Question

Explain why the process of dividing by a rational number is the same as multiplying by its reciprocal.

Answer 1

Think of when you are given an equation in which a number is being divided by a fraction. Any time you do so, you are taught to flip the fraction (the denominator) and then multiply, instead of divide, by the numerator:
 
5/(2/3) --> 5 x (3/2) --> 15/2 

A rational number, by definition, is not a fraction. However, you can easily turn any whole number into a fraction by placing it over 1. So, if you were to divide a number by a fraction whose denominator is 1, you would then flip the fraction (the denominator) and multiply it by the numerator. By definition, you would be multiplying by the reciprocal of the previous fraction, because the reciprocal of any whole number is that number over 1:

5/4 = 5/(4/1)
5/(4/1) --> 5 x (1/4)

In this case, 1/4 is the reciprocal of 4. 

Answer 2

That's why you're being told to flip the second fraction. You're recognizing that dividing by a number is the same as multiplying by the reciprocal. ... Now to multiply fractions, we simply multiply the numerators to get the new numerator and multiply the denominators to get the new denominator.