Multiplication by seven

The rule for multiplying by seven is very much like that for six:

Double the number and add half the neighbor; add 5 if the number is odd.

Suppose we wish to multiply 4,242 by 7. There are no odd digits in this number, so we shall not need to add the extra 5. For this example, the work goes the same as for six, except that now we double:

First step: 0 4 2 4 2 x 7 = _ _ _ _4 (double the number 2 )

Second step: 0 4 2 4 2 x 7 = _ _9 4 (double 4 plus half the neighbor)

Third step: 0 4 2 4 2 x 7 = _ 6 9 4 (double 2 plus half the neighbor)

Fourth step: 0 4 2 4 2 x 7 = 9 6 9 4

Last step: 0 4 2 4 2 x 7 = 2 9 6 9 4 (doubling zero we still have zero, but add half of neighbor)

Here is an example containing odd digits. The 3 and 1 are both odd:

First step: 0 3 4 1 2 x 7 = _ _ _ _ 4 (double 2; there is no neighbor)

Second step: 0 3 4 1 2 x 7 = _ _ 8 4 (double 1, plus 5 – because 1 is odd –is 7, and add half of 2)

Third step: 0 3 4 1 2 x 7 = _ _8 8 4 (not odd: double 4 and add half of 1)

Fourth step: 0 3 4 1 2 x 7 = _ 3 8 8 4

Last step: 0 3 4 1 2 x 7 = 2 3 8 8 4 (doubling zero is zero, but add half of 3, plus the dot)

The correct mental steps are:

1) Say “1” for the dot, if there is a carried 1.

2) Look at the next number to work on and notice whether it is odd. If it is, add 5 to the carried 1, saying “6”, or say “5” if there was no dot.

3) Looking at the number and doubling it mentally, we say the sum of the 5 and this doubled figure. If the figure is 3, for instance, we say “5”, then we say “11”, because doubling the 3 to get 6 and adding it to the 5 can be done in one step.

4) Looking at the neighbor, say 6 for example, we add half of it to what we have already. We were saying just now that we had 11. If the neighbor is 6, we next say “14”.

Let us take this process a little at a time. The mental training involved in doing this sort of thing is very valuable because it develops the ability to concentrate, and concentration is practically the whole secret of success.