New here? You’ve stumbled upon January’s Review of Math Basics here at Math for Grownups. This week, we’re doing a quick refresher of fractions. Monday was multiplication, so if you missed that, you might want to take a quick look before reading further. Math concepts build, you know.
Psst! Wanna know a secret? Sure you do. So here you go: There’s a debate among math educators about whether dividing with fractions is useful at all. There. I said it. But don’t tell your kids or they might rebel.
But yes, I’m being somewhat serious here. Among math teachers who really, really think about these things — perhaps too much and I’m often in that camp — dividing with fractions is pretty much unnecessary. Okay, so you might need to divide with fractions (like when you’re halving a recipe). But while the process is stupidly simple (trust me), there are other ways to think about it that may make more sense.
Let’s take a look at that rule:
Dividing by a fraction is the same thing as multiplying by its reciprocal.
If you know what all of those words mean, you can recognize that this is pretty darned easy. But if your days in elementary school are long past, you might have forgotten what the reciprocal is. Luckily, this is no big deal. The reciprocal of a fraction is formed when you switch the numerator and denominator. In layman’s terms, you turn the fraction upside down. Like this:
It couldn’t be easier, right? So let’s put it all in context with an example.
See what we did there? We turned the second fraction over and multiplied instead of divided. This is called the “invert and multiply” process. Now, all we need to do is simplify the answer.
Notice how the 4 and 6 are both divisible by 2? Well, that means the fraction can be simplified. On a 4th-grade math test, this means your teacher wants you to do more work. In the real world, it just means that the fraction will be easier to work with or even understand. (When you see the result, you’ll know what I mean.)
Doesn’t 2/3 seem a lot easier to understand than 4/6? Think of recipes. Do you have a 1/6-cup measure in your cabinet? (I don’t.)
So let’s consider how this works (or why, if you’d rather) by considering a really basic division problem: 1 ÷ 1/2.
How many ½s fit into 1? That’s the question that division asks, right? Think about those measuring cups. If you had two ½ cup measuring cups, you would have the equivalent of 1 cup. In other words:
Make sense? Now here’s another way to look at it:
Let me summarize: 2 ½s fits into 1. In other words, 1÷ ½ is 2. And that turns out to be the same thing as multiplying by the reciprocal of ½, which is 2.
That’s a lot to take in, and you don’t have to know it by heart – or even fully understand. It just explains why this crazy rule works. And here’s another secret – there are lots of other ways to divide fractions. You can do it in your head. (It’s pretty easy to solve this problem without any arithmetic: ½ ÷ ¼. Right?) Or you could even find a common denominator (more on that Friday) and then just divide the numerators. (I’ll leave that process for you to figure out if you’re so inclined.)
The thing is, there aren’t many times in the real world that dividing by fractions is really necessary. Here’s an example to explain what I mean. Let’s say I’m cutting a recipe in half. The recipe calls for ¾ cup of sugar. How much will I actually need? Well, I can look at the question in a couple of different ways. (See which one jumps out at you.)
I would bet – and I can’t prove it – that most of you thought about the second option. That’s because you’re cutting the recipe in half, not dividing the recipe by 2.
In short, dividing by fractions is pretty darned simple, compared to other things you have been required to do in math. Too bad it doesn’t show up much in the real world, right?
Just for fun, try these problems on for size – using whatever method works for you. (No need to show your work!) Bonus points if you can simplify your answer, when necessary. (And no, there are no bonus points, because there are no points.)
The answers to Monday’s problems: ⅓, 4/35, 15/8 or 1⅞, 5¼, 9⅔. How did you do? ETA: Me? Not so good. I made a careless error with the last problem. The correct answer is 3 ⅔, which is explained by the comments below.