Verbal Reasoning : Working out Number Relationships (Type K)

The Number Relationships question type will give you 3 groups of numbers with 3 numbers in each group. The 2 numbers on the right and left side come together to make the middle number, this may not be obvious at the beginning, but you are required to work out the formula.

There will be one group which will have a number missing in the middle usually surrounded by brackets [ ].

You will be required to work out what that missing number is.

Now, in order to do that, you will be expected to look at the other 2 groups which have the middle numbers already filled in. From there, you have to work out what the formula is by using the number on the right and the number on the left (not the middle number), and determining whether you need to add, multiply, divide or subtract.

You will be able to test that formula by checking to see if it gives you the middle numbers of the groups that have the middle number already filled in, you can then apply it to the third group in order to work out the missing number.

A question like this is very tricky because you are not only expected to work backwards, but you could potentially go down different routes whilst trying to get to the original formula – some may seem to work, others may not work at all, which eats into your time. The skills that a question like this looks to test are:

• Quick Maths skills
• Identification and application of rules / formulas
• Trial and error
• Identifying the mathematical relationship, pattern and similarity
• Working out under pressure
• Exploring solutions
• Analysis and deciphering of an existing solution
• Ability to break down an already complete solution in order to understand how they got there

As mentioned before, this question type is very difficult because you’re working backwards. You have to narrow down your potential solutions into the one solution, formula or equation that works. To do so you have to apply a process of trial and error, which is an area where you can lose the most time.

The one key thing they are expecting to really get out of you are your Maths skills – the more familiar and comfortable you are with Maths, the more natural it will be when trying to find a formula that works.

What you will be expected to demonstrate here is the ability to think broadly, logically, quickly and creatively whilst coming to a conclusion without the full picture (which is the missing number). Try to ask yourself:

• How do these 2 numbers come together?
• What can I do to do these 2 numbers to give me the final number?
• Is it multiplication, division, addition or subtraction?
• What patterns can I see, what possible solutions are there that get me close to that middle number?

What does the Number Relationships Question Type look like?

As you can see from the Gaggle example below, the Number Relationships question type is in the following format:

• General example with instructions
• 2 groups of numbers which are completely filled – there is a number on the left, middle and right for each group
• 1 group of numbers that is missing the middle number (one you need to find the solution for)
• 1 mark per question
• Multiple choice options on the answer sheet

How do I answer this question?

It is really important you apply a process with a question like this because it is open-ended. What this means is that the equation / formula is up to you to decide, this means you could end up spending a lot of time on one question testing out multiple solutions, which will waste your time. You need to understand you have to explore, this means coming up with your own rules, formulas and equations.

The success of this process is only as good as your Maths skills, and because you are under time pressure, you want to be able to multiply, divide, add and subtract quickly. By doing so, you can work out which solutions you should leave and which one is your answer.

By applying a process you aim to work systematically, in order, and as you go through your process, it should bring you closer to the correct rule. When we say rule we also mean formula, equation, basically what do the 3 groups need to follow in order to get to the right answer for each group of numbers.

Remember these basic rules

1. Both numbers have to be put together at some point, at the beginning or end for example both left and right need to be added together, subtracted, multiplied or divided
2. Something could happen to each individual number before they are put together for example a number could be added, subtracted, multiplied or divided to the left or the right
3. Something could happen after both numbers have been put together for example another number is added, subtracted, multiplied or divided by
4. What happens to the right doesn’t have to happen to the left and vice versa for example if the right number has 2 added to it, this doesn’t mean the left number must have 2 added to it
5. You can work left to right, but also right to left

Lets look at what the basic process is and what kind of questions you should ask yourself whilst going through it:

1. Look at the numbers in the first group (on your left). This group of numbers should be complete meaning you have the number on the left, number on the right and the middle number is what they should equate to
2. Read the number in the middle, this is the number you want to get to in one way or another
3. If you add the number on the left to something and the number on the right to something do you get the the middle number?
4. If you multiply the number on the left to something and the number on the right to something do you get the the middle number?
5. If you divide the number on the left to something and the number on the right to something do you get the the middle number?
6. If you subtract the number on the left to something and the number on the right to something do you get the the middle number?
7. Do you need to have a mixture of addition, subtraction, multiplication or division to get to the number in the middle?
8. Is it left number first then right?
9. Is it right number first then left?
10. Work out how the 2 numbers on the left and right equal the middle number in Group 1
11. Apply the same rule for Group 2
12. If the rule works for both Groups (it gives you the middle number) then work out the middle number for Group 3

REMEMBER: Do not feel like you are restricted to only the numbers in the question. You can do whatever you want with them. This means you can multiply them by any number, subtract them with any number or add them to any number. You have already been given the answer which is the middle number, the task is for you to work out how they got to the middle number.

Lets apply the process to an example.

Working Example of Number Relationships

Question 1:

6 [ 10 ] 14

3 [ 19 ] 20

4 [ __ ] 9

If you look at the example above we notice a few things. First of all there are 3 groups of numbers, 2 groups have all the numbers filled out and the last group has the middle number missing.

We need to use Groups 1 and 2 to understand what the relationship is between the 2 numbers on the left and right side of the brackets. This will help us to get to the rule / equation or formula for the number in the brackets.

Take Group 1 which is 6 [ 10 ] 14

6 is on the left, 14 is on the right, they somehow equal to 10 (in the middle). We can then use the same rule to work out the middle number in the last group.

We have already done steps 1 and 2 in the process so lets move on to steps 3 and further.

If I add 6 to 14 i get 20, if I then divide 20 by 2 I get 10.

So I have made up a rule, which is based on putting the numbers together first and then doing something after that:

Add the left number to the right number and divide by 2 to get the middle number

Lets try that for Group 2. If I add 3 to 20 I get 23. If I divide 23 by 2 I get 11.5, which is not the middle number for Group 2. So that rule doesn’t work, I have try find another rule or relationship.

Lets try another rule which is based on doing something to the numbers individually then putting them together:

If I divide the left by 2, divide the right by 2, then add the left and right together

So if I divide 6 by 2 I get 3 and divide 14 by 2 I get 7. Add 7 and 3 together I get the middle number 10!

Lets test if this works by applying it to Group 2:

3 divided by 2 is 1.5 and 20 divided by 2 is 10. If I add 1.5 to 10 I get 11.5, this isn’t the number in the middle which is 19 so the rule isn’t right.

Now as you can tell we can carry on coming up with different rules and testing them out to see if they work. Its up to you to come up with your own rule, so lets try something else but start simple.

This time, the rule is based on putting the numbers together but from right to left:

Minus the right number from the left number

So lets give it a shot with Group 1. Minus 6 from 14 and you get 8, which is only 2 off from the actual number so lets change the rule a bit and say you have to add 2 after you have done the subtraction.

Now lets do the same to Group 2.

20 minus 3 is 17, then if you add 2 you get 19!

We have found a rule which works on both groups, we can now apply it to the third group.

9 minus 4 is 5, add 2 is 7 so your answer is 7!

Why did you choose to minus the right from the left?

The reason why I chose to minus the right number from the left is because all my other solutions weren’t working but I followed the main patterns. The pattern that I applied was putting the numbers together then doing something after that. Part of this pattern was to flip the solution on its head.

Where am I coming up with these patterns / rules – well, I am trying to use both the numbers on the left and right to get to the middle number – I am allowed to do whatever I want to them.

This was part of my trial and error, I didn’t know it was going to work, but I did know that I can apply any rule I wanted. Other things that stuck out to me were that the right number was always bigger than the left, this made it harder for me at the beginning.

I also realised that it wasn’t a simple add, subtract or multiply. I had to do a few things which was minus 2 numbers then add another number.

I didn’t know I had to add 2 as well as the end, it was just my trial and error approach that made me think that by adding 2 i would get to my final middle number.

REMEMBER: This question type is very difficult, you are expected to analyse and trial out different solutions until you get to one that works. At the beginning this may be very easy but as you do more questions the harder and more complex the rules may become. Don’t get frustrated, take it easy, and work through your process which is: add, subtract, multiply, divide and then do a mix. Always remember to look at it from different angles, not everything is left to right only.