Quantitative Reasoning

Overview of the section

The Quantitative Reasoning section is the closest to a maths section that the UCAT exam has. To succeed in this section, you need basic GCSE maths skills and problem-solving capabilities. The section consists of 36 questions which have to be completed in 25 minutes. There will be various types of question and scenarios that you will need to problem solve, and the data will come in different formats, including numbers, shapes, tables and graphs.

The answer format is multiple-choice, where you will need to select the correct answer from five different options.

An on-screen calculator is provided, allowing you to carry out simple calculations if you do not wish to do them in your head. Another option is to use your whiteboard to do quick maths.

In what follows, we will lay out some guidance and tips we think are important to follow in the Quantitative Reasoning section. We then provide a dedicated section on tax and interest, as these are frequent areas of confusion and mistakes in the UCAT, but are topics which appear regularly. Finally, the rest of the guide is dedicated to the key maths required for this section. Learn the content of this guide and you will be ready to perform very highly in the Quantitative reasoning section of your UCAT.

Some key tips

Excluding options that are definitely incorrect

– You not have much time in this section, so, when approaching questions, it can be helpful to begin by immediately excluding any obviously incorrect answers.
– This may be possible based on the ‘magnitude’ or size of the answer: if one answer is clearly much larger or much smaller than the others, it is likely that it is an outlier that can be immediately ruled out.

Rounding

– Instead of using valuable time typing long numbers into the onscreen calculator, or calculating them on your whiteboard, round them up or down to simpler numbers.
– With practice, you can learn to round effectively so that the calculations are easier and quicker to make, but so that answers can still be effectively differentiated.

Paying attention to units

– While this won’t make up the main calculation of a question, a sneaky unit conversion may be required in order to reach the correct answer.
– Later in this guide, we have listed some important unit conversions to memorise so you can complete this step quickly.
– Any conversions involving uncommon units, i.e. yards or inches, will be provided for you. You just need to ensure you are able to use the information given in order to calculate the conversion correctly.

Skim reading

– Sometimes, questions will follow after long swathes of text that will not always need to be read in detail. We recommend reading the actual question at the bottom of the text/information before reading the text itself. This technique allows you to tackle the text knowing precisely what you are looking for, and thus saving valuable time.
– Of course, at times the question will require a full understanding of all the information given and the whole text will need to be worked through, but hopefully you should have saved more time for these questions by being fast in the more straightforward questions.

Tax and interest questions
Often in the Quantitative Reasoning section, questions are asked which require you to have a good understanding of tax and interest calculations. These questions can seem daunting, so we will now walk you through the knowledge required to help you tackle these subjects confidently, hopefully helping to make you feel more comfortable when these questions come up.

What is a tax bracket?
– A tax bracket is an income range to which a specific tax (or interest) rate applies.
– In practice, and in most economies, this means that as you earn more, you are taxed more. Or if you have more money in your bank account, you will gain more interest on it.

For example:

Question: How much tax would a person pay if they earned £17,000 per year?

An easy mistake to make here is to reason that if a person earns £17,000, then they pay 20% tax on all £17,000 (because of the fact that £17,000 falls within the 20% tax bracket of £15,000-£30,000), giving a tax bill of £3,400 (£17,000 x 30% = £3,400). However is incorrect! This would be the correct answer if you were calculating simple tax, but this question relates to tax brackets and as such includes staggered tax rates.

Instead of working out 20% of £17,000, you need to calculate how much is earned within each individual tax bracket, then add the individual totals together:

• 15,000 at 10% tax = £1,500
• 17,000-15,000 = £2,000 in the next tax bracket, so 2,000 at 20% tax = £400
• So, the total amount of tax paid on £17,000 would be £1,500 + £400 = £1,900

Interest calculations also frequently come up

Simple interest

• For example, you are asked to calculate the interest that will accrue on £500 at 10% simple interest per year over 5 years.
• This means that each year 10% of the initial investment is added as interest. 10% of £500 is £50, so £50 interest per year is added. So, after 5 years, £250 interest would have accrued.
• To calculate simple interest you can follow this simple formula:

I = AxRxY
Interest = Amount of money x interest Rate x number of Years

Compound interest
• Compound interest differs from simple interest in that the calculation each year is based on the initial amount plus the interest that has accrued in the previous year. Some people like to refer to compound interest as ‘interest on interest’. It is best explained with an example.

If a sum of £1,000 receives compound interest of 20% each year for three years, what is the final sum?

Here is a formula that can calculate compound interest:

F = O(1+R)Y
Final amount = Original amount (1 + interest Rate)no of Years

The interest rate should be used in this equation as a decimal.

So, in this case the answer = 1000(1+0.2)3 = £1728

If you don’t want to use the formula, you can calculate the final amount from a series of calculations:

1000 x 1.2 = £1200 at the end of year 1
1200 x 1.2 = £1440 at the end of year 2
1440 x 1.2 = £1728 as the final amount after 3 years

Percentage increase/decrease

• As seen in the above example, being able to calculate the percentage increase/decrease of a number quickly is vital for working through the questions in this section speedily.
• When increasing a number by a percentage, you can do this in one of several ways:What is a 5% increase on 100?

1) Directly calculating 5%:
5% = 0.05
Therefore, 100 x 0.05 = 5
So, 100 + 5 = 105

2) Reducing the steps:
5% = 0.05
The final percentage will = 1.05 (5% addition on the original amount = 100% +5%)
So, 100 x 1.05 = 105

3) Calculating 1% first:
1% = 100/100 = 1
And, 1 x 5 = 5
Therefore 5% of 100= 5
So, 100 + 5 = 105

• To calculate percentage decreases, repeat whichever method you prefer from the above options, then simply subtract your final percentage from the original amount.
• You can use the same time saving trick as described in option 1 for a percentage decrease:

What is a 10% decrease on 200?

Here, the final amount will represent 90% of the original amount (100- 10 = 90)

So, 200 x 0.9 = 180

Some key calculations/formulae:

• Mean
  Sum all of the numbers together and then divide by the total individual numbers there are.

E.g. What is the mean of 2,4,6 & 3?
2 + 4 + 6 + 3 = 15
15 4 = 3.75

Top Tip – mean is the meanest calculation! Therefore, you can remember means by knowing that they will be the hardest calculations – the one with the most steps!

• Mode
  The mode is the number which appears most frequently in a set of numbers.

E.g. What is the mode of the following numbers: 7, 9, 11, 4, 2, 7, 34, 2, 7?

Place the numbers in order: 2, 2, 4, 7, 7, 7, 9, 11, 34

7 appears most therefore the mode is 7

Top Tip – think ‘mode’ sounds like ‘most’, so it is the number which occurs the most often.

• Median
  The median represents the middle value of a set of numbers.

E.g. What is the median of the following numbers: 3, 6, 6, 6, 7 ,9, 11, 11, 11?

The number in the middle is the median. If already ordered, you just need to tick off the number on either side until you reach the middle which in this case is 7. Otherwise, you first need to list the numbers in order.

If the median lies between two numbers, you find the halfway point.

Top Tip – ‘median’ sounds loosely similar to middle, hence, it is the middle number, or halfway point.

• Probability
  Probability questions often account for a large proportion of the Quantitative Reasoning section.

Probability is often calculated when you wish to find out the chance of something happening. Here is a helpful formula for calculating probability:

E.g. What is the probability of rolling a 6 on a dice?

⅙ can also be written as a decimal = ⅙ = 0.167 (the 6 is recurring so round to the nearest decimal).

This can be converted to a percentage by multiplying by 100 = 16.7%

• Percentages
Percentages are another high yield topic in this section of the UCAT.

Some simple equations are all you need to remember for these questions:

Percentage = Given amount/Total amount x 100
Percentage change = Difference/original x 100

Some percentages and decimals you should consider memorising to save you time using the calculator in your test are:

1/10 = 0.1 = 10%
1/2 =0.5 = 50%
1/3 = 0.3333 = 33.33%
1/4 = 0.25 = 25%
1/5 = 0.2 = 20%
1/6 = 0.167 = 16.7%
1/7 = 0.142 = 14.3%
1/8 = 0.125 = 12.5%
1/9 = 0.1111 = 11.11%

Highlighted are the ones we think you definitely should know of by heart.

• Ratios and proportions

A proportion is the amount something is compared to the whole or total, whereas a ratio is the amount compared to another amount.

E.g The ratio of boys to girls in a class might be 6:4. Meaning there iare 6 boys for every 4 girls in the class. However, as a proportion this means that there are 6/10 boys in the class (or 60%) and 4/10 girls (or 40%).

When calculating the proportion from a ratio you add the two numbers together:

6:4 -> 6 +4 = 10

While the total number of boys and girls in the class may not be 10, it will be a multiple of 10, so when calculating the proportion, you can use this number.

To prove this, imagine there are 12 boys and 8 girls. The ratio is still 6:4 as you can divide to the smallest numbers (ie. a ratio of 12:8 = 6:4 = 3:2 = 1.5:1)

12 boys out 20 total gives 12/20 = 0.6 (x100 = 60%)
8 girls out of 20 gives 8/20 = 0.4 = 40%

• Time and distance calculations
A common theme throughout the UCAT is the need to use the following formula:

Speed = Distance/Time

This can be rearranged to calculate out any of the variables.

One common way people do this is by using the triangle method:

For each part of the triangle, the equation can be altered to give the required result:

1. If you want to calculate the speed, focus on the S and you can see the D is over the T like a fraction. To calculate the speed, you divide D by T, so:
   Speed = Distance/Time
2. If you wish to calculate the time, it is the D that is over the S, so, Time = Distance/Speed.
3. Finally, to find the Distance, S and T are next to each other like a multiplication equation: Distance = Speed x Time.

Top Tip: One common way this speed and distance questions come up in the Quantitative Reasoning section is through the use of bus/train timetables. Make sure you practise reading these!

• Rates of change

Often questions will come up which test your ability to calculate a rate, be it the rate of change of speed (acceleration/deceleration) the rate of distance over time (speed) or rates of flow. Some tips:

– It is important to be able to calculate these things and comfortably handle the units they are measured in.
– Calculating speed is discussed above.
– Calculating acceleration = the change in velocity over the change in time and is often measured in m/s2
– Calculating flow rate = volume flowing / the time and is measured in m3/s or L/min.

• Unit conversions
Often you will be expected to convert quickly between different units. Here are some you should know going into your exam:

1m = 100 cm = 1,000mm
1km = 1,000m = 100,000cm
1km2 = 1,000,000m2
1g = 1,000mg
1kg = 1,000g
1 (metric) tonne = 1,000 kg = 100,000g
1dL = 100mL
1L = 10dL = 1,000mL
1m3 = 1,000L
1L = 1000cm3

• Shapes: areas, perimeters, surface area and volume
You should learn all the basic key geometry equations ahead of your UCAT exam, as questions relating to these often come up.

Note about π – this can be considered to equal 3.14. However, often in the UCAT the answer will incorporate π so you don’t actually need to calculate it as a number (e.g. if calculating the perimeter of a circle and the diameter = 4cm, the answer will almost certainly simply be 4πcm, rather than 12.56cm).

2D shapes:

– Square

– Perimeter: 4 x length of sides
– Area: length of sides2

– Rectangle

– Perimeter: (2 x length) + (2 x height)
– Area: Length x height

– Circle

– Perimeter: 2 x π x radius OR diameter x π
– Area: πr2

– Triangle

– Perimeter: side 1 + side 2 + side 3
– Area: ½ x base x height
– Pythagoras theorem: a2+ b2= c2

– Parallelogram

– Perimeter: side 1 + side 2 + side 3 + side 4
– Area: base x height

3D shapes:

– Sphere

– Surface area: 4π x radius2
– Volume: 4/3 π x radius3

– Cube

– Surface area: 6 x length2
– Volume: length3

– Cuboid

– Surface area: 2(length x width) + 2(length x height) + 2(width x height)
– Volume: Length x width x height

– Cylinder

– Surface area: 2 x π x radius2 + 2 x π x radius x height
– Volume: π x radius2 x height

– Pyramid

– Surface area: Area of the square + 3 x area of the triangle (use above equations)
– Volume: (length x width x height) / 3

• Permutations – not common but worth knowing about

Some of the questions you can be asked to solve in the Quantitative Reasoning section may involve permutations. These are best explained with an example:

If you are told there are five students sitting in a classroom, how many different ways can the students arrange the way in which they are sitting?

– In this case there cannot be repetition, as there are only five individual students, so any arrangement of the students can’t include more than one of each student.
– i.e. An arrangement could be students 1,3,4,5 and 2 sat in that order.
– However, the order couldn’t be students 2, 4, 4, 1, 5 as this does not include student 3 and there are not two of student 4.
– In order to calculate the number of possible permutations, we use the factorial function – which in maths is represented by the exclamation mark (!).
– What the ‘!’ does is to calculate the total number of permutations by multiplying the series of numbers with each other.
– In the case of our five students sitting in a classroom, the calculation is:

5! = 5 x 4 x 3 x 2 x 1 = 120

– So, in a class of five students, there are 120 different ways in which they can arrange their seating.

An alternative type of permutation question will ask how many different arrangements of something there may be, where, unlike the previous example, there can be repetition. An example would be:

On a bike lock with five different numbers, where the choice for each individual number is 0-9, how many different combinations are there?

– In this case, the combination could be five different numbers, i.e. 7,2,4,9,1, but of course it can also include repeated numbers, i.e. 4,4,4,4,4. There are lots of possible combinations!
– So, to calculate the answer, you need to multiply the series of individual numbers, or ‘wheels’ on the lock, by the number of choices available on each individual wheel:

10 x 10 x 10 x 10 x 10 = 100,000

– A simpler way of writing this is n^r where n is the number of ‘options’ and r is the number of times this option occurs. In the above example, therefore, there are 10 individual options (as each ‘wheel’ can be set to any number between 0-9), so n = 10, and there are five times each option can occur (the five ‘wheels’ on the lock), so r = 5.
Therefore:

105 = 100,000

A slightly more complex example:

in how many different ways can four boys and three girls be seated in a row so that they alternate, one boy sat next to one girl, etc?

– In order to calculate this, you need to combine the permutations:
– 4 boys can be seated in 4! ways and 3 girls can be seated in 3! Ways.
– To reach the answer, therefore, you simply multiply the factorials:

N = 4! x 3!
So, N = (4 x 3 x 2 x 1) x (3 x 2 x 1)
So, N = 24 x 6
So, N = 144

Finally, a slightly different mathematical problem is called a combination. As ever, this is best explained with an example:

In how many different ways can two people be chosen from a room of 20 people?

– The relevant equation = C (n,r) = n!/(r!(n-r)!)
– The answer to the question, and the number of ways that this scenario can occur is represented by C(n,r)

Where n = the number of options (in the case of this question, 20) and r = the specific number we are interested in (in the case of this question, 2). So, using the equation, we can calculate that:

C (20,2) = 20!/(2!(20-2)!) = 20!/(2! x 18!) = (20 x 19 x 18 x 17 x 16 ect.)/((2×1) x 18 x 17 x 16 ect) = (20 x 19)/(2 )= 190

A final note
Most of the maths and problem-solving knowledge required to succeed in the Quantitative Reasoning section is basic. Having to complete 36 questions in 25 minutes, however, creates considerable time pressure for most people. It is vital, therefore, that you practice. Plenty of practice using the equations and processes we have described in this Guide will help you to complete all the questions in a timely manner when it comes to the exam. Once you have prepared adequately, this should be a section you feel confident to complete well on test day!