Why do they change the division sign to multiplication and change the divisor to its reciprocal?

Answer 1

This process is to make life easier, in ways.

For example:

1/3

----

1/3

As with multiplication, the numerators stick together and denominators stick together

In this case

1/1 (numerators) and 3/3 (denominators)

In this case, this question works out nicely to one (1/1=1, 3/3=1, therefore the question equals 1/1 or 1)

However, this is not always the case.

An example of this would be:

1/3

----

1/2

Obviously, here we get decimal places in the denominator, which is not allowed in math (neither are decimals allowed in numerators).

Lets go back to the first question.

What if we changed the division sign to multiplication and made the second fraction a reciprocal?

We get:

1/3 1 3

---- := --- x ---

1/3 3 1

Then do the multiplication

1x3 = 3

3x1 = 3

Then calculate the fraction

3/3 = 1

Well what do you know! It's the same as the previous answer.

Does this work all the time? Yes.

Why does it work? Well, that's because reciprocal is just the other end of the spectrum when calculating. For example, when dividing a number by two, you could make that into a reciprocal (now 1/2) and multiply them, getting the same answer.