IMAT Logical Reasoning · Numerical Reasoning

Percentages: Change, Reverse and Parts of Parts

Turn every percentage change into a multiplier; multiply for successive changes, divide to go back to the original, and keep percent apart from percentage points.

All Numerical Reasoning notes

Why this matters

Percentages are the most frequent single topic in this chapter: discounts, price offers and reverse percentages run through the Cambridge papers, and the ministry papers asked successive discounts in 2024 and a percentage of a remainder in 2026.

Concept 1 of 3: Percentage change and the multiplier

A percentage is a fraction out of 100, so a change of p percent is the same as multiplying by one number. A rise of 15% multiplies by 1.15; a fall of 15% multiplies by 0.85. Thinking in multipliers turns every percentage question into a single multiplication or division.

Definition

p% of a quantity is the quantity times $\dfrac{p}{100}$ .
An increase of p% multiplies by $1 + \dfrac{p}{100}$ ; a decrease of p% multiplies by $1 - \dfrac{p}{100}$ .
Percentage change compares the change with the original value, never with the new one.

Percentage change

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

$\text{original}$ the value before the change (the base of the percentage)
$\text{new}$ the value after the change

Worked example

A bicycle costs €240 and its price rises by 15%. A scooter's price falls from €500 to €420. Find the new price of the bicycle and the percentage fall in the price of the scooter.

Practice this conceptself-check · 4 quick reps

Percentage change divides by the original

A rise from 12,500 to 13,250 is 750 out of 12,500, which is 6%. Dividing by the new value, 13,250, gives about 5.7%, and that wrong value is usually among the options.

Concept 2 of 3: Successive changes and reverse percentages

A second percentage change acts on the value after the first change, not on the original. So the multipliers multiply. Going back to the original is the same idea in reverse: the new value is the original times the multiplier, so the original is the new value divided by the multiplier.

Definition

Successive changes: the overall multiplier is the product of the separate multipliers. 20% off and then 15% off is $0.80 \times 0.85 = 0.68$ , a single 32% discount.
A rise of p% followed by a fall of p% always leaves less than you started with.
Reverse percentage: if a value after a change is known, the original is that value divided by the multiplier. Never take the percentage of the new value.

Successive changes and the original value

\text{final} = \text{original} \times m_1 \times m_2 \qquad \text{original} = \frac{\text{final}}{m}

$m_1, m_2$ the multipliers of each change, e.g. 0.80 for 20% off
$m$ the overall multiplier from the original to the final value

Worked example

A coat is reduced by 20% in a sale, and then a further 15% is taken off the sale price. What single discount is this equal to? A jacket in the same sale costs €91 after a 30% reduction. What was its original price?

Practice this conceptself-check · 3 quick reps

Successive percentages do not add

20% off followed by 15% off is a 32% discount, not 35%, because the second discount is taken from a smaller price. In the same way, two 20% rises make 44%, not 40%.

To find the original, divide by the multiplier

If a price after an 8% rise is €2,160, subtracting 8% of €2,160 gives €1,987.20, which is wrong. The 8% was 8% of the old price, so divide:

2160 / 1.08 = 2000

Concept 3 of 3: Percentages of a part, and percentage points

Many questions take a percentage of a group and then a percentage of what is left. Write each step as a fraction of the whole and multiply. Separately, when the quantity being changed is itself a percentage (a rate, a share of the vote), there are two ways to describe a change, and they give very different numbers.

Definition

Percentage of a part: "25% of the remaining 60%" is $0.25 \times 0.60 = 0.15$ , so 15% of the whole.
If a known number of people is a known percentage of the whole, the whole is that number divided by the percentage as a decimal.
Percentage points measure the plain difference between two percentages: 4% to 5% is a rise of 1 percentage point.
The percentage change of the same rate is $\dfrac{1}{4} = 25\%$ . Both statements are true; they answer different questions.

Part of a part

\text{fraction of the whole} = \frac{p_1}{100} \times \frac{p_2}{100}

$p_1$ the percentage that picks out the first group
$p_2$ the percentage of that group that is wanted

Worked example

In a school, 40% of the students come by bus. Of the other students, 25% cycle. What percentage of all the students cycle? If 270 students cycle, how many students are in the school?

Practice this conceptself-check · 3 quick reps

Percentage points are not percent

A rate moving from 8% to 6% falls by 2 percentage points, but by 25% of its old value. An option saying "it fell by 2%" uses the wrong unit and is wrong.

"Of the rest" means of the remainder, not of the whole

If 40% take the bus and 25% of the rest cycle, the cyclists are 25% of 60%, which is 15% of the school, not 25%.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Percentage change and the multiplier
Percentage change
$\text{percentage change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%$
Successive changes and reverse percentages
Successive changes and the original value
$\text{final} = \text{original} \times m_1 \times m_2 \qquad \text{original} = \frac{\text{final}}{m}$
Percentages of a part, and percentage points
Part of a part
$\text{fraction of the whole} = \frac{p_1}{100} \times \frac{p_2}{100}$

Watch out for (5)

Percentage change divides by the original
→ Percentage change and the multiplier
Successive percentages do not add
→ Successive changes and reverse percentages
To find the original, divide by the multiplier
→ Successive changes and reverse percentages
Percentage points are not percent
→ Percentages of a part, and percentage points
"Of the rest" means of the remainder, not of the whole
→ Percentages of a part, and percentage points

Drill the past-year questions on this topic

Open the bank with this topic pre-filtered.

Open in the bank