Difference Between 5/16 And 1/4

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 function of something. Fractions are useful, since they let united states tell exactly how much we accept of something. Some fractions are larger than others. For example, which is larger: 6/8 of a pizza or vii/8 of a pizza?

In this image, we can come across that vii/viii is larger. The analogy makes it easy to compare these fractions. But how could we take done it without the pictures?

Click through the slideshow to learn how to compare fractions.

Earlier, we saw that fractions accept two parts.
Ane part is the height 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 us how many of those parts nosotros have.
When fractions take the aforementioned denominator, it means they're divide into the same number of parts.
This ways we can compare these fractions merely past looking at the numerator.
Here, 5 is more than 4...
Hither, 5 is more than four...so nosotros can tell that 5/6 is more than four/six.
Let's look at another instance. Which of these is larger: ii/8 or half dozen/8?
If yous thought half-dozen/8 was larger, y'all were right!
Both fractions accept the aforementioned denominator.
So we compared the numerators. 6 is larger than 2, so half dozen/8 is more than than 2/8.

Equally y'all saw, if ii or more fractions accept the same denominator, you lot tin can compare them by looking at their numerators. Every bit you can encounter below, 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 lesser numbers, or denominators . Merely you know that fractions can have any number equally 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/v? Information technology's difficult to tell only by looking at them. After all, 2 is larger than 1, but the denominators aren't the same.

If you lot expect at the picture, though, the difference is clear: 2/3 is larger than i/five. 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 moving-picture show?

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

Let's compare these fractions: 5/viii and iv/half dozen.
Earlier nosotros compare them, we need to change both fractions and then they take the same denominator, or bottom number.
Kickoff, we'll notice the smallest number that can be divided by both denominators. We call that the lowest mutual denominator.
Our showtime pace is to discover numbers that can exist divided evenly by 8.
Using a multiplication table makes this easy. All of the numbers on the viii row tin be divided evenly past viii.
Now let'southward wait at our second denominator: 6.
We tin can employ the multiplication tabular array again. All of the numbers in the 6 row can be divided evenly by 6.
Let's compare the two rows. It looks like there are a few numbers that can exist divided evenly by both 6 and eight.
24 is the smallest number that appears on both rows, so information technology's the lowest common denominator.
At present nosotros're going to change our fractions and then they both accept the aforementioned denominator: 24.
To do that, we'll have to change the numerators the same way we inverse the denominators.
Permit's look at five/eight over again. In order to change the denominator to 24...
Allow's look at 5/eight again. In order to change the denominator to 24...we had to multiply viii by 3.
Since we multiplied the denominator by iii, we'll also multiply the numerator, or top number, by iii.
5 times iii equals 15. So we've changed 5/eight into 15/24.
Nosotros tin can do that considering any number over itself is equal to 1.
And so when nosotros multiply 5/8 by 3/iii...
So when we multiply five/eight by 3/3...nosotros're really multiplying 5/8 by 1.
Since any number times i is equal to itself...
Since whatsoever number times 1 is equal to itself...nosotros tin can say that 5/8 is equal to 15/24.
Now nosotros'll do the same to our other fraction: 4/half dozen. We also inverse its denominator to 24.
Our old denominator was 6. To get 24, we multiplied 6 by iv.
So we'll also multiply the numerator by 4.
4 times 4 is 16. So 4/6 is equal to sixteen/24.
At present that the denominators are the aforementioned, we can compare the two fractions by looking at their numerators.
16/24 is larger than 15/24...
16/24 is larger than fifteen/24... so 4/6 is larger than v/eight.

Reducing fractions

Which of these is larger: 4/8 or ane/ii?

If you did the math or even just looked at the picture, you might take been able to tell that they're equal . In other words, 4/8 and 1/2 mean the same thing, even though they're written differently.

If 4/viii means the same thing every bit 1/2, why not simply call it that? Half is easier to say than four-eighths, and for most people it'south also easier to understand. After all, when y'all eat out with a friend, you lot split the bill in half, non in eighths.

If y'all write iv/viii as 1/ii, yous're reducing it. When we reduce a fraction, we're writing it in a simpler form. Reduced fractions are always equal to the original fraction.

We already reduced 4/eight to ane/2. If yous look at the examples below, you lot can meet that other numbers tin can exist reduced to one/2 as well. These fractions are all equal.

five/ten = 1/two

eleven/22 = ane/2

36/72 = 1/2

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

4/12 = 1/iii

xiv/21 = ii/three

35/50 = 7/10

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

Allow'south try reducing this fraction: xvi/20.
Since the numerator and denominator are even numbers, y'all can divide them by 2 to reduce the fraction.
First, we'll divide the numerator by 2. 16 divided by 2 is 8.
Next, nosotros'll dissever the denominator by ii. 20 divided by 2 is 10.
We've reduced 16/20 to 8/ten. We could too say that 16/20 is equal to 8/10.
If the numerator and denominator can notwithstanding exist divided by 2, we can continue reducing the fraction.
eight divided by 2 is iv.
10 divided by ii is v.
Since in that location's no number that 4 and 5 can exist divided by, we tin can't reduce 4/five any farther.
This ways 4/v is the simplest form of 16/20.
Let's try reducing some other fraction: 6/9.
While the numerator is even, the denominator is an odd number, so we can't reduce past dividing by 2.
Instead, we'll need to find a number that 6 and ix can be divided by. A multiplication tabular array volition make that number piece of cake to observe.
Let'south discover 6 and 9 on the same row. As you can see, 6 and 9 tin both be divided by 1 and 3.
Dividing past 1 won't change these fractions, so we'll use 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 can also call it the greatest mutual factor, or GCF.)
three is the GCD of 6 and 9 because it'southward the largest number they tin be divided past.
So we'll dissever the numerator by 3. half-dozen divided by iii is two.
Then we'll divide the denominator by iii. 9 divided by 3 is 3.
Now nosotros've reduced 6/9 to ii/3, which is its simplest form. We could also say that vi/9 is equal to 2/3.

Irreducible fractions

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

Some fractions that have larger numbers can't exist reduced either. For instance, 17/36 tin can't be reduced because there'due south no number that both 17 and 36 can exist divided by. If you can't notice any common multiples for the numbers in a fraction, chances are it'south irreducible .

Attempt This!

Reduce each fraction to its simplest form.

Mixed numbers and improper fractions

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

Another way to write this would exist 5/three, or five-thirds. These two numbers look different, but they're actually the same. 5/3 is an improper fraction. This just means the numerator is larger than the denominator.

There are times when you may prefer to use an improper fraction instead of a mixed number. It's easy to alter a mixed number into an improper fraction. Allow's learn how:

Let's convert i 1/4 into an improper fraction.
First, we'll demand to find out how many parts brand up the whole number: i in this example.
To do this, we'll multiply the whole number, i, by the denominator, four.
1 times four equals iv.
Now, let's add that number, 4, to the numerator, 1.
4 plus 1 equals v.
The denominator stays the same.
Our improper fraction is five/4, or five-fourths. And then we could say that 1 1/4 is equal to 5/4.
This means in that location are five 1/4s in 1 1/iv.
Let's convert another mixed number: two 2/five.
Outset, we'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.
As always, the denominator will stay the same.
And then 2 2/5 is equal to 12/5.

Effort This!

Try converting these mixed numbers into improper fractions.

Converting improper fractions into mixed numbers

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

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

Let's turn 10/4 into a mixed number.
You lot can think of whatever fraction as a sectionalization problem. Just treat the line between the numbers like a division sign (/).
So nosotros'll divide the numerator, 10, by the denominator, iv.
10 divided by four equals 2...
10 divided by 4 equals two... with a residual of ii.
The answer, two, will get our whole number because ten can exist divided by four twice.
And the residue, 2, will become the numerator of the fraction considering nosotros have 2 parts left over.
The denominator remains the same.
So 10/4 equals two 2/4.
Let's try another example: 33/3.
We'll divide the numerator, 33, past the denominator, 3.
33 divided past iii...
33 divided by 3... equals 11, with no remainder.
The answer, 11, volition go our whole number.
There is no residue, so we can see that our improper fraction was actually a whole number. 33/3 equals 11.