Study Guide

Comparing Fractions

Comparing Fractions

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, be sure to reduce your fractions. That way, we're all speaking a common language. ¿Entiendes?

Comparing Fractions

Another solid skill we'll need is the ability to compare different fractions, even when their denominators aren't the same.

Pick any fraction you like. Actually, you look like you've got your hands full at the moment. We'll do it for you.

Sample Problem

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 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.

Because we don't deal with fractions like 5/18, 42/53, etc. on a daily basis, 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.

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 do we 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.

Sample Problem

Suppose that, after the party was over and the damage was done, 3/5 of one cake remained, and 2/3 of the other was left over. (Maybe you should've just gotten one cake.) Which partial cake is bigger?

The first cake was originally cut into fifths, so let's now cut the remaining 3 slices into thirds: 3/5 × 3/3 = 9/15.

With the second cake, which was originally cut into thirds, let's now cut the remaining 2 slices into fifths: 2/3 × 5/5 = 10/15.

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. The first cake has 9 slices remaining, and the second cake has 10 slices remaining, with all slices being the exact same size. So:

That means  is bigger. Not Jabba the Hutt 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.

Sample Problem

We can multiply the top and bottom of the first fraction by 175 and the top and bottom of the second fraction by 200, 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 that 200 = 8 × 25, and 175 = 7 × 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 both be equal at 1400, and we'll be ready to compare the numerators.

We get 77 and 56, respectively, for our two numerators. 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.

  • Least Common Denominator

    This process of cutting numbers down to size in order to get the pieces we want is referred to as finding the "lowest (or least) common denominator (LCD)." Yes, the "L" can stand for either "lowest" or "least," which mean the same thing. It could also stand for "littlest," we suppose, but that doesn't sound very professional.

    The LCD is basically the same thing as the LCM (least common multiple) of two denominators. When comparing two fractions, the LCD is the smallest number that's a multiple of both denominators. Another way to say this is that the LCD is the smallest number divisible by both denominators. Another way to say this is...oh, you know what, you already have enough ways to say it.

    To find the LCD quickly (because you never know when you'll only have 10 seconds to find one so that you can defuse a bomb in time), we use prime factorizations again.

    Notice that, to find the LCD, it doesn't matter what the numerators of the fractions are. Usually, after finding the LCD, we replace both fractions with the equivalent versions whose denominator is the LCD. Having common denominators trumps having reduced fractions. If your teacher complains, you tell her we said so.

  • Addition and Subtraction of Fractions

    It's easy to add and subtract fractions when the pieces involved are the same size. Who doesn't like "easy"?

    Sample Problem

    Translation: "We have 1 piece out of 4 and add 1 more piece out of 4."

    Sample Problem

    "We have 3 pieces out of 5, take 1 away, and we're left with 2 pieces out of 5."

    Sample Problem

    "We have 3 pieces out of 5, try to take 4 away, are left owing 1 piece out of 5." Eh, put it on our tab.

    Sample Problem

    See what we did there? If the pieces are different sizes (i.e. if the denominators are different), we use the same trick we used when comparing fractions. We find the LCD, turn both fractions into something with the same denominator, and then add them up the easy way.

    Sample Problem

    When mixed numbers are involved, we turn the mixed numbers back into fractions and carry on as normal. Just act natural, and if anyone asks, deny everything. Remember that mixed numbers are themselves simply abbreviations for addition. But now that we've turned that mixed number into an improper fraction, how do we finish up the addition?

    This will require one extra step, but we're up to the task. The LCD of these two fractions is 10, so now all we have to do is put the fractions over their LCD and then total 'em up:

    If you want, you can now convert this back into a mixed number (8 1/10), but don't bother if you aren't asked to do so. You've got better things you can be doing with your time. Like painting rainbows on your fingernails.

  • Multiplication of Fractions

    You might think of multiplication as a slightly more complex process than 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 × b, we can picture a box with side lengths a and b. The area of this box is the product of a and b, since a rectangle's area is just length times width. So far, so good.

    We're going to show you a visual representation of how this works with fractions, but don't freak out. You won't actually need to draw all these boxes going forward. This is just to help you understand the nitty-gritty of what you're technically doing when you're multiplying fractions.

    To find the product of  and , 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 

  • Equivalent Fractions and Multiplication by 1


    Multiplication by 1. We bet you like the sound of that. And well you should. It's as simple as it sounds.

    Mathematicians call 1 the multiplicative identity, meaning that any number multiplied by 1 gets to keep its identity. Any number multiplied by a number other than one is immediately given an alias and escorted by a U.S. Marshal to a farmhouse in Iowa as a member of the Witness Protection Program.

    When we find equivalent fractions, what we're really doing is multiplying by a cleverly disguised form of 1. We're talking fake mustache, glasses, a wig made out of real human hair—the works.

  • Multiplication by Clever Form of 1


    This super handy trick lets you simplify an expression without changing its value. Which is good, because fractions should not be subject to inflation.

    Sample Problem

    Here's a horrible fraction. A really horrible fraction. This thing is getting coal for Christmas.

    If we multiply it by 1, we won't change its value. If we multiply it by anything equivalent to 1 (such as ), we also won't change its value. Since each of our two mini-fractions has a 3 in its denominator, what happens if we multiply the whole thing by ?

    Our new numerator is , and our new denominator is . Now the fraction's value hasn't changed, and it looks way more manageable. It might even get an Xbox under the tree this year.

    Sample Problem

    We can also use this trick to rewrite decimal division problems:

    3.4/7.8 × 10/10 = 34/78

    We sort of hate it when decimals and fractions are all mixed up like that, so we just multiply the whole fraction by 10/10 (a.k.a. a clever form of 1), which turns the numerator and denominator both into nice, whole numbers. Ah, much better.

  • Multiplicative Inverses

    Every fraction with a nonzero numerator has a multiplicative inverse, which is simply the number we can multiply our fraction by to get 1. Basically, we're striving to get back to 1 or to "achieve oneness." It's like the math version of meditation.

    Sample Problem

    What's the multiplicative inverse of ?

    In other words, what can we multiply this fraction by to turn it into 1? Since () × (-4) = 1, our multiplicative inverse is just -4.

    Sample Problem

    What's the multiplicative inverse of  ?

    It's , since  ×  = 1.

    The multiplicative inverse of a fraction is called the reciprocal, and it's the upside-down version of that fraction. Don't take us too literally here—you don't need to stand on your head or anything. Instead, you're just pulling the old switcheroo on the numerator and denominator.

    To find the multiplicative inverse of an integer or mixed number, write the integer or mixed number as a fraction first, and then make the switch. It's like one of those bad movies on ABC Family where someone wakes up in someone else's place and has to figure out how to get back.

    Why can't a fraction with a zero in its numerator like  have a multiplicative inverse? Here's why: , so  There's no way to multiply  by something and get 1 as an answer. It has no multiplicative inverse. It's like it has a case of multiplicative amnesia. It should probably go to a multiplicative hospital. Okay, we've officially beaten that word to death. We'll get it to a multiplicative morgue, stat.

  • Division of Fractions

    In the world of fractions, division is really just another way of saying "multiplication by the reciprocal."

    The phrase "a divided by b" means "a multiplied by the reciprocal of b."

    For example:

     

    Which should make you happy, since you already know how to multiply by a reciprocal. This should really free up your weekend.

    You won't always see a fraction division problem written in that left-to-right format. Sometimes it will appear as a repulsive, nightmarish "fraction over a fraction." Don't sweat it though. Keep in mind that the line between the two mini-fractions means "divided by" and you'll be fine. In other words, you can take something unappealing like this:

    And rewrite it like this:

    Now it doesn't look so bad, and you know what to do with it.

    There's another option, and you should probably know how to do it, just in case you're asked. Never hurts to have a second method in your back pocket. Unless the method in question is jagged or covered in spikes. Ouch.

    Method Numero Dos:

    Multiply the numerator and denominator by the reciprocal of the fraction in the denominator. That sentence has a lot of long words in it, but break it down and it's not so bad.