Ratios, Proportions, and Rates

Ratio, proportion, and rate questions appear frequently in the UCAT Quantitative Reasoning subtest. They require you to scale quantities, divide amounts fairly, convert between units, and work with rates such as speed, density, and flow. This lesson covers the essential techniques with worked examples.

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Simplifying Ratios

A ratio compares two or more quantities. To simplify, divide all parts by their highest common factor (HCF).

Original Ratio	HCF	Simplified
12 : 8	4	3 : 2
45 : 30 : 15	15	3 : 2 : 1
100 : 250	50	2 : 5

Ratios from Different Units

If quantities are in different units, convert to the same unit first:

Example: Express 2 hours to 45 minutes as a ratio.

• Convert: 2 hours = 120 minutes
• Ratio: 120 : 45
• Simplify (HCF = 15): 8 : 3

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Sharing in a Ratio

When a total quantity is divided in a given ratio:

1. Add the parts of the ratio
2. Divide the total by the sum to find the value of one part
3. Multiply each part of the ratio by this value

Worked Example

Question: A hospital trust allocates £450,000 to three departments in the ratio 5 : 3 : 1. How much does each department receive?

Solution:

• Total parts: 5 + 3 + 1 = 9
• Value of one part: £450,000 ÷ 9 = £50,000
• Department A: 5 × £50,000 = £250,000
• Department B: 3 × £50,000 = £150,000
• Department C: 1 × £50,000 = £50,000

Verification: £250,000 + £150,000 + £50,000 = £450,000 ✓

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Direct Proportion

Two quantities are directly proportional if when one doubles, the other doubles too. The ratio between them stays constant.

Formula: If a/b = c/d (constant ratio), then a × d = b × c (cross-multiplication)

Worked Example

Question: A recipe requires 300g of flour to make 12 biscuits. How much flour is needed for 20 biscuits?

Solution (unitary method):

• Flour per biscuit: 300 ÷ 12 = 25g
• Flour for 20: 25 × 20 = 500g

Solution (ratio method):

• 300/12 = x/20
• x = 300 × 20 / 12 = 6,000 / 12 = 500g

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Inverse Proportion

Two quantities are inversely proportional if when one doubles, the other halves. Their product stays constant.

Formula: a × b = constant. So if a₁ × b₁ = a₂ × b₂

Worked Example

Question: If 4 builders can complete a job in 15 days, how long would it take 6 builders (working at the same rate)?

Solution:

• Work = 4 × 15 = 60 builder-days
• With 6 builders: 60 ÷ 6 = 10 days

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Speed, Distance, and Time

The most important rate relationship in the UCAT:

Speed = Distance / Time Distance = Speed × Time Time = Distance / Speed

Key Unit Conversions for Speed Problems

Conversion	Method
km/h to m/s	Divide by 3.6 (or × 1000 / 3600)
m/s to km/h	Multiply by 3.6
mph to km/h	Multiply by 1.6 (approximately)
km/h to mph	Divide by 1.6 (approximately)

Worked Example 1: Basic Speed

Question: A car travels 210 km in 3 hours. What is its average speed?

Solution: Speed = 210 / 3 = 70 km/h

Worked Example 2: Finding Time

Question: How long does it take to travel 450 km at 90 km/h?

Solution: Time = 450 / 90 = 5 hours

Worked Example 3: Average Speed (Trap Question)

Question: A cyclist rides 30 km at 15 km/h and then 30 km at 10 km/h. What is the average speed for the entire journey?