Episode 137: January 11, 2013
by Jason Marshall
What’s 3-1/2 + 2-1/3? How about 3-1/2 – 2-1/3? Once you understand the last episode about adding and subtracting mixed fractions, these questions should be relatively easy to answer. And they naturally lead to a few additional questions. In particular, they may lead you to wonder about how the other two big arithmetic operations—multiplication and division—work when dealing with mixed fractions. In other words, what’s 3-1/2 x 2-1/3? Or 3-1/2 / 2-1/3? Are there any tricks to make this type of problem easier to solve? Stay tuned because those are precisely the questions we’ll be answering today.
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Review: Adding and Subtracting Mixed Fractions
As we’ll soon see, the techniques that make it easy to multiply and divide mixed fractions are very similar to the techniques we used when adding and subtracting them. In fact, not only are they very similar, they’re pretty much the same. Which means that if you haven’t yet nailed down how to add and subtract mixed fractions, you’ll definitely want to check out last week’s episode on that topic.
To help jog your memory, the basic method we used to add and subtract mixed fractions was to start by turning all mixed fractions into improper fractions. In doing, we turn problems about mixed fractions into problems about regular fractions…something we know how to solve! Which means that adding and subtracting mixed fractions becomes no harder than adding and subtracting improper fractions.
How to Multiply Mixed Fractions
So far I’ve claimed that once you know this trick, adding and subtracting mixed fractions is not too tough. And I’ve also claimed that the technique we used to add and subtract mixed fractions—the very same technique I just said was “not too tough” to use—is very similar to the one we’re going to use to multiply and divide them. If you put these two things together, you might be led to the conclusion that multiplying and dividing mixed fractions is not too tough, too. But is that true? Let’s find out.
Let’s say we need to solve a problem like 3-1/2 x 2-1/3. How do we get started? Well, you might not be surprised that the easiest way to get started is to convert the mixed fractions into improper fractions. I’ll let you work out the details, but when you do that you should find that 3-1/2 is equal to 7/2 and that 2-1/3 is equal to 7/3. Which means that the problem 3-1/2 x 2-1/3 is the same as the problem 7/2 x 7/3. What do we do now? All we have to do is use the technique we learned long ago for multiplying fractions. Namely multiply the two numerators to find that the numerator of the answer is 7 x 7 = 49, and multiply the two denominators to find that the denominator of the answer is 2 x 3 = 6. Which means that 3-1/2 x 2-1/3 = 49/6. Or, if we want or need to, we can convert this mixed fraction back into an improper fraction to find that 3-1/2 x 2-1/3 = 8-1/6.
Multiplying mixed fractions is very similar to adding or subtracting them.
How to Divide Mixed Fractions
As we’ve now seen—and as we’d previously speculated—multiplying mixed fractions is very similar to adding or subtracting them. Which, to answer our earlier question, means that once you know the trick, mixed fraction multiplication is indeed “not too tough.” But what about division with mixed fractions?
Well, let’s say you need to solve the problem 3-1/2 / 2-1/3. How do you get started? By now you might be catching on to our trick and you’ll hopefully realize that all we need to do to solve this problem is convert the mixed fractions into improper fractions and then carry out normal fraction division. As we saw earlier, the mixed fractions 3-1/2 and 2-1/3 are equivalent to the improper fractions 7/2 and 7/3. Which means that 3-1/2 / 2-1/3 is equal to the problem 7/2 / 7/3. As we’ve learned, we can divide a pair of fractions like this by following the invert-and-multiply rule. This rule tells us that 7/2 / 7/3 is equal to 7/2 x 3/7. Which, after a bit of arithmetic, gives us the final answer that 3-1/2 / 2-1/3 = 3/2 or 1-1/2.
Just like multiplication, dividing a pair of mixed fractions is not too tough…once you know the trick of converting the mixed fractions into improper fractions. That’s not to say that these problems are always quick and super easy to do in your head. By “not too tough,” I mean that the method you use to solve these problems is straightforward and it’s always the same. Which means that once you know how to do it, you should never have trouble adding, subtracting, multiplying, or dividing mixed fractions again.
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