Non-Vedic Maths: Half and five for the odds

In my series of Non-Vedic maths I will present mental arithmetic techniques that allow you to speed up your number-crunching. In contrast to Vedic maths, I will also explain why these techniques work. I believe that being able to add and multiply quickly does not make a good mathematician. But understanding how things work and being able to develop your own techniques is more important.

Enough already! I'm repeating myself!

Today's technique teaches how to multiply any number by 5, 50, 500 etc. The technique is based on the simple fact that

5 = 10/2

In this way multiplication by 5 is the same as multiplication by 10 and the division by 2. By using a technique for quickly dividing by 2 the method is accelerated further.

So let's start with an example. Say you want to multiply 3465 by 5. We start with multiplying by 10, i.e. adding a zero at the end

3465 × 5 = 34650 / 2

Dividing by two can be done in two ways. If you know your multiples of two up to 9×2=18 by heart then the traditional method is probably best. Just divide every digit from left to right including the carry over from the remainder. Note that the carry over is either 0 or 1, so the largest number that you will ever have to divide is 19. In our example we get

34650 / 2 = 17325

This is our result!

Some people are not as comfortable or quick with dividing numbers above 10 by 2. For these people the following alternative method might be faster. First divide every digit by 2 and forget about the remainder or carry over,

34650
12320

Next we shift the number to divide one place to the right (this was our original number) and replace every even digit with 0 and every odd digit with 5.

3465
5005

This second step effectively accounts for the carry overs that we missed in the first step. Now add the results of these two operations together

12320
 5005
-----
17325

Because all the digits of the first number result from division of numbers less than ten, none of the digits can be larger than 4. For this reason we never have to worry about carry-overs when adding the two numbers.

For a slightly more mathematical explanation to what we have done here, let's write 34650/2 in the following way

34650/2 = (24640 + 10010)/2
        = 24640/2 + 10010/2
        = 12320 + 5005
        = 17325

To summarise, in order to multiply a number by five

• append a zero at the right of the number
• divide every digit by two, ignoring any remainders
• add five to every digit to the right of an odd digit of the number that you got after the first step.