🧮 Aptitude & Reasoning · Chapter 1

Percentages, and the base you forgot

Every percentage trick question works the same way: it changes the base halfway through the sentence and hopes you don't notice.

Study this lesson free in the app →

The basics

A percentage is a fraction with 100 as the denominator: 18% means 18 out of every 100 of something.

The problem it solves: comparing quantities of different sizes. A ₹200 rise means a lot on a ₹1,000 shirt and nothing on a ₹10 lakh car; 20% and 0.02% make that difference visible.

The mental model: every percentage is attached to a base — the number after "of" or "than". Change the base and the same 20% becomes a different amount. Nearly every trick question in an aptitude round moves the base mid-sentence and waits for you to miss it.

The working method: turn every change into a multiplier (+20% → ×1.2, −20% → ×0.8) and multiply. Never add percentages from different steps.

Explained

A shirt costs ₹1,000. The price rises 20%, then falls 20%.

Most people say ₹1,000. It is ₹960.

The rise was 20% of 1,000. The fall was 20% of 1,200. Different bases, different sizes — and that single idea is most of this topic.

The two staircases

You climb 20 steps up a staircase, then 20 steps down a different staircase whose steps are taller.

You do not end up where you started.

A percentage is not a fixed distance — it is a distance measured against wherever you currently are. Go up first and the down-step is measured from a higher, bigger number.

So before computing anything, ask one question: 20% of what?

Work in multipliers, never in additions

Stop adding and subtracting percentages. Convert everything to a multiplier and multiply.

increase by 20%   →  × 1.20
decrease by 20%   →  × 0.80
increase by 8%    →  × 1.08
decrease by 35%   →  × 0.65

Chains become trivial:

+20% then −20%       1.20 × 0.80 = 0.96   → 4% loss
−25% then +25%       0.75 × 1.25 = 0.9375 → 6.25% loss
+10% three years     1.10³        = 1.331  → 33.1% growth
two discounts 20,10  0.80 × 0.90 = 0.72   → a single 28% discount

Notice every chain ends below what naive addition suggests. That is not a coincidence — it happens whenever you multiply a number by (1+a)(1−a), which is always less than 1.

Reverse the comparison

Anil earns 25% more than Bela.

Where it costs real money

A dashboard shows conversion rising from 4% to 5%.

The growth team writes: "conversion up 1%". Marketing reads that and concludes the campaign barely moved anything.

Both are describing a 25% relative increase — a very good quarter — reported as if it were nearly nothing.

Percentage points measure the gap between two percentages: 5% − 4% = 1 percentage point. Percent measures the relative change: 1/4 = 25%.

Interviewers use this deliberately, and so do reports. When someone says "up 1%", ask which one they mean — that question alone signals numeracy.

Underline the word after "than" or "of". That is the base.

Multiply multipliers. Never add percentages.

+x% then −x% always loses. Percentage points ≠ percent.

Gotchas

• Adding successive percentages — 20% then 10% is 28%, not 30%
• Percentage points confused with percent — 30% → 35% is 5 points but a 16.7% rise
• Reversing without recomputing — the base flips with the comparison
• Profit vs discount bases — profit % is on cost price, discount % is on marked price. Most profit-and-loss questions hide exactly this
• Working in fractions under time pressure — convert to a decimal multiplier immediately

Speed, out loud

Aptitude rounds are timed, so the marks are in the setup, not the arithmetic. Three habits:

1. Assume 100. Percentage questions are scale-free — pick the base that makes the mental maths free
2. Say the base aloud: "20% of the increased price, so 1200". Saying it prevents the mistake
3. Use the successive-change shortcut for two-step questions: a + b + ab/100. For +20 and −20: 20 − 20 − 4 = −4% in about two seconds

And in a data role this is not just an aptitude round — you will be asked to sanity-check a growth number in a meeting, and being the person who spots a base error is a genuinely visible skill.

Words to know

In two minutes

• A percentage is always a percentage of something — the base, found right after 'of' or 'than'.
• Convert every change to a multiplier: +x% is ×(1 + x/100), −x% is ×(1 − x/100).
• Successive changes multiply, so +20% then −20% is 1.2 × 0.8 = 0.96, a 4% loss.
• Reversing a comparison flips the base: 25% more becomes 20% less, by x/(100+x).
• Percentage points measure the gap between two percentages; percent measures relative change.
• Profit % is on cost price; discount % is on marked price — most P&L traps hide that switch.
• Assume a base of 100 whenever the question doesn't fix one; percentages are scale-free.

Test yourself

There are 8 questions, 3 puzzles and 6 recall cards for this lesson in the app — the recall cards come back on later days, just before you would forget.