What a Percentage Is

Percentages · 20 min

Aptitude · Percentages

A fraction wearing a disguise, and one word that fixes the base

Percentages are the most reusable chapter in the syllabus — profit, interest, ratio and data interpretation all run on them. Almost everything rests on two question types and on reading which number the word “of” points at.

Switch between the two question types on the same numbers →
Per cent means per hundred. So x% is literally x/100, and the number after “of” is always the base.

01 The idea

Two question types, and the word that separates them

Per cent means per hundred, so 65% is nothing more than the fraction 65/100. Every percentage question is a fraction question that has been dressed up, and undressing it is the first move.

There are only two shapes. Type 1 asks for a share of a number: “what is 65% of 40?” Strip the sign, put 100 underneath, simplify, multiply. 65/100 becomes 13/20, and 13/20 of 40 is 26.

Type 2 runs it backwards: “26 is what per cent of 40?” Put the part over the base and scale to a hundred — 26/40 = 13/20, times 100 is 65%. Same relation, different unknown.

The word that decides everything is of. Whatever follows it is the base and goes underneath. “60 is what per cent of 80” is 75%; “80 is what per cent of 60” is 133.33%. Same two numbers, and swapping them is not a small error — it is a different question.

The number immediately after “of” is the base and belongs in the denominator. Get that right and Type 2 questions stop being guesswork.
Per centPer hundred. x% means x/100, which is why every percentage can be handled as a fraction.
BaseThe whole that the percentage is taken of — the denominator. Named by the word “of”, and choosing it wrongly is the chapter’s commonest error.
The two typesType 1 is “what is x% of y”, giving a value. Type 2 is “x is what % of y”, giving a percentage. Recognise which before calculating.

02 Worked example

65% of 40, and then 26 out of 40

One pair of numbers, both question types. (i) What is 65% of 40? (ii) 26 is what per cent of 40? (iii) And what per cent of 26 is 40?

1
Type 1: strip the signSixty-five per cent is 65 over 100. Simplify before multiplying.65% = 65/100 = 13/20
2
Multiply by the baseThirteen twentieths of forty. The twenty cancels into the forty, leaving 13 × 2.13/20 × 40 = 13 × 2 = 26
3
Type 2: put the part over the baseThe word “of” points at 40, so 40 is the denominator.26/40 = 13/20
4
Scale to a hundredA fraction becomes a percentage by multiplying by 100.13/20 × 100 = 65%  ✓ consistent with part (i)
5
Now swap the base and watch it change“What per cent of 26 is 40” makes 26 the base instead.40/26 × 100 = 153.85% — a different question, a different answer

Notice that parts (ii) and (iii) use the same two numbers and give 65% and 153.85%. Neither is wrong; they answer different questions. Before computing any Type 2 percentage, point at the word “of” and put whatever follows it underneath. That one second of discipline removes most of the errors available in this chapter.

03 The method

The method, and why fractions beat decimals

Both types are one line. The habit that makes them fast is refusing to work in decimals.

Type 1: x% of y = (x/100) × y. Type 2: x is (x/y) × 100 per cent of y.
Simplify the fraction before you multiply. 65% of 40 as a decimal is 0.65 × 40, which needs actual multiplication. As 13/20 × 40 the twenty cancels and you are left with 13 × 2. This matters more as the numbers get uglier: 14.28% of 343 looks impossible until you recognise 14.28% as 1/7, at which point it is 49 in three seconds.
The questionTypeRoute
What is 65% of 40?113/20 × 40 = 26
26 is what % of 40?226/40 × 100 = 65%
60 is what % of 80?23/4 × 100 = 75%
80 is what % of 60?24/3 × 100 = 133.33%
14.28% of 343?11/7 × 343 = 49
Working in decimalsslower, and rounds0.1428 × 343
Guessing the basedifferent question75% vs 133.33%

05 Cheat sheet

The basics on one page

Two types, one word, and the fractions worth knowing by sight.

CaseRouteExample
Type 1: x% of y(x/100) × y65% of 40 = 26
Type 2: x as % of y(x/y) × 10026/40 = 65%
Finding the basethe number after 'of'“of 40” → base 40
Fraction to percentage× 10013/20 → 65%
Percentage to fraction÷ 100, then simplify65% → 13/20
Swapping the basedifferent question75% vs 133.33%
Decimalsslower and roundsuse fractions instead
Simplify before multiplying13/20 × 40 cancels to 13 × 2. The decimal route needs real multiplication and introduces rounding for no benefit.
A percentage above 100 is normal80 is 133.33% of 60. Nothing is wrong — the part is simply larger than the base, which happens whenever you compare upward.
“Of” names the denominatorIt is the only word in the sentence that fixes the base, so find it before you write anything down.

06 Where & why

Where this shows up

Percentages are the load-bearing chapter of the whole syllabus, so this lesson pays off far outside its own module.

Every quantitative paper
Direct percentage questions

Two or three per paper on their own, plus percentages embedded in almost everything else.

Profit, loss and interest
The base is the whole question

Profit per cent is on cost, discount on marked price, interest on principal. Each is this lesson’s base rule with a specific base named.

Data interpretation
Reading a chart

Nearly every DI question is Type 2 — this value as a percentage of that one. The base is usually the trap.

Ratio and proportion
Percentages as ratios

A ratio of 9 : 8 is a 12.5% difference. Converting freely between the two forms is what makes both chapters fast.

If you build one habit here, make it pointing at the word “of” before you calculate. It costs nothing and it prevents the single most common mistake in quantitative aptitude.

07 Interview questions

What gets asked

Nine, from the definition to the reason fractions beat decimals under time pressure.

What does per cent actually mean?
Per hundred. So x% is the fraction x/100, and every percentage question is a fraction question in disguise. That is why converting to a fraction is almost always the first useful step.
What is 65% of 40?
Twenty-six. Sixty-five per cent is 65/100, which simplifies to 13/20, and 13/20 of 40 lets the 20 cancel into the 40, leaving 13 × 2 = 26.
26 is what per cent of 40?
Sixty-five per cent. Put the part over the base: 26/40 = 13/20, and multiplying by 100 gives 65%. It is the same relation as the previous question with a different unknown.
How do you know which number is the base?
It is the one immediately after the word “of”. In “26 is what per cent of 40”, the base is 40 and it goes in the denominator. That word is the only thing in the sentence that fixes it.
60 is what per cent of 80, and 80 is what per cent of 60?
75% and 133.33% respectively. Same two numbers, opposite bases, completely different answers. Neither is wrong — they answer different questions, which is exactly why the base has to be read rather than assumed.
Can a percentage exceed 100?
Yes, whenever the part is larger than the base. 80 is 133.33% of 60. It is only impossible when the quantity is inherently bounded — you cannot have 120% of the students in a class pass, but you can certainly have 120% of last year’s revenue.
What is 14.28% of 343?
Forty-nine. 14.28% is 1/7, and a seventh of 343 is 49. This is the whole argument for learning the fraction table: as a decimal the question needs real multiplication and gives a rounded answer, while as a fraction it takes three seconds and is exact.
Why work in fractions rather than decimals?
Two reasons. Fractions cancel against the other number, so 13/20 × 40 becomes 13 × 2 with no multiplication worth the name. And they are exact — 1/7 is precise where 0.1428 is not, which matters when the answer options are close together.
How do you convert a fraction to a percentage and back?
Multiply by 100 to go one way and divide by 100 to come back, simplifying as you go. So 13/20 becomes 65% and 65% becomes 13/20. Being fluent in both directions is what lets you spot that an awkward percentage is really a tidy fraction.

08 Practice problems

Six on the basics

Convert every percentage to a simplified fraction before multiplying, and point at the word “of” in each Type 2 question.

Type 1

Easy
Find 45% of 180, working in fractions rather than decimals.
Follow-up
Simplify 45/100 before touching 180 and the multiplication becomes small. If you found yourself computing 0.45 × 180, you took the slower route.
Show the hint
45/100 cancels down to 9/20.

Type 2

Easy
84 is what per cent of 240? Then state what per cent of 84 is 240.
Follow-up
The same two numbers both ways round. Doing both fixes the base rule in place, and the two answers should be very different.
Show the hint
For the first, the base is 240; simplify 84/240 before scaling.

Recognise the fraction

Medium
Find (a) 16.66% of 480, (b) 12.5% of 344, and (c) 11.11% of 549, in each case by recognising the percentage as a unit fraction.
Follow-up
All three are ugly as decimals and trivial as fractions. If any of them takes long multiplication, you have not spotted the fraction yet.
Show the hint
16.66% is 1/6, 12.5% is 1/8, and 11.11% is 1/9.

Find the base

Medium
35% of a number is 147. Find the number. Then find what per cent of 147 the original number is.
Follow-up
The first part runs Type 1 backwards, which is a division rather than a multiplication. The second part deliberately reverses the base so you have to re-read rather than reuse.
Show the hint
If 35% is 147 then 1% is 147/35 — and 35% simplifies to 7/20.

Percentages above 100

Medium
A shop’s sales rose from ₹48,000 to ₹60,000. Express the new sales (a) as a percentage of the old, and (b) the old as a percentage of the new. Explain why the two answers are not 125% and 75%.
Follow-up
The two percentages have different bases, so they are not complements. This is the same asymmetry that makes a 25% rise need only a 20% fall to undo — the idea the next lesson is built on.
Show the hint
Simplify 60000 : 48000 to a small ratio first, then take each direction.

Build the fraction table

Hard
(a) Write out 1/n as a percentage in exact fractional form for n from 2 to 12 — for example 1/6 = 16⅔%. (b) Use your table to compute 8.33% of 636, 9.09% of 671 and 7.69% of 481 by inspection. (c) Explain why the percentages for n = 7, 11 and 13 are non-terminating decimals while those for n = 8 and n = 20 terminate, in terms of the prime factors of n.
Follow-up
Part (c) is the reason the table exists at all. A fraction terminates as a decimal exactly when its denominator’s only prime factors are 2 and 5 — so sevenths and elevenths can never be written exactly as decimals, which is precisely why competitive exams quote them as 14⅞% and expect you to think in sevenths.
Show the hint
For (c), ask which denominators divide some power of 10.