Unit Conversions and Compound Measures

Unit conversion questions appear regularly in the UCAT Quantitative Reasoning subtest. You need to convert between metric units, handle currency conversions, and work with compound measures like speed, density, and pressure. This lesson provides a comprehensive reference and worked examples.

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Metric Conversions: Length

Conversion	Factor
1 km = 1,000 m	× 1,000 to go from km to m
1 m = 100 cm	× 100 to go from m to cm
1 cm = 10 mm	× 10 to go from cm to mm
1 m = 1,000 mm	× 1,000 to go from m to mm

Key Principle: Multiply When Going to Smaller Units, Divide When Going to Larger Units

• 3.5 km → metres: 3.5 × 1,000 = 3,500 m
• 4,200 m → km: 4,200 ÷ 1,000 = 4.2 km
• 150 cm → metres: 150 ÷ 100 = 1.5 m

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Metric Conversions: Mass

Conversion	Factor
1 kg = 1,000 g	× 1,000
1 g = 1,000 mg	× 1,000
1 tonne = 1,000 kg	× 1,000

Worked Example

Question: A medication dose is 250 mg. A box contains 30 tablets. What is the total mass of medication in the box, in grams?

Solution: 250 × 30 = 7,500 mg = 7,500 ÷ 1,000 = 7.5 g

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Metric Conversions: Volume and Capacity

Conversion	Factor
1 litre = 1,000 ml	× 1,000
1 litre = 1,000 cm³	1 ml = 1 cm³
1 m³ = 1,000 litres	× 1,000
1 m³ = 1,000,000 cm³	× 1,000,000

Worked Example

Question: A swimming pool has dimensions 25 m × 10 m × 2 m. What is its capacity in litres?

Solution:

• Volume = 25 × 10 × 2 = 500 m³
• 500 m³ = 500 × 1,000 = 500,000 litres

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Area Conversions

Critical point: Area conversions use the square of the linear conversion factor.

Conversion	Factor
1 m² = 10,000 cm²	(100 cm)² = 10,000 cm²
1 km² = 1,000,000 m²	(1,000 m)² = 1,000,000 m²
1 cm² = 100 mm²	(10 mm)² = 100 mm²
1 hectare = 10,000 m²	—

Common error: Converting 3 m² to cm² by multiplying by 100 (getting 300 cm²). This is WRONG. The correct conversion is 3 × 10,000 = 30,000 cm². Think of it as a 1 m × 1 m square = 100 cm × 100 cm = 10,000 cm².

Worked Example

Question: A field is 0.5 km by 0.3 km. What is its area in m²?

Solution:

• 0.5 km = 500 m, 0.3 km = 300 m
• Area = 500 × 300 = 150,000 m²

Alternative: 0.5 × 0.3 = 0.15 km² = 0.15 × 1,000,000 = 150,000 m² ✓

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Volume Conversions

Volume conversions use the cube of the linear conversion factor:

Conversion	Factor
1 m³ = 1,000,000 cm³	(100)³ = 1,000,000
1 cm³ = 1,000 mm³	(10)³ = 1,000

Worked Example

Question: A container is 50 cm × 40 cm × 30 cm. What is its volume in m³?

Solution:

• Volume = 50 × 40 × 30 = 60,000 cm³
• 60,000 ÷ 1,000,000 = 0.06 m³

Or convert first: 0.5 m × 0.4 m × 0.3 m = 0.06 m³ ✓

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Currency Conversions

UCAT questions sometimes involve currency conversions. The exchange rate will always be provided in the question.

Worked Example

Data: Exchange rate: £1 = €1.15

Question 1: Convert £450 to euros.

Solution: 450 × 1.15 = €517.50

Question 2: Convert €800 to pounds.