4 6 As A Fraction

Lesson 2: Comparison 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 let the states tell exactly how much we have of something. Some fractions are larger than others. For case, which is larger: half dozen/8 of a pizza or 7/8 of a pizza?

In this paradigm, we can come across that 7/viii is larger. The illustration makes information technology like shooting fish in a barrel to compare these fractions. Simply how could nosotros take done it without the pictures?

Click through the slideshow to acquire how to compare fractions.

Earlier, nosotros saw that fractions have two parts.
1 part is the top number, or numerator .
The other is the bottom number, or denominator .
The denominator tells united states how many parts are in a whole.
The numerator tells united states how many of those parts we have.
When fractions accept the aforementioned denominator, it ways they're carve up into the same number of parts.
This means we can compare these fractions just by looking at the numerator.
Hither, 5 is more than than 4...
Here, 5 is more than than 4...and then nosotros can tell that 5/6 is more 4/half-dozen.
Allow'southward look at some other example. Which of these is larger: 2/8 or vi/viii?
If you lot thought half-dozen/eight was larger, y'all were right!
Both fractions have the same denominator.
So nosotros compared the numerators. 6 is larger than two, and so six/viii is more than 2/8.

As you saw, if ii or more fractions have the same denominator, you lot can compare them past looking at their numerators. As you can encounter below, 3/4 is larger than one/4. The larger the numerator, the larger the fraction.

Comparing fractions with different denominators

On the previous page, we compared fractions that have the same bottom numbers, or denominators . But you lot know that fractions tin take whatever number as a denominator. What happens when you need to compare fractions with different bottom numbers?

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

If you look at the flick, though, the divergence is clear: ii/3 is larger than 1/5. With an illustration, it was piece of cake to compare these fractions, just how could we have done it without the moving-picture show?

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

Let'due south compare these fractions: 5/eight and four/half-dozen.
Before we compare them, we demand to alter both fractions and so they have the same denominator, or lesser number.
First, we'll find the smallest number that tin can be divided by both denominators. Nosotros call that the everyman common denominator.
Our first step is to find numbers that can exist divided evenly by 8.
Using a multiplication tabular array makes this easy. All of the numbers on the 8 row tin can exist divided evenly past 8.
Now let's look at our 2d denominator: 6.
Nosotros can use the multiplication table again. All of the numbers in the 6 row tin be divided evenly by 6.
Allow's compare the ii rows. It looks similar in that location are a few numbers that tin can be divided evenly by both 6 and 8.
24 is the smallest number that appears on both rows, so it'southward the everyman common denominator.
Now we're going to alter our fractions so they both have the same denominator: 24.
To do that, we'll accept to modify the numerators the same way we changed the denominators.
Allow's look at 5/8 over again. In order to alter the denominator to 24...
Allow'southward await at five/8 once more. In order to change the denominator to 24...we had to multiply viii by 3.
Since we multiplied the denominator past three, we'll too multiply the numerator, or tiptop number, by 3.
5 times three equals fifteen. Then we've inverse 5/8 into 15/24.
We can do that because any number over itself is equal to i.
And so when we multiply v/8 past 3/3...
And then when nosotros multiply five/viii by iii/3...we're really multiplying 5/eight past 1.
Since any number times 1 is equal to itself...
Since whatever number times 1 is equal to itself...we can say that five/8 is equal to fifteen/24.
Now we'll do the same to our other fraction: iv/vi. We also changed its denominator to 24.
Our former denominator was 6. To get 24, nosotros multiplied 6 by four.
Then we'll also multiply the numerator by iv.
4 times 4 is 16. Then four/vi is equal to 16/24.
Now that the denominators are the same, nosotros tin compare the two fractions past looking at their numerators.
16/24 is larger than xv/24...
16/24 is larger than 15/24... so 4/6 is larger than five/8.

Reducing fractions

Which of these is larger: 4/viii or ane/2?

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

If 4/viii ways the same thing as 1/2, why not but call it that? One-one-half is easier to say than four-eighths, and for most people it's also easier to empathize. After all, when y'all eat out with a friend, yous dissever the beak in half, not in eighths.

If you write four/8 as ane/2, you're reducing information technology. When we reduce a fraction, nosotros're writing it in a simpler form. Reduced fractions are e'er equal to the original fraction.

We already reduced iv/8 to 1/two. If you look at the examples beneath, you can see that other numbers tin can be reduced to 1/two as well. These fractions are all equal.

5/10 = 1/2

11/22 = 1/ii

36/72 = one/two

These fractions take all been reduced to a simpler form too.

4/12 = 1/three

14/21 = 2/three

35/fifty = vii/10

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

Let's attempt reducing this fraction: 16/20.
Since the numerator and denominator are fifty-fifty numbers, you can dissever them by 2 to reduce the fraction.
Offset, we'll divide the numerator by two. 16 divided past 2 is 8.
Adjacent, we'll dissever the denominator by ii. 20 divided by 2 is 10.
We've reduced sixteen/xx to 8/10. We could also say that 16/20 is equal to 8/x.
If the numerator and denominator can yet be divided by 2, nosotros tin can continue reducing the fraction.
8 divided by 2 is 4.
10 divided by 2 is 5.
Since there's no number that iv and 5 can exist divided by, we can't reduce 4/5 whatsoever further.
This means 4/v is the simplest grade of 16/xx.
Let'south try reducing another fraction: 6/ix.
While the numerator is even, the denominator is an odd number, so we tin't reduce past dividing past 2.
Instead, we'll need to notice a number that 6 and nine can be divided by. A multiplication table volition make that number easy to find.
Let'south notice 6 and nine on the aforementioned row. Equally yous can run across, half-dozen and nine can both be divided by 1 and 3.
Dividing by one won't change these fractions, and then nosotros'll use the largest number that half dozen and 9 can be divided by.
That's 3. This is chosen the greatest common divisor, or GCD. (You tin also call it the greatest mutual factor, or GCF.)
3 is the GCD of 6 and nine because it's the largest number they tin can exist divided by.
So we'll divide the numerator by 3. 6 divided by 3 is 2.
Then we'll split up the denominator past iii. ix divided past 3 is 3.
Now nosotros've reduced 6/9 to ii/three, which is its simplest course. We could also say that 6/nine is equal to 2/iii.

Irreducible fractions

Not all fractions tin can be reduced. Some are already as simple every bit they can exist. For instance, yous tin can't reduce 1/two because at that place'due south no number other than 1 that both 1 and ii can be divided past. (For that reason, you lot can't reduce any fraction that has a numerator of 1.)

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

Try This!

Reduce each fraction to its simplest course.

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 1 2/three. You'd read i 2/3 like this: i and 2-thirds.

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

At that place are times when you may adopt to use an improper fraction instead of a mixed number. It'due south easy to alter a mixed number into an improper fraction. Let'due south learn how:

Permit'south convert 1 1/four into an improper fraction.
First, we'll need to find out how many parts make upwardly the whole number: 1 in this instance.
To exercise this, we'll multiply the whole number, 1, by the denominator, 4.
1 times 4 equals 4.
At present, allow's add that number, iv, to the numerator, 1.
4 plus 1 equals five.
The denominator stays the same.
Our improper fraction is 5/4, or 5-fourths. So nosotros could say that 1 1/4 is equal to 5/4.
This means there are five one/4s in 1 1/4.
Permit'due south convert some other mixed number: 2 2/5.
First, we'll multiply the whole number by the denominator. 2 times 5 equals 10.
Next, we'll add 10 to the numerator. x plus 2 equals 12.
As ever, the denominator volition stay the same.
So two 2/5 is equal to 12/5.

Endeavour This!

Try 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 larn later. However, they're likewise more difficult to read and empathise than mixed numbers. For example, it'south a lot easier to picture 2 four/seven in your head than 18/7.

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

Permit's turn x/four into a mixed number.
You tin can call up of any fraction as a division problem. Just treat the line between the numbers like a division sign (/).
So we'll divide the numerator, ten, past the denominator, 4.
10 divided by four equals two...
ten divided by 4 equals ii... with a balance of 2.
The answer, 2, will go our whole number considering ten can be divided by 4 twice.
And the rest, 2, volition become the numerator of the fraction because we accept two parts left over.
The denominator remains the aforementioned.
So x/4 equals ii 2/4.
Let's try another case: 33/three.
We'll carve up the numerator, 33, by the denominator, three.
33 divided past 3...
33 divided past 3... equals 11, with no residual.
The answer, eleven, will become our whole number.
There is no remainder, then we tin see that our improper fraction was actually a whole number. 33/three equals eleven.