Is 2/10 Greater Than 1/2

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 permit us tell exactly how much nosotros take of something. Some fractions are larger than others. For example, which is larger: 6/viii of a pizza or 7/8 of a pizza?

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

Click through the slideshow to larn how to compare fractions.

Before, we saw that fractions accept two parts.
One part is the peak number, or numerator .
The other is the bottom number, or denominator .
The denominator tells us how many parts are in a whole.
The numerator tells united states how many of those parts nosotros have.
When fractions accept the aforementioned denominator, it ways they're split into the aforementioned number of parts.
This means we can compare these fractions merely by looking at the numerator.
Here, 5 is more than than 4...
Here, v is more iv...and so we can tell that five/6 is more than iv/6.
Allow'due south look at another example. Which of these is larger: 2/eight or 6/8?
If y'all thought 6/8 was larger, you were correct!
Both fractions have the same denominator.
So we compared the numerators. 6 is larger than two, and then 6/8 is more than than ii/viii.

As you saw, if ii or more fractions have the same denominator, yous can compare them by looking at their numerators. As you can encounter 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 accept the same bottom numbers, or denominators . Just yous know that fractions tin accept whatever number as a denominator. What happens when you need to compare fractions with different bottom numbers?

For case, which of these is larger: 2/3 or one/5? It'southward hard to tell just by looking at them. Subsequently all, 2 is larger than 1, but the denominators aren't the same.

If yous look at the picture, though, the difference is clear: two/3 is larger than 1/5. With an illustration, it was easy to compare these fractions, simply how could nosotros have done it without the movie?

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

Allow's compare these fractions: 5/viii and four/half dozen.
Before nosotros compare them, we need to change both fractions so they take the aforementioned denominator, or bottom number.
First, we'll find the smallest number that can be divided by both denominators. We phone call that the lowest common denominator.
Our get-go footstep is to detect numbers that can be divided evenly past viii.
Using a multiplication tabular array makes this easy. All of the numbers on the 8 row tin be divided evenly past 8.
Now let'south wait at our 2d denominator: 6.
We tin can use the multiplication table over again. All of the numbers in the six row can exist divided evenly by six.
Let'due south compare the two rows. It looks like in that location are a few numbers that can exist divided evenly past both half-dozen and viii.
24 is the smallest number that appears on both rows, so it's the everyman common denominator.
Now nosotros're going to change our fractions then they both accept the aforementioned denominator: 24.
To practise that, we'll have to alter the numerators the same way we inverse the denominators.
Let'south look at 5/8 once again. In order to alter the denominator to 24...
Permit's look at five/8 again. In order to change the denominator to 24...we had to multiply eight by 3.
Since nosotros multiplied the denominator past three, we'll as well multiply the numerator, or meridian number, by three.
5 times iii equals xv. So we've changed five/viii into 15/24.
We tin can do that because any number over itself is equal to ane.
So when we multiply 5/8 by three/3...
And then when we multiply 5/8 by 3/3...we're actually multiplying v/8 by 1.
Since whatsoever number times ane is equal to itself...
Since whatsoever number times 1 is equal to itself...we can say that 5/8 is equal to xv/24.
At present nosotros'll exercise the aforementioned to our other fraction: 4/6. Nosotros also changed its denominator to 24.
Our erstwhile denominator was vi. To go 24, nosotros multiplied six by four.
Then we'll also multiply the numerator by 4.
4 times 4 is 16. Then 4/6 is equal to 16/24.
Now that the denominators are the same, we can compare the ii fractions by looking at their numerators.
16/24 is larger than 15/24...
16/24 is larger than 15/24... so 4/six is larger than v/8.

Reducing fractions

Which of these is larger: iv/eight or one/ii?

If you did the math or even just looked at the moving-picture show, you might have been able to tell that they're equal . In other words, iv/8 and 1/2 mean the same matter, even though they're written differently.

If 4/viii means the same affair as 1/2, why not just call it that? Half is easier to say than 4-eighths, and for nigh people it'southward also easier to understand. Subsequently all, when you consume out with a friend, you split the bill in one-half, not in eighths.

If you lot write iv/8 as i/two, you're reducing information technology. When we reduce a fraction, we're writing information technology in a simpler form. Reduced fractions are always equal to the original fraction.

Nosotros already reduced four/8 to 1/2. If you wait at the examples below, you can see that other numbers tin be reduced to ane/2 as well. These fractions are all equal.

5/10 = i/2

11/22 = i/2

36/72 = one/2

These fractions have all been reduced to a simpler form equally well.

four/12 = 1/3

14/21 = two/3

35/50 = vii/ten

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

Let's try reducing this fraction: 16/xx.
Since the numerator and denominator are even numbers, you lot can divide them by two to reduce the fraction.
Get-go, we'll divide the numerator by two. xvi divided by two is 8.
Side by side, we'll divide the denominator past two. 20 divided by 2 is x.
We've reduced sixteen/20 to 8/10. We could also say that xvi/20 is equal to 8/10.
If the numerator and denominator tin can withal exist divided by 2, we tin continue reducing the fraction.
eight divided by 2 is 4.
x divided by two is five.
Since at that place's no number that four and 5 can exist divided past, we can't reduce 4/5 any further.
This ways 4/5 is the simplest form of 16/twenty.
Allow's try reducing some other fraction: six/ix.
While the numerator is even, the denominator is an odd number, so we can't reduce by dividing past ii.
Instead, we'll need to find a number that 6 and 9 tin can be divided by. A multiplication table will make that number easy to find.
Permit'due south find half-dozen and 9 on the same row. As you lot can see, 6 and 9 can both be divided past 1 and 3.
Dividing by 1 won't change these fractions, then nosotros'll use the largest number that 6 and nine tin can be divided by.
That's three. This is called the greatest common divisor, or GCD. (You can also call it the greatest common cistron, or GCF.)
three is the GCD of 6 and 9 considering it's the largest number they can be divided by.
So we'll divide the numerator by 3. 6 divided past iii is 2.
Then we'll split the denominator by 3. 9 divided by 3 is iii.
At present we've reduced six/ix to ii/3, which is its simplest course. We could also say that vi/ix is equal to 2/3.

Irreducible fractions

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

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

Try This!

Reduce each fraction to its simplest course.

Mixed numbers and improper fractions

In the previous lesson, y'all learned virtually mixed numbers. A mixed number has both a fraction and a whole number. An case is one two/3. Yous'd read 1 2/3 like this: one and two-thirds.

Another fashion to write this would be five/3, or v-thirds. These ii numbers look different, simply they're actually the same. 5/3 is an improper fraction. This only means the numerator is larger than the denominator.

There are times when y'all may prefer to employ an improper fraction instead of a mixed number. Information technology's piece of cake to change a mixed number into an improper fraction. Let's learn how:

Let'due south catechumen ane 1/4 into an improper fraction.
Starting time, nosotros'll demand to find out how many parts make upwardly the whole number: 1 in this instance.
To do this, we'll multiply the whole number, 1, by the denominator, iv.
1 times 4 equals 4.
At present, let'south add that number, 4, to the numerator, i.
4 plus 1 equals 5.
The denominator stays the same.
Our improper fraction is 5/four, or five-fourths. And then we could say that ane 1/iv is equal to five/4.
This means there are five ane/4s in 1 1/4.
Let'due south convert another mixed number: ii 2/v.
First, nosotros'll multiply the whole number by the denominator. 2 times 5 equals 10.
Next, we'll add 10 to the numerator. 10 plus 2 equals 12.
Equally always, the denominator will stay the same.
So 2 two/5 is equal to 12/v.

Try This!

Effort converting these mixed numbers into improper fractions.

Converting improper fractions into mixed numbers

Improper fractions are useful for math bug that use fractions, as you'll learn later. However, they're also more difficult to read and empathise than mixed numbers. For example, it's a lot easier to picture 2 4/7 in your head than 18/7.

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

Let's turn 10/four into a mixed number.
You lot can think of any fraction as a division problem. Only care for the line betwixt the numbers similar a division sign (/).
So nosotros'll divide the numerator, x, by the denominator, iv.
ten divided past 4 equals 2...
10 divided by four equals 2... with a residuum of 2.
The reply, 2, will become our whole number because ten tin can be divided by 4 twice.
And the rest, ii, will get the numerator of the fraction because we accept 2 parts left over.
The denominator remains the same.
So 10/4 equals 2 2/iv.
Permit'due south try some other example: 33/3.
We'll split up the numerator, 33, past the denominator, three.
33 divided past 3...
33 divided by 3... equals eleven, with no residuum.
The respond, xi, will become our whole number.
At that place is no remainder, then we tin see that our improper fraction was really a whole number. 33/3 equals 11.