# Numerical Reasoning – from 11 plus through GCSE to careers

Numerical reasoning – what is it, what type of questions?

Working in the Maths tutoring arena, I’m hearing more about the topic of numerical reasoning and therefore so will pupils and parents. This blogpost helps you understand it and try lots of examples yourself. Numerical reasoning is more than addition, multiplication, and division. But equally it’s not about quadratic equations or calculus.

In general, its more about real-life problem-solving and in particular about interpretation of numerical charts, graphs, tables, data sets, trends and series. Leading to a conclusion often requiring choosing an answer to a multiple choice question, and the wording of the question often needs careful attention.

The specific underlying maths skills needed are quite limited in their topic-scope and are mostly KS3 level. Principally:

• accurate, quick mental arithmetic backed up with calculator if allowed
• BIDMAS order of operations and execution thereof
• fractions decimals, percents including operations with percents
• ratios
• pie charts, bar charts, line graphs
• two way tables of data
• money calculations from simple pounds and pennies to basic sales units, value, costs and profit, and currency conversion
• series, sequences and patterns
• speed time and distance formula triangle
• units of measure, conversions, multiples of 10

In fact, it’s the generic skill requirements which differentiate numerical reasoning questions. Interpreting data involves understanding the data, regardless of the various presentation formats, or sometimes column-headings not seen before. Then being able to manipulate the data, perhaps involving combinations of the maths techniques above, or realising a two-step process is needed to calculate missing information to finally answer the question.

Comprehension of the data, deciphering patterns, performing estimates and determining relevant information are other requirements – for instance quickly realising the question is pointing you to just one bar in a bar chart or just one row and column in a table.

Because the actual maths topics, like percentages, are fairly basic, then the surprising result is that you could find the same numerical reasoning question in almost any age-group test. In theory at least these questions should be accessible to anyone who’s been to school, not necessarily a high performing school or top maths set. So it is an “equalising” method. All you need is generic maths intuition, rather than specific difficult techniques, or so it is claimed! (I have my doubts – if you struggle with percents and ratios you will probably struggle with interpretational maths skills too).  Which leads us to the question, where could you come across these questions?

Which tests and exams feature particular types of numerical reasoning questions?

Most selective independent or grammar schools will have some kind of Maths test as part of the admissions process. And most of these tests will contain at least a few numerical reasoning questions to back up the basic numbers-only questions.

An example would be this:

Q1. A purely numerical “fractions” question might be, what is ½ plus ¼?
(Ans. ¾ )

Q2. A version of this, shall we say on its way to numerical reasoning style, might be: if I have £10 to spend, and spend a quarter on sweets and a half on drinks, how much change do I get? (Ans. £2.50)

Selective schools typically use exam Boards such as ISEB, GL, CEM and Ukiset for admissions testing. All of these will contain numerical reasoning to a certain extent, but two stand out.

First CEM, which has a specific paper called “Numerical Reasoning”.  A typical question might be:

Rick is 1.8m tall and John is the same as Rick. Peter is taller than Rick. Carol is 57cm shorter than Peter. John is 45cm taller than Carol. What height is Peter?

Ans. Carol must be 1m 80cm less 45 cm = 1m 35 cm. So Peter must be 1m 35cm plus 57 cm = 1m 92 cm

Or a sequence question like:

Q3. This picture represents a sequence of triangle numbers. How many blocks would be in the next pattern? Ans. 10

Second, Ukiset, the UK Independent Schools’ Entry Test, (particularly for international students) This is a test nearest to what the purists would regard as classic numerical reasoning. (Like many exam Boards, it also contains verbal and non-verbal reasoning, but we’ll not cover those here). A score is generated after the test which can be benchmarked to indicate potential.

Here are some examples of typical style of Ukiset questions

Q4. From the graph above, what is the percentage increase in Hare population between 1970 and 1980? Choose from A 10%   B 20%   C 25%   D 30%

Ans C 25%.  (10,000 – 8,000) / 8,000   x 100%

Q5. The total attendance for three South Coast football teams was 1,200,000 in 2018 and 1,000,000 in 2019. Using the Pie charts above how much greater was the attendance for Portsmouth in 2019 than 2018

Choose from A 25,000    B 50,000   C 75,000   D 100,000

Ans B 50,000
2018 = 60/360 x 1,200,000 = 1/6 x 1,200,000 = 200,000
2019 = ¼ x 1,000,000 = 250,000
Difference = 50,000

Q6a. From the table below, showing which sports 100 male and female pupils play at school. What is the ratio of male to female pupils at the school, expressed in its simplest form?
A 4:6   B 6:4   C 2:3   D 3:2

Ans. C 2:3. Male total = 40, female total = 60 so ratio = 40:60 = 2:3

Q6b. Amongst the males only, what percentage of them play soccer?
A 25%    B 40%   C 50%   C 75%

Ans C 50%.  20/40 x 100% = 50%

Q7.

A business sells two products and the units sold in thousands are shown above by year. Some financial details for 1980 are shown below.

Profit is calculated as sales value (sales price per unit x number of units sold), less total costs.

How much more profit was made in \$ in 1980 for product 1 than product 2?
A \$ 20,000    B \$40,000   C \$60,000   D \$80,000

Ans. C \$60,000

Product 1 Sales value  =  10,000 x £10 = £100,000 so profit after deducting £20,000 costs = £80,000.
Product 2 Sales value  = 1,000 x £50 = £50,000 so profit after deducting £10,000 costs = £40,000
Difference = £40,000 x 1.5 = \$ 60.000

The UKMT U.K. Maths Challenge from Junior to Senior provides a good source of numerical reasoning practice. Here is an example:

Q8. To paint a room, half of a 3 litre can of paint was used for the first coat then 2/3 of the remainder was used for the 2nd coat. How much paint remained?

Ans 0.5 litre = 500 ml.  ½ of 3 = 1.5 and 2/3 of 1.5 – 1.0 so 0.5 left.

Let us move on to GCSE Maths exams.

In the O-Level years we had Pure Maths and Applied Maths, with numerical reasoning very much in the latter category. While it is clear that in recent years examiners have aspired to introduce more “real-world” and “wordy” Maths questions to GCSE 9-1 graded exams, we can be a bit nore specific and identify certain questions in the style of some of the above numerical reasoning questions. All of the below are based on actual recent questions.

Q9. A firm has a total of 160 vehicles. They are cars and lorries.
The number of cars : the number of lorries = 3 : 7
Each car and each lorry uses electricity or diesel or petrol.
1/8 of the cars use electricity.
25% of the cars use diesel.
The rest of the cars use petrol.

How many cars use petrol? You must show all your working.

Ans 30. 160 vehicles = 10 shares so 1 share = 16
Cars = 3 x 16 = 48 so electricity = 1/8 x 48 = 6
Diesel = ¼ of 48 = 12 so petrol = 48 – 6 – 12 = 30

Q10. In Europe, Rick pays 27 euros for 18 litres of petrol. In the U.K., Malcolm pays £40 for 8 gallons of the same type of petrol.
1 euro = £0.85 and 4.5 litres = 1 gallon
Rick thinks that petrol is cheaper in Spain than in U.K. Is he correct?

Ans.No:
Many ways to prove it for instance: in Europe 1.5 Euros = 1 litre =£1.275
In U.K. £5 = 1 gallon = 4.5 litres so 1 litre = £1.11 So cheaper in
U.K.

Q11. Ellie makes cakes in a restaurant using potato, cheese and onion so that
weight of potato : weight of cheese : weight of onion = 9 : 2 : 1
Ellie needs to make 6000 g of cakes.
Cheese costs £2.25 for 175 g.

Work out the cost of the cheese needed to make 6000 g of cakes.

Ans.£12.86
12 shares = 6000 so 1 share = 500 g so cheese = 2 x 500 = 1000g
So cheese = 1000 / 175  x 2.25 = £12.86

Q12. A sign on a motorway says the time to reach a junction 30 miles way is 26 minutes. The driver thinks they would have to drive faster than the speed limit of 70 miles per hour to do that. Are they right?

Ans. No . Speed = distance / time = 30 / (26/60) = 1800 /26 = 69.2 mph

Q13. In a survey of 10 people their fruit preferences are shown in the table below

In a second survey of more people the preferences are shown in the pie chart below. Explain how the second survey shows a lower preference for bananas.

Ans. First survey: 5/10 is a half. Second survey: yellow slice is less than 1/2

A-level

Many questions at Maths A-Level feature “real world” scenarios but do not qualify as classic numerical reasoning questions because they require high level mathematical techiques such as calculus and standard deviation, not accessible to most people. However, there is a set question each year which contains the spirit of numerical reasoning, namely the large data set.

Typically the data set and graph questions at GCSE and earlier contain rolled up summaries of accumulated data. But this annual A-Level question pre-releases a large data set of original detail in Microsoft Excel from which the student can make summaries themselves and draw conclusions. An example is shown.

Question Q14: does the data and graph prove that the amount of salt consumed reduced greatly in the period shown: Ans: not completely because “greatly” is ill- defined and purchasing does not necessarily correlate with consumption

In the new worlld of AI, programming, algorithms, information technology and even socila media, the use of Maths in general and numerical reasoning in particular is becoming a more important skill. Even the Prime Minister says so! And the use of graphs and statistics in the pandemic encourged non-mathemticians to evaluate data.

So it is not surprising that numerical reasoning is becoming part of the job application process – not just in the technical sector but also in a variety of other areas such as private companies like Amazon and many banks, consultancies and energy companies. Also the public sector such as Civil Service, Police and to focus upon one, the Military. Officer applications must undertake Aptitude tests including spatial awareness, verbal, non-verbal and numerical reasoning, for which some typical questions are shown below:

Q15. A worker who has to work for 8 hours a day is entitled to three 20-min­ute breaks, and an hour for lunch during the working day. If they work for 5 days per week for 4 weeks, how many hours will they have actually worked?
Ans 120 hours. Working day = 8 – 1 – 1 = 6 hours x 5 x 4 = 120

Q16a.
Below is a table listing the percentage changes in profit from 2014 to 2016 for five different companies.

Using the above table, if company Q earned £412,500 in 2014, how much profit did they make in 2016?

Ans: £485,100
412,500 x 1.12
= 462,000
x 1.05   = 485,100

Q16b

Q17a

A service’s costs are shown in total for the hours a customer uses. How much would the customer pay for 8 hours?

Ans: £24 . Double £12 : or £12 = 4 hours so £3 = 1 hour so 8 hours = 8 x 3 = £24

Q17b

In fact the customer uses 2 hour of the service. The bus stop to the service is exactly at their home but on reaching the terminus a half hour walk is needed to the service and then half hour back to the terminus. Using the timetables, what is the latest bus the customer can catch from home to be sure of getting home after the service for 3.45 pm?

Ans: 10 am bus . This would reach terminus at 11 am, arrive at service at 11.30, take 2 hour service to 1.30 pm, walk back to Terminus by 2pm, catch 2.30 pm bus back and arrive at home at 3.30 pm. This is 15 minutes in hand but later bus would be 45 minutes late. The 9am bus would be unnecessarily early.

Edexcel’s Pearson is in the Job applicant arena too.

Pupils will be familiar with the Pearson Edexcel GCSE papers. Well, Pearson have a job applicant’s test too called NDIT (numerical data intepretation test). They say that “NDIT measures candidate ability to manipulate and interpret numerical information from dashboards and reports. These skills are rated as “important” for nearly 300 jobs ranging from sales managers to executives”.

The questions could range from the simple ones like this:
Q18. What number must replace N to make this correct?

7N
+ N
=88

Choice
A5 B6 C8 D9

Ans D9. There is an element of reasoning in this because if the question is algebraic then the answer might be N = 11 since 8 times N would be 88. However, that’s not an available answer so it must be a simple sum of
79 + 9 = 88 so N = 9.

And then typically business oriented questions like this:

Q19. A consultancy’s operating costs to turnover ratio is 3:20 each year. If the company’s turnover is £213,250 in Year 1, £268,460 in Year 2, and £328,915 in Year 3 what are the total operating costs for the three-year period?

Ans: £121,594

There are many different Job Application reasoning tests. Another is “EPSO” for e.g. EU applicants and a typical question is this:

Q20 .In 2012, Belgium represented 2,5% of the total EU output value of the agricultural industry. In 2013, the total EU output value is expected to grow by 15%. Given France is expected to increase its total share by 5 percentage points, what is France’s expected output value in 2013?

Ans 117,300

Belgium is 2.5% so 2012 total EU is 40 x 8500 = 340.000
France share = 85,000/340,000 = 25% in 2012 so 30% in 2013

2013 total = 340,000 x 1.15 = 391,000
So France = 0.3 x 391,000 = 117,300

Although the test contains many conventional numerical reasoning tests it does assume quite an advanced maths knowledge such as Surds. And some follow a very specific style as follows:

Question 21: Does 2x + 8 = 12?

Then comes the standard format:

You are given two statements:

In this case the statements are::

1. 2x + 10 = 14
2. 3x + 8 = 14

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

The answer is option 4 since in both cases x = 2, which fits the original.

Tips

Here is some advice – my top ten tips – for approaching Numerical Reasoning tests in general: first before the Test:

• Know which test you are taking. This sounds obvious but this can be vital in identifying the style and typical format of the questions. Then you can hopefully find some free practice pehaps on the internet, or at worst invest a small amount of money in a license, or a book, or a little tutoring.
• Practice the key topic areas outlined at the beginning such as BIDMAS and ratios – not just within your specific test but in general e.g. from past GCSE papers. Of all the techniques, probably percentage is the most important to be tight on,  including for instance “10% of…” , or a “5% increase…”
• Practice both mental arithmetic and calculator skills – assume you can use a calculator but if, as is likely the case you are told not to, then there are certain key operations to perfect which might be quicker in your head anyway, such as “1/2 of…” or “180 degrees of a pie chart circle….” or “50 % of…”.
• Practice working at speed as this can be a key differentiator
• Whether out and about or listening the news or reading newspapers, whenver a graph or data table appears then you or your children should take the opportunity to understand it

Then during the test:
• Understand the question but do not over complicate or be intimidated: for instance if a column heading in a data table contains a term you are unfamilar with – let us say job title “actuary” – then don’t worry simply use the number beneath it. Look out for key words or phrases like “more than” or “cumulative” or “value”
• When looking at graphs ensure you look at the units of measure carefully on the axes such as “thousands of units”
• In tables of data identify the correct row and column or intersection depending on the question. Note also that in some cases some data may be irrelevant. Hence a quick look at the question before you look at the table or graph may be useful.
• In multiple choice questions it is sometimes possible to eliminate obviously wrong answers. Sometimes a quick estimate will point to perhaps only two possible answers.
• Work at a good pace and remember that a blank answer scores zero. Don’t get stuck on one question to the detriment of others.

And in conclusion

Numerical reasoning tests are a great leveller. In principle anyone from 9 (admitedly pretty good at maths) to 90 could take a test and get a similar score. Indeed, the same numerical reasoning question could appear in an 11 plus exam as a job application aptitude test.

The large majority of maths techniques needed to suceed are not of the “quadratic equation” type, rather the “basics” type.
But you do need to be very comfortable with the basics, that is a prerequisite; however, that in itself is not enough because higher order skills are needed such as understanding the question, processing and analysing the data, determing your approach – and all very quickly.

The rise of such tests is welcome in terms of the numerical skills needeed for the future of work, – see the Barclays Life Skills video – not just in technical roles but in for example sports,  managerial and military, and (it would be nice!) among local and national politicians too.

In terms of inclusion of numerical reasoning questions beginning to appear in for example GCSE exams, this is good, but it would be naïve to believe pupils are desperate for real world, wordy, problems – most simply want the most straight forward in line with their standard revision. So examiners, be kind!

Appendix.