Percentage

What is a percentage?

A percentage expresses a number as a fraction of 100. To turn a score of $a$ out of $b$ into a percentage, compute $\dfrac{a}{b} \times 100$ .

When combining scores from more than one test, add up all the marks scored and all the marks available first, and only then convert. Averaging each test's percentage gives a different answer whenever the tests are out of different totals.

Worked example: combining scores

A student scored 42 out of 50 in a physics test, 65 out of 80 in chemistry and 88 out of 120 in mathematics. Marks scored: $42 + 65 + 88 = 195$ . Marks available: $50 + 80 + 120 = 250$ . Combined percentage: $\dfrac{195}{250} \times 100 = 78\%$ .

Averaging the three test percentages instead (84%, 81.25% and about 73.33%) gives about 79.53%, which overstates the result because the largest test, mathematics, had the lowest score.

Shortcuts for common percentages

Common percentages as quick operations

Percentage	Shortcut	Example
10%	divide by 10	10% of 340 = 34
5%	half of 10%	5% of 340 = 17
1%	divide by 100	1% of 340 = 3.4
50%	divide by 2	50% of 96 = 48
25%	divide by 4	25% of 96 = 24
20%	divide by 5	20% of 65 = 13

Build other percentages from these: 15% of 260 is $26 + 13 = 39$ , and 35% of 180 is $54 + 9 = 63$ . And $X\%$ of $Y$ always equals $Y\%$ of $X$ : 16% of 25 is 25% of 16, which is 4.

Increase, decrease and finding the percentage

To increase a number by $X\%$ , multiply by $\left(1 + \dfrac{X}{100}\right)$ ; to decrease it, multiply by $\left(1 - \dfrac{X}{100}\right)$ . Increasing 520 by 15% gives $520 \times 1.15 = 598$ , and decreasing it by 15% gives $520 \times 0.85 = 442$ . To find what percentage 39 is of 52: $\dfrac{39}{52} \times 100 = 75\%$ .

Worked example: successive changes

A share price rises by 30% one month and falls by 30% the next. Is it back where it started?

Solution: Multiply the factors: $1.3 \times 0.7 = 0.91$ . The price ends 9% lower, because the 30% fall is taken on the larger, increased price.

Worked example: reverse percentage

After a 20% increase, a fee is 720. What was it before?

Solution: $720$ is $1.2$ times the original, so the original is $720 \div 1.2 = 600$ . Taking 20% off 720 instead would give 576, which is wrong.

Profit and loss

Profit and loss is percentages applied to money. With S.P. the selling price and C.P. the cost price:

\text{Profit} = \text{S.P.} - \text{C.P.}, \qquad \text{Profit \%} = \dfrac{\text{Profit}}{\text{C.P.}} \times 100

When S.P. is less than C.P., the same formula gives a loss. Profit and loss percentages are always taken on the cost price, never the selling price.

Profit percent. A car bought for ₹25,000 is sold for ₹28,000. The profit is ₹3,000, so the profit percent is $\dfrac{3000}{25000} \times 100 = 12\%$ .

Selling at a loss. An article bought for ₹450 is sold at a 20% loss. The loss is $0.20 \times 450 = 90$ rupees, so the selling price is $450 - 90 = 360$ rupees.

Finding the cost price (a reverse percentage). An article sold for ₹500 at a 25% profit was sold at 125% of its cost, so $\text{C.P.} = \dfrac{500}{1.25} = 400$ rupees. Dividing by 1.25 is right; taking 25% off ₹500 (which gives ₹375) is the classic mistake.

Discount. A 12% discount on a marked price of ₹150 leaves $150 \times 0.88 = 132$ rupees to pay. A discount is taken on the marked price, while profit is measured on the cost price.

Percentage points vs percent

If an exam's pass rate rises from 30% to 36%, it has risen by 6 percentage points, but by $\dfrac{36-30}{30} \times 100 = 20\%$ in relative terms. Questions that say 'by what percent' usually want the relative change.

Common mistakes

Averaging percentages from tests with different totals. Add the raw marks first.
Adding successive changes. +30% then −30% is not 0%; multiply the factors.
Reversing a change by subtracting the same percentage. Divide by the factor instead.
Mixing up percentage points and percent. From 30% to 36% is 6 points, but a 20% rise.
Taking profit percent on the selling price. Profit and loss percent are always on the cost price: profit ÷ C.P. × 100.

Where this comes up in exams

The General Aptitude section of every GATE 2027 paper lists numerical computation and estimation, including ratios and percentages, in its Quantitative Aptitude syllabus.

Checked against GATE 2027 information brochure (General Aptitude syllabus). Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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