Multiplying Fractions

Multiplying fractions. To many people, it might not be clear what something like two thirds multiplied by one quarter means. So before we talk about multiplying fractions, let's think a bit about multiplication itself. Normally, we think of multiplication as involving whole numbers. You are trying to solve problems like two times three, which equals six or seven times four, which is 28. If you buy three packs of canned soda, how many cans do you have? There are three packs of six cans, three times six. We can count them all up. There are 18 cans of soda. But let's think about multiplication in a different way than we do when buying cans of soda. Let's think of it as the area of two numbers. Here's a house. It's a pretty small house, only one room, and we need to buy some carpet for it. How many square feet of carpet do we need to buy to cover the bare floor? This house is only ten feet by 15 feet. So how many square feet of carpeting do we need to pie? The way to find the area of a rectangle is to multiply the width by the height. In this case, it's 15 by ten feet, which is equal to 150 square feet. The answer to a multiplication problem can be thought of as the area of a rectangle whose width and height are the numbers being multiplied. We aren't necessarily talking about floor plans or cans of soda now. We are just trying to come up with a useful way of thinking about how to solve a multiplication problem. You're probably thinking, This is all great, but what does it have to do with multiplying fractions? Let's start with a square. This square is one unit wide and one unit tall, which makes its area one square unit. We aren't going to worry about what size units are. It could be inches, meters, kilometers or light years. Personally, I'm going to think of these units as square meters, but you do you. The important fact, though, is that this square is wide, one tall, and this is one whole square. Now that we've got that settled, let's bring back our fraction multiplication problem from earlier, two thirds times one fourth. Let's take a marker and divide our square into three equal sized parts. One, two, three. Since the square hasn't changed size, the height of these sections we've marked off are still one, but the widths are now one third of the width of the whole square. So each section is one third wide and one tall. To get the area of each section, we just multiply the width by the height. One times one third is one third, because one times anything is the same thing. Since our first fraction is two thirds, we can paint the first two of the one third sections red. Look at that. We've got our two thirds. That's the first fraction of our multiplication problem. We can make our second fraction the same way. But this time, let's make the marker lines go horizontal instead of vertical. Each of these new horizontal sections are one quarter tall and one wide. And if we paint one of them, say, yellow, we can see that the bottom section is one wide and one quarter in height, which means that the yellow area is one quarter in area, one times one quarter is one, and that's our second fraction. Look, as the paints dried, some of the red paint has seeped into the yellow paint. We've got an orange section, as well as red and yellow sections. That orange section is the overlap between our two fractions. And it looks to me like the width is two thirds and the height is one fourth. That's our two fractions, which means that the orange rectangles are what we get when we multiply our two fractions. But how do we know what the area is in terms of a fraction? Well, those dividing lines we drew with the marker has divided the one unit square into equal sized rectangles. And there are one, two, three, 12 of them. So each one of those rectangles are one 12th of the one unit square. And there are two of the little rectangles in the orange area, which means that the orange areas area is drumroll please. 2/12. And that's the answer to our fraction multiplication problem. Two thirds times one fourth is 2/12. So does this mean that every time we need to multiply fraction, we've got to make a square, get our markers and paint, and then wait for the paint to dry? No, look at the problem. Two times one is two, three times four is 12. To multiply two fractions, all we have to do is multiply the top numbers, the numerators together. This gives us the numerator for the answer. Then we multiply the bottom numbers, the denominators together. This gives us the denominator for the answer, and that's a lot less work.