Percentages and Percentage Change

Percentage questions are the most frequently tested topic in the UCAT Quantitative Reasoning subtest. This lesson covers every type of percentage calculation you might encounter, with worked examples using the kind of data presentations you will see on test day.

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Finding a Percentage of a Number

The most basic percentage calculation: finding X% of Y.

Formula: X% of Y = (X / 100) × Y

Mental shortcut: Use the 10%, 5%, 1% building-block method from Lesson 2.

Worked Example

Data: A hospital has 1,400 staff members. 35% are nurses.

Question: How many nurses work at the hospital?

Solution:

• 10% of 1,400 = 140
• 30% = 3 × 140 = 420
• 5% = 140 ÷ 2 = 70
• 35% = 420 + 70 = 490 nurses

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Expressing One Number as a Percentage of Another

Formula: (Part / Whole) × 100

Worked Example

Data:

Department	Staff
Cardiology	84
Neurology	56
Oncology	70
Paediatrics	42
Total	252

Question: What percentage of total staff work in Neurology?

Solution: (56 / 252) × 100 = 22.2% (to 1 d.p.)

Calculator tip: Enter 56 ÷ 252 × 100 directly. No need to simplify the fraction first.

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Percentage Increase

Formula: Percentage increase = [(New - Old) / Old] × 100

Worked Example

Data: In 2022, a GP surgery had 8,400 patient registrations. In 2023, this rose to 9,660.

Question: What was the percentage increase?

Solution:

• Increase = 9,660 - 8,400 = 1,260
• Percentage increase = (1,260 / 8,400) × 100
• = 0.15 × 100
• = 15%

Common error: Using the new value as the denominator instead of the old value. Always divide by the original value.

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Percentage Decrease

Formula: Percentage decrease = [(Old - New) / Old] × 100

Worked Example

Data: A clinic's waiting time dropped from 45 minutes to 36 minutes.

Question: What was the percentage decrease?

Solution:

• Decrease = 45 - 36 = 9
• Percentage decrease = (9 / 45) × 100 = 20%
• 20% decrease

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Reverse Percentages

Reverse percentage questions give you the value after a percentage change and ask you to find the original value.

After an Increase

If a value increased by X% and is now Y:

Original = Y / (1 + X/100)

After a Decrease

If a value decreased by X% and is now Y:

Original = Y / (1 - X/100)

Worked Example 1 (Increase)

Question: After a 20% increase, a hospital's budget is £1,440,000. What was the original budget?

Solution:

• Original × 1.20 = £1,440,000
• Original = £1,440,000 / 1.20 = £1,200,000

Common error: Calculating 20% of £1,440,000 and subtracting it. This gives 80% of £1,440,000 = £1,152,000, which is WRONG. The 20% increase was applied to the original, not to the new value.

Worked Example 2 (Decrease)

Question: After a 15% reduction, a drug costs £25.50. What was the original price?

Solution:

• Original × 0.85 = £25.50
• Original = £25.50 / 0.85 = £30.00

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Compound Percentage Changes

When multiple percentage changes are applied successively, you cannot simply add them. Each percentage change is applied to the result of the previous one.

Worked Example

Question: A share price starts at £200. It increases by 10% in Year 1 and decreases by 10% in Year 2. What is the final price?

Solution:

• After Year 1: £200 × 1.10 = £220
• After Year 2: £220 × 0.90 = £198