Is 3/8 More Than 1/3

Lesson 2: Comparing and Reducing Fractions

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Comparing fractions

In Introduction to Fractions, we learned that fractions are a way of showing part of something. Fractions are useful, since they allow united states tell exactly how much we have of something. Some fractions are larger than others. For example, which is larger: half-dozen/eight of a pizza or 7/8 of a pizza?

In this image, we tin can run across that seven/8 is larger. The analogy makes it easy to compare these fractions. But how could nosotros accept done information technology without the pictures?

Click through the slideshow to learn how to compare fractions.

Earlier, we saw that fractions have two parts.
Ane part is the top number, or numerator .
The other is the bottom number, or denominator .
The denominator tells u.s. how many parts are in a whole.
The numerator tells us how many of those parts nosotros accept.
When fractions accept the aforementioned denominator, it means they're split into the same number of parts.
This ways we can compare these fractions just by looking at the numerator.
Hither, 5 is more than 4...
Here, 5 is more than 4...and so we tin tell that 5/6 is more than 4/half dozen.
Let'southward look at another example. Which of these is larger: 2/8 or vi/viii?
If you idea vi/8 was larger, you lot were right!
Both fractions have the aforementioned denominator.
Then we compared the numerators. six is larger than ii, and then 6/viii is more 2/viii.

Equally you saw, if two or more fractions have the same denominator, you lot tin can compare them by looking at their numerators. As you tin can see beneath, 3/4 is larger than 1/4. The larger the numerator, the larger the fraction.

Comparing fractions with dissimilar denominators

On the previous page, we compared fractions that have the aforementioned bottom numbers, or denominators . Just y'all know that fractions can have any number as a denominator. What happens when you need to compare fractions with different bottom numbers?

For example, which of these is larger: ii/3 or 1/5? It's difficult to tell merely by looking at them. Afterwards all, 2 is larger than 1, merely the denominators aren't the aforementioned.

If y'all look at the picture, though, the difference is clear: 2/three is larger than 1/v. With an illustration, it was like shooting fish in a barrel to compare these fractions, but how could we have done information technology without the picture?

Click through the slideshow to learn how to compare fractions with different denominators.

Let's compare these fractions: five/8 and 4/half dozen.
Before we compare them, we need to change both fractions so they have the aforementioned denominator, or bottom number.
Commencement, we'll notice the smallest number that can be divided by both denominators. We telephone call that the lowest mutual denominator.
Our kickoff step is to discover numbers that can be divided evenly by 8.
Using a multiplication table makes this easy. All of the numbers on the eight row can exist divided evenly by viii.
At present let's look at our second denominator: half dozen.
We tin utilise the multiplication table again. All of the numbers in the 6 row tin can be divided evenly past 6.
Let'south compare the two rows. It looks like there are a few numbers that can exist divided evenly by both vi and 8.
24 is the smallest number that appears on both rows, so it's the lowest mutual denominator.
Now nosotros're going to change our fractions so they both have the same denominator: 24.
To practice that, we'll accept to change the numerators the same way we inverse the denominators.
Let'due south expect at five/8 again. In guild to change the denominator to 24...
Let'south look at 5/8 once again. In order to change the denominator to 24...we had to multiply 8 by three.
Since we multiplied the denominator by iii, we'll too multiply the numerator, or tiptop number, by three.
5 times 3 equals 15. So we've changed five/8 into fifteen/24.
We tin can exercise that because any number over itself is equal to 1.
So when we multiply 5/viii past 3/3...
So when we multiply 5/8 by 3/three...we're really multiplying 5/8 by 1.
Since any number times 1 is equal to itself...
Since whatsoever number times 1 is equal to itself...we tin say that v/viii is equal to 15/24.
Now nosotros'll do the same to our other fraction: 4/half-dozen. We likewise changed its denominator to 24.
Our onetime denominator was 6. To get 24, we multiplied 6 past 4.
Then nosotros'll also multiply the numerator past 4.
4 times iv is 16. So iv/six is equal to 16/24.
Now that the denominators are the same, we tin can compare the two fractions by looking at their numerators.
16/24 is larger than 15/24...
xvi/24 is larger than fifteen/24... so 4/half-dozen is larger than 5/8.

Reducing fractions

Which of these is larger: 4/8 or 1/2?

If you did the math or even just looked at the motion-picture show, you lot might have been able to tell that they're equal . In other words, iv/8 and 1/two mean the aforementioned thing, even though they're written differently.

If 4/8 means the same matter as 1/2, why not but phone call it that? One-one-half is easier to say than four-eighths, and for most people it's also easier to sympathise. After all, when you eat out with a friend, you lot split the bill in one-half, non in eighths.

If yous write iv/8 equally 1/two, you lot're reducing information technology. When we reduce a fraction, nosotros're writing it in a simpler course. Reduced fractions are always equal to the original fraction.

We already reduced 4/8 to 1/2. If you look at the examples below, you can encounter that other numbers can be reduced to i/2 as well. These fractions are all equal.

5/10 = 1/2

11/22 = 1/2

36/72 = 1/2

These fractions have all been reduced to a simpler course also.

iv/12 = 1/iii

fourteen/21 = ii/3

35/fifty = 7/10

Click through the slideshow to learn how to reduce fractions past dividing.

Let's endeavor reducing this fraction: 16/20.
Since the numerator and denominator are even numbers, you can divide them past ii to reduce the fraction.
First, we'll divide the numerator past 2. sixteen divided by ii is viii.
Next, we'll divide the denominator by ii. 20 divided by ii is x.
We've reduced 16/20 to 8/10. We could too say that 16/20 is equal to viii/x.
If the numerator and denominator can still be divided by 2, nosotros can continue reducing the fraction.
8 divided by ii is four.
x divided past 2 is five.
Since there's no number that four and 5 can be divided by, nosotros can't reduce 4/5 any further.
This means iv/5 is the simplest form of 16/20.
Permit'due south endeavour reducing another fraction: 6/ix.
While the numerator is fifty-fifty, the denominator is an odd number, so we can't reduce by dividing by 2.
Instead, we'll demand to find a number that half dozen and 9 can be divided by. A multiplication tabular array will make that number like shooting fish in a barrel to detect.
Let'due south find vi and nine on the same row. Every bit you lot tin encounter, vi and 9 can both be divided by 1 and 3.
Dividing past 1 won't change these fractions, and then we'll apply the largest number that 6 and ix tin can be divided by.
That's 3. This is called the greatest common divisor, or GCD. (You tin can also call information technology the greatest common factor, or GCF.)
three is the GCD of 6 and 9 considering it's the largest number they tin can be divided by.
So we'll dissever the numerator by three. 6 divided by 3 is two.
Then nosotros'll split up the denominator by three. 9 divided past three is 3.
At present we've reduced 6/ix to ii/3, which is its simplest form. We could besides say that vi/ix is equal to two/three.

Irreducible fractions

Not all fractions can be reduced. Some are already as simple as they tin be. For example, y'all can't reduce 1/ii because there'south no number other than ane that both ane and 2 tin can be divided by. (For that reason, you can't reduce whatsoever fraction that has a numerator of one.)

Some fractions that have larger numbers tin can't be reduced either. For example, 17/36 can't exist reduced considering there'southward no number that both 17 and 36 can be divided by. If you lot tin can't find whatever common multiples for the numbers in a fraction, chances are it's irreducible .

Try This!

Reduce each fraction to its simplest grade.

Mixed numbers and improper fractions

In the previous lesson, you learned nigh mixed numbers. A mixed number has both a fraction and a whole number. An example is 1 2/3. You'd read 1 2/iii like this: one and two-thirds.

Another way to write this would exist 5/3, or five-thirds. These ii numbers await unlike, but they're actually the aforementioned. 5/3 is an improper fraction. This just means the numerator is larger than the denominator.

There are times when y'all may adopt to use an improper fraction instead of a mixed number. Information technology'due south easy to modify a mixed number into an improper fraction. Let's acquire how:

Let'due south convert ane one/iv into an improper fraction.
Outset, we'll need to notice out how many parts make up the whole number: one in this example.
To exercise this, we'll multiply the whole number, 1, by the denominator, iv.
1 times 4 equals 4.
Now, let's add together that number, four, to the numerator, one.
4 plus 1 equals 5.
The denominator stays the same.
Our improper fraction is 5/iv, or five-fourths. And so we could say that 1 1/4 is equal to 5/4.
This ways in that location are v 1/4southward in i 1/four.
Allow's convert another mixed number: 2 two/5.
Get-go, we'll multiply the whole number by the denominator. 2 times v equals 10.
Side by side, nosotros'll add together ten to the numerator. ten plus ii equals 12.
Every bit always, the denominator will stay the same.
So two two/five is equal to 12/five.

Try This!

Attempt converting these mixed numbers into improper fractions.

Converting improper fractions into mixed numbers

Improper fractions are useful for math problems that use fractions, as you'll acquire subsequently. However, they're also more difficult to read and sympathize than mixed numbers. For example, it's a lot easier to picture ii 4/vii in your head than 18/7.

Click through the slideshow to learn how to change an improper fraction into a mixed number.

Permit's turn ten/iv into a mixed number.
You can recollect of any fraction as a sectionalization problem. Only care for the line between the numbers like a partitioning sign (/).
So we'll carve up the numerator, 10, past the denominator, four.
10 divided past 4 equals two...
x divided by 4 equals ii... with a remainder of 2.
The answer, 2, will become our whole number because x can be divided by 4 twice.
And the balance, two, will become the numerator of the fraction considering nosotros have two parts left over.
The denominator remains the same.
So ten/4 equals 2 2/4.
Allow's try some other example: 33/3.
We'll separate the numerator, 33, by the denominator, iii.
33 divided by 3...
33 divided past 3... equals 11, with no remainder.
The answer, 11, will go our whole number.
There is no remainder, and then we can see that our improper fraction was actually a whole number. 33/three equals eleven.