Percentage Calculations

Percentage questions are the most frequently tested topic in UCAT Quantitative Reasoning. They appear in nearly every data set — calculating percentage increases, finding what percentage one value is of another, working backwards from a percentage, and handling compound changes. This lesson covers every type of percentage calculation you will encounter, with efficient methods for each.

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Type 1: Finding X% of Y

The most basic percentage calculation.

Formula: X% of Y = (X ÷ 100) × Y

The Building-Block Method

Break any percentage into components you can calculate instantly:

Component	How to Find It
50%	Halve the number
10%	Divide by 10
5%	Half of 10%
1%	Divide by 100
25%	Divide by 4 (or halve 50%)

Example: Find 35% of 840

• 10% of 840 = 84
• 30% = 3 × 84 = 252
• 5% = 84 ÷ 2 = 42
• 35% = 252 + 42 = 294

Example: Find 17% of 6,000

• 10% = 600
• 5% = 300
• 2% = 120
• 17% = 600 + 300 + 120 = 1,020

The Decimal Multiplier Method

Convert the percentage to a decimal and multiply:

• 35% of 840 = 0.35 × 840
• For simple percentages, the building-block method is faster
• For calculator work, the decimal multiplier is more direct

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Type 2: Expressing One Value as a Percentage of Another

Formula: (Part ÷ Whole) × 100

Worked Example

Data:

Category	Number of Patients
Emergency	234
Elective	486
Outpatient	780
Total	1,500

Question: What percentage of patients are Emergency patients?

Solution: (234 ÷ 1,500) × 100

• Simplify: 234/1,500 → both divisible by 6 → 39/250
• 39/250 = 39 × 4 / 1,000 = 156/1,000 = 0.156 = 15.6%

Or estimate: 234/1,500 ≈ 225/1,500 = 15%. Answer is close to 15%, so 15.6% is correct.

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Type 3: Percentage Increase

Formula: New Value = Original × (1 + Percentage/100)

Or equivalently: Increase = Original × (Percentage/100), then New Value = Original + Increase

The Multiplier Shortcut

Instead of calculating the increase and then adding, use a single multiplier:

Percentage Increase	Multiplier
5%	×1.05
10%	×1.10
15%	×1.15
20%	×1.20
25%	×1.25
50%	×1.50
100%	×2.00

Example: A flat costs £185,000 and increases in value by 12%.

New value = £185,000 × 1.12

• 185,000 × 1 = 185,000
• 185,000 × 0.1 = 18,500
• 185,000 × 0.02 = 3,700
• New value = 185,000 + 18,500 + 3,700 = £207,200

Calculating the Percentage Increase

Formula: Percentage Increase = ((New − Original) ÷ Original) × 100

Example: Sales rose from 450 to 540. What is the percentage increase?

• Increase = 540 − 450 = 90
• Percentage = (90 ÷ 450) × 100 = 20%
• Check: 90/450 = 1/5 = 0.2 = 20% ✓

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Type 4: Percentage Decrease

Formula: New Value = Original × (1 − Percentage/100)

Percentage Decrease	Multiplier
5%	×0.95
10%	×0.90
15%	×0.85
20%	×0.80
25%	×0.75
50%	×0.50

Example: A car worth £24,000 depreciates by 15%.

New value = £24,000 × 0.85

• 24,000 × 0.8 = 19,200
• 24,000 × 0.05 = 1,200
• New value = 19,200 + 1,200 = £20,400

Calculating the Percentage Decrease

Formula: Percentage Decrease = ((Original − New) ÷ Original) × 100

Common Mistake: Always divide by the original value, not the new value. The original is the "base" for calculating the change.

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Type 5: Reverse Percentages

You are given the value after a percentage change and must find the original value.

After an Increase

Formula: Original = New Value ÷ (1 + Percentage/100)

Example: After a 20% increase, a product costs £360. What was the original price?

• 360 ÷ 1.20 = £300

Common Trap: Candidates often calculate 20% of £360 = £72, then subtract to get £288. This is wrong because 20% of the original is not the same as 20% of the new value.

After a Decrease

Formula: Original = New Value ÷ (1 − Percentage/100)

Example: After a 25% discount, a jacket costs £90. What was the original price?

• 90 ÷ 0.75 = £120