📖Definition
In these questions, the missing number in the third set of brackets is formed from the numbers either of side of it in the same way as those in the first two sets of brackets are formed. You have to find the missing number.
✏️Example
( 4 [ 20 ] 5 ) ( 2 [ 16 ] 8 ) ( 3 [ ? ] 7 )
Knowledge Required
You need to know your times tables and be able to do mental arithmetic accurately. Also, being able to spot patterns quickly will help.
✅How To Answer
Some of these questions are straightforward in that the middle number in each set is formed by a single operation on the outer two numbers.
The operations will be: + - ÷ x.
Looking at the example above, in the first two sets, the numbers inside the brackets are bigger than those on the outside. The operations that make (positive whole) numbers bigger are + and x so look at those first.
4 + 5 = 9
2 + 8 = 10
Addition isn't the answer.
Try multiplication.
4 x 5 = 20
2 x 8 = 16
That's it! So, to form the number inside the brackets, you multiply the numbers on the outside by one another.
That means our answer will be: 3 x 7 = 21
✏️Example
( 16 [ 15 ] 31 ) ( 3 [ 14 ] 17 ) ( 12 [ ? ] 19 )
Here, in the first two sets, the numbers inside the brackets are smaller than all those on the outside. The operations that make (positive whole) numbers smaller are - and ÷ so look at those first.
The last number in each case is greater than the first number so take the first number away from the first one.
31 - 16 = 15
17 - 3 = 14
So, the middle number is obtained by subtracting the first number from the last one.
Our answer will be 19 - 12 = 7
Now let's look at a more complicated example.
✏️Example
( 10 [ 4 ] 2 ) ( 5 [ 6 ] 13 ) ( 7 [ ? ] 2 )
Here, it's clear to see that adding the numbers won't work, neither will multiplication.
And neither will division or subtraction.
So, we need to try a combination of operations.
Let's try multiplying and see if a pattern emerges.
10 x 2 = 20. Is 20 related to 4, the number in the brackets?
Yes! 20 = 4 x 5.
Let's try that on the second pair.
First multiply the outer numbers: 5 x 13 = 65.
Does 65 equal the inner number, 6 times 5? No!
So, start again and try something else. Addition.
10 + 2 = 12
Is 12 related to 4? Yes, 12 = 4 x 3.
Now try the second set.
5 + 13 = 18.
AND 18 = 6 x 3.
That's our pattern. Add the outer numbers and then divide by 3.
Our missing number will be [ 7 + 2 ] ÷ 3 = 9 ÷ 3 = 3.
Variances
One type of pattern that comes up often is adding the outer numbers together and dividing by 2. That is, the middle number is the Average of the outer numbers.
✏️Example
7 [ 11 ] 15 19 [ 16 ] 13 8 [ ? ] 12
7 + 15 = 22, 22 ÷ 2 = 11
19 + 13 = 32, 32 ÷ 2 = 16
So, finding the average in the final set, 8 + 12 = 20, 20 ÷ 2 = 10.
✏️Example
And occasionally, there'll be an operation performed on the average.
So, the above example could have been:
( 7 [ 8 ] 15 ) ( 19 [ 13 ] 13 ) ( 8 [ ? ] 12 )
Once you've eliminated a single +, -, ÷ or x, then it's worth checking the average.
The average of 7 and 15 is 11, and of 19 and 13 is 16.
Look at the numbers in the brackets and see how they relate to these:
8 ... 11 and 13 ... 16
Can you see the pattern? The number in the brackets is 3 less than the average so our answer will be 10 - 3 = 7.
💡Tips/hints
If the outer numbers add up to an odd number, then averages won't be involved as the average of two numbers, A and B is [ A + B ] ÷ 2 and these questions don't use decimals.
If you're stumped, look at the two middle numbers you've been given and see if they have any common factors (these will usually be either 2 or 3). If they do, then divide them each by that factor and then see if you can then find a pattern.