Rates
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You will likely see many Quantitative questions involving rates and percentages.
You may have learned that a rate of change is a comparison of two quantities, with one changing in relation to another. For example, miles per gallon is a rate of miles driven for each gallon of fuel used.
In everyday maths, and in the UCAT, we can define rates more broadly: a rate is a comparison of two quantities that have different units.
This allows for endless possible rates that compare numbers with different units. For example, you could have data transferred per second (in download speeds) or songs per kilometre (listening to a music playlist whilst driving or jogging).
A percentage is a type of rate measured as a part per hundred.
A common rate in everyday life – and in UCAT QR – is speed.
You are no doubt familiar with the speed formula:
Like many frequently tested maths concepts in QR, speed is expressed in a three-part formula. This means the formula includes three possible variables.
You must select values from the data for two of the parts, then plug into the formula to solve for the unknown value.
Be prepared to rearrange the speed formula as needed to answer any questions in Quantitative Reasoning.
- Distance = Speed × Time
- Time = Distance ÷ Speed
- Speed = Distance ÷ Time
You may prefer to use the Speed Triangle to quickly and accurately visualise the three parts:
Cover the unknown part of the Speed Triangle to remember if the other two values are written as a fraction (T = D ÷ S or S = D ÷ T) or side-by-side multiplication (D = S × T).
Practise rearranging the speed formula until solving for any variable becomes second nature.
Walkthrough question
Speed
Try this QR question that uses the speed formula.
When you work with the speed formula, keep an eye on the units in case the units change from the data to the correct answer.
This can be more obvious, or it could be quite subtle:
- The data may include data labels in each cell, or in the row and column headings.
- You may have to read the text above or below the visual data to confirm the units used in the data.
- Look out for units in the question stem but not with each answer choice.
- You may find units beside each answer choice, which could be the only indication that the answers do not match the original units.
Walkthrough question
Speed with changing units
Now attempt this question with changing units.
Rate questions measure how one value changes relative to another and can involve any units.
You will often convert while you work through the steps of solving, particularly when it is fastest to do so. Other times, it is simpler and quicker to wait and convert as the final step.
Always double check for matching units before you click your answer. You may spot a wrong answer trap waiting for someone who forgets the final unit conversion.
You should not need to know how to convert between metric and imperial units. If you must convert from miles to kilometres or litres to gallons, the UCAT will provide the conversion factor.
However, you are expected to convert between metric units and convert between common lengths of time without a conversion factor. The UCAT will not specify how to convert seconds to minutes, hours, days or weeks, or how to convert years to quarters, months, weeks or days.
- 60 seconds = 1 minute
- 60 minutes = 1 hour
- 24 hours = 1 day
- 7 days = 1 week
- 4 quarters = 1 year
- 12 months = 1 year
- 52 weeks = 1 year
- 365 days = 1 year (the UCAT may or may not specify non-leap year)
Practise these conversions for speed and accuracy, so that you can do them almost instantly – or work briskly with the calculator – on Test Day.
The word ‘per’ indicates division, just like ‘the rate of A to B’ means ‘A divided by B’.
The following words and symbols indicate the same maths:
- Rate of A to B
- A per B
- A by the B
- A for every B
- A:B
- A ÷ B
This means that any division or any fraction can effectively describe a rate. Further proof that rates can involve any units and could help in solving many QR questions.
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Walkthrough question
Rates
Let's work through a QR question that involves a rate.
Some questions may ask about rate more subtly by focusing on trends in the visual data.
Since a rate indicates relative change of one value against another, you can find rates represented in graphs. For example, in a line graph, changes in the gradient of a line indicate that the rate is increasing (sloping upwards) or the rate is decreasing (sloping downwards).
Use the steepness of the gradient as a further hint. A line sloping upwards more steeply has a faster rate of increase than one sloping upwards less steeply. The inverse applies to a line sloping downwards more steeply, as it must have a faster rate of decrease than another line sloping downwards less steeply.
Walkthrough question
Trends in visual data
Try this QR question using rates with visual data.