Learn what it really means to multiply two fractions and how to efficiently perform the multiplication.
In the last article, “Is Multiplication Repeated Addition?” we talked about what it really means to multiply two numbers. We found that the conventional meaning of multiplication—“repeated addition”—breaks down when multiplying fractions, and that we should instead think of multiplication as a process that scales one number by some other amount. As we’ll discuss in a minute, multiplication is fairly straight-forward to do with integers. But admittedly, it’s a little trickier to do with fractions. Though by the end of this article, you’ll be an expert at multiplying fractions.
Review: What is Multiplication? What are Fractions?
Okay, let’s start off by reviewing the various players in our story to make sure everybody is up to speed. As we discussed at length in the last article, we can picture the meaning of multiplication by thinking about the number line. For example, 5 x 2 can be thought of as the number you get when you stretch a 5-unit long stick lying along the number line until it’s twice its original length—that is, until it’s a 10-unit long stick (so, 5 x 2 = 10). Things get a little strange, however, when we talk about fractions. Remember, fractions are just numbers that exist between the integers along the number line. As such, it’s clear we can still stretch sticks along the number line that have fractional lengths by some other fractional amount. For example, 1/2 x 1/3 can be thought of as stretching (or in this case squeezing) a 1/2-unit long stick until it’s 1/3 its original size—and the new length will be 1/6-unit. But how does this work in general? How can we easily figure out the final “length” when multiplying any two fractions together?
The Relationship Between Fractions and Division
Well, let’s start by recalling the very important relationship between fractions and division. Take the fraction 1/2, for example. We can think of 1/2 in two different—but ultimately equivalent—ways:
The length of a 1/2-unit long stick laying along the number line;
The length of an initially 1-unit long stick after it has been divided by two.
These may seem identical, but they’re not. The first describes the typical meaning of a fraction as being part of a whole; the second instead views the fraction as meaning “the number you get by dividing 1 by 2.” Or, for the fraction 3/4, “the number you get by dividing 3 by 4.” As you’ll see in a moment, this interpretation that uses the connection between fractions and division is key to understanding how to multiply fractions!
How to Multiply a Fraction and an Integer
Before we go all out and multiply two fractions together, let’s first talk about how to multiply one fractional number by one integer—say, a problem like 2 x 1/2. According to our picture of stretching sticks along the number line, this is just asking us to squeeze a 2-unit long stick until it’s half its original length. Of course, the answer is 1—but what’s the general method to solve problems like this? Well, this is where the relationship between fractions and division we talked about before comes in handy. Since the fraction 1/2 means “one divided by two,” the problem 2 x 1/2 can be interpreted as meaning “two times one divided by two.” In other words, when multiplying an integer by a fraction, simply multiply the integer by the numerator of the fraction, and then divide this result by the denominator of the fraction. So the problem 2 x 1/2 (“two times one-half”) is equivalent to the problem 2 x 1 / 2 (“two times one divided by two”). In other words, first multiply 2 by 1, giving 2, and then divide this result by 2. So, 2 / 2 = 1.
How to Multiply Fractions
Finally, we’re now ready to multiply two fractions together. Actually, you may not have realized it, but we’ve already done it! Because any integer, such as 2, can actually be thought of as a fraction since the fraction 2/1 has the same value as 2. So the problem 2 x 1/2 can actually be thought of as 2/1 x 1/2. Using the relationship between fractions and division, this becomes 2 / 1 x 1 / 2 (“two divided by one times one divided by two”). No surprise—the answer is still 1.
There’s also a handy mental algorithm based on this logic that’ll help you to quickly multiply fractions. The quick and dirty tip is to multiply all of the numerators of the fractions in your problem together to obtain the numerator of the resulting fraction, and to multiply all of the denominators of the fractions in your problem together to obtain the denominator of the resulting fraction. So, for a problem like 1/8 x 3/5, the numerator of the resulting fraction is given by 1x3 (that’s the 1 from 1/8 and the 3 from 3/5), which equals 3, and the denominator of the resulting fraction is given by 8x5 (that’s the 8 from 1/8 and the 5 from 3/5), which equals 40. So, the answer to 1/8 x 3/5 = (1x3) / (8x5) = 3/40. That’s all there is to it! It’s not magic, it’s not due to some obscure formula that someone pulled out of a hat and told you to use, it’s simply a result of the logic that follows from what we’ve been discovering about math.
Okay, that’s all the math we have time for today.
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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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