Medic Mind Quantitative Reasoning

Quantitative Reasoning UKCAT Course Book

Theory & Technique

Mock Questions

Step-by-Step Guides

Detailed Explanations

Motivate. Mentor. Maximise.

Estimating vs. Working Precisely LESSON 2

What is the benefit of estimating? In the UKCAT you will face a variety of questions, each requiring a different level of precision. In an ideal world you would use your calculator to work out each answer to several decimal places, and then check your answer. But this is just not possible due to the timing constraints. Estimating will save you time in several ways: • You can use rounded figures that are easier to work with quickly • It can make it easier to do mental maths • You can sometimes answer by inspection, without having to do any calculations.

What is the benefit of working precisely? Working precisely, bearing in mind time constraints, is the most ideal method: • It will get you closer to the actual answer. • You avoid missing the answer due to the inaccuracy of your estimated answer.

When should you estimate? Estimate when: • The answer options are spaced out. For example, if the answers are to the nearest thousand, and the data is given to the nearest ten, you can round the data to the nearest hundred. • The question asks for the ‘approximate answer’, or it begins with ‘estimate’. It might sound simple, but many students miss these subtle signs when they read the question.

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Work precisely when: • The answer options are close together. For example, if the answer options are 0.1, 0.2, 0.3, 0.4 and 0.5, you have to work precisely and at least to the nearest 1dp. • The question asks for the ‘exact answer’.

What is the exact value of the drink?

What is the approximate value of the cake? A. B. C. D.

A. B. C. D. E.

vs.

£20 £28 £39 £44

£0.25 £0.26 £0.27 £0.28 £0.29

Example Scenario 1 The following table is information on football clubs in the Premier League and their stadiums for the 2016/2017 season. Team

Average Attendance

Lowest Attendance

Highest Attendance

Total Attendance

Stadium Capacity

Middlesborough

30,466

27,395

32,704

272,200

33,746

Arsenal

59,999

59,962

60,039

479,993

60,335

Bournemouth

11,167

11,029

11,355

100,508

11,464

Southampton

30,466

27,395

32,704

274,200

32,689

Crystal Palace

24,974

23,503

25,643

224,773

26,309

Liverpool

52,873

51,232

53,218

330,116

54,074

Manchester United

75,284

75,251

75,326

602,278

75,731

Tottenham

31,473

31,211

31,868

283,259

32,000

Chelsea

41,522

41,424

41,622

332,177

41,841

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1. Which of these clubs fills on average 93.1% of their stadium? A. B. C. D. E.

Arsenal Bournemouth Middlesborough Southampton Liverpool

2. Which club has the greatest ratio between highest attendance and lowest attendance? A. B. C. D. E.

Tottenham Liverpool Southampton Crystal Palace Manchester United

3. Wembley stadium has a capacity of 90,000. It was filled with Bournemouth and Arsenal fans for a match. What proportion of Wembley’s capacity was filled? Use the average attendance figures. A. B. C. D. E.

79.05% 79.06% 79.07% 79.08% 79.09%

4. On average, which club fills its stadium capacity the most? A. B. C. D. E.

Chelsea Bournemouth Arsenal Crystal Palace Tottenham

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5. Which of these clubs have the greatest difference in average attendance? A. B. C. D. E.

Liverpool and Arsenal Tottenham and Crystal Palace Chelsea and Southampton Middlesborough and Bournemouth Manchester United and Liverpool

6. Arsenal sell their tickets at £40. What price would Liverpool have to sell at to make the same sales revenue for an average match? A. B. C. D. E.

£30 £35 £45 £50 £55

Estimating with graphs and pie charts With graphs and pie charts you can often estimate depending on the question. If a graph only has axis labels to the nearest 10, don’t worry too much about reading the plot to the nearest single unit.

Overestimate vs. Underestimate If you are estimating, you should check whether the final answer is an underestimate or overestimate. For example, if instead of multiplying 41.25 x 82.3, you did 40 x 80, your answer would be an underestimate, so the correct answer is slightly higher. For example, if instead of dividing 80 by 8.4, you did 80 divided by 8, your answer would be an overestimate, so the correct answer is slightly lower.

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Estimating to check your answer If you are unsure about an answer, you can do a quick estimate to check if you followed the correct general direction. Use approximate figures to get an estimate, and check whether your actual answer should be an overestimate or underestimate. e.g. If the interest paid on a loan is paid in terms of compound interest at 3% each year, what is the amount owed on a £1560 loan after 3 years? The method involves doing 1560 x (1.03) 3 = £1794.65 Do a quick estimate using simple interest to make sure you are going along the right lines. 1560 x 1.09 = £1700, and you know this has to be an underestimate. This is a good check for certain questions (not every one!) to make sure you haven’t gone completely off track.

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Reading Question Information LESSON 3

Only Read When You Have To With Verbal Reasoning we saw that it was best not to read all of the information, but instead to skip straight to the passage. With Quantitative Reasoning, the approach is more dependent on the specific type of question you are tackling: • For questions with a simple data (e.g. simple graph or pie chart), it may be worth reading the data properly if it doesn’t take too much time. • For questions with complex data (e.g. multi-columned tables), it is worth reading the column labels to understand what the table is showing, but you should not spend time reading every single column. Instead, look at the question and find the relevant data when searching for the answer.

Beware of Bullet Points Often tables will have bullet points underneath relating to the information shown. For example, the table may show the prices of a Fun Fair on different days of the week, and a bullet point below might tell you that ‘for children the price is discounted by 10% on Tuesday’. It is easy to miss this because your natural instinct is to jump straight to the data set, but it is important to be wary of information such as this.

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Example Scenario 2 The table below shows the prices at Claire’s local cinema. • • • • •

Student ID required for student tickets Government ID required for Child and Senior prices Children tickets only for under 12s. Group discounts do not apply on weekends On Tuesday there is a 10% discount on all prices Ticket Type

Price (£)

Child

6

Student

8

Adult

12

Senior

5

Family (Group of 4)

32

7. John (47) is going to the cinema with his daughter, Lucy (11), on Monday, and again by himself on Tuesday. He is paying for all the tickets, for both himself and Lucy. How much does he have to pay? A. B. C. D. E.

£27.50 £28 £28.80 £30 £32

8. Deraine (68) is taking her family to the cinema on a Friday. She has a son (30), a granddaughter (6) and her daughter’s friend (8). Deborah’s brother, Erald (56) is also considering coming. How much extra would it cost if Erald came? A. B. C. D. E.

£6 £7 £8 £9 £10

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9. Tony goes to the cinema by himself once a month from his 10th Birthday until his 27th Birthday. He was a student at Leicester University for 3 years. How much have all his tickets cost? A. B. C. D. E.

£177 £1432 £1980 £2124 £2268

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Units in the UKCAT LESSON 4

Beware of Units UKCAT students often lose marks over units, in several ways: • Unit variation between scenario and question. Sometimes the data will have different units to the question, and you may forget to convert. For example, the scenario information might tell you the distance in kilometres, but the question asks ‘how many miles?’. In this question they would give you the unit conversion. • Unit variation amongst answer options. Sometimes the various answer options will have different units, so be careful to select the correct one. You may work out that the answer to a question is 10kg, and then see 10g in the answer and choose it, when in fact you should have chosen 10,000g. • Spotting units in tables. For tables the units will be shown in the row and column labels, rather than in the table itself. In this case it is particularly easy to miss the units and make silly mistakes. If the question gives you a unit conversion, then it is very likely that you have to use it. So be wary of the units in the data, question and answers.

When do you convert the units? For certain question it is fine to convert units at the end of the question. However, if there are several calculations and different units in a question (e.g. in speed - km/h), then you will find it beneficial to convert from the offset. Jack runs the London Marathon in 4 hours and 20 minutes. The Marathon is 41.4 km long. Calculate his speed in metres per minute. Converting units at the end Speed = Distance / Time = 41.4 / 4.333 = 9.56 km / h You need to multiply by 1000 to convert into metres = 9560 m / h You need to divide by 60 to convert to minutes = 159 m / m

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Converting units at the start Speed = Distance / Time = (41.4 x 1000) / (4.33 x 60) = 159 m / m You could point out that, with the UKCAT calculator, you would have to do 3 calculations anyway. But with the second method you are less likely to slip up, especially for even longer calculations.

Example Scenario 3 The table below presents the temperature on a particular day in different countries.

Country

Temperature (oC)

Australia

38

England

22

Thailand

29

Chile

35

Canada

23

Temperature in F = Temperature in C x 1.8 + 32 10. Which country has a temperature of 71.6 Fahrenheit? A. B. C. D. E.

Australia England Thailand Chile Canada

11. The thermometer is not always accurate, so the real temperature could be 2 C more or less. What is the maximum difference possible, in Fahrenheit, between two countries? A. B. C. D. E.

60.8 64.4 68.0 71.6 82.5

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Weighted Means LESSON 5

Calculating Weighted Means Weighted mean questions can come up in the UKCAT, and often students have not revised how to answer them. They can be slightly difficult to get your head around at first, but with some practice they should be fairly simple. Weighted mean = Σ (each value x individual weight)

Example: Simple Mean We will walk through the topic step-by-step, starting with the bread and butter: Simple Mean. What is the mean of 1, 2, 3 and 4? When we calculate a simple mean (or average), we give equal weight to each number. Normally we add up the numbers, and divide by the number of figures: Mean

=

1+2+3+4

=

10 / 4

=

2.5

Applying weights… We could think that each of those numbers has a "weight" of ¼ (because there are 4 numbers). Remember, for weighted means, we multiply each value by its individual weight, and then sum: Mean = (¼ × 1) + (¼ × 2) + (¼ × 3) + (¼ × 4)

=

0.25 + 0.5 + 0.75 + 1

=

2.5

Same answer. Changing the weights… Now let's change the weight of 3 to 0.7, and the weights of the other numbers to 0.1 so the total of the weights is still 1: Mean = (0.1 × 1) + (0.1 × 2) + (0.7 × 3) + (0.1 × 4)

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=

0.1 + 0.2 + 2.1 + 0.4

=

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2.8

This weighted mean is now a little higher - it has been “pulled" there by the weight of 3.

Worked Example: Weighted Means

In the below example, the weights are not given as % or fractions. Therefore you have to set up your own fraction to work out the weight.

Worked Example: Weighted Mean There are 10 boys and 20 girls in a group. The average age of the boys is 17 years and the average age of girls is 20 years. What is the average age of the group? A. B. C. D. E.

18 18.5 19 19.5 20

Identify which figures you will use to calculate your mean. These are 17 and 20. • •

17 has a weighting of 1/3 20 has a weighting of 2/3

Weighted Mean = (17 x 1/3) + (20 x 2/3) = 19 years

Wait! Why don’t you just add up the total years and divide by total students? For this particular question this approach can also work, but in longer questions it can be quite laborious, so it is good to practice the above approach.

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Example Scenario 4 Sathu wants to buy a new camera, and rates each camera based on Resolution, Battery Life, and Ease of Use. However, he weights each factor differently: Resolution = 50% weighting Battery Life = 30% weighting Ease of Use = 20% weighting The cameras scored as follows: Camera

Resolution /10

Battery Life /10

Ease of Use /10

Cony

8

6

7

Sanon

9

4

6

12. Which camera is the best? A. Cony B. Sanon C. Neither

13.Sathu’s friend, KJ, is going to buy a camera too. He cares about price too, and weighs the quality score (made up of resolution, battery life, ease of use) at 40%, and price at 60%. The updated scores are as follows: Camera

Resolution /10

Battery Life /10 Ease of Use /10

Price /10

Cony

8

6

7

5

Sanon

9

4

6

8

Which camera is better for KJ? A. Cony B. Sanon C. Neither

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Example Scenario 5 Jude usually works 7 days a week at Endsleigh Court, but sometimes he only works 1,2 or 5 days. This year Jude worked: • • • •

For 1 day per week 2 times For 2 days per week 14 times For 5 days per week 8 times For 7 days a week 32 times

14. What is the mean number of days Jude works per week at Endsleigh Court? A. B. C. D.

3.75 4.25 5.25 6.75

15.Jude gets paid £60 per day, which he can spend in his home town in Thailand. How much does he earn a year in Thai Currency (£1 = 44B)? A. B. C. D.

13,860 16,380 137,280 720,720

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Area Questions LESSON 6

Units in Area Questions When you get area questions, before multiplying together the dimensions, remember to convert into the correct units. For example, if a square was 10cm x 10cm: Area in cm = 10 x 10 = 100cm2 Area in m = 0.1 x 0.1 = 0.01 m2 If you forgot to convert, you might have converted 100cm2 to 1m2. Don’t fall into this trap!

Finding shortcuts Wherever possible, you should try to cut down the number of calculations you have to do. This is especially important considering the calculator you are given in the exam.

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Example Scenario 6

Claude has decided to paint his bedroom, which measures 16.5 x 9.9 feet. He has different colours for painting his floor: white, cream and violet. The prices are shown below:

Colour of Paint

Price per square metre (£ / m2)

Cost of labour to install (£ / m2)

White

40

6

Cream

20

8

Violet

6

4

1 metre = 3.3 feet 16. What would be the cost of buying cream to cover the entire bedroom floor? A. B. C. D. E.

£100 £280 £300 £360 £3267

17. What is the different in cost (for both labour and materials) between using White Paint and Cream Paint to cover the floor? A. B. C. D. E.

£270 £300 £450 £475 £2940

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18. Claude buys a benchpress, wardrobe and rug for his bedroom. The base of the benchpress measures 140cm x 200cm, the base of the wardrobe measures 50cm x 100cm, and the base of the rug measures 75cm x 40cm. Calculate the percentage of the floor that will be covered by these items. A. B. C. D. E.

22% 24% 25% 27% 28%

19. There is 20% off the original price to buy the Cream Paint. Claude has a budget of £1500 to redecorate. If the benchpress costs £400, the wardrobe £250 and he hires help to add the Cream Paint, how much money will he have left to spend on a rug? A. B. C. D. E.

£325 £370 £400 £460 £490

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Quantitative Reasoning Formulae LESSON 7

Memorising Formulae Often the question will give you the formula required, but in certain questions they may not. Therefore it is best to learn these formulae below:

Area of circle

πr2

Area of square

a2

Area of parallelogram

bh

Area of triangle

1/2 x b x h

Surface area of a cylinder

2πrh + 2πr2

Surface area of sphere

4πr2

Volume of cube

a3

Volume of sphere

4/3 x π x r3

Volume of cylinder

πr2h

Volume of cone

πr2 x h/3 lwh / 3

Volume of pyramid

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Ratios LESSON 8

What is a ratio? A ratio is a way to compare relative amounts of something. Recipes, for example, are often given as ratios. To make a cake you may need to mix 2 parts flour to 1 part sugar. This means the ratio of flour to sugar is 2 : 1.

Working out parts If a cake is 2 parts flour to 1 part sugar, then there are 3 parts (2 + 1) altogether. Two thirds of the cake is flour; one third sugar. If the mass of the mixture is 45, one part is 15. If a wooden chair is made up 20% by timber, and 80% by forest wood, the ratio is 1:4. There are 5 parts, 1 part timber and 4 parts forest wood. If the volume of the chair is 250cm3, then 1 part is 50cm3.

Ratios in Simplest Form Whenever you get given a ratio, you should aim to convert it to its simplest form. This means reducing each value to the smallest possible integer (whole number). For example, 6 : 3 can be simplified to 2 : 1.This can be done by inspection. For more complicated questions, it is harder to use inspection, so divide each number by the highest common factor (HCF). For example, 44 : 4 have a HCF of 4, so dividing each side by 11 gives 11 : 1. Having a ratio in its simplest form makes them easier to work with. However, if it is time consuming and difficult, it is not essential to simplify.

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Example Scenario 7 20.There are 16 students in a class. They all study one subject each in the ratio of 3:4:1 for Maths, English and German. How many students study Maths? A. B. C. D. E.

4 5 6 7 8

Example Scenario 8

21.164 students in a school choose one sport each. The ratio of Rugby, Cricket, Football and Netball is 24 : 44 : 40 : 56. What fraction of students are playing Netball? Give your answer in the simplest form. A. B. C. D. E.

14 / 41 56 / 164 6 / 41 11 / 41 10 / 41

Ratios with decimals When you are setting up ratios, you should have whole numbers on both sides. For example, if the ratio of milk to cheese in a pastry is 1.5 : 3.5, you should multiply both sides by 2 to get the ratio of 3 : 7. You can also estimate to set up a simple ratio if the question does not require an exact answer. If there are 125 children in a year group, 34 boys and 91 girls, you can simplify the ratio of 34 : 91 to 3 : 9 and then on to 1 : 3.

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Ratio in maps Maps scales can be written in ratios, which tell you how many units of the real world are equal to one unit on the map.


Example Scenario 9 22. A group of students are looking at a map of Berry Town. The scale on the map is 1 : 20 000. The distance from Earfield to King’s Men House is measured as 6cm on the map. How many km is this equivalent to in real life? A. B. C. D. E.

0.2 km 1.2 km 200 km 1200 km 120 000 km

23. The Berry Town Hall has undergone construction, and been redeveloped for 2 years. The actual project took 8 months less than originally anticipated. The Town Hall measures 2cm wide on the map, and 8m long. What is the actual area of the Berry Town Hall? The scale on the map is still 1 : 20 000. A. B. C. D. E.

0.064 km2 0.64 km2 1.6 km2 3.2 km2 6.4 km2

24. The group are using a new map for their hotel, Hot Springs Spa & Resort. This map has a scale of 1 : 50 000. At the Spa a rectangular pool has an area of 44m2. What is the area of the Spa on the map in in2? 1m = 38.37 inches A. B. C. D. E.

0.0007 in2 0.00014 in2 0.3344 in2 0.1364 in2 6.82 in2

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Adapting to Test Conditions LESSON 9

Using the Calculator

Using the Whiteboard In the exam you will be given a whiteboard to use. Here are some tips on how to use it effectively for Quantitative Reasoning: • Use it for multi-step calculations. Write down your working for multi step calculations because there is no Answer Function • Jot down as you eliminate incorrect options. When you knock out answers write down the letter of the incorrect answers as you go along because it is easy to forget which ones you crossed out. E.g. if you cross out A, make a note on your board.

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• Make algebraic formulae. Always try to make an algebraic formula wherever possible, because it can break down a complex question to make it much simpler for you. • Draw Venn Diagrams. Use Venn Diagrams for problems that need it- e.g. a question about the subjects three different classes in a school take. • Classify flagged questions as ‘guessed’ or ‘uncertain’. As with Verbal Reasoning, make a list of the question numbers which you had to completely guess. Then use the flagging function for questions you completely guessed, and questions you were unsure about. This way you can return to the numbers noted on your whiteboard first, and then the other flagged ones after if you have enough time.

Reducing Human Error When performing calculations in your calculator, for example summing a long list of numbers, read the numbers directly from the monitor screen / test paper instead of your rough working. It is easy to misread your scribbles. Also it cuts out one opportunity for human error (the incorrect writing down of numbers onto paper).

Doing an answer check Especially when working with graphs, do a quick check of your answer. You do not need to re-do the whole calculation, but you should check that you have answered what is being asked, have a go at estimating a range for where a sensible answer should lie. This helps your ensure that you haven’t done anything silly, or got your units wrong. You can still do a sense-check if your answer matches one of the multiple choice options. Distractors are often generated from common mistakes. For example, if a question has 4 steps: • • • • •

A mistake at step 1 might lead to incorrect answer A A mistake at step 2 might lead to incorrect answer D A mistake at step 3 might lead to incorrect answer B A mistake at step 4 might lead to incorrect answer C A perfect calculation with no mistakes leads to correct answer E. 


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Fractions LESSON 10

Fraction - Percentages - Decimals You should try to memorise the decimal and percentage equivalents of common fractions: 1/2 1/3 1/4 1/5 1/6

= = = = =

50% 33% 25% 20% 17%

= = = = =

0.5 0.33 0.25 0.2 1.67

1/7 1/8 1/9 1/10

= = = =

14% 13% 11% 10%

= = = =

0.14 0.125 0.11 0.1


Fractions of 7 For fractions of 7, memorise the sequence of numbers 1 4 2 8 5 7. • 1/7 starts with the smallest number out of these six (0.142857). • 2/7 starts with the second smallest number out of these six (0.2857) • 3/7 starts with the third smallest number out of these six (0.42857) … and so on. This is a useful tip to work out the sevenths.

Fractions of 8 For fractions of 8, use multiples of 0.125: • 1/8 is 0.125 • 2/8 is 0.25 • 3/8 is 0.375

… and so on.

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Fractions of 9 For fractions of 0, use multiples of 0.11. • 1/9 is 0.11 • 2/9 is 0.22 • 3/9 is 0.33

… and so on.

Working out percentage of a percentage When working out the % of a %, set up two fractions and multiply them together. e.g. 25% of 11% e.g. 14% of 66%

1/4 1/7

x x

1/9 2/3

= =

1/36 2/21

Example Scenario 10

There are 100 beetles in a box. 60 are green and 40 are red. 70 are of Species A, 20 are of Species B and 10 are of Species C. 80 have black spots, 5 have grey spots and 15 have white spots. 25. What is the smallest number of green, Species A, black spotted beetles that there could be in the box? A. B. C. D. E.

0 10 30 31 60

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26. Tom is taking one beetle out of the box, and is aiming to pick a Red, Species B, Grey Spotted one. However, before this Jackie removed 10 red coloured, white spotted Species C beetles from the box. What is the probability of picking Tom’s desired beetle after Jackie has taken her 10? Give your answer as a fraction in its simplest form. A. B. C. D. E.

2 / 27 1 / 243 1 / 162 2 / 486 2 / 324

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Interest Rates Lesson 11

Simple Interest Simple interest is when the amount of interest earned is fixed over time. For example, if you saved £100 at 30% simple interest, you would earn £30 per year. To calculate the interest earned over 5 years - (100 x 0.03) x 5 = £15 Simple Interest = Loan x Annual Interest Rate x No. of Years (Annual Interest Rate should be in the form of a decimal, so 5% would be 0.05)

Example Scenario 11

27. Tony took out a loan from Lakefield, his local bank. The interest rate was 5% per annum on the original amount year on year. If Tony borrowed £20,350 from Lakefield, how much interest does he pay over 4 years? A. B. C. D. E.

£1,017.50 £4070 £4386 £5000 £5350

Compound Interest Compound interest is when the interest is paid based on the amount already earned, leading to greater and greater amounts of interest over years. For example, £100 at 30% compound interest would earn £30 in the first year, but in the second year you would earn 30% of the new amount of £130 which would be £39. Compound interest is the most common type of interest used in real life. In the UKCAT you need to identify which type of interest they are asking you to calculate.

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How do you calculate compound interest? If you remember our ‘Percentages’ Tutorial, you calculate New by multiplying Original by Base Value. To calculate compound interest, you put the base value to the power of the number of years. New = Original x base value n

(n = no. of years)

In other words; Compound Interest

=

Loan

x

Annual Interest Rate

no. of years


Example Scenario 11 cont.

Tony has negotiated a new simple interest rate of 3.7% for another loan of £5,000.. Meanwhile, Tony’s brother, Rush, took out the same loan from a different bank, The Harvey Group, which charges compound interest 3.5%. 28. How much does Rush have to pay back after 3 years? A. B. C. D. E.

£5,175 £5,356 £5,544 £5,738 £5,938

29. After how many years will the interest paid be the same (to the nearest £5)? A. B. C. D. E.

1 year 2 years 3 years 4 years 5 years

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Tax Questions LESSON 12

Tax Brackets Banks tend to give you more interest as you deposit more into your account. This means that there are interest brackets. The same can apply for income tax rates.


Example Scenario 12

The table shows the total tax paid (£) on annual taxable income. For example, a person earning 12,000 will pay 10,000 at 10% and 2,000 at 15%.

Annual Taxable Income Bracket (£)

Tax Rate

0 - 10,000

10%

10,000 - 20,000

15%

20,000 - 35,000

25%

35,000+

35%

30. Kate Ward has an annual income of £24,000. How much income tax, to the nearest £, does she pay? A. B. C. D. E.

£2,000 £2,500 £3,500 £4,500 £6,000

31. James pays £450 of income tax per month. To the nearest £500, what is his annual income? A. B. C. D. E.

£30,000 £30,500 £31,000 £31,500 £32,000

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Venn Diagrams LESSON 13

When a question gives you a group of people and several categories, it can be very useful to set up a Venn Diagram.

Example Scenario 13

Rickin asked 60 people which sports they liked from rugby, football and cricket. 8 people like all three sports. 17 people like rugby and football. 13 people like football and cricket. 19 people like rugby and cricket. 35 people like football. 27 people like cricket 30 people like rugby. 32. How many people liked neither rugby, football or cricket? A. B. C. D. E.

6 7 8 9 10

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Percentages LESSON 14

The Challenge of Percentages Percentages can often trick students out, and as such are a common feature of the UKCAT almost guaranteed to come up on Exam Day! The Medic Mind technique for tackling Percentages is different to the norm, but we believe it is the fastest and most efficient way to tackle these questions quickly. So bear with it, even if it seems unusual at first!

Medic Mind Method: Percentages 1. Identifying the Three Variables in Every Question Every percentage change question will centre around three variables - original value, new value and percentage change. The question will give you two of the variables, and ask you to work out the third. Original Value → New Value

x % change

We can therefore make a triangle formula, just like we do for Speed, Distance, Time questions. 2. Using a Base Value Now it gets slightly tricky, so take it slowly. To represent % change we will use the base value. This is equivalent to the decimal you use for percentages on your calculator. First let us work on correlating base value and % change: 10% increase = base value of 1.10 10% decrease = base value of 0.90 40% increase = base value of 1.40 110% increase = base value of 2.10

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330% increase = base value of 4.30 You get the idea! The base value is there to save you time in the exam because then you do not always have to multiply or divide by 100 when using percentages. 3. Applying the triangle formula Now let us try applying it to the formula with the worked examples below.

Worked Examples What is the % change from 60 to 75? New / Original = Base

New = 75

Original = 60

Base = ?

75 / 60 = 1.25 1.25 = 25% increase

Tom got a 10% discount when purchasing a book, so had to pay £8. What was the original price? 10% decrease means that the base value is 0.9. New / Base = Original

New = 8

Original = ?

Base = 0.9

8 / 0.9 = £8.88

Percentage Question Types To re-cap, we have established that every question takes this format: Original Value → New Value

x% change

There are three parts- the original, new and % change. Each question type will give you two parts and ask for the third. Let’s do some practice: ••••••••••••••••••••••••••••••••••••••••••••••••• UKCAT

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Example Scenario 14

The table below shows the price of clothing worn by various members in the Medic Mind offices. All clothing is provided for free to the staff by Medic Mind.

Clothing Type

Small (£)

Medium (£)

Large (£)

White Shirt

15.50

17.50

21.00

Blue Shirt

16.50

18.00

22.00

Black Trousers

14.00

16.50

17.00

Blue Trousers

16.00

18.50

19.00

Navy Blazer

43.00

39.50

45.00

Black Blazer

40.00

42.00

41.00

Black Skirt

23.00

29.00

26.00

White Female Shirt

14.00

17.00

22.00

Black Leggings

16.25

18.00

19.70

Blue Leggings

22.70

14.50

17.40

• The males in the Accounts department wears White Shirts, Black Trousers, and any colour Blazer. The females can wear what they wish. • The males in the UKCAT department wears White Shirts, Blue Trousers, and Navy Blazers. The females wear White Shirts, Black Skirts and Black Leggings • The Student Support workers can all wear what they wish. • Females can wear one each of skirts, leggings, and shirts. Men can wear one each of shirts, blazers, and trousers.

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33. There are 8 males and 2 females in the Accounts department. What is the total cost of all the male clothing if they all wear Navy Blazers and are all Medium? A. B. C. D. E.

£73.50 £588 £604 £608 £735

34. What is the percentage increase from buying a female, small-sized UKCAT teacher outfit, to buying a male, medium-sized UKCAT teacher outfit. A. B. C. D. E.

38% 42% 46% 48% 52%

35. The price of all shirts increase by 10%, and the price of all blazers fall by 10%. What is the percentage change between the old and new price of a large male white shirt and large navy blazer. A. B. C. D. E.

6.8% decrease 2.4% decrease 3.6% decrease 3.2% increase 3.8% increase

36. There are 6 males and 12 females in the Student Support department. What is the maximum cost their clothing can be? A. B. C. D. E.

£149.70 £151.70 £159.70 £165.80 £182.50

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Speed, Distance, Time LESSON 15

Key Formulae 
 In the UKCAT they will often give you two elements of a three part formula, and ask you to work out the third. Speed

=

Distance / Time

Distance

=

Speed x Time

Time

=

Distance / Speed





 Common traps 
 Units - you should be very careful about the units you are working with because there can often be traps with units. 
 Doubling distances - if a question talks about travelling somewhere and coming back, don’t forget to double the distance!

Acceleration
 Formula
 Acceleration (m/s2)

=

Change in Velocity / Time Taken

For example, a car takes 5 seconds to accelerate from 25m/s to 35m/s. Its velocity changes by 35 - 25 = 10m/s.
 So its acceleration is 10 / 5 = 2m/s2.

Graphical representation On a speed time graph, the gradient will be equal to the acceleration. So a rising line means a constant acceleration and an increasing speed. The distance is the area under the graph.

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Example Scenario 15

The graph below shows the velocities of three different bikers on a 3 hour journey.

Bike 1

Bike 2

Bike 3

160

Velocity (km/h)

120

80

40

0 0

1

2

3

Time (h)

Acceleration (km/h2) = Change in velocity (km/h) / Change in time (h)

37. How much greater is the overall acceleration of Bike 1 than Bike 2? A. B. C. D. E.

0 53.33 36.66 80 110

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38. What was the difference between the distance covered by Bike 3 and Bike 1? A. B. C. D. E.

30km 60km 75km 90km 100km

39. What is the distance covered by Bike 2? A. B. C. D. E.

220km 240km 300km 400km 450km

40. What is the average speed of Bike 2? A. B. C. D. E.

20 km / h 30 km / h 40 km / h 100 km / h 150 km / h

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Tips from the Experts LESSON 16

1. Do Not Become Engrossed In The Data “It is tempting to read every part of the table, or every bullet point below a price list, but this wastes so much time!” In Quantitative Reasoning you need to extract the information that is required to answer the specific question being asked. You can use a keyword approach even though it is not Verbal Reasoning - for example if they ask you for the price of T shirts in a sale, find the column in the table about T shirts and ignore the columns about shorts, jeans and shoes.

2. The UKCAT Calculator Is A Game Changer “Having an on screen calculator makes the section much harder. I found it hard to use it on screen as I had always praticed using my handheld scientific one!” The calculator is hard to use, and therefore you should minimise the number of steps wherever possible. Also you need to practice with the calculator to get used to how it works and find the method that best suits you. Do not practice with a scientific calculator, as it will skew your timing and pace, leaving you for a shock on test day.

3. Practice Makes Perfect for Quantitative “After lots of practice you realise that they are the same questions with different numbers” The tutorials in this section cover the main difficult topics in Quantitative Reasoning, and if you keep practicing eventually each question type will become second nature to you. For example, if you have done 50 percentage questions in practice, in the exam it will be the exact same question with the numbers changed slightly. Practice makes perfect.

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Quantitative Reasoning - Test Day LESSON 17

When will this section be? Quantitative Reasoning is the second section on test day. Before this section you may want to ask for an extra whiteboard, as they often let you have on if you ask during the exam. You may also want to jot down some formulae on your whiteboard to help save time in the section.

Preparing for Quantitative Reasoning 4 weeks to go - Consolidating Techniques Each tutorial for Quantitative Reasoning is quite distinct, so make sure you have mastered each technique, method and formula before entering intense question practice. Especially for less common questions such as Tax Questions and Weighted Means, consolidate the theory first. 3 weeks to go - Practicing Questions Using UKCAT Calculator With some sections we tell you to adjust to the UKCAT conditions close to the exam. With Quantitatrive Reasoning, we want you to be using the official calculator from day one. With maths-based questions, they tend to repeat again and again with just different numbers each time, so practice does make perfect. 2.5 weeks to go - Focus on Timing Quantitative Reasoning is very time pressured, like most sections of the UKCAT. Begin to focus more on timing with 2.5 weeks to go - work on cutting down steps in a method, using mental maths, and picking out only the important parts of data. 1.5 weeks to go - Revisit Theory Use the online tutorials and course booklet to re-visit theory before course day. You should now memorise the formulae required for course day, and consolidate the step-by-step method for each question type. 1 week to go - Building Your Concentration Practice several Quantitative Reasoning mocks back to back to develop your concentration skills for test day.

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3 days to go - Adapt to Test Conditions As we described in detail in ‘Verbal Reasoning - Test Conditions’, you should replicate the exact test day with a whiteboard and an on-screen test. If you have been using your phone as the UKCAT calculator, now switch to the official one. One thing to bear in mind is that if you are doing questions on paper, such as in this book, you have the benefit of annotating the data they give you. In the UKCAT you do not have this, so bear this in mind and practice using the whiteboard. 1 day to go - Consolidate and Relax Consolidate the techniques by scanning through the Quantitative Reasoning tutorials. And then relax! Test Day - Read the Formulae Again On the day of the test you should just flick through the formulae table once more, and you should be ready for the test!

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Quantitative Reasoning Mock


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Scenario 1 Phillip is raising money for charity and ran a half marathon race. He completed this in 2 hours 26 minutes. In training, it took him an average of 2 hours 37 minutes. His quickest time was 2 hours 15 minutes, and longest time 3 hours 37 minutes. (5.28mph) A full marathon is 26.2 miles. 1. What was Phillip’s average speed on his race day? A.. 11.3 mph B. 10.8mph C. 5.38 mph D. 5.56 mph E. 5.22 mph 2. How much slower was Phillip in his race compared to his quickest time in training? A. 0.36mph B. 0.44mph C. 0.66mph D. 0.22mph E. 5.38mph 3. What was the percentage decrease of his time in the race compared to his average time in training? A. 4.6% B. 11.4% C. 5.3% D. 6.2% E. 7% 4. On one occasion, Phillip ran at 6.7 miles per hour, but only managed to complete 3/5ths of the half-marathon because he ran out of energy. How long did he run until he ran out of energy? A. B. C. D. E.

8 hours 26 minutes 1 hour 10 minutes 51 minutes 2 hours 11 minutes 1 hour 7 minutes

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5. If Phillip ran for the same length of time he did in his longest time at a speed of 5 miles per hour, how much further would he run than in his half-marathon? A. 3.16666 miles B. 18.08 miles C. 4.02 miles D. 4.98 miles E. 5.66 miles

Scenario 2 Tahmeed, Jeffrey and Ben went to watch Adele live. Tahmeed bought tickets for everyone, and they were £99.95 each, plus a booking fee of £13.56 on each ticket. He managed to get a 25% student discount on the ticket prices, excluding the booking fee. However he couldn’t get discount for Ben since he had graduated. 6. How much did Tahmeed pay for all three tickets? A. B. C. D. E.

£291 £266 £284 £277 £202

7. How much more did Ben’s ticket cost compared to Jeffrey’s? A. B. C. D. E.

£24.99 £25.01 £25.03 £24.96 £23.44

8. Anthony buys a student ticket. What percentage of the total cost is the booking fee? A. B. C. D. E.

15.2% 15.3% 15.5% 15.8% 14.8%

At the concert, Jeffrey bought Tahmeed a meal consisting of medium fries costing £1.65, a fish-o’fillet costing £3.79 and a diet coke for £4.76.

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9. How much does Jeffrey still owe Tahmeed after buying the meal? A. B. C. D. E.

£88.52 £78.32 £95.55 £83.55 £76.11

10. How many meals, including the one he has already purchased, would Jeffrey have to buy to pay back his ticket? A. B. C. D. E.

8 5 6 9 10

Scenario 3 This graph shows the sales of T-Shirts and Jeans in Timothy’s TrendCloset shop.

T-Shirts

Jeans

100

75

50

25

0

April

May

June

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July

August

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11. Which month had the largest total sales? A. B. C. D. E.

April May June July August

12. Which month had the largest difference between T-shirt and jeans sales? A. B. C. D. E.

April May June July August

T-shirts are priced at £5, and jeans are priced at £3 13. How much money was generated in June? A. £655 B. £1000 C. £470 D. £485 E. £560 14. Which months generated the most profit? A. B. C. D.

April and August July and June May and June April and May

15. If the price of T-shirts increased to £6, and jeans now cost £2 - what would be the change in money generated over all 5 months? A. £250 increase B. £310 decrease C. £80 increase D. £60 decrease E. £40 increase

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Scenario 4 This pie-chart shows the percentage share of total visitors to the local library since 2013. In 2016 there were 334475 visitors.

2013

2014

2015

2016

2017

9% 11% 38% 12%

31%

16. How many unique visitors did the most popular year have? A. B. C. D. E.

1,109,466 996,546 1,124,666 1,155,459 1,225,443

In 2015, the library invested in tablets due to the popularity of e-books. This cost them £778,765. 17. How much did the library invest per visitor? A. B. C. D. E.

£2.13 £2.33 £2.57 £2.22 £2.88

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18. If one in every three visitors in 2016 bought an e-book costing £6.99 would the library have made their money back from the cost of the tablets? A. B. C. D. E.

£533 loss £566 profit £561 loss £505 loss £562 loss

19. At the start of 2017, the average cost of an e-book was reduced to £6.50. What is the change in money made by the library from the e-books from 2016 to 2017, assuming one in every three visitors is still purchasing one? A. £44,698 more B. £141,694 more C. £44,698 less D. £141,694 less E. £186,392 less 20. If the average person spends 24 minutes in the library, what was the total time spent in 2014? A. B. C. D. E.

377,074 hours 262,267 hours 345,653 hours 456,213 hours 576,111 hours

Scenario 5 Class 11C took a biology school trip to a farm in Wales. A map of this farm showed a rectangular field 3cm by 6.5cm. The scale of the map was 1 : 20,000 1 hectare = 10,000m2 = 2.47 acres 21. What is the area of the field in hectares? A. B. C. D. E.

39 780000 7800 78 78000

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22. A new map was printed with a scale of 1 : 27,000. What was the area of the field in this map? F. 25.22cm2 G. 27.22cm2 C. 26.33cm2 D. 26.69cm2 E. 27.56cm2 On the trip, they spotted 13 sheep, 26 cows, 11 pigs and 39 horses. 23. If the animals are evenly spaced out, how much of the field does each have to itself? A.0.87 hectares B. 0.88 hectares C. 0.89 hectares D. 0.90 hectares E. 0.91 hectares 24. If each animal occupied 0.5 hectares, what percentage of the occupied land did the cows occupy? A. B. C. D. E.

25% 26% 27% 28% 29%

25. The same trip took place the year after but there were 36% animals more in total. On average, how many pigs were there? A. B. C. D. E.

11 12 15 17 19

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Scenario 6 This table shows average temperatures per year, in degrees celsius, over the course of 5 months. Temperature

January

February

March

April

May

2013

11

14

15

16

17

2014

8

12

12

13

15

2015

10

14

13

15

16

2016

11

13

14

17

19

2017

9

10

13

14

16

26. Which year had the biggest range in temperature? A. B. C. D. E.

2013 2014 2015 2016 2017

27. The temperatures were recorded for 9 months in 2013 and 2014, and 12 months in all other years. Taking this into account, what was the average temperature across all months recorded? A. B. C. D. E.

13.1 13.2 13.3 13.4 13.5

28. What is the difference in percentage increase of temperature from February to April in 2016 compared to 2013? A. B. C. D. E.

16.22% 16.48% 16.66% 17.22% 18.11%

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29. If the temperature was to follow the same percentage decrease from February 2017 to 2018 as February 2016 and 2017, what would be the temperature in February 2018? A. B. C. D. E.

7.7 7.9 6.7 7.4 7.5

30. When their average temperature is combined, which of these had an average temperature of 12.5 degrees in 2017? A. B. C. D. E.

January and May February and April February and March April and May February and April

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Quantitative Reasoning Answers and Explanations


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Tutorial Questions Answers and Explanations

Scenario 1 Question 1 - D For this question you need the percentage to 1dp. You can save time by working to the nearest 100. Use values in 1000s (i.e. 60,000 = 60) to save time when using the UKCAT calculator: Team

Average / Capacity

Percentage

Arsenal

60.0 / 60.3

99.5%

Bournemouth

11.2 / 11.5

97.3%

Middlesborough

30.5 / 33.7

90.5%

Southampton

30.5 / 32.7

93.3%

Liverpool

52.9 / 54.1

97.8%

Question 2 - E Again, for this question you should work to 1dp for the attendances. If you find any ratios to be particularly close, you can always re-calculate at 2dp for these two teams to find the answer. The ratio is calculated by dividing the highest attendance figure by the lowest attendance figure. The ratio is given to 2dp to help distinguish between them properly. You are looking for the greatest ratio. This means we need a large value for highest attendance and small value for lowest attendance in order to gain the greatest ratio. Team

Highest / Lowest

Ratio

Tottenham

31.9 / 31.2

1.02 : 1

Liverpool

53.2 / 51.2

1.04 : 1

Southampton

32.7 / 27.4

1.19 : 1

Crystal Palace

25.6 / 23.5

1.09 : 1

Manchester United

75.3 / 75.3

1:1

Question 3 - C In this question the answer values are very close together. There aren’t many steps in this calculation, so work using exact values.

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Step 1: Add together the average attendances for Arsenal and Bournemouth. 59,999 + 11,167 = 71,166. Step 2: Percentage = Value / Total = 71,166 / 90,000 = 79.07% = C

Question 4 - C Again we can work to 1dp for this question, and if need be re-calculate to 2dp for close answers. For each question, divide the Average Capacity / Stadium Capacity Team

Average / Capacity

Proportion

Chelsea

41.5 / 41.8

99.3%

Bournemouth

11.2 / 11.5

97.4%

Arsenal

60.0 / 60.3

99.5%

Crystal Palace

25.0 / 26.3

95.1%

Tottenham

31.5 / 32.0

98.4%

Medic Mind Tip: If you scan the calculations, you can see that some are definitely too low. Some values, such as Arsenal, are 60 / 60.3, whereas Crystal Palace has a 1.3 difference (25.0 / 26.3). So from inspection you can see that it is not worth calculating for Crystal Palace. Medic Mind Tip: It is a good idea to write down your calculations before you do them. It avoids making silly mistakes when referring back from table to screen. Also, it means you can eliminate before you calculate, as we just discussed in the tip above.

Question 5 - E You cannot be sure of how precise to be here, so it is safe to work to 2dp (using 000s). By writing down the calculations first, you can see that it is not worth doing the first three. Teams

High Average - Low Average

Difference (000s)

Liverpool & Arsenal

60.00 - 52.87

7.13

Tottenham & Crystal Palace

31.47 - 24.97

6.50

Chelsea & Southampton

41.52 - 30.47

11.05

Middlesborough & Bournemouth

30.47 - 11.17

19.30

Manchester United & Liverpool

75.29 - 52.87

22.42

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Question 6 - C The answers are quite spaced out, but the calculation is very short, so work to at least 1dp. For this question we can set up a simple algebraic formula. 53.9 Y = 2400 Y = 2400 / 53.9 = £44.53 = C Team

Sales Revenue

Value (000s)

Arsenal

60.0 x 40

2400

Liverpool

53.9 x Y

53.9Y

Scenario 2 Question 7 - C Step 1: Identify that John is an Adult, and Lucy a Child. Step 2: Scan the bullet points for any important points. Spot that Tuesday has a 10% discount. Step 3: Add together Monday Tickets (6 + 12) and Tuesday ticket (0.9 x 12) to give £28.80 = C.

Question 8 - C You need to work out the cheapest possible method for both scenarios: Without Erald - it is cheaper to buy individual tickets rather than the Family ticket. There are two children, one adult and one senior = 12 + 12 + 5 = 29 With Erald - with Erald, there are two children, two adults and one senior. Buying individually would cost 29 from before, plus 12 for Erald = 41. However, it is much cheaper to buy a £32 Family Ticket and then buy an extra Senior for Deraine = 32 + 5 = 37 37 - 29 = £8 difference = C

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Question 9 - A Tony spends: • 2 years as a Child = 24 x 6 = 144 • 3 years as a Student = 36 x 7 = 252 • 12 years as a Adult = 144 x 12 = 1728 144 + 252 + 1728 = £2124 = A

Scenario 3 Question 10 - B Do not fall into the trap of working out the Fahrenheit temperature for each country.. Instead, work backwards using algebra. 71.6 = 1.8 C + 32 39.6 = 1.8 C C = 39.6 / 1.8 = 22 = B Question 11 - C Step 1: The country with the maximum temperature is Australia (38C). The country with the lowest temperature is England (22C). To get the biggest difference, we want to maximise Australia to 40C and minimise England to 20C. Step 2: Work out the difference: 0 - 20 = 20C Step 3: Convert to F: (20 x 1.8) + 32 = 68F = C Medic Mind Tip: if you convert to F before, then you would have to do two conversions. By converting later you only have to do one conversion. There is no multiplication of units (such as in Area Questions), so you can leave conversion until the end.

Scenario 4 Question 12 - A Cony = (8 x 0.5) + (6 x 0.3) + (7 x 0.2) = 7.2 Sanon = (9 x 0.5) + (4 x 0.3) + (6 x 0.2) = 6.9 Cony is the best. ••••••••••••••••••••••••••••••••••••••••••••••••• UKCAT

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Question 13 - B From the previous question we know that the Cony scored 7.2, and Sanon scored 6.9. We need to use these Quality Scores and combine them with the Price Score. Cony = (7.2 x 0.4) + (5 x 0.6) = 5.88 Sanon = (8 x 0.4) + (8 x 0.6) = 8

Scenario 5 Question 14 - B Again, identify the figures which will make up your mean - here it will be 1, 2, 5 and 7. Now give each number a weighting, and multiply by the weighting: (1 x 2/56) + (2 x 14/56) + (5 x 8/56) + (7 x 32/56) = 5.25 • For this particular question it may be better to do (1 x 2) + (2 x 14) + (5 x 8) + (7 x 32) and then divide by 56. • You can estimate the answer before you begin to help you see if your answer is in the correct general direction. Jude works for 7 days for the majority of the weeks, so the answer should be closer to 7 than 1. Question 15 - D From the previous question we know he works on average 5.25 days per week. He therefore earns 5.25 x £60 per week = £315 He earns £315 x 52 per year = £16,380 In Thai Currency it is 16380 x 44 = 720,720 = D

Scenario 6 Question 16 - C The room measures 5 x 3 metres = 15m2 Cream Paint costs £20 per m2

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Total price = 15 x 20 = £300 = C Type of Flooring

Price per square metre (£ / m2)

Cost of hiring a painter (£ / m2)

White

50 / 40

5 /6

Cream

25 / 20

10 / 8

Violet

5/6

3/4

Question 17 - A The room measures 5 x 3 metres = 15m2 Cost of using cream = (20 + 8) x 15 = £420 Cost of using white = (40 + 6) x 15 = £690 Difference in price = 675 - 450 = £270 = A There are 3 different ways to tackle this question: Method

Steps

No. of Steps

Slow Method

Working out the material cost and labour cost separately for each of Cream and White

7

Common Method (as above)

Working out the total sum cost of material and labour for each of Cream and White

3

Fast Method

If you realise that the difference in cost (both material and labour) per m2 between White (46) and Cream (28) is 18, you realise that you save £15 per m2. Hence just do one sum: 18 x 15 = 270

1

Question 18 - B The room measures 5 x 3 metres = 15m2 Total area of furniture = (1.4 x 2) + (0.5 x 1) + (0.75 x 0.4) = 3.6m2 Percentage of floor covered = (3.6 / 15) x 100 = 24% = B ••••••••••••••••••••••••••••••••••••••••••••••••• UKCAT

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Question 19 - E Cost of fitting Cream Paint = ((0.8 x 20) + 8) x 15 = £360 Total price of Furniture and Cream Paint = 400 + 250 + 360 = £1010 Money remaining = 1500 - 1010 = £490 = E

Scenario 7 Question 20 - C There are 8 parts in total (3 + 4 + 1), and 16 students. 1 part is equal to two students. Therefore 3 parts equates to 6 students. Alternatively, you could say that 3 / 8 of the cohort study Maths, and then: 3 / 8 x 16 = 6 = C

Scenario 8 Question 21 - A The ratio numbers add up to 164, and 56 do Netball. So the fraction is 56 / 164. The highest common factor for both is 4, so divide both sides by this to get 14 / 41 = A

Scenario 9 Question 22 - B 1 cm on the map represents 20 000cm. Therefore, 6cm on the map represents 120 000cm. To convert from cm to m, divide by 100 = 1200m To convert from m to km divide by 1000 = 1.2km = A

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Question 23 - C The first two lines of information are distractors to waste your time. You need to convert to real units and then to kilometres before you multiply together the units. 2cm in map = 40 000 cm in real life = 0.4 km 8cm in map = 160 000 cm in real life = 1.6 km 0.4 x 1.6 = 0.64 km2 = C TIP: If you converted later, you would have to remember that 1km2 = 10,000,000,000 cm2. So it is easier to convert before incase you forget this. TIP: In the answer options if you see three answer options to be factors of 10 away from each other, such as A, B and E, then it is a strong hint that i) one of these is the correct answer and ii) there is a units trap that many candidates fall into.

Question 24 - D 1m = 38.37 inches Step 1: Work out the area of the pool in the map, in cm2. Remember that 1m2 = 10,000 cm2. So 44m2 = 44,000 cm2. In the map it is 44,000 / 50,000 = 0.88 cm2. Step 2: Convert the area from cm2 to inches squared. 1m = 38.37 in 1cm = 0.38 in 1cm2 = 0.155 in2 0.88 cm2 = 0.155 x 0.88 = 0.1364 in2 = D

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Scenario 10 Question 25 - B There are three factors, so lets look at two first - colour and species. There are 70 Species A, so we want to maximise the number that are not green. Therefore we can say that 40 of Species A are red, leaving 30 green Species A. Out of these 30 green Species A, a maximum of 20 (5+15) can have grey or white spots, meaning that 10 have black spots = B

Question 26 - B Step 1: Write down what is left in the box after Jackie removes her 10. • 60 green, 30 red • 70 Species A, 20 Species B • 80 black spots, 5 grey spots, 5 white spots Step 2: Set up fractions to work out probabilities • Red - 30 / 90 = 1 / 3 • Species B - 20 / 90 = 2 / 9 • Grey Spotted - 5 / 90 = 1 /18 Multiplying the fractions together gives you 2 / 486 = 1 / 243 = B

Scenario 11 Question 27 - B The interest rate is based on the original amount, so it is simple interest. Simple Interest = Loan x (Annual Interest Rate - 1) x No. of Years Simple Interest = 20350 x 0.05 x 4 = £4,070 = B

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Question 28 - C Compound Interest = Loan x (Annual Interest Rate) number of years Compound Interest = 5000 x 1.035 3 = £5,544 = C For reference, the amounts after each year are 1 - 5,356, 2 - 5,525, 3 - 5,544, 4 - 5,624, 5 - 5,738.

Question 29 - D The method for this question is to work out the interest for each bank year-by-year. Length of Loan

Tony

Rush

Year 1

5,185

5,175

Year 2

5,370

5,356

Year 3

5,555

5,544

Year 4

5,740

5,738

The answer is 4 years = D

Scenario 12 Question 30 - C To answer this, break down how much tax she pays in each bracket. • Kate pays £10,000 at 10% = £1,000 • Kate pays £10,000 at 15% = £1,500 • Kate pays £4,000 at 25% = £1,000 Her total income tax is £3,500 = C.

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Question 31 - D We first need to work through each bracket to identify which bracket James earns his tax in. Be careful, the question says that he pays £450 per month, so he must pay £5,400 per year. • He must be beyond the first tax bracket in which he pays £1,000 (10% of £10,000) • He is also beyond the second tax bracket in which he pays £1,500 (15% of £10,000) • The maximum for the third tax bracket is £3,750, meaning that James would have to pay more than £6,250 to be beyond this. He is not, so we know his income is in this bracket. Now work out how much tax he pays in the third bracket: 5400 - 1500 - 1000 = 2900 Now set up an algebraic formula to work out how much he earns (x) in the third bracket: 0.25x = 2900

x = 11600

We can then add 11,600 to 20,000 to get his total income = £31,600 = D

Scenario 13 Question 32 - C

The answer is 8 = C ••••••••••••••••••••••••••••••••••••••••••••••••• UKCAT

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Scenario 14 Question 33 - B For a question like this you should NOT read the whole table - it is time consuming and wasteful. Instead, read the question and then scan for the information you need. For this question we need to look at the price of White Shirts, Black Trousers, and Navy Blazers. To save time add up the price of one outfit, and then multiply by 8, rather than multiplying each item by 8. 17.50 + 16.50 + 39.50 = 73.50 73.50 x 8 = £588 = B

Question 34 - B Male - White Shirt, Blue Trousers, Navy Blazers Female - White Shirt, Black Skirts, Black Leggings

Year

Temperature

Weighting

2013

15

9 / 54

2014

12

9 / 54

2015

13

12 / 54

2016

14

12 / 54

2017

13

12 / 54

Male - 17.50 + 18.50 + 39.50 = 75.50 Female - 14.00 + 23.00 + 16.25 = 53.25 Base Value = New / Original 75.50 / 53.25 = 1.42 = 42% increase = B

Question 35 - C ••••••••••••••••••••••••••••••••••••••••••••••••• UKCAT

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Step 1: Work out the new prices New = Base Value x Original The base values are 1.1 (shirt) and 0.9 (blazer). Original Prices White Shirt = 21.00 Navy Blazer = 45.00 Total price = 66.00 New Prices White Shirt = 21 x 1.1 = 23.1 Navy Blazer = 45 x 0.9 = 40.5 Total price = 23.1 + 40.5 = 63.6 Step 2: Work out the % change from old to new price Base Value = New / Original 63.6 / 66.0 = 0.964 = 3.6% decrease = C Question 36 - C We have to take this question step by step. Men’s shirts- large Blue shirts are the most expensive at £22.00. Men’s blazers- large Navy blazers are the most expensive at £45.00. Men’s trousers- large Blue trousers are the most expensive at £19.00. Female shirts- large White shirts are the most expensive at £22.00. Female leggings- small Blue leggings are the most expensive at £22.70. Female skirts- medium black skirts are the most expensive at £29.00. 22 + 45 + 19 + 22 + 22.70 + 29 = £159.70 = C

Scenario 15 ••••••••••••••••••••••••••••••••••••••••••••••••• UKCAT

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Question 37 - B For bike 2, the change in velocity over 3 hours is 160 km / h. For bike 1, the velocity is constant so there is no acceleration. Bike 2 Acceleration = 160 / 3 = 53.33 km / h2 = B

Question 38 - B Speed = Distance / Time Distance = Speed x Time = Area under graph The average velocity for Bike 1 is 50 km/h. The average velocity for Bike 3 can be calculated by doing (60 + 0) / 2 = 30 km/h. This only works because Bike 3 has a straight line - it is not possible to do this for Bike 2. Distance Bike 1 = 50 x 3 = 150 km Distance Bike 3 = 30 x 3 = 90 km The difference is 60km = B

Question 39 - C Bike 2 160

3

Velocity (km/h)

120

2 80

1

5

40

0

0

1

2

3

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Bike 2 does not have a straight line, so you have to work out the distance under the line segment by segment. Area of Triangle = base x height x 1/2 Area of Square = length x width Triangle 1 = 1 x 120 x 0.5 = 60 Triangle 2 = 1 x 20 x 0.5 = 10 Triangle 3 = 1 x 60 x 0.5 = 30 Square 4 = 1 x 100 = 100 Square 5 = 1 x 100 = 100 Area under graph = 60 + 10 + 30 + 100 + 100 = 300km = C

Question 40 - D You can do this question by inspection. Three out of four plots are 100 km/h and above, so the average is not going to be as low as 40 - eliminate A, B and C. E is also incorrect by inspection as the maximum speed recorded at one out of four points is only just above this. 100 km / h seems plausible, so you can choose this without even doing any working. If you did want to calculate, here is the method: You can work out average speed by working out the gradient. However, the line is not straight, so instead you can calculate a gradient (= speed) for each hour, and then work out an average. • For the first hour the bike travels at 60 km/h average: (120 + 0) / 2. • For the second hour the bike travels at 110 km/h average: (120 + 100) / 2 • For the third hour the bike travels at 130 km/h average: (100 + 160) / 2 Average of journey = (60 + 110 + 130) / 3 = 100 km/h = D

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Mock Test Questions Answers and Explanations

Scenario 1 Question 1 - C Speed = Distance / Time The distance is 13.1 miles. It is a half marathon not a full one. The time is 2 hours 26 minutes. Be careful not to get mixed up with his training times and his actual time. At this stage you should check the answer options to see which units you need to work in. We need miles per hour, so converting the time into hours gives us 2.433. Speed = 13.1 / 2.433 = 5.384 mph = C Medic Mind Tip: When you look at the answer choices, it seems odd that almost half the answers are more than double the other half. This could hint that there is a units trap in this question. In other questions you might see some answers with a difference of a power to 10, which is a huge hint that there is a units trap. Question 2 - B Difference = Quickest speed - Race speed We can re-use his race speed from the previous question. To calculate his quickest speed you should do Distance / Time = 13.1 / 2.25 = 5.82mph Difference = 5.82 - 5.38 - 0.44mph = B

Question 3 - E Base Value = New / Original It is easier to work in minutes for this question: Base Value = 146 / 157 = 0.93 = 7% decrease = E Question 4 - B Time = Distance / Speed ••••••••••••••••••••••••••••••••••••••••••••••••• UKCAT

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The distance is 3/5 x 13.1 = 7.86 miles. Time = 7.86 / 6.7 = 1.17 hours We have been working in mph so the units is in hours. Converting 0.17 into minutes gives 10 minutes, so the answer is 1h 10m = B Question 5 - D Step 1: Work out the speed he runs in his longest time. The distance is 13.1 miles and the time is 3.62 hours. 13.1 / 3.62 = 3.62 mph Step 2: Work out the difference in speed. 5 - 3.62 = 1.38mph Step 3: Work out the difference in distance. Effectively Phillip is running at 1.38mph faster for 3.62 hours. So the extra distance covered is the speed x time = 1.38 x 3.62 = 5.00 miles

Scenario 2 Question 6 - A Step 1: Calculate Tahmeed and Jeffrey Price (Discounted Ticket Price + Booking Fee) x 2 = Tahmeed + Jeffrey Price When you work out the ticket price take off the 25% discount - 0.75 x 99.95 = £74.96. The answer choices are close together, so work precisely using exact units. (74.96 + 13.56) x 2 = £177.04 Step 2: Add in Ben’s price Remember that Ben doesn’t get a discount. 177.04 + 99.95 + 13.56 = £290.55 Question 7 - A This is a question where you can see the value of writing things down on your board. If you hadn’t written working down, then you would have to re-calculate Ben’s price as per the previous question. ••••••••••••••••••••••••••••••••••••••••••••••••• UKCAT

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Ben: 99.95 + 13.56 = £113.51 Jeffrey: (99.95 x 0.75) + 13.56 = £88.52 Difference: 113.51 - 88.52 = £24.99 = A Question 8 - B This is a proportion question, not a percentage change question. Proportion = Booking Fee / Total Price Proportion = 13.56 / ((99.9 x 0.75) + 13.56) = 0.1532 = 15.3% = B Question 9 - B Jeffrey Ticket Cost - Meal Cost = Amount Owed Jeffrey Ticket Cost (99.95 x 0.75) + 13.56 = £88.52 Meal Cost 1.65 + 3.79 + 4.76 = £10.20 Amount Owed = 88.52 - 10.2 = £78.32 = B Question 10 - D From the previous question we know the following: Amount Owed After 1 Meal = £78.32 Cost of One Meal = £10.20 Therefore the number of extra meals required is 78.32 / 10.20 = 7.68 Remember to add the 1 meal already bought to give 8.68. You cannot buy two thirds of a meal, so he needs to buy 9 to fully pay back his debt = D.

Scenario 3 Question 11 - D To answer this question you have to add up the two bars for each By inspection we can tell that only July and August will be in contention here.

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July = 95 + 60 = 155 August = 65 + 85 = 150 The answer is July = D Medic Mind Tip: You only need to work to the nearest 5, and it is likely that it is not more precise than this because they only give you axis markings every 25 units. Medic Mind Tip: For questions involving graphs try to answer as much as you can via inspection. Here you have saved 3 sums by eliminating April, May and June by inspection. Medic Mind Tip: At the start of this question it might save time to write the number value for each bar in the chart (on your whiteboard). It will save you time for later questions Question 12 - A Again for this question you can use inspection to eliminate all but April and July. April = 55 - 15 = 40 July = 95 - 60 = 35 The answer is April = A Question 13 - D To work this out you have to individually work out Jeans and T-shirt revenue. Jeans Revenue = £3 x 70 = £210 T-shirt Revenue = £5 x 55 = £275 Adding the two together gives 210 + 275 = £485 = D

Question 14 - B April and August April = 15 + 55 = 70 August = 65 + 85 = 150 April and August = 150 + 70 = 220 July and June July = 95 + 60 = 155 June = 55 + 70 = 125 July and June = 280

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The answer is July and June = B

Question 15 - D Step 1: Calculate the difference for T-shirts The price difference is 6 - 5 = £1 increase Quantity for the year is 15 + 25 + 50 + 95 + 65 = 250 Price Difference x Quantity = Change in Sales Revenue £1 x 250 = £250 increase Step 2: Calculate the difference for Jeans The price difference is 3 - 2 = £1 decrease Quantity for the year is 55 + 40 + 70 + 60 + 85 = 310 Price Difference x Quantity = Change in Sales Revenue £1 x 310 = £310 decrease Step 3: Calculate the difference 250 - 310 = £60 decrease = D

Scenario 4 Question 16 - D Step 1: Calculate the numerical equivalent of 1% By inspection we can see that the most popular year is 2013. However, we do not know what figure equates to 38%, so we need to look at the information given. We are told that in 2016 there were 334,475 visitors, which was 11%. If 11% is 334,475, then 1% is 334475 / 11 = £30,407

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Step 2: Use the numerical equivalent to calculate the 2013 value 1% is £30,407 38% is 30,407 x 38 = £1,155,459 = D Question 17 - A Step 1: Calculate the number of visitors in 2015 1% is £30,407 12% is £364,884 Step 2: Calculate the investment per visitor Investment per Visitor = Total Investment / No. of Visitors 778,765 / 364,884 = £2.13 = A Question 18 - B Step 1: Calculate the number of visitors in 2016 1% is 30,407 11% is 334,477 Step 2: Calculate the number that book an ebook 1/3 bought of the 334,477 visitors bought an ebook = 111,492 Step 3: Calculate the difference between cost and revenue 111,492 bought an ebook at £6.99 each = £779,331 revenue The original investment was 778,765. The difference is 779,331 - 778,765 = £566 profit = B

Question 19 - E Step 1: Calculate the visitors in each year 2016

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1% is 30,407 11% is 334,477 2017 1% is 30,407 9% is 273,663 One third of the visitors bought which is 111,492 for 2016 and 91,221 for 2017. Step 2: Calculate the revenue for each year Revenue = Price x Quantity 2016 111,492 x 6.99 = £779,329 2017 91,221 x 6.50 = £592,937 592,937 - 779,329 = - 186,392 = E

Question 20 - A Step 1: Calculate the visitors in 2014 1% is 30,407 31% is 942,617 Step 2: Calculate the time spent To work out the time spent you should multiply the number of visitors by the average time.Remember to work in hours. Time spent = 942,617 x (24/60) = 377,047 hours

Scenario 5 Question 21 - D The scale of 1 : 20,000 means that a unit of 1 in the map is equal to 20,000 units in real life. ••••••••••••••••••••••••••••••••••••••••••••••••• UKCAT

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The dimensions of the field in the map are 3 x 6.5 cm. Remember what we learnt in The Units Tutorial - convert units to the final form before multiplying. We can work in metres as we are given the conversion from m2 Step 1: Convert the dimensions to metres on the map The dimensions of the map in metres are 0.03 and 0.065. Step 2: Convert the metres dimensions to real life values The dimensions of the field in real life are: 0.03 x 20,000 = 600m 0.065 x 20,000 = 1300m Step 3: Find the area in m2 in real life 600 x 1300 = 780,000 m2 Step 4: Convert the area to hectares 780,000 / 10,000 = 78 hectares = D Question 22 - C The area of the field would have increased proportionally by 27,000 / 20,000. On this occasion it doesn’t matter whether you convert to real form before multiplying the dimensions or after, because no unit changes are involved. Step 1: Work out the area in the old map 3 x 6.5 = 19.5 cm2 Step 2: Adjust the area for the size of the new map The new area is 19.5 x 27/20 = 26.33 cm2 = C Question 23 - B Step 1: Work out the total number of animals 13 + 26 + 11 + 39 = 89 animals Step 2: Work out the individual area each animal has Divide the area of the field in real life by the number of animals

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From the first part of this question we know the area is 78 hectares. 78 / 89 = 0.88 hectares = B Question 24 - E If each animal occupies the same amount, then we do not have to use the 1.5 hectares figure, we can just compare the actual number of animals - 26 cows out of a total of 89. 26 / 89 = 29% = E Question 25 - C Again, we do not need to look at the areas, we just need to do a percentage increase of 36% for the pigs, of which there are currently 11. New = Base Value x Old = 1.36 x 11 = 14.96 You can’t have half a sheep, but on average the closest value to 14.96 is C = 15.

Scenario 6 Question 26 - D The range is the difference between the smallest and highest value: Year

Calculation

Range

2013

17 - 11

6

2014

15 - 8

7

2015

16 - 10

6

2016

19 - 11

8

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Year

Calculation

Range

2017

16 - 9

7

The answer is 2016 = D Medic Mind Tip: For this question it is not worth your time writing out the calculations like we did for Question 1 in the Tutorials (about the football attendances). The calculations are simple, so don’t bother using your calculator or whiteboard much. Question 27 - D This is a weighted means question. Step 1: Identify which numbers will make up your mean The numbers we are using to find the mean are the temperature values in March of each year: 15, 12, 13, 14, 13. Step 2: Give each of these numbers a weighting First work out the total number of months recorded across all 5 years: 9 + 9 + 12 + 12 + 12 = 54 months Step 3: Work out the weighted mean Weighted mean = Σ (each value x individual weight) It is easier to multiply first and then divide by 54 at the end: (15 x 9) + (12 x 9) + (13 x 12) + (14 x 12) + (13 x 12) = 723 Now divide by 54… 723 / 54 = 13.4 degrees celsius = D Question 28 - B The answer values are very close together and to 2dp, so work precisely to at least 3dp for percentages. Step 1: Work out the percentage increase in 2016 Base Value = New / Old 17 / 13 = 1.30769 = 30.769% increase Step 2: Work out the percentage increase in 2013 ••••••••••••••••••••••••••••••••••••••••••••••••• UKCAT

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16 / 14 = 1.14285 = 14.285% increase Step 3: Work out the difference 30.8 - 14.3 = 16.484% = B Question 29 - A Step 1: Work out the % decrease from 2016 to 2017 Base Value = New / Old 10 / 13 = 0.769 = 23.1% decrease Step 2: Work out the 2018 value New = Old x Base Value 0.769 x 10 = 7.7 degrees = A Medic Mind Tip: For questions such as this do not bother reading the whole table properly before you start any questions. Only skim read, and then find the important information required for each specific question. Question 30 - A By inspection you can knock out D. Then going through each of the options one by one, you will stumble upon A first: (9 + 16) / 2 = 12.5 degrees = A

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Introduction to Quantitative Reasoning LESSON 1

Quantitative Reasoning Number of scenarios: 9 Number of Qs per scenario: 4

Total number of Qs: 36 Total time: 24 minutes


What is Quantitative Reasoning? Quantitative Reasoning subtest assesses your ability to use numerical skills to solve problems. Data is presented in various forms, such as graphs, charts, text or diagrams.

Why do they test it? Doctors and dentists are constantly required to review data and apply it to their own practice. On a practical level drug calculations based on patient weight, age and other factors have to be correct. At a more advanced level, clinical research requires an ability to interpret, critique and apply results presented in the form of complex statistics. Universities considering applicants need to know they have the aptitude to cope in these situations.

What are the most common topics tested? • Percentage Change- there are three types of percentage change question, and it is a question type that tricks many students out • Areas and Ratios- the UKCAT often has area and ratio questions with unit conversion tricks that many candidates lose marks on. • Reading Graphs- you will get tested on your ability to read data of graphs and spot trends • Weighted Means- many students have forgotten how to do this since GCSE, so we will revisit the algebraic techniques.

• Triangle Formulae- application of formulae such as Speed, Distance, Time.

General Information • Again, this section is time pressurised, and often people struggle with the mathematical intricacies of the questions. • The level of maths will be relatively simple to do, but under time pressure it can be daunting to answer certain questions. • For certain topics, you need to adapt your usual methods to improve your timing. For example, for Percentages our method is completely different to the national method commonly used for GCSE syllabuses. We have developed this method specifically for the UKCAT.

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Quantitative Reasoning Lessons Lesson 1: Introduction to Quantitative Reasoning Lesson 2: Estimating vs. Working Properly Lesson 3: Reading Question Information Lesson 4: Units in the UKCAT Lesson 5: Weighted Means Lesson 6: Area Questions Lesson 7: Quantitative Reasoning Formulae Lesson 8: Ratios Lesson 9: Adapting to Test Conditions Lesson 10: Fractions Lesson 11: Interest Questions Lesson 12: Tax Questions Lesson 13: Venn Diagrams Lesson 14: Percentages Lesson 15: Speed, Distance, Time Lesson 16: Quantitative Reasoning Mock Answers and Explanations