set a percentage. The top-left readout shows the percentage you have selected; the top-right readout shows the same fraction of $\textsf{₹}1000$ in rupees. At $30\%$ the rupee amount is $\textsf{₹}300$; at $75\%$ it is $\textsf{₹}750$. The relationship between the two is just multiplication by $10$, because $\textsf{₹}1000$ has been chosen so that one percent is exactly $\textsf{₹}10$.

The reverse question — X is what percentage of Y? — uses the same formula, solved for X. If a discount of ₹240 has been taken from a price of ₹800, the discount as a percentage is \tfrac{240}{800} \times 100 = 30\%.

Ratios

A ratio is a comparison written as a : b, read as "a to b." If a class has 12 boys and 18 girls, the ratio of boys to girls is 12 : 18. Like a fraction, a ratio can be reduced by dividing both sides by their highest common factor — here both are divisible by 6, so the simplified ratio is 2 : 3.

The way to read a : b is: for every a of the first kind, there are b of the second. The total is a + b "parts," and each side is its share of those parts. So in a 2 : 3 ratio, the first quantity is \tfrac{2}{5} of the total and the second is \tfrac{3}{5} of the total. The denominator is the sum of the two ratio terms, not either one of them on its own — and forgetting this is the most common ratio mistake.

A ratio with three quantities works the same way: a : b : c has a + b + c parts, and each side gets its own fractional share.

Class of thirty students split in the ratio two to three boys to girlsA horizontal bar representing thirty students total, divided into five equal parts. The first two parts (twelve students) are shaded in one colour and labelled boys. The remaining three parts (eighteen students) are shaded in another colour and labelled girls. Below the bar, the ratio two to three is shown along with the actual count twelve to eighteen.2 parts (boys)3 parts (girls)12 students18 studentsratio 2 : 3 → total 5 parts → 30 students
A class of $30$ students split in the ratio $2 : 3$ (boys to girls). The total is $5$ "parts," each part is $30 / 5 = 6$ students, and the two shares are $2 \times 6 = 12$ boys and $3 \times 6 = 18$ girls. Notice that the ratio $2 : 3$ does not say there are five students in the class — it says the proportions are in those terms, regardless of the actual size.

Proportion and the unitary method

When two ratios are equal, the equation is called a proportion:

\frac{a}{b} = \frac{c}{d}

Cross-multiplying gives the equivalent form ad = bc — the product of the "outer" pair equals the product of the "inner" pair. This is the rule that lets you solve "if a of these costs b rupees, how much do c of them cost?" by setting up a proportion and solving for the unknown.

But the cleanest, most general way to handle proportional problems is the unitary method, which has just one rule: first find what one unit is worth, then scale up to whatever quantity you need.

Take a concrete example. Five mangoes cost ₹120. How much do eight mangoes cost?

Find one. If five mangoes cost ₹120, then one mango costs \textsf{₹}120 \div 5 = \textsf{₹}24.

Scale up. Eight mangoes cost 8 \times \textsf{₹}24 = \textsf{₹}192.

That is the entire technique. You went from "five for ₹120" to "one for ₹24" to "eight for ₹192" with no algebra. The unitary method works for any proportional relationship — speed and time, recipe ingredients, exchange rates, work and labour, fuel and distance — as long as the rule "twice as much input gives twice as much output" holds. Almost all real-world percentage and ratio problems are unitary-method problems in disguise.

The unitary method going from five mangoes for one hundred twenty rupees to eight mangoes for one hundred ninety-two rupeesThree boxed quantities arranged in a row. The first box says five mangoes equals one hundred twenty rupees. An arrow labelled divide by five leads to a middle box which says one mango equals twenty-four rupees. A second arrow labelled multiply by eight leads to a final box which says eight mangoes equals one hundred ninety-two rupees.5 mangoes= ₹120÷ 51 mango= ₹24× 88 mangoes= ₹192"first find what one is worth, then scale up"
The unitary method has two moves and they always come in the same order. Divide by the original quantity to find the value of one unit, then multiply by the new quantity. The middle step — the cost of *one* — is the bridge that converts any proportional question into pure multiplication.

Two worked examples

The two examples below cover the most common real-world patterns: a percentage chain (discount followed by tax) and a ratio split.

Example 1: A kurta marked at ₹1500 has a 20% discount, after which 18% GST is added. What do you pay?

This is a chain of two percentages, and the order matters: the discount comes off the marked price, and the GST is then added on top of the discounted price (not on top of the original).

Step 1. Apply the 20\% discount.

The discount amount is \tfrac{20}{100} \times 1500 = 300. So the discounted price is

1500 - 300 = 1200

Why: 20\% of 1500 is 300, found by the formula \tfrac{X}{100} \times Y. You then subtract the discount because the customer pays less.

Step 2. Apply 18\% GST on the discounted price.

The GST amount is \tfrac{18}{100} \times 1200 = 216. So the final price is

1200 + 216 = 1416

Why: GST is computed on the price you actually pay, after any discount — that is the standard tax convention. So the base for the tax is the discounted price ₹1200, not the original ₹1500. Then you add the tax because the customer pays more.

A faster way using multipliers. Each percentage step can be written as a multiplication. A 20\% discount multiplies the price by 0.80 (since 100\% - 20\% = 80\%), and an 18\% tax multiplies by 1.18. So the whole chain is a single product:

1500 \times 0.80 \times 1.18 = 1416

The order of the two multiplications doesn't matter (multiplication is commutative — see Operations and Properties), so you would get the same answer if the GST were applied first and the discount second. But the rupee amount of each step would be different, even though the final total is the same.

Result. You pay ₹1416.

Three-bar visualisation of fifteen hundred discounted to twelve hundred and then increased to fourteen sixteenThree horizontal bars stacked, each scaled to its rupee amount. The top bar is fifteen hundred rupees long and labelled marked price. The middle bar is twelve hundred rupees long and labelled after twenty per cent discount. The bottom bar is fourteen sixteen rupees long and labelled after eighteen per cent GST. Each bar shows the change from the previous one — three hundred subtracted, then two sixteen added.₹1500 (marked price)₹1200 (after −20% discount)−₹300₹1416 (after +18% GST)+₹216final price = 1500 × 0.80 × 1.18 = ₹1416
The marked price falls by ₹$300$ (the $20\%$ discount), then rises by ₹$216$ (the $18\%$ GST on the discounted price). The two changes are not symmetric — even though both are "$20\%$ and $18\%$," they apply to different bases — and the final amount is *not* the same as $1500 \times 0.98$. The multiplier form $1500 \times 0.80 \times 1.18 = 1416$ does the same calculation in one line.

Example 2: Three friends share ₹4500 in the ratio 2 : 3 : 4. How much does each get?

This is a clean ratio-splitting problem and the unitary method handles it directly. The two-line plan: find the total number of parts, then find the value of one part, then scale up to each share.

Step 1. Add the ratio terms to find the total number of parts.

2 + 3 + 4 = 9

So the ₹4500 is being split into 9 equal parts.

Step 2. Find the value of one part.

\text{one part} = \frac{4500}{9} = 500

Why: this is the unitary step. The total amount divided by the total number of parts gives the value of a single part, and from there you can multiply up to any share.

Step 3. Multiply to find each friend's share.

\text{first friend} = 2 \times 500 = 1000
\text{second friend} = 3 \times 500 = 1500
\text{third friend} = 4 \times 500 = 2000

Step 4. Check: the three shares should add up to the original total.

1000 + 1500 + 2000 = 4500 \,\,✓

Result. The three friends get ₹1000, ₹1500, and ₹2000 respectively.

Bar of forty-five hundred rupees split in the ratio two to three to fourA horizontal bar representing forty-five hundred rupees divided into nine equal parts. The first two parts are shaded in one colour and labelled the first friend's share of one thousand rupees. The next three parts are shaded in a second colour and labelled the second friend's share of fifteen hundred rupees. The last four parts are shaded in a third colour and labelled the third friend's share of two thousand rupees.2 parts3 parts4 parts₹1000₹1500₹20009 parts × ₹500 each = ₹4500
The ₹$4500$ split visually into nine equal parts, distributed to the three friends in the ratio $2 : 3 : 4$. The figure makes the unitary step obvious: each "part" is one ninth of the bar, worth ₹$500$, and each friend simply gets their share of those parts.

Common confusions

Going deeper

If you came here to handle the everyday percentage and ratio calculations of school and shopping, you have it. The rest of this section is for readers who want to see how the same patterns power some larger ideas — compound growth, comparative pricing, and financial returns.

Multipliers and compound growth

Every percentage operation can be rewritten as a single multiplication, and that rewriting is the key to handling repeated percentage changes cleanly.

So a 5\% annual interest rate compounded for 10 years multiplies your money by 1.05^{10} \approx 1.629 — your principal grows by about 63\% over the decade, not 50\%. The difference is the interest on the interest, and it is exactly the difference between adding percentages and multiplying multipliers.

The exponent in 1.05^{10} is the topic of Exponents and Powers, and the way that exponent shapes the long-run behaviour of compounding is one of the most important consequences of the laws of exponents.

Why "average percentage" is almost always wrong

If a stock goes up 50\% one year and down 50\% the next, what is your average return? Naively you might say "zero" — but the correct answer is a loss. Start with ₹100. Up 50\% takes you to ₹150. Down 50\% takes you to ₹75. You lost a quarter of your money, despite the two percentages "averaging out."

The reason is that the two percentages are computed on different bases. The right average for compounded multiplicative changes is not the arithmetic mean (the sum-divided-by-count) but the geometric mean (the product-to-the-one-over-count). For the example, the geometric mean of 1.5 and 0.5 is \sqrt{1.5 \times 0.5} = \sqrt{0.75} \approx 0.866, which corresponds to an average loss of about 13\% per year — and indeed, 0.866^2 \approx 0.75, recovering the actual outcome.

This is why finance, demography, and any field that deals with compounded change quotes "geometric returns" instead of arithmetic averages. It is also why the shopkeeper's "20% off and 20% off" isn't the same as "40% off" — the two percentages are operating on changing bases, and the right combination is multiplicative.

Ratios as fractions in disguise

Every ratio can be rewritten as a fraction, and vice versa. The ratio a : b corresponds to the fraction \tfrac{a}{a + b} for the share of the first part out of the total. The proportion equation \tfrac{a}{b} = \tfrac{c}{d} is just two fractions being equal — which is the same as the equation you would set up in Operations and Properties, with cross-multiplication being the do-the-same-thing-to-both-sides rule applied twice.

So when you learn ratios, you are learning fractions in a different costume. And when you learn the unitary method, you are learning a particular way to solve linear equations of the form \tfrac{x}{a} = \tfrac{y}{b} — which is the simplest possible algebraic equation. Almost every "real-world" arithmetic problem reduces to the same template.

Where this leads next

Percentages and ratios feed directly into the financial and quantitative chapters that follow.