Hmmm… this fraction has 25% more fiber than this fraction…

Oooo! But this fraction has trisodium phosphate!

Well… it’s not quite like that.

Comparing fractions just means telling which one is bigger.

You know, just like we do with regular numbers when we use the greater-than, less-than, and equal-to signs.

That sounds easy, right? But unfortunately, unlike regular numbers,

it’s not always easy to tell which fraction is bigger just by looking at them.

That’s because the value of a fraction depends on both the top AND bottom numbers and how they relate to each other.

For example, if you have to compare these two fractions, 1 over 3 and 1 over 10,

some of you might be tempted to say that 1 over 10 is bigger because you know that 10 is bigger than 3, right?

But we need to remember that the fraction is really a number written like a division problem,

and its value depends on that division.

So in this case, the 1 over 3 is really the bigger fraction

because its decimal value (what you get when you divide) is 0.333 but the value of 1 over 10 is only 0.1

Okay, so comparing fractions isn’t quite as easy as comparing regular numbers,

but that doesn’t mean it’s going to be that hard.

We’re going to learn two methods for comparing fractions that make it very easy.

The first method is called cross-multiplying,

and it takes advantage of the fact that it’s easy to compare fractions with the same bottom numbers.

If two fractions have the same bottom numbers, then we can just compare the top numbers.

That’s because we are comparing the same size parts.

We’re comparing fourths to fourths, eighths to eighths, tenths to tenths, and so on…

And the top number just tells us how many of those parts we have,

so it’s easy to see that 5 eighths is more than 3 eighths.

But many times, you’ll have to compare fractions that have different bottom numbers. (or different size parts)

Fortunately, there’s a trick we can do to make the comparison easy.

In the Math Antics Videos about Common Denominators,

we learn a simple method for changing “unlike fractions” (with different bottom numbers)

into “like fractions” (with the same bottom number).

Basically, it shows how you can multiply two unlike fractions

by wholes fractions made from the different bottom numbers,

so you end up with the same bottom number.

This will give you two new ‘equivalent’ fractions that you can easily add, subtract, or compare.

But, there’s a shortcut for comparing fractions.

As long as we know that the bottom numbers of our fractions are the same,

we don’t really need to know what that number is.

We just need to know what the top numbers will be, since those are the ones that we’ll actually compare.

So instead of multiplying each fraction by a whole fraction,

we can just multiply the top number of each fraction by the bottom number of the other fraction.

This is called ‘Cross Multiplying’ because if you draw a diagram of what you’re multiplying,

it forms a criss-cross pattern.

After you cross multiply, you will have two numbers that would be the new top numbers

if you had made ‘like’ fractions, and those numbers will show you which fraction is greater.

Let’s try this cross-multiplying method on an example or two.

Let’s compare the fractions: 7 over 8 and 4 over 5.

We start by multiplying the second fraction’s bottom number (5) by the first fraction’s top number (7)

and that gives us 35 for the new top number on this side.

You’ll always keep the answer on the side of the top number that you multiplied.

…now for the other side.

The bottom number (8) times the top number (4) gives us 32 for its new top number.

Ah-ha! Now it’s easy to see that the fraction 7 over 8 is greater than the fraction 4 over 5

because its new top number (35) is greater than the other new top number (32).

Let’s do one more comparison by cross multiplying.

Let’s compare 6 over 11 to 9 over 15.

First we’ll multiply 15 by 6 to get the new top number of the first side, which is 90.

Now you can use a calculator to do the multiplications if you need to.

Next, we multiply 11 by 9 to get the second new top number, which is 99.

So, that tells us that the second fraction (9 over 15) is greater than the first fraction because its new top number (99) is bigger.

Pretty simple, huh?

Okay, cross multiplying is pretty cool,

but there’s another way to compare fractions that you need to know about.

But this one is only really good if you can use a calculator.

Remember, the reason that fractions are tricky to compare is because they’re really division problems.

But if we want to, we can just do the division and get the answer,

which is the decimal value of the fraction.

So if you have two fractions to compare, you can just do the division

(preferably using a calculator) and then compare the decimal values.

For example, let’s say I offered to give you either 5/12 of a pizza or 7/15 of a pizza.

Now, you happen to be really hungry, so you want to choose the biggest amount,

but it’s not very easy to tell just by looking which is bigger: 5/12 or 7/15

This is were decimal values can really help you out.

If you convert the fractions to decimals by doing division,

it will make it much easier to see which one is bigger.

5 divided by 12 is about 0.42

and 7 divided by 15 is about 0.47

Yep, that makes comparing them much easier.

Since 0.47 is greater than 0.42, it means that 7/15 is greater than 5/12.

And that means that you’d rather have 7/15 of the pizza!

Sometimes when you compare fractions this way,

you’ll find two fractions that look different, but have the same decimal value: like 3/8 and 15/40.

If you convert each fraction to a decimal, you’ll see that they both have the value 0.375

Two fractions that have different top and bottom numbers, but the same value are called ‘equivalent fractions’.

If two fractions are equivalent, then you can just use the equal sign to show the comparison between them, like this…

Alright, so those are two great methods you can use to compare fractions.

Cross multiplying is simple and works great, even if you don’t have a calculator.

And comparing the decimal vales by dividing is easy if you do have a calculator.

As always, practice makes perfect,

so spend some time doing the exercises for this section, and I’ll see you next time.

Learn more at www.mathantics.com