Why is multiplying a number by 200 the same as multiplying it by 100 and then by 2?

Answer 1

When you multiply three numbers together you can either multiply the first two together and multiply the result of that by the third, or multiply the last two together and multiply the result of that by the first.1

200 = 100 x 2

So, for example:

5 x 200 = 5 x (100 x 2)

= 5 x 100 x 2

= (5 x 100) x 2

Meaning that multiplying by 200 is the same as multiplying by 100 and then by 2.

1This is the "Associative Law". Using smaller numbers as an example:

Suppose you have 2 x 3 x 4.

It says:

2 x 3 x 4 = (2 x 3) x 4

= 6 x 4

= 24

or:

2 x 3 x 4 = 2 x (3 x 4)

= 2 x 12

= 24

As multiplication is short hand for repeated addition, this can be seen by using only additions:

2 x 3 is the same a 3 + 3 (adding 3 to itself two times, that is twice 3) or 2 + 2 + 2 (adding 2 to itself three times, that is 2 thrice), the result of all three being 6:

2 x 3 = 6

3 + 3 = 6

2 + 2 + 2 = 6

Now, 2 x 3 x 4 can be calculated as (2 x 3) x 4, that is do 2 x 3 first and then multiply the result of that by 4. In terms of addition, this becomes:

(2 x 3) x 4 = (3 + 3) x 4

= (3 + 3) + (3 + 3) + (3 + 3) + (3 + 3)

= 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3

= 24

Or, it can be calculated as 2 x (3 x 4), that is do 3 x 4 first and then multiply the result of that by 2. In terms of addition, this becomes:

2 x (3 x 4) = 2 x (3 + 3 + 3 + 3)

= (3 + 3 + 3 + 3) + (3 + 3 + 3 + 3)

= 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3

= 24

In both cases the result is the same - adding 3 to itself eight times to get 24.