Mental multiplication of odd numbers by 5

There are some very interesting, magical (initially) tricks used in mental mathematics which once practiced can serve you well. Once I am over the apparent magical quality of a procedure I cannot help myself but try to fathom out why it actually works. This blog post looks at just such a trick. The multiplication of any odd number by 5. Now some people might think multiplication by 5 is not so difficult as it is half of a multiplication by 10. However take the following problem and consider how to calculate using only mental methods.

mental1

Here is a procedural set of rules that will get you to the result very quickly.

Subtract 1 from the number being multiplied by 5 (231 – 1 = 230)

Half this (115)

Place a 5 at the end of this making a 4-digit number (1155)

Result 231 x 5 = 1155

Before we go further try using the same procedure to solve 417 x 5 to see if you get 2085 as the result.

Why does this method work?

We start by looking at the two whole numbers either side of 231, that is, 230 and 232. Both of these numbers are even and therefore when multiplied by 5 will both be numbers that end in a ‘zero’. We deduce from this reasoning that 231 when multiplied by 5, as it is an odd number will end with a 5. Keep this thought in your head. We can now temporarily forget multiplying 231 by 5 and instead concentrate on multiplying 230 by 5.

Why choose 230 instead of 232?

Simply because 230 x 5 is less than 231 x 5, but only 5 less. Therefore when we have found the result of 230 x 5 we simply need to add 5 to the result. So we can now start to think about 230 multiplied by five but in a cunning way! If instead we multiply it by 10 then divide by 2 we will get the same result.

divide by2

The right hand side of this equation (identity really) results in 2 300 ÷ 2 which is 1 150. Great that is what we thought it would be! Look at the digits however and we can see that the ‘units’ place holder is 0 and as the multiplication is 5 less than the original question asked we simply add the 5 to it effectively replacing the 0 with a 5 in the number (1 155) and always getting a correct result.

Recap– Multiply any odd number by 5

Take any odd number and subtract 1 from it, half the result (leaving so many 10s) and add 5 at the end.

Test it out on these

a) 205 x 5,      b)49 x 5      c) 111 x 5      d) 3 641 x 5

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