Percentage change: increase, decrease and successive change

A 20% rise means multiplying by 1.2, and a 20% fall by 0.8. Use this single idea to solve successive change, spending, savings and pass-fail questions.

The recharge puzzle

Say your ₹200 mobile recharge becomes 10% costlier, so it now costs ₹220. A few months later the company gives 10% off. Do you pay ₹200 again?

Most people say, "Yes. Up 10%, down 10%, so we are back to the start." But the discount is on ₹220, and 10% of 220 = 22. So you pay 220 − 22 = ₹198, not ₹200.

Both changes were 10%, yet they did not cancel out, because the second 10% was taken on the new price. That is what this lesson is about: what does a percentage change act on, and how do you solve such questions quickly without a calculator? The answer is one simple idea.

Every change is a multiplication

Think of a percentage as paise in a rupee: ₹1 = 100 paise. When an amount rises or falls, every single rupee in it changes in the same way:

This 1.2 or 0.8 is called the multiplier. The rule is short:

• Up by r%: multiplier = (100 + r) ÷ 100
• Down by r%: multiplier = (100 − r) ÷ 100

Change	100 becomes	Multiplier	As a fraction
up 10%	110	1.1	11/10
up 20%	120	1.2	6/5
up 25%	125	1.25	5/4
up 50%	150	1.5	3/2
down 10%	90	0.9	9/10
down 20%	80	0.8	4/5
down 25%	75	0.75	3/4
down 50%	50	0.5	1/2

Old to new, new to old

Old to new: multiply

The usual way takes two steps: 20% of ₹500 = 100, so the new price = 500 + 100 = ₹600.

The shortcut takes one step: 500 × 1.2 = 600. The answer is the same, because the multiplier already does the adding.

By what percentage did it change? This is the previous lesson's "A is what percentage of B", with the change in place of A and the old value in place of B:

Percentage change = (new − old) ÷ old × 100

If a price goes from ₹40 to ₹50, then (50 − 40) ÷ 40 × 100 = 25, so it rose by 25%. With the multiplier: 50 ÷ 40 = 1.25, and 1.25 means up 25%. Always divide by the old value, because you are comparing with the old value.

Questions may also call it a "percentage increase" or a "percentage decrease".

Example 1 · Easy

New to old: divide

Now the reverse question: after a 20% rise the price is ₹600. What was it before?

The usual way (from the previous lesson): the new price is 120% of the old one. 120% = 600, so 1% = 5 and 100% = 500.

The shortcut: old × 1.2 = new, so to go back, divide by the same multiplier: 600 ÷ 1.2 = 500. A fraction makes it even easier: 1.2 = 6/5, so 600 × 5/6 = 500.

flowchart TD
  A["Old price: ₹500"] -->|"forward: up 20%, × 1.2"| B["New price: ₹600"]
  B -->|"back: ÷ 1.2"| A

To go forward, multiply by the multiplier. To come back, divide by the same multiplier. Taking 20% of 600 off 600 is wrong, because the 20% was taken on the old price, not on 600.

Example 2 · Easy

25% more = 20% less: it is all about the base

In the previous lesson you learnt to pick the right base: the value you compare with. In change questions, this base quietly switches.

In "A is 25% more than B", the base is B. But in "B is how much less than A", the base becomes A. The gap stays the same, the base changes, so the percentage changes too.

See it in parts. 25% = 1/4, so if B has 4 parts, A has 4 + 1 = 5 parts:

Who	Parts	Taking B as 100
B	4	100
A	5	125
Gap	1	25

• Compared with B: a gap of 1 part on a base of 4 parts is 1/4, that is 25% more.
• Compared with A: the same 1 part on a base of 5 parts is 1/5, that is 20% less.

More by	The other way, less by	As fractions
10%	$9\frac{1}{11}\%$	1/10 → 1/11
20%	16⅔%	1/5 → 1/6
25%	20%	1/4 → 1/5
33⅓%	25%	1/3 → 1/4
50%	33⅓%	1/2 → 1/3
100%	50%	1/1 → 1/2

Read the table backwards too: 20% less = 25% more, and 50% less = 100% more.

When the fraction is not easy, use the formula. If A is r% more than B, then B is this many per cent less than A:

And if A is r% less than B, then B is this many per cent more than A:

For 25% more: 25 ÷ 125 × 100 = 20, so 20% less.

Example 3 · Medium

Successive change: two changes in a row

When one and the same thing changes twice, one change after the other, it is called successive change. If a question says one thing changed "by 20% and 10% successively", this is what it means.

When a question gives no actual value, start with 100: then every change shows up directly as a percentage. Run the recharge puzzle from 100:

Step	Value	Change
Start	100	—
up 10%	110	+10 (10% of 100)
down 10%	99	−11 (10% of 110)
Total	99	−1, that is 1% less

It went up by 10 but down by 11, because the second time the base was 110. With multipliers it takes one line: 1.1 × 0.9 = 0.99, that is 1% less. Swap the order and 0.9 × 1.1 = 0.99 as well. Rise first or fall first, the result is the same.

Shortcut formula: for two changes of a% and b% (write a rise as + and a fall as −), the total percentage change is:

For the recharge: 10 + (−10) + (10 × (−10)) ÷ 100 = −1, so 1% less.

Example 4 · Medium

Two years in a row: growth or fall

If a population, a crop or a price changes by r% every year, that is also successive change, with both a and b equal to r. Keep multiplying by the yearly multiplier:

• Up 10% a year: 1.1 × 1.1 = 1.21, so up 21% in 2 years, not 20%.
• Down 10% a year: 0.9 × 0.9 = 0.81, so down 19% in 2 years, not 20%.

The second year's percentage also acts on the first year's change, so the answer is not simply r + r. To go back 2 years, divide by both multipliers.

Example 5 · Medium

A chain of comparisons: A to B, B to C

A multiplier is not only for changes over time. "A is 50% more than B" also means A = B × 1.5. So when three things are compared in a chain (A with B, and B with C), multiply the multipliers.

Why? Each percentage acts on its own base. The comparison of A uses B as its base, and the comparison of B uses C. Percentages on different bases cannot simply be added, but their multipliers can be multiplied, just like two changes in a row.

Example 6 · Medium

Price goes up, spending stays the same

Spending = price × quantity. If the price rises and you want to spend the same amount, you must buy less, that is, cut your consumption.

A 25% rise makes the price multiplier 5/4. To keep spending the same, the quantity must be multiplied by the flipped multiplier, 4/5, because 5/4 × 4/5 = 1: spending stays the same only when the price multiplier times the quantity multiplier is 1. And 4/5 means 1/5 less, that is 20% less. It is the same "25% more = 20% less" pair again.

So the rule is:

• Price up r%: cut consumption by r ÷ (100 + r) × 100 per cent.
• Price down r%: the same money buys r ÷ (100 − r) × 100 per cent more.

Example 7 · SSC GD level

Income, spending and savings

A related question uses savings = income − spending. Income and spending rise by different percentages, and you are asked by what percentage the savings changed.

The usual mistake is to subtract one percentage from the other. The income percentage acts on the income and the spending percentage acts on the spending, so the two have different bases. Work in rupees instead: take the income as 100, then find the savings before and after. Always measure the change in savings on the old savings, because that is what you are comparing with.

Example 8 · SSC GD level

Pass and fail questions

The full marks of the paper are called the "maximum marks"; some questions say "total marks".

Think of the pass marks as a line. A student who failed stayed below the line; a student who passed went above it:

• "Failed by X marks": pass marks = marks scored + X
• "Passed by X marks": pass marks = marks scored − X

Once you have the pass marks, they are the pass percentage of the maximum marks. Then use the previous lesson's method: find 1% first, then 100%.

Example 9 · SSC GD level

Now the two-student question. One student is below the pass line and the other is above it. The distance between them is the sum of the two distances, and it is also the gap between their percentages:

flowchart BT
  A["Student with 25%"] -->|"21 marks short"| B["Pass line"]
  B -->|"24 marks above"| C["Student with 40%"]

Example 10 · SSC GD level

Where students often go wrong

Learn to spot these mistakes, so they do not cost you marks in the exam.

Try it yourself

Solve each question in your notebook first. If you get stuck, read the hint under the question. Then pick an option and press "Check answer". If you get it wrong, do not worry: the explanation shows where that mistake comes from.

Quick reference

What next: now try the practice set below. It has questions from both lessons. Then take the mock test on percentage: 20 questions in 15 minutes. Each right answer gets 2 marks and each wrong answer costs 0.25 marks, as per the 2026 notice.