Types of Numbers
Your Teacher May be a Complex Person, but She Likes Her Fractions Simplified
As you now know, there are infinitely many ways to express any particular fraction. Even more if you speak Spanish.
However, infinity is quite a lot. So, to keep your teacher from going batty attempting to decode your homework as well as that of your classmates, each of whom has expressed the solution to every problem using a different expression of the correct answer, be sure to reduce your fractions. That way, we're all speaking a common language. ¿Entiendes?
Pick any fraction you like. Actually, you look like you've got your hands full at the moment. We'll do it for you.
Now make the numerator bigger. You can use a bicycle pump, helium tank - whatever gets the job done. Your new fraction is now larger than your original fraction, because you have more pieces of the same size. More pieces equals more size. That's why you never hear of any pirates looting for pieces of nine. Too heavy.
Now go back to your original fraction. This time, make the denominator bigger and keep the numerator the same size. Is your new fraction bigger or smaller than your original fraction?
The new fraction is smaller. Meaning that we have the same number of pieces, but each piece is now smaller. What a rip-off. Good thing we held onto the receipt.
Try this with a couple other fractions to convince yourself of these rules:
1. If you keep the denominator the same and make the numerator bigger, the fraction gets bigger.
2. If you keep the numerator the same and make the denominator bigger, the fraction gets smaller.
If we make both the numerator and the denominator bigger, there's no telling what might happen. Everyone had better stand back, just in case. Could get ugly.
So does the new fraction become smaller than, equivalent to, or bigger than the original fraction? Really, it depends on how much you increase each the numerator and denominator by.
Because we don't deal on a daily basis with fractions like 5/18, 42/53, etc., it can sometimes be a difficult task to compare one of these guys to another fraction and instantly be able to tell which is larger. That's why it's always nice to have matching denominators. They're like socks - when they don't match, it's going to make your life so much harder. Especially if that bully who's always picking on you notices.
If we have two fractions with the same denominator, they're easy to compare. Whichever has the bigger numerator is the bigger fraction. So what to do with a troublesome pair like 3/5 and 2/3?
To compare them, let's think about the two Star Wars cakes you had at your last birthday. (Wow - talk about giving that bully some extra ammunition.)
Suppose that, after the damage was done, 3/5 of one cake remained, and 2/3 of the other was left behind. (Maybe you should have just gotten one cake.) The first cake was originally cut into fifths - let's now cut the remaining 3 slices into thirds. With the second cake, which was originally cut into thirds, let's now cut the remaining 2 slices into fifths.
So the 3 slices of the first cake become 9 smaller slices...
...and the 2 slices of the second cake become 10 smaller slices.
Now it's easy to compare the fractions. Each cake is now divided up into fifteenths (1/3 x 1/5). The first cake has 9 slices remaining, and the second cake has 10 slices remaining, with all slices being the exact same size. So:
is bigger. Not Jabba the Hut bigger, but bigger nonetheless.
Although this method will always work when comparing two fractions, sometimes it isn't the most efficient way. Like when you don't have a couple of cakes and spatulas handy. Consider the following example.
We can cross-multiply, but yikes. We'll have matching denominators, but at what cost? Our numbers are going to be massive, and while that's a good quality to have in a cruise ship, not so much in the "solving fraction problems" department. So if there's a way to break this down into smaller numbers somehow, that would be peachy keen.
Our other choice is to recognize (yes - recognition is a choice) that 200 = 8 x 25, and 175 = 7 x 25. The number 25 divides evenly into both of these suckers! So if we multiply the numerator and denominator of by 7 and multiply the numerator and denominator of by 8, the denominators will be equal, and we'll be ready to compare the numerators! We get 77 and 56 respectively - clearly, 77 is bigger, which means that is bigger than . How did we arrive at this conclusion? By using the idea of Least Common Denominators! Since every one of those three words starts with a capital letter, it must be important. So important that we hereby dedicate the next section to it.