This is not a math shortcut, but an explanation to help those confused by the usual process of dividing fractions.
Consider this problem:
3/4 ÷ 1/4 =
We are taught to flip the last term (the divisor) and change the process from division to multiplication to find the answer:
3/4 x 4/1 = 3.
Is this a magic trick? Why does this work?
Let’s rewrite the problem as fraction over fraction:
3/4
____
1/4
To get rid of the problem’s bottom fraction, we can multiply top and bottom fractions by the inverse of the bottom. This effectively multiplies the whole problem by 1, which does not change the problem’s value. However, the bottom fraction is cancelled out. The problem’s form is changed from fraction / fraction to fraction x fraction, as shown:
3/4 4/1
___ x ___ = 3/4 x 4/1
1/4 4/1
This gives 3/4 x 4/1, which is the form we wanted explained.
As you can see, the process of flipping the last term and changing the process from division to multiplication is simply skipping steps- not a weird unexplainable process.