Episode 130: November 23, 2012
Real World Math
by Jason Marshall
Quick, what's 1/4 + 3/4? If you remember your math from school, you probably know that the answer is 1. How about 1/5 + 3/4? Not so easy this time, right? Adding 1/4 to 3/4 is fairly straightforward because the denominators of both fractions are the same. But adding 1/5 to 3/4 is not so simple, and that's because the denominators are different. So how do we solve problems like this? We start by finding what's called a common denominator. Which is exactly what we're going to learn how to do today.
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Numerators and Denominators
Before we learn how to add and subtract fractions, we need to learn how to find a common denominator. And before we do that, we need to make sure we're up to speed on some key fraction fundamentals. In particular, when we first talked about numerators and denominators, we learned that the denominator of a fraction tells you how many equal parts a whole—and that could be a whole pie, a whole year, a whole iPhone, or a whole anything else—is broken into, and the numerator of a fraction tells you how many of those equal parts the fraction contains.
So the 4 in a fraction like 3/4 means that we're breaking a whole number of somethings up into 4 equal parts. And the 3 in 3/4 therefore tells us that this fraction represents the amount we get when we take 3 of those 4 equal pieces of something. Okay, easy enough. Now let's take a look at what this all means when doing simple addition and subtraction of fractions.
Adding and Subtracting Fractions
To start with, what happens if we want to take that 3/4 of something from before and add it to 1/4 of that something? Well, common sense—and, as we'll thankfully see, math—tells us that we end up with 1 whole something. How about if we start with 1 whole something and subtract 1/4 of that something from it? Of course, we end up with 3/4 of whatever that something was we started with.
These basic facts have essentially become common sense to us after the hours and hours we’ve spent dealing with and thinking about things like pizzas, pies, and everything else in the world that can be broken into fractions. As most of us learned way back in the day, if we start with a whole pie and remove 1/4 of it (presumably with our mouths), we're left with 3/4 of a pie. If we instead start with 3/4 of a pie and then somehow (I’m not sure I want to know how) add 1/4 of a pie back to it, we not surprisingly get back our whole pie.
What Are Common Denominators?
It’s easy to add fractions when they’re written in terms of the same common denominator.
What makes problems like this fairly intuitive and even easy to think about is the fact that all of the fractions are written in terms of what's called a common denominator. In other words, they’re written in terms of the same kind of “somethings”—be it equal portions of apples, pies, or whatever. In situations like this, all you have to do to add or subtract fractions is add or subtract their numerators (since that tells us the total number of somethings you have) and then write this over the original denominator (to put the answer back in terms of those somethings).
For example, since both fractions in 3/4 + 1/4 share the denominator 4, we find the answer by adding the numerators, 3 + 1 = 4, and then writing this over the original denominator, 4, to find that 3/4 + 1/4 = 4/4. As we learned when talking about simplifying fractions, this fraction, 4/4, is equal to 1. For the problem 2/3 – 1/3, since both fractions are written in terms of the same common denominator, we can subtract the numerators to get 2 – 1 = 1, and therefore find that 2/3 – 1/3 = 1/3.
The key thing here (which is why I keep repeating it!) is that it’s easy to add fractions if they’re written in terms of the same common denominator. Which naturally leads to the question: How do you rewrite fractions in terms of a common denominator?
How to Find Common Denominators
The easiest way to find a common denominator for a pair of fractions is to multiply the numerator and denominator of each fraction by the denominator of the other. So, if you're trying to rewrite 1/3 and 1/6 in terms of the same common denominator, all you have to do is multiply the top and bottom of 1/3 by 6 (which is the denominator of 1/6) and the top and bottom of 1/6 by 3 (which is the denominator of 1/3) to find that 1/3 = (1 x 6) / (3 x 6) = 6/18 and 1/6 = (1 x 3) / (6 x 3) = 3/18.
Why does this work? Because, as we know from our experience with simplifying fractions, 1/3 and 6/18 are equivalent, as are 1/6 and 3/18. So the trick we learned is nothing more than a method for quickly finding equivalent forms of fractions written in terms of the same common denominator. As usual, notice that once we've rewritten these fractions in terms of a common denominator, we can add or subtract them with ease. So 1/3 + 1/6 = 6/18 + 3/18 = 9/18 (which can be simplified to 1/2), 1/3 – 1/6 = 6/18 – 3/18 = 3/18 (which can be simplified to 1/6), and so on.
While this method of finding a common denominator will always work, it's not necessarily the best method to use in every problem. Why? Well, the short answer is that it will often leave you with a lot of simplifying to do. And, as we'll find out next time, much of that simplifying can be avoided entirely by being a bit more clever about how you choose your common denominator.
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 firstname.lastname@example.org.