Verbal Reasoning : Number Series (Type P)

The Number Series question type requires you to complete the series by working out the missing number in that sequence. This question will provide you with a series, which is a set or sequence, of numbers from left to right.

The aim for you is to work out, from the series, what the rule is from one number to another. By establishing a rule you can then use the same rule to work out the missing number. So there are 2 objectives here:

1. Identify the general rule by understanding the change from number to number
2. Recognise which number is missing
3. Apply the rule to get your missing number

The changes from number to number can happen in a variety of ways these include:

1. Addition by a set number e.g. +2 each time
2. Subtraction by a set number e.g. -2 each time
3. Multiplication by a set number e.g. x2 each time
4. All of the above but with increases to the change amount e.g. +2, +3, +4
5. All of the above but with decreases to the change amount e.g. -10, -9, -8

The above are the most common example and changes you will come across but in order to work out the rule you have to make sure you confirm the changes from number to number in the existing series. By doing so you can confidently apply the rule – rushing too much or not double-checking could easily lose you marks.

Remember in a question like this all the answers are on the paper, all you have to do is work it out.

By taking a step back and looking at the question type you can see that it aims to test the following skills:

• Logical and systematic process which includes double-checking
• Being able to work with numbers or data and identify patterns / trends
• Systematically identifying and proving a rule
• Being able to apply that rule and confidently state the next number in the sequence
• Strong mathematical skills in order to successfully work out the changes between numbers
• After working out the changes in the series of numbers, recognising the differences or similarities in those changes to create a rule
• Being able to apply a rule at any point in the sequence by understanding the overall rule

When you look at initial questions this question type does look straightforward, it can be become tricky when more complex rules and patterns are applied. If you mix that with missing numbers which aren’t always at the end of the series you can understand how it can become difficult very quickly.

Essentially the skills it is testing boils down to your mathematics ability and knowledge. Not only should you be practising them seperately so addition, subtraction, multiplication and division. But you should also be comfortable mixing them up and doing longer maths so for example doing a multiplication followed by a subtraction.

You should also be familiar with odd, even and prime numbers.

This will help your mind to quickly work through multiple operations and make you more comfortable in doing so.

You also need to be aware of how patterns and sequences are established – that includes how things follow, are the same, or similar to other things and why. Being able to spot number patterns quickly will give you an edge here from which you can create a rule to apply.

To continue the skills above, this question type is here to test the following abilities:

• Basic mathematics knowledge and working out ability
• Being able to work with data that hasn’t been presented to you before
• Ability to recognise similarities and differences between numbers
• Creating a rule after recognising a pattern or trend in how things change
• Applying that rule in different circumstances / situations e.g. missing numbers in different places

If you really hone down and focus on the abilities above you should be able to adapt your skills to any question that comes up.

What does the Number Series question type look like?

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

• General example with instructions
• 1 set / sequences of numbers per question
• A missing number in either the beginning, middle or end of the sequence
• 1 mark per question
• Multiple choice options on the answer sheet

How do I answer this question?

As always, like our skills and abilities, there is always a systematic process we can apply that will help us identify the pattern / changes and then create a rule off the back of that.

By regularly practising will you then be confident in applying a process, remember, it is just as important to be familiar and quick in applying a process as much as it is to know the process.

Lets breakdown at what the process to answer the Number Series question type looks like:

1. Read the full series of numbers and identify where the missing number is (number you have to fill)
2. Are the numbers ascending or descending? You can use this to decide whether to add or subtract
3. Which number before the missing number do you need to look at to work out the missing number?
4. By knowing whether the number that is missing is at the beginning, middle or end you can apply the process in slightly different ways to help you work out the pattern and rule
5. Missing number at the beginning
   • Look at the numbers after the missing space
   • Go from each number to the next and work out the difference between the two numbers e.g. is it going up by 5? Or going down by 2?
   • Keep a record of the changes on paper so you can quickly look at them
6. Missing number in the middle
   • Look at the numbers before the missing number
   • Go from each number to the next and work out the difference until you get to the missing number
   • Look at the numbers after the missing number, go from each number to the next and work out the difference between numbers
   • If you struggle to do that look at the last number in the series and work backwards
   • Keep a record of the changes on paper so you can quickly look at them
7. Missing number at the end
   • Go from the numbers left to right
   • Between each number and the next work out the difference
   • Keep a record of the changes on paper so you can quickly look at them
8. Depending on where the missing number is what have you noticed? Are the numbers in the existing series going up by 2? Are they going down by 3? Is the change between numbers increasing each time e.g. +2 then +3 then +4
9. Work out the differences in the changes that are occurring
10. Create a rule based on what you see
11. Work out the amount to be added, taken away or multiplied based on the changes you have recognised and also based on the position the missing number is in
12. Double check that your answer makes sense according to the rule you have identified

Working Example

Before we start its important to understand that the process focuses on:

• Reviewing the question
• Identifying the series and the missing number
• Working out the differences and changes so we can create a rule
• Applying the rule to find the missing number

Now we have broken down the process and understood each focus lets apply it.

Question 1

Work out the missing number in the following sequence:

85, 84, 80, [ ], 63, 50, 34

So lets look at the question from left to right.

First of all there are 7 numbers in the series.

The missing number is in Position 4.

The numbers are decreasing starting at 85 in the series and ending at 34 – so already I can tell that the change occurring between numbers will be subtraction and I need to work out the 4th number.

Now I know the middle number is missing let me work out the changes in the numbers before Position 4.

So I need to look at number 80 and apply the rule to number 80 to then work out my missing number. So the current pattern in number changes are:

85 to 84 = -1

84 to 80 = -4

Now because the number that is missing is in the middle I will go to the other side and work out the changes / differences from number to number on the other side.

63 to 50 = -13

50 to 34 = -16

So here are what the changes look like from number to number including the missing number:

• 85 to 84 = -1
• 84 to 80 = -4
• 80 to Missing Number = -?
• Missing Number to 63 = -?
• 63 to 50 = -13
• 50 to 34 = -16

Now this is where it may become a bit complex, in order to create a rule I need to look at the differences in the changes because the changes aren’t all the same so:

• To get from -1 to -4 you have to -3
• To get from -13 to -16 you have to -3

Now we see a pattern, the series starts off with the first change being by -1, but each change after that decreases by -3. So lets apply that pattern:

• 85 to 84 = -1
• 84 to 80 = -4
• 80 to Missing Number = -7
• Missing Number to 63 = -10
• 63 to 50 = -13
• 50 to 34 = -16

This looks like it makes sense, we will work it all out and see if we get complete whole numbers in the series:

• 85 – 1 = 84
• 84 – 4 = 80
• 80 – 7 = 73
• 73 – 10 = 63
• 63 – 13 = 50
• 50 – 16 = 34

There are a few indications that we have worked out the right rule:

1. We applied -3 consistently to each change e.g. -1 to -4 we -3, -4 to -7 we -3, -7 to -10 we -3
2. Going from the missing number (Position 4) to the next number (63) and applying a -3 meant that we got 63 which is already part of the series

We have worked out that we:

1. Need to subtract every time we go from number to number in the series
2. Each subtraction number increases by 3 from number to number so it starts at -1, then -4, then -7, then -10 and so on
3. From position 3 to position 4 that the change would have increased to -10
4. That the missing number is 73!

Make sure to mark the answer correctly on the answer sheet or the box given to you.