Ratios and Rates

Ratios and rates appear frequently in UCAT DM data interpretation questions. They test whether you can compare quantities that are expressed in different units or formats, scale values proportionally, and identify when a direct comparison is invalid. This lesson covers the essential techniques.

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What Is a Ratio?

A ratio compares two or more quantities using the same unit. It can be written as:

• 3:2 (colon notation)
• 3/2 (fraction notation)
• "3 to 2" (verbal notation)

Simplifying Ratios

Divide all parts by their highest common factor.

Original ratio	Simplified
15:10	3:2
24:36:12	2:3:1
100:250	2:5

Sharing in a Ratio

A research grant of £120,000 is split between three departments in the ratio 3:2:1. How much does each department receive?

• Total parts = 3 + 2 + 1 = 6
• One part = £120,000 / 6 = £20,000
• Department A = 3 × £20,000 = £60,000
• Department B = 2 × £20,000 = £40,000
• Department C = 1 × £20,000 = £20,000

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What Is a Rate?

A rate compares two quantities with different units. Common rates include:

Rate	Units	Example
Speed	km per hour	60 km/h
Heart rate	beats per minute	72 bpm
Mortality rate	deaths per 100,000 population	45 per 100,000
Incidence rate	cases per 1,000 per year	8 per 1,000 per year
Cost rate	£ per item	£3.50 per unit

Unit Rates

A unit rate is a rate expressed per single unit (e.g., per 1 person, per 1 hour).

Example: 150 patients are treated in 6 hours. The unit rate is 150/6 = 25 patients per hour.

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Comparing Quantities with Different Bases

A common UCAT trap is presenting two quantities with different bases and asking for a comparison.

Worked Example

Hospital A treats 500 patients and has 20 complications. Hospital B treats 300 patients and has 15 complications. Which hospital has the lower complication rate?

Hospital A: 20/500 = 0.040 = 4.0%

Hospital B: 15/300 = 0.050 = 5.0%

Hospital A has the lower complication rate (4.0% vs 5.0%), even though Hospital B has fewer complications in absolute terms.

Key Lesson: Always convert to the same basis (usually a rate or percentage) before comparing. Absolute numbers alone can be misleading.

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Scaling

Scaling involves adjusting quantities proportionally to answer "what if" questions.

Direct Proportion

If two quantities are in direct proportion, doubling one doubles the other.

Example: If 5 nurses can complete a vaccination programme in 8 hours, how long would it take 10 nurses?

Assuming the work scales linearly: 10 nurses do twice the work per hour, so the time halves.

$\text{Time} = \frac{8}{10/5} = \frac{8}{2} = \textbf{4 hours}$Time=10/58​=28​=4 hours

Inverse Proportion

If two quantities are inversely proportional, doubling one halves the other. The above example is inverse proportion between number of nurses and time.