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.

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.

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