When Dividing Two Fractions, Why Do We Actually Multiply by the Reciprocal?

Recently, I was watching the Studio Ghibli movie Only Yesterday and there was a scene that aroused my curiosity regarding a common math “trick” for dividing two fractions. In the scene, a young girl had failed a math quiz and was being scolded by her family, resulting quite naturally in her feeling lots of shame, and perhaps even worse the beginnings of the idea that she was stupid was beginning to take root in her mind. However, as the older and “smarter” sister of this young girl sits down and tries to explain how to divide two fractions, it becomes clear, to me anyway, who is actually the smart one.

Repeatedly, the older sister confidently exclaims “when dividing two fractions, just flip the top and bottom of the second fraction and then multiply”. The younger curious girl sits there, silently wanting to know - but why. The older one proceeds to do dozens of examples, executing the procedure to perfection, but the younger one just sits there even more lost and withdrawn. Eventually the younger one sincerely asks “Why would you divide a fraction by a fraction?”. Grabbing pencil and paper she starts drawing a picture of an apple and says “dividing two thirds of an apple into quarters means you take two thirds of the apple and split it into four ways…how much apple does each person get?”. She concludes 1/6 which in a sense is understandable given how she has been taught division. But the actual answer, 8/3, does not make ANY sense to her, especially since division has always been associated for her with resulting in a smaller number, not bigger.

The scene affected me and highlights the difference between certain types of people. Some learners are content to execute a procedure without any understanding as to how it works and why. Others seem to have a thirst for genuine understanding and need to understand clearly why something works. I tend to fall in the latter camp, so I paused the movie and, like the young girl, grabbed pencil and paper and started thinking deeply.

Before sharing how I think about it, I am curious how many people reading this can honestly say they understand why you actually multiply by the reciprocal of the divisor when dividing two fractions. Everyone of my friends who I posed this question to eventually admitted they had no idea. So, let’s dive in.

The first question I asked myself is - what does it mean to “divide” two numbers. To begin this inquiry, I wrote down a/b = c. In words, we can interpret this as there is a number a such that when you “divide” it by b it produces the number c. Working from a/b = c, I can multiply b to both sides of this equation resulting in a = c*b. Therefore, one, emphasis on one, interpretation of division is that when we divide two numbers it produces a number such that when I multiply that number by the divisor (in kid language, the “bottom” number) it equals the dividend (the “top” number). By means of a simple example, if we are trying to solve12/4 we must determine a number such that when I multiply that number by 4, the result is 12. What number multiplied by 4 results in 12? With a little thought, we see the answer is 3. So, 12/4 = 3 because 3*4 = 12.

With this in mind, let’s finally explore dividing two general fractions a/b ÷ c/d. We know the result of this division operation is some number, let’s call it e, such that when I multiply that number by the divisor, c/d, it equals the dividend, a/b. In symbols, this translates to a/b = e*c/d. Since we are interested in determining the number e by itself, we can multiply both sides of the previous equation by d/c to isolate e. This results in, a/b * d/c = e. In other words, we are multiplying by the reciprocal. But rather than starting blindly from that place and simply obeying orders, we are actually ending there as a result of a deeper understanding of what it means to divide two numbers.