Episode 132: December 7, 2012
Real World Math
by Jason Marshall
In the last episode, I kicked things off by asking you to find the lowest common denominator of a pair of fractions—let’s say 1/2 and 3/4. That question led us down a path in which we discovered the fascinating fact that groups of fractions have not just one but an infinite number of common denominators. And it also led us to discover that calculating lowest common denominators requires that we first know how to calculate least common multiples.
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Now that we’ve got all of that stuff figured out and safely secured in our bag of mathematical tricks, we’re ready to move on to the big question: How do you find the lowest common denominator of a pair or a group of fractions? Stay tuned because that’s exactly the problem we’ll be learning to solve today.
Recap: What Is the Least Common Multiple?
If you’re not exactly sure what the least common multiple of a group of numbers is and how to calculate it, I highly recommend taking a look at last week’s show about that very topic before continuing on. Pretty much everything we’re going to do today relies upon that idea, so I promise you it’ll be time well spent. To make sure you’re good-to-go on this topic, here’s a quick question to test yourself: What’s the least common multiple of the numbers 3, 5, and 10? I’ll give you a second to think about it…
So, did you come up with 30? The quick and dirty way to find the least common multiple of a group of numbers is to first figure out what the multiples of each individual number look like. In this case, the multiples of 3 are all the numbers like 3, 6, 9, 12, and so on; the multiples of 5 look like 5, 10, 15, 20, and so on; and the multiples of 10 look like 10, 20, 30, and so on. The least common multiple of a group of numbers is the smallest multiple that all the numbers have in common. Which, in this case, is 30.
What Is the Lowest Common Denominator?
With that idea now firmly planted in our brains, we're ready to move on to calculating lowest common denominators. As a refresher, the lowest common denominator is exactly what it sounds like: Out of the infinite number of possible common denominators, the lowest common denominator of a group of fractions is the particular one that is smallest. So how do you find it? Well, the lowest common denominator of a group of fractions is equal to the least common multiple of their denominators.
This is actually a lot simpler than it sounds. To see how it works, let's go back and look at the problem we started last time: What’s the lowest common denominator of the fractions 1/6 and 2/3? As you may recall, we can use our tried and trusty method of calculating regular old common denominators (but not necessarily lowest common denominators) to figure out that these fractions can be written in terms of the common denominator 18. We find that the fraction 1/6 is equivalent to 6/18 and the fraction 2/3 is equivalent to 12/18. But 18 is not exactly a tiny number—which sort of leads you to wonder if it might not be the lowest common denominator. So, how do we find out?
The LCD of a group of fractions is equal to the LCM of their denominators.
How to Find the Lowest Common Denominator (LCD)
As we’ve learned, to find the lowest common denominator of a group of fractions, we need to find the least common multiple of their denominators. In our example, those denominators are 6 from 1/6 and 3 from 2/3. What are the multiples of 6? Well, they're all the numbers like 6, 12, 18, and so on. How about the multiples of 3? They're all the numbers like 3, 6, 9, 12, 15, 18, and so on. It’s clear that although the common denominator 18 that we used earlier is a common multiple, it's definitely not the least common multiple. Which means that it’s not the least common denominator. What is the least common multiple…and therefore the lowest common denominator? It’s 6.
How to Write Fractions In Terms of the LCD
To write the original fractions 1/6 and 2/3 in terms of this lowest common denominator, all we have to do is multiply the top and bottom of each fraction by whatever number is needed to create the equivalent fraction that has that lowest common denominator. I know that’s a lot of words and it might sound a little confusing, but it’s actually pretty easy. In this case, since 1/6 is already written in terms of the lowest common denominator, 6, we don't need to do anything. In order for 2/3 to be written in terms of the lowest common denominator, we need to multiply its top and bottom by 2. That gives us a numerator of 2 x 2 = 4 and a denominator of 2 x 3 = 6…for a total of 4/6. Which means that 1/6 and 2/3 are equivalent to 1/6 and 4/6 when written in terms of their lowest common denominator.
What if we’re dealing with more than two fractions? For example, how do you write the trio of fractions 1/2, 2/3, and 3/4 in terms of their lowest common denominator? Well, the first thing we have to do is find their lowest common denominator. Which means that we first need to find the least common multiple of their denominators. As a little work will show you, the least common multiple of the numbers 2, 3, and 4 is the number 12…so that’s also the lowest common denominator. To rewrite the fraction 1/2 in terms of this common denominator, we need to multiply its top and bottom by 6 to get 6/12. To rewrite 2/3, just multiply its top and bottom by 4 to get the equivalent fraction 8/12. And 3/4 can be rewritten in terms of the common denominator 12 by multiplying its top and bottom by 3 to get 9/12.
Why Are Lowest Common Denominators Useful?
But why do we need to go through all of this trouble rewriting fractions in terms of their lowest common denominator when we already have a perfectly good way of writing them in terms of a perhaps-not-lowest-but-still-perfectly-good common denominator? The short and sweet answer is that doing this little bit of work up front to rewrite fractions in terms of their lowest common denominator usually saves you from doing more work later on.
If you don’t work in terms of the lowest common denominator, you’ll inevitably end up needing to simplify the fractions you calculate—which, in truth, can be kind of a pain. But if you instead make a habit of working in terms of the lowest common denominator from the outset, you'll usually finish your problem and find that your answer is already nicely written in its reduced form. No, it's not absolutely essential to always do things this way, but doing so can make math—and your life—a lot easier. Which is exactly what this show is all about!
Okay, that's all the math we have time for today. Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via Facebook, Twitter, or email at email@example.com.