The Alpha State

Dear Mr. C:

So, multiplying fractions seems to be pretty simple. You just multiply across the top and then multiply across the bottom. So I assume you're just going to do the same thing with dividing fractions, right? Just divide across the top and divide across the bottom? 

Thanks,
Joanie Mahoney

Joanie, whatever you do, stop before you do any more problems dividing fractions, otherwise you're going to get them wrong! Believe me, there is a twist in the problem-solving process that you're going to want to know.

The good news is that, as far as the simpler problems go, they are almost every bit as simple.

 

To divide fractions, you're going to need to remember one thing before you begin. You are going to need to take the reciprocal of the second fraction. We'll talk more about reciprocals later, but for now, just remember this means you will need to flip the second fraction over. 

Then, as strange as it may seem, you will multiply across the top, and multiply across the bottom!

 

The answer you get is two! 

Now, I know what you're thinking. How did I divide two fractions and then end up with a whole number? A whole number is bigger than both fractions, and I divided. It makes no sense! 

Or does it? 

I posed this very question way back when I was in graduate school to my math teacher. I told him that I actually thought this was not even based on any mathematical logic, and that it was just some kind of rule that some math people made up. 

His name was Dr. Goldberg. He asked me, "Jeff, what is 56 divided by 7?" 

"Eight," I said. Duh.

 

Then he asked, "How many times can you fit 7 into 56?"

"Eight," I replied.

"Now, how many times can you fit 1/4 into 1/2?"

"Oh. Two."

"Yes," he said, "now sit back down and take notes."

So yeah, I was humbled, but that may be also when I started loving math for the first time.  

Let's look at another one.

 

Don't forget, you have to take the reciprocal of the second number.

Then we're going to multiply the numerators and the denominators.

Looks like we're top-heavy again. Let's turn it into a division problem!

Of course we know that we can take that leftover fraction and simplify it a little bit by dividing the top and bottom both by 3.

And here's our answer. Of course, there are multiple ways to do this, but I just wanted to show you my favorite way. There are about six or seven side-lessons I could throw in here, but I've written them all down for future posts.