understanding how multiplying a number by itself and than using the result for division gives you a consistent result

Question

hey guys i am basically a programmer and just came across a peice of mathematical calculation that i was curious to understand , have a look below ::

1200 / 40 * 40 = 1200

1200 / 30 * 30 = 1200

1200 / 20 * 20 = 1200 

how come when you multiply a number by the same number and than devide a number by the result , you get back the number that you devided ? this is a bit pusszling to me .

can somebody explain what this work and is there any kind of terminology for this formula ?

Thank you.

Alexander .

Answer 1

$\frac ab \times b=a $

Answer 2

The problem is that you think the multiplication is happening first, but the division is actually happening first. Then you get the scenario as in the other answer here. Try putting parenthesis around the multiplication and see what you get that way.

Answer 3

The formality is that division ( by any number other than 0) " undoes" the effects of multiplication. More formally, you can say that instead of dividing by $40$, you are multiplying by $\frac {1}{40}$ , so that, for any number $x$ : $$ x * 40 * \frac {1}{40}= x * \frac {40}{40}=x *1 =x $$, and we say that , or any nonzero number $n$, that $\frac {1}{n} $ is the multiplicative inverse of the number $n$ in any mathematical object called a field.