Percentage change and comparisons — notes, question types & practice · Data Interpretation

01

Percentage change: new against old

\% \text{ change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100

Sales of 250 rising to 320 grew by $\dfrac{70}{250} \times 100 = 28\%$ .

Rule: The base is the earlier (or original) value. Rise and fall use the same formula; a fall just gives a minus sign.

02

Change over a long span

From 160 to 320 over five years: $\dfrac{320-160}{160} \times 100 = 100\%$ . The value doubled.

Watch: A question asking "by 2023 from 2019" wants one jump, not the sum of yearly rises. 25% then 30% is not 55%.

03

Share of a total

One item as a percent of the whole table:

\text{share} = \frac{\text{item}}{\text{total}} \times 100

Spending table: Rent 12000, Food 15000, Travel 5000, Education 4000; total 36000. Rent's share $= \dfrac{12000}{36000} = 33\dfrac{1}{3}\%$ .

Tip: Add the full column first and keep the total visible. Every share question reuses it.

04

Points versus percent

A rise from 60 points to 72 points is 12 points. As a percent it is $\dfrac{12}{60} \times 100 = 20\%$ .

Watch: Options offer both 12 and 20. Read whether the question says points or percent.

05

Which year grew fastest?

Compute each year's rise on its own base, then compare.

Sales 160, 200, 260, 250, 320: rises are $25\%$ , $30\%$ , about $-3.8\%$ , $28\%$ . The fastest growth year is 2021, even though 2023 added more units.

Tip: Fastest growth is about the rate, not the amount. A small base inflates the rate.

06

Falls follow the same rule

A dip computes exactly like a rise. Sales slip from 260 to 250: $\dfrac{250-260}{260} \times 100 \approx -3.8\%$ . The minus sign is the only difference.

Watch: The base is still the earlier value, 260. Dividing 10 by 250 gives 4%, a planted wrong option.

07

One item as percent of another

"A is what percent of B" divides A by B, no subtraction:

Sales of A are 200 when B sells 250. A is $\dfrac{200}{250} \times 100 = 80\%$ of B. Turned around, B is $\dfrac{250}{200} \times 100 = 125\%$ of A.

Tip: The value after "of" is the base. Swapping the two swaps the answer between 80 and 125.

08

Approximate, then aim

Ugly tables reward rough work. Round each value to one significant figure, compute, and match the closest option. Only compute exactly when two options sit close together.

Example: A rise from 199 to 251 is about $50/200 = 25\%$ ; scan the options and only one will be near 25.

09

Percentage of a percentage

"A's sales are 25% more than B's" does not make B 25% less than A. If B is 200, A is 250; then B is $\dfrac{200}{250} = 80\%$ of A, which is 20% less.

Example: Reverse the sentence, reverse the base. 250 on base 200 is +25%; 200 on base 250 is −20%.

10

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Single-step percentage rise

How to spot it:

A table of yearly values; the percent change between two named years is asked.

\frac{\text{new}-\text{old}}{\text{old}} \times 100\%

Method

Subtract the earlier year from the later one.
Divide by the earlier year.
Multiply by 100.

Why it works:

Change only means something against its starting base.

Try this

Sales of a store (in lakh rupees):

Year	2019	2020	2021	2022	2023
Sales	160	200	260	250	320

The percentage increase in sales from 2022 to 2023 is:

Show solution

Rise $= 320 - 250 = 70$ .
$\dfrac{70}{250} \times 100 = 28\%$ .

Answer

28%

Type 2very common2 practice Q

Percentage change over a span

How to spot it:

Growth from the first year to the last year is asked.

\frac{V_{last} - V_{first}}{V_{first}} \times 100\%

Method

Take only the first and last values.
Find the difference.
Divide by the first value.

Why it works:

A span has one start and one end; middle years are noise.

Try this

Sales of a store (in lakh rupees):

Year	2019	2020	2021	2022	2023
Sales	160	200	260	250	320

The percentage increase in sales from 2019 to 2023 is:

Show solution

Difference $= 320 - 160 = 160$ .
$\dfrac{160}{160} \times 100 = 100\%$ .

Answer

100%

Type 3common2 practice Q

Share of a total

How to spot it:

One item of an expense or production table as a percent of the table's total.

\frac{\text{item}}{\text{total}} \times 100\%

Method

Total the whole column.
Place the item over the total.
Multiply by 100.

Why it works:

A share is a fraction of a fixed whole.

Try this

Monthly expenses of a family:

Item	Rent	Food	Travel	Education
Amount (₹)	12000	15000	5000	4000

Rent is what percent of total expenses?

Show solution

Total $= 12000 + 15000 + 5000 + 4000 = 36000$ .
Rent share $= \dfrac{12000}{36000} \times 100 = 33\dfrac{1}{3}\%$ .

Answer

33 $\frac{1}{3}$ %

Type 4common2 practice Q

Points or percent

How to spot it:

A table of scores or points; the question asks for the rise in points or in percent.

\text{points} = a-b; \quad \%\text{age} = \frac{a-b}{b} \times 100

Method

Decide the unit asked: points or percent.
Points: subtract.
Percent: divide the difference by the earlier score.

Why it works:

Same numbers, two answers; the unit decides.

Try this

Points scored by a team across three seasons:

Season	1st	2nd	3rd
Points	60	72	81

The percentage increase in points from the 1st to the 2nd season is:

Show solution

Rise $= 72 - 60 = 12$ points.
$\dfrac{12}{60} \times 100 = 20\%$ .

Answer

20%

Type 5common2 practice Q

Fastest growing year

How to spot it:

Which year saw the highest percentage growth?

\max_i \frac{V_{i+1}-V_i}{V_i}

Method

Compute each year's rise.
Divide each by that year's base.
Pick the largest rate.

Why it works:

Growth speed is a rate on each year's own base.

Try this

Sales of a store (in lakh rupees):

Year	2019	2020	2021	2022	2023
Sales	160	200	260	250	320

In which year was the percentage growth in sales the highest?

Show solution

Rises: $40/160 = 25\%$ , $60/200 = 30\%$ , $-10/260 \approx -3.8\%$ , $70/250 = 28\%$ .
Largest rate is $30\%$ .
Year 2021.

Answer

2021

11

Formula sheet

Percentage change

\frac{\text{new}-\text{old}}{\text{old}} \times 100\%

Base = the old / original value.

Share

\frac{\text{part}}{\text{whole}} \times 100\%

Points vs percent

\text{points} = a - b; \quad \%\text{age} = \frac{a-b}{b} \times 100

12

Shortcuts that save time

⚡ The 1% probe

Find 1% of the base first; then any percentage is that times n. 1% of 250 is 2.5, so 70 units is 28%.

Example

Sales rose from 250 to 320. Find the percentage rise.

Show solution

Rise $= 70$ units; $1\%$ of 250 $= 2.5$ .
$70 \div 2.5 = 28$ .

Answer

28%

⚡ Anchor on the double

A value that doubles is +100%; halves is −50%. Spot these before computing: they frame every other answer.

Example

Sales went from 160 to 320 over five years. Total growth?

Show solution

$320 = 2 \times 160$ .
Doubling is $+100\%$ .

Answer

100%

13

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Dividing by the new value for a percentage rise.

Divide the rise by the old value.

Mistake 02

Adding yearly percentage rises for a span.

Use only the first and last values for a span.

Mistake 03

Reporting 12 when 20 is asked (points vs percent).

Check the unit the question wants.

Mistake 04

Calling +25% reversible, so a −25% fall returns the start.

Bases differ: +25% then −25% of the new value loses money.

Mistake 05

Using the largest rise in units as the fastest growth.

Growth rate divides by that year's own base.

14

Quick revision

Read this the night before the exam.

$\%\text{ rise} = \dfrac{\text{rise}}{\text{old}} \times 100$ .
Span questions: first and last values only.
Share: part over whole, after totalling the column.
Points are differences; percent divides by the base.
Fastest growth year = highest rate on its own base.
Doubling is +100%; halving is −50%.

15

Practice: 14 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.