Multiplication of Fractions
You may generally think of multiplication as being a slightly more complex process than that of addition, but when it comes to fractions it's actually not as big a pain in the neck. That's because we don't need to convert our fractions so that they have common denominators. Bonus.
To find the product of two whole numbers a x b, we can picture a box with side lengths a and b. (If your imagination isn't that great, you can just look at the actual drawing of a box with side lengths a and b provided below.) The area of this box is the product of a and b.
We're going to show you a visual representation of how this works with fractions, but don't freak out. You won't actually have to draw all these boxes going forward. This is just to help you understand the nitty-gritty of what you are technically doing when you're multiplying fractions.
To find , you'd draw a box with side lengths and , inside a big box with side lengths 1 and 1.
Since 1 × 1 = 1, the area of the big box with side lengths 1 and 1 is 1. Notice that the box is cut into 12 smaller boxes, which is just 3 times 4. By counting the tiny boxes, we see that the area of the smaller box is , so . If that seems convoluted, that's because it is. You will never, ever again have to draw all these boxes to multiply fractions for as long as you live. So hopefully you didn't enjoy it too much.
Notice that, to take the product of and , we simply multiplied the numerators together, multiplied the denominators together, and then simplified:
Well that was way easier. That's almost like magic. Yep. Ta-da.
This is all we ever have to do to multiply any two fractions. We just multiply the numerators together, multiply the denominators together, and simplify. No more big, clumsy boxes. That should free up a lot of space.
Be Careful: When it comes to fractions, the word "of" means multiply. For example, "half of 6" means