Why do you not need to find a common denominator when you multiply fractions but you must find a common denominator when you add them?

Answer 1

Oh, boy, you've asked a hard one. It's a concept that is very difficult to explain qualitatively, and in plain English.

The simplest explanation is this . . . .

You can think of fractions as simple division problems. One half (1/2) can be thought of as one divided by two. Two thirds (2/3) is two divided by three, and so on. Therefore, when you multiply two fractions, you are mixing multiplication and division, which are two compatible, complementary operations. It doesn't matter whether you multiply first or divide first, since the order of operations is not important.

Let's try an example. Let's multiply 1/2 by 3/4.

Think of it as one divided by two times three divided by four, which equals three eighths.

Now, let's change the order of operation. You can multiply the numerators (the top numbers) and divide by the product of the bottom ones (the denominators). So it's [1 times 3] divided by [2 times 4]. Be careful with that one, though. You have to do the operations inside the square brackets first. So, it's [3] divided by [8], which equals three eighths (3/8), just like before.

Addition is another kettle of fish. Adding fractions is a mixture of division and addition, which are not compatible operations. Further, it is not possible to add unlike terms. You cannot add five Oranges to ten apples and say you have 15 apples or 15 oranges. And it's the same for fractions. Think of it this way: 1/2 is one half. It's not one third or one quarter or one apple or one orange. 3/4 is three fourths (or if you prefer, quarters). It's not three halves or three apples or three oranges.So, if you're adding one half and three quarters, it's like you're trying to add apples and oranges to get just apples or just oranges. No good! To do the math, you must find a way to convert the apples to oranges or the oranges to apples (or perhaps both fruits into another fruit altogether) and THEN add them together. The way we do that with fractions is to find the least common denominator. By finding the LCD and then finding the equivalent fractions for the ones we started with, we can then add them together.

In the example above, we can't add 1/2 and 3/4 directly. We must find the LCD, which is 4 in this case. The 1/2 converts to 2/4, which we can then add to 3/4 to get 5/4, which equals 1 1/4 (one and one quarter).

In English, I had one half (that is, one apple) and wanted to add it to three quarters (that is, three oranges), so I converted the one apple to two oranges (1/2 = 2/4) and then added the oranges together, giving me five oranges.

Well, that should be clear as mud. So eat an apple -- or an orange. It'll make you feel better.