Why does inverting an operation and taking a reciprocal give an equivalent answer?

[M10]

An example of what I mean:

8 x 4 = 32
8 / (1/4) = 32

32 / 4 = 8
32 x (1/4) = 8

Id just like to understand the logic why reversing the operation as well as taking the repicrocal of the divisor/multiplier is equivalent to the original problem. If anyone can help, thanks.

 

[Steven G]

An example of what I mean:

8 x 4 = 32
8 / (1/4) = 32

32 / 4 = 8
32 x (1/4) = 8

Id just like to understand the logic why reversing the operation as well as taking the repicrocal of the divisor/multiplier is equivalent to the original problem. If anyone can help, thanks.

Whenever you want to change the way something looks all you need to do is multiply by 1. Now please understand that 1= (1/2)/(1/2)= (3/5)/(3/5)=...
Also anything divided by 1 equals the anything.

a/b = (a/b)*1 =(a/b)*[(1/b)/(1/b)]=[a*(1/b)]/[b*(1/b)]=[a*(1/b)]/1 = [a*(1/b)]