Mental Arithmetic – Basic Mental Maths Hacks

See also: Multiplication

Mental arithmetic is the invaluable maths skill of doing calculations in your head, without the use of any tools, such as a calculator or pen and paper, or fingers! It can come in handy in countless everyday situations, from working out the best multi-buy deal in the supermarket, to calculating how long you will need to wait for the next train.

People who need to use maths in their jobs, whether accountancy, retail or engineering for example, often do quite complex and fast estimations in their heads, so that they have a good idea of what the answer will be before they take the time to do a more complex calculation.

Mental arithmetic also helps to develop a real understanding of the mathematical methods of arithmetic, rather than just doing calculations through a process of memorization.

Practising mental arithmetic might seem like hard work, and to some people who find maths difficult it might even seem like a scary prospect. But as with all things, the more you do it, the easier it gets. This page gives you some helpful hints and tips to make the process quicker, easier and a whole lot less scary.

Everyone can learn mental maths hacks! They are not just for maths wizards.


Multiplying numbers by 10, 100 and 1000 and their multiples

To do simple multiplications, you need to have a basic understanding of place value. For more on this, see our page on Numbers. Two things to remember here are:

  • Zeros are important
  • Decimal points always separate the whole numbers from the ‘bits’.

To mentally multiply any number by 10:

Keep the decimal point where it is. In your head, move all the digits one place to the left and add a zero to the end if needed.

24 × 10 = 24.0 × 10 = 240
175 × 10 = 175.0 × 10 = 1750
3.56 × 10 = 35.6

You can move the decimal point instead of the digits, but only do one or the other!


Some people find it easier to think of the decimal point moving, rather than the digits moving. In the example above, the decimal point stays in the same place and all of the digits shift to the left.

This is the same as moving the decimal point to the right!

24 × 10 = 24.0 × 10 = 240
175 × 10 = 175.0 × 10 = 1750
3.56 × 10 = 35.6

To multiply any number by 100:

Either
Keep the decimal point where it is. Move the digits two places to the left, adding zeros to the end if needed:
845 × 100 = 845.00 ×100 = 84500
37.64 × 100 = 3764

OR
Move the decimal point two places to the right:
56.734 × 100 = 5673.4

To multiply any number by 1000:

Use either of the two methods as before and move three places:
Move the digits to the left:
23.476 × 1000 = 23476
Or move the decimal point to the right:
8.45692 × 1000 = 8456.92

Multiplying by multiples of tens, hundreds and thousands or more:

The basic idea: If you need to multiply a number by 200, multiply by 2 first, then move the digits. You can do this with any quantity. For example, if you need to multiply something by 5000, multiply your number by 5 first, then move three decimal places.

The number of places you move is always the same as the number of zeros.

For example, multiply 25 by 5000. This seems quite hard to do in your head, but the trick is to break it down into easy calculations.

First multiply 25 by 5:
25 × 5 = 125

Then move the digits three places to the left (or the decimal point three places to the right):
125 × 1000 = 125000.

Dividing by 10, 100, 1000 and multiples

This process is exactly the same as with multiplication, but in reverse.

To divide by 10 you either

keep the decimal point where it is and move your digits one place to the right,

or

move your decimal point one place to the left.

For 100, you move two places.
For 1000, you move three places, and so on.

Examples:

785 ÷ 100 = 7.85
56 ÷ 1000 = 0.056

Remember there must always be a zero to the left of your decimal point if your answer is less than 1.0

450 ÷ 1000 = 0.450 = 0.45

You can remove any zeros to the right of the numbers after the decimal point. However, you CANNOT do this if the zeros come before the decimal point, or between the decimal point and other numbers.

Diving by multiples of tens, hundreds or thousands (or more):

The basic idea: If you need to divide by 7000, first divide by 7, then move your digits three spaces.

For example, 56 ÷ 7000:
56 ÷ 7 = 8
8 ÷ 1000 = 0.008

Is your answer what you would expect?


If you are worried you won’t remember whether you are mentally moving your digits left or right, take a look at your answer.

If you are multiplying your original number by number greater than 1, then you would expect your answer to be larger than the number you started with.

Likewise, if you are dividing by a number greater than 1, then your answer will be smaller. If it isn’t, then you know you’ve got it the wrong way round!


Adding and subtracting in your head

In the same way as you did with mental multiplication and mental division, you can learn some tricks for making mental addition and subtraction easier.

As before, these tricks don’t involve any maths wizardry, it is simply a case of breaking the problem down into smaller parts that are easier to tackle in your head.

The best way to do this is with some examples.

Example 1:

Splitting the subtraction into hundreds, tens and units (or more).

Calculate 352 – 13 in your head.
Split this into two easier subtractions: Taking away 13 is the same as taking away 10, then taking away 3.
352 – 10 = 342
342 – 3 = 339


Example 2:

You can apply the same principle as illustrated in Example 1 to a harder subtraction:

Calculate 4583 – 333 in your head.
First take away 300, then 30, then 3:
4583 – 300 = 4283
4283 – 30 = 4253
4253 – 3 = 4250


Example 3:

Dealing with awkward numbers that are close to 10:

Calculate 77 – 9 in your head.
Taking away 9 is the same as taking away 10, then adding 1.
77 – 10 = 67
67 + 1 = 68


Example 4:

Dealing with awkward numbers that are close to 100:

Calculate 737 + 96 in your head.
Adding 96 is the same as adding 100, then taking away 4.
737 + 100 = 837
837 – 4 = 833


Example 5:

Dealing with awkward numbers that are close to 1000 (or even bigger):

Calculate 5372 – 985 in your head.

This one looks even harder than the others, but no matter how large the numbers involved, you can still break the calculation down in to simple chunks.

Subtracting 985 is the same as subtracting 1000, then adding 15 (because 1000 – 985 = 15). You can even add the 15 in stages, by adding 10 then adding 5.

5372 – 1000 = 4372
4372 + 10 = 4382
4382 + 5 = 4387


Adding and multiplying in your head

Sometimes you might have a really tricky calculation to do in your head and it just seems impossible. However, if you look at how it can be split up using the skills you have learned in the examples above, something really tricky can become much simpler.

For example, calculate 97 × 7 in your head.

There are two ways you could tackle this one and you might find one way easier than the other:

Method 1:

97 is the same as (100 − 3), so you can think of the calculation as
7 × (100-3)
This is the same as
(7 × 100) – (7 × 3)

Now you have replaced the difficult multiplication with two simple multiplications and a subtraction:

7 × 100 = 700
7 × 3 = 21
700 – 21 = 700 – 20 – 1 = 679

Therefore 97 × 7 = 679

Method 2:

97 is almost 100, so you can begin by working out 7 × 100 = 700.
The next step is to take account of the difference between 97 and 100, which is 3.
So 7 lots of 3 is 21.

700 – 21 = 679


Applying Mental Maths Skills to Money and Percentages


As you have found out from the examples above, mental maths skills are all about breaking a problem down into numbers that are easy to deal with in your head. Sometimes we need to turn the calculation around and think of it in a different way.

Two examples of when you might need your mental maths skills are when you are dealing with money, or when you need to calculate a percentage, both of which occur often when you are out shopping.

When dealing with money, it can help to round the amount up to the nearest whole pound, then deal with the pennies separately. You often see prices marked in a way to make you think they are cheaper than they actually are. £24.99 for example, is only one penny away from £25, but the seller wants you to think that it’s nearer £24. When you are doing mental maths calculations, £25 is a lot easier to deal with than £24.99.

A useful mental maths hack for percentages is to remember that they are reversible, so 16% of 25 is the same as 25% of 16. Invariably, one of those will be much easier to work out in your head… try it!

Conclusion

Mental arithmetic can seem quite scary, but with practice, you can use these mental maths hacks to break a difficult problem down into smaller chunks that are easier to think about. There is no wizardry involved, it is just a matter of seeing the problem in a different way.


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