How to Turn Mixed Fractions Into Improper Fractions
What do mixed fractions like 2-3/5 and 5-7/8 look like when written as improper fractions? How do you quickly convert between these two workhorses of the fraction world? Read on to find out!
Hot on the heels of our first foray into the always exciting world of mixed fractions, this week we’re going to jump right back into that topic and clean up a few rough edges. In particular, after a quick review of the stuff we learned last time—namely what proper, improper, and mixed fractions are and how to turn improper fractions into mixed fractions—today we’re going to talk about how you can turn all of those mixed fractions you’ve just created back into improper fractions. Why would you want to do that? That’s a great question! And it’s exactly the question we’ll be answering over the next few weeks.
Review: Mixed Fractions
As we learned last time, proper fractions are all the fractions with numerators that are smaller than their denominators—they’re the normal, everyday fractions like 1/3, 2/5, 7/13, and so on—and improper fractions are all the fractions with numerators that are larger than their denominators—fractions like 3/2, 5/3, and 13/7. As we also learned, every improper fraction can be written as what’s called a “mixed fraction.” As the name implies, mixed fractions are a mixture of a whole number and a proper fraction. For example, 2-3/8, 4-1/16, and 1-1/2 are all mixed fractions.
Last time around, we also learned that turning an improper fraction into a mixed fraction is relatively easy. To find the whole number part of the mixed fraction, you just need to figure out how many times the denominator of the improper fraction goes into its numerator. For example, since the denominator of the improper fraction 8/3 goes into its numerator 2 times—in other words, since 8 ÷ 3 = 2 remainder 2—we know that the whole number part of the mixed fraction representing the improper fraction 8/3 is 2. That’s half the answer. Now, what about the proper fraction part? That’s equal to the remainder of the division we just did over the denominator of the original improper fraction. In this case, that’s 2/3. Which means that the improper fraction 8/3 is equivalent to the mixed fraction 2-2/3.
How to Turn Mixed Fractions Into Improper Fractions
Now that we’ve had some practice converting improper fractions into mixed fractions, you might be wondering how to go the other way. In other words, how do you undo everything we’ve done and convert a mixed fraction back into an improper fraction? The trick is to start by multiplying the whole number part of the mixed fraction by the denominator of the proper fraction part. For the fraction 2-1/4, that means you first need to multiply 2 (the whole number part) by 4 (the denominator of the proper fraction part) to find that 2×4=8. Next, add this number to the numerator of the proper fraction part. For 2-1/4, that means we need to add the number we got—which was 8—to the numerator 1 to get 8+1=9. This number is actually the numerator of the improper fraction you’re seeking. What’s its denominator? It’s just the denominator of the proper fraction part of the original mixed fraction. So for 2-1/4, it’s 4. Which means that the mixed fraction 2-1/4 is equivalent to the improper fraction 9/4.
How about the mixed fraction 4-2/3? Well, the first step is to multiply its whole number part—that’s the 4—by the denominator of its proper fraction part—that’s the 3—to get 4×3=12. Next, add this number to the numerator of the proper fraction part—that’s the 2—to get 12+2=14. Finally, we write this number over the denominator of the proper fraction part of 4-2/3—that’s the 3—to find that the mixed fraction 4-2/3 is equivalent to the improper fraction 14/3.
Why Does this Conversion Work?
At this point, this all kind of looks like some sort of magic trick, right? While I enjoy magic, I never enjoy making math look like magic…because it’s important to remember that it’s not! So let’s take a minute to figure out exactly what’s going on. First, let’s think about what a mixed fraction like 2-1/4 really means. Notice that 2-1/4 is really just another way of writing 2 + 1/4. In other words, as we’ve talked about before, every mixed fraction is the sum of a whole number and a proper fraction. For reasons that will become clear in a moment, let’s rewrite the number 2 here as the fraction 2/1. And then, just for fun, let’s write the two fractions in the sum 2/1 + 1/4 in terms of their lowest common denominator. Which is…? Well, as you should check for yourself, the lowest common denominator of 2/1 and 1/4 is 4.
Okay, let’s now write these fractions in terms of this lowest common denominator. Since 1/4 is already in terms of it, we only need to worry about finding the equivalent of 2/1. One way to do that is to multiply its top and bottom by 4 to get the equivalent fraction 8/4. So the mixed fraction 2-1/4 can be rewritten as 8/4 + 1/4. Since the denominators of these fractions are the same, all we have to do to add them up is add their numerators and write the result over their common denominator. What does this tell us? It tells us that the mixed fraction 2-1/4 is equivalent to the improper fraction 9/4. Look familiar? It should since it’s exactly the same answer we got before. Which means that the trick for turning mixed fractions into improper fractions really isn’t a trick at all—it’s just a speedy way of rewriting certain fractions in terms of their lowest common denominator and adding them up. Once again, it’s not magic…it’s math!
Why Convert Mixed Fractions Into Improper Fractions?
But the question still remains: Why do we need to bother with any of this conversion stuff? Can’t we just leave all the fractions in our lives alone and let them remain in the same mixed or improper form that we find them in? As we briefly touched upon last time, the answer is that although we never absolutely have to convert from one kind of fraction to another, doing so can make life a lot easier. Why? Because many problems are much easier to solve when the fractions in them are written in their improper form. What kind of problems? Well, unfortunately we’re all out of time for today. So the answer to that question is going to have to wait until next time.
Wrap Up
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
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About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.
