Episode 134: December 21, 2012
by Jason Marshall
What do you get when you add a whole number like 1, 2, 3, or anything else to one of the good old-fashioned proper fractions that you’ve come to know-and-love such as 1/2 or 1/3? You get what’s called a mixed fraction. What exactly do mixed fractions look like? How can you convert them into normal everyday fractions? And when do you need to bother doing so? Stay tuned because those are exactly the questions we’ll be answering today.
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Review: Proper and Improper Fractions
As we’ve seen, fractions come in a number of different flavors. The typical kind that you’re used to dealing with on an everyday basis—the vanilla flavored guys—are called “proper fractions.” A fraction is dubbed “proper” if its numerator is smaller than its denominator. So 1/2, 2/3, and 3/4 are all proper fractions. On the other hand, the more exotically flavored fractional beasties are numbers like 3/2, 4/3, and 27/11 whose numerators are all greater than their denominators. What are these not-so-proper guys called? Rather logically, fractions that fall into this camp are called “improper fractions.”
What Are Mixed Fractions?
So we now know that all fractions can be broken up into two categories: those whose numerators are smaller than their denominators (proper fractions), and those whose numerators are larger than their denominators (improper fractions). Since that covers every possible fraction, you might be wondering why we’re talking about an apparently completely different type of fraction—so called mixed fractions—today? The truth is that mixed fractions aren’t actually a different type of fraction. Every mixed fractions is just an improper fraction written a bit differently. More specifically, a mixed fraction is simply an improper fraction written as the sum of a whole number and a proper fraction.
For example, the improper fraction 3/2 can be written as the equivalent mixed fraction 1-1/2 (read aloud as “one-and-a-half” or “one-and-one-half”). By the way, the dash here isn’t a subtraction sign; it’s there to make it clear that this is the mixed fraction 1-1/2 and not the improper fraction 11/2! A bit of thought should convince you that both of these numbers—1-1/2 and 3/2—do indeed represent the exact same quantity. This fact is fairly obvious if you think about the two fractions in terms of pizzas or pies. The first, 1-1/2, represents 1 whole pizza plus 1/2 of another pizza stuck together. The second, 3/2, represents 3 different pizza halves stuck together. In the end, both represent the same total quantity of pizza.
Every mixed fraction is just an improper fraction written a bit differently.
How to Turn Improper Fractions Into Mixed Fractions
As we’ve seen, every improper fraction can also be written as a mixed fraction. But how exactly do you do this conversion? Fortunately, it’s pretty easy. To find the whole number part of a mixed fraction that corresponds to some improper fraction, just figure out how many times the denominator of the improper fraction goes into its numerator. For example, the denominator of the improper fraction 7/2 goes into its numerator 3 times. In other words, since 7÷2 is equal to 3 plus some remainder, we know that the whole number part of the mixed fraction representing the improper fraction 7/2 is 3.
What about the proper fraction part? How do you find that? Well, the numerator of the proper fraction is just the remainder from the division problem we did before. In our example, since 7÷2 is equal to 3 remainder 1, the numerator of the proper fraction is 1. The denominator of the proper fraction part is just the denominator of the original improper fraction—in this case 2. Which, when you put it all together, means that the improper fraction 7/2 is equivalent to the mixed fraction 3-1/2. How about the improper fraction 15/7? What’s its mixed fraction representation? Well, since 15÷7 = 2 remainder 1, we find that the improper fraction 15/7 is equivalent to the mixed fraction 2-1/7.
How to Turn Mixed Fractions Into Improper Fractions
How about the other way around? What if you want to convert a mixed fraction into an improper fraction? Again, the process is fairly straightforward. Start by multiplying the whole number part of the mixed fraction by the denominator of the proper fraction part. Now add the number you get to the numerator of the proper fraction part. The number you get is the numerator of the improper fraction you’re seeking and the denominator of the improper fraction is just the denominator from the original mixed fraction. It’s probably a little easier to see how this works with an example so let’s use this technique to convert the mixed fraction 2-3/8 into an improper fraction.
Start by multiplying the whole number part—that’s the 2 in 2-3/8—by the denominator of the proper fraction part—that’s the 8—to get 16. Now add this number to the numerator of the proper fraction part—that’s the 3 in 2-3/8—to get 19. Finally, write this number over the denominator of the proper fraction—that’s 8—to get the final value of the improper fraction: 19/8. For the mixed fraction 4-2/3, first multiply 4 x 3 = 12. Then add this to 2 to get 12 + 2 = 14. And finally write this over the denominator 3 to find that 4-2/3 is equivalent to 14/3. Easy enough! But why exactly does this trick work? Give it some thought and see if you can make sense of it. If you’re not sure at this point, don’t worry—the reason will definitely be clearer after we talk about adding and subtracting mixed fractions next time.
Should You Use Mixed or Improper Fractions?
At this point, you might be wondering why we need two different ways to write the same thing. After all, every mixed fraction is equivalent to an improper fraction—so why should we bother with mixed fractions at all? The answer is simple: they’re a lot easier for us humans to think about. As we’ll see in the upcoming episodes, while it’s usually more convenient to add, subtract, multiply, and divide fractions when they’re in their improper form, it’s much easier to think about and intuitively grasp the meaning of fractions in mixed form. For example, which is easier to understand: 16/5 or 3-1/5? For me, it’s easy to picture the size of 3-1/5 since I immediately know that its a number that’s a little larger than 3. On the other hand, the size of the improper fraction 16/5 isn’t nearly as easy to picture.
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