How to carry out calculations crisply

By Mark Dawes, June 2025

Here is a 3-mark question from the 2025 KS2 mathematics SATS, paper 3:

The mark scheme gives all three marks for the correct answer, and suggests that this involves long multiplication and then long division:

There is an interesting alternative way to carry out this calculation that involves no long multiplication or long division.  I think it would be helpful for Year 7s to explore this.

First, let’s look at a calculation that KS2 pupils probably wouldn’t carry out from left to right:

I expect many children would realise there is no point in adding 57 and then later subtracting it, and would recognise that the answer is the same as 76 + 12.

To make our crisp question easy in a similar way, the pupils need three important ideas/skills:

1) Division can be written as a fraction. 

2) Fractions can be simplified by dividing the numerator and denominator by the same thing:

3) The multiplication table can be used to rewrite numbers as a product.  For example, 56 = 8 x 7

This means

can be rewritten as:

35 = 7 x 5, 48 = 6 x 8 and 56 = 8 x 7, so we can write

We can now simplify this to give 5 x 6 and the answer is therefore 30.

What did we need to do here?  Recognise that 35, 48 and 56 all appear as answers on the multiplication table, write each of them as a product, and then divide common factors.  While this is conceptually more challenging that carrying out long multiplication and then long division, it is procedurally more straightforward.  (It is also perhaps interesting that the context to the question doesn’t interest me at all, whereas the maths behind it is more exciting!)

In the same way that in 76 + 57 + 12 – 57 we made things easier for ourselves by spotting that it included +57 and –57, here we are recognising that the 35 and 56 both include a 7, and that when we multiply by 48 (which is 6x8) and then divide by 56 (which is 7x8) we needed multiply and divide by the 8s.

This leads to an idea for a lesson, perhaps in Year 7, where pupils could first carry out calculations like this the long way and could then explore some conceptual short-cuts.

How easy is it to find other questions that are structured like this? 

What about:

This can be written as 

which can be seen to be 35.

Is there a way to be more systematic about this?  And what ‘rules’ are we going to follow? 

Let’s just pick three numbers that appear in the multiplication table.  (I will use 12 by 12, but I think 10 by 10 would also be fine for this purpose.)  I will follow the structure of the original crisps question and will multiply two of the numbers together and then divide by the third one.

If the denominator is 56, then we know that this is 7 x 8.  (It is also 14 x 4 and other things, but these don’t appear on this version of the multiplication table).  This means we need one of the two numbers in the numerator to be a multiple of 7 and the other to be a multiple of 8.  Can we pick anything from the 7 row and anything from the 8 row?  Yes, we can!

Here are a few more that involve dividing by 56:

What would need to happen if we divide by 42?

Because 42 = 6 x 7, one of the two initial numbers would come from the 6x table and the other from the 7 times table.

That means we can pick anything from each of the brackets and could have:

(Some are clearly not very interesting, like choosing 42 from either of them.)

If the answer to a question of this form is a whole number, can we always break it up in this way?

Yes … though there can occasionally be some issues that crop up.  For example, if there is more than one way to break the numbers up then we might pick an unhelpful one. 

21 x 40 ÷ 24 works very well if 24 is treated as 3 x 8  and 40 is treated as 5 x 8 because it gives

but is less helpful if we use just one of 24 = 4 x 6 or 40 = 4 x 10, when we might get

and while this still simplifies to the same result, the manipulation is slightly more challenging than in our previous questions. 

If we continue to write our numbers as multiplications (without ever using a 1) then there isn’t a problem.

So: 24 = 3 x 8 and the 8 can be rewritten as 8 = 2 x 4 and the 4 is 2 x 2.

This gives us 24 = 3 x 2 x 2 x 2, which is perhaps a useful introduction to the product of prime factors.

Returning to 21 x 40 ÷ 24, this becomes

and this simplifies nicely.

Are there any others that are problematic?  If we go beyond the multiplication table then we can have some issues, but other than that it will always work.  (An example is 39 x 14 ÷ 21)

Can pupils explain why it is the case that if we can’t do a question like this in the quick way then the answer will include a decimal?

Can pupils tell which of these questions give an integer answer just by looking at them and without needing to work them out?

I am looking forward to trying this out with Year 7 at the start of next year!

Source: https://www.gov.uk/government/publications/key-stage-2-tests-2025-mathematics-test-materials