Is 1/4 Bigger Than 7/16

Lesson two: Comparison and Reducing Fractions

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Comparing fractions

In Introduction to Fractions, we learned that fractions are a fashion of showing office of something. Fractions are useful, since they allow the states tell exactly how much nosotros have of something. Some fractions are larger than others. For example, which is larger: 6/8 of a pizza or seven/8 of a pizza?

In this epitome, we tin come across that vii/eight is larger. The illustration makes it piece of cake to compare these fractions. But how could we have done it without the pictures?

Click through the slideshow to learn how to compare fractions.

Earlier, nosotros saw that fractions have two parts.
Ane part is the acme 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 of america how many of those parts we accept.
When fractions have the same denominator, it means they're carve up into the same number of parts.
This means we can compare these fractions just by looking at the numerator.
Here, v is more than four...
Here, five is more than iv...and so we can tell that 5/6 is more than 4/half dozen.
Let'south await at another example. Which of these is larger: 2/8 or 6/8?
If you thought 6/8 was larger, yous were correct!
Both fractions take the aforementioned denominator.
So nosotros compared the numerators. 6 is larger than 2, then 6/eight is more than 2/8.

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

Comparing fractions with unlike denominators

On the previous folio, we compared fractions that have the same bottom numbers, or denominators . But yous know that fractions tin have any number equally a denominator. What happens when you need to compare fractions with dissimilar lesser numbers?

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

If you expect at the motion picture, though, the departure is articulate: 2/3 is larger than i/5. With an analogy, it was easy to compare these fractions, just how could nosotros have done information technology without the picture?

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

Let'southward compare these fractions: 5/8 and iv/6.
Before we compare them, nosotros need to change both fractions so they accept the same denominator, or bottom number.
First, we'll observe the smallest number that can be divided by both denominators. We call that the everyman common denominator.
Our first footstep is to find numbers that can be divided evenly by 8.
Using a multiplication table makes this easy. All of the numbers on the 8 row can exist divided evenly by 8.
Now let'southward expect at our second denominator: six.
We can use the multiplication table again. All of the numbers in the 6 row can exist divided evenly by vi.
Let's compare the two rows. It looks like in that location are a few numbers that can exist divided evenly by both six and eight.
24 is the smallest number that appears on both rows, and then it'due south the everyman common denominator.
Now we're going to change our fractions so they both accept the aforementioned denominator: 24.
To practise that, nosotros'll have to change the numerators the aforementioned way nosotros changed the denominators.
Let'due south look at 5/8 again. In order to modify the denominator to 24...
Let'southward look at five/8 again. In club to modify the denominator to 24...nosotros had to multiply 8 by three.
Since we multiplied the denominator by 3, we'll likewise multiply the numerator, or top number, by iii.
v times three equals 15. So we've changed 5/viii into 15/24.
We can practise that considering whatsoever number over itself is equal to 1.
And then when we multiply 5/viii by three/3...
So when we multiply five/8 by 3/3...we're actually multiplying 5/8 by 1.
Since any number times 1 is equal to itself...
Since any number times ane is equal to itself...nosotros tin can say that 5/eight is equal to 15/24.
Now we'll do the same to our other fraction: iv/6. We also inverse its denominator to 24.
Our old denominator was 6. To get 24, nosotros multiplied 6 past 4.
So we'll too multiply the numerator by 4.
4 times 4 is xvi. So 4/half-dozen is equal to xvi/24.
Now that the denominators are the same, nosotros tin compare the 2 fractions past looking at their numerators.
16/24 is larger than 15/24...
16/24 is larger than 15/24... and then 4/6 is larger than 5/8.

Reducing fractions

Which of these is larger: iv/8 or ane/two?

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

If 4/viii means the same thing every bit 1/two, why not just phone call information technology that? Half is easier to say than iv-eighths, and for nearly people it's also easier to empathize. Afterwards all, when you consume out with a friend, you lot split the bill in half, non in eighths.

If you write 4/eight equally 1/2, you're reducing it. When we reduce a fraction, we're writing it in a simpler course. Reduced fractions are e'er equal to the original fraction.

We already reduced 4/8 to 1/two. If you look at the examples below, you lot tin meet that other numbers can be reduced to 1/2 too. These fractions are all equal.

5/10 = 1/2

11/22 = i/2

36/72 = ane/2

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

iv/12 = 1/3

14/21 = 2/3

35/50 = 7/10

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

Let's try reducing this fraction: 16/twenty.
Since the numerator and denominator are even numbers, you lot can divide them by 2 to reduce the fraction.
First, we'll dissever the numerator past 2. xvi divided by 2 is 8.
Next, we'll carve up the denominator by 2. 20 divided by ii is 10.
We've reduced 16/xx to eight/ten. We could as well say that xvi/20 is equal to eight/10.
If the numerator and denominator can still exist divided by two, we can keep reducing the fraction.
8 divided by 2 is iv.
10 divided by 2 is 5.
Since there's no number that four and 5 can exist divided by, nosotros tin can't reduce 4/5 whatsoever further.
This ways 4/5 is the simplest form of xvi/20.
Let's try reducing another fraction: six/9.
While the numerator is fifty-fifty, the denominator is an odd number, so we can't reduce past dividing past ii.
Instead, we'll demand to discover a number that 6 and 9 can be divided by. A multiplication table will make that number easy to discover.
Let's find six and 9 on the same row. As you lot can meet, half-dozen and 9 tin can both be divided by one and 3.
Dividing by one won't change these fractions, so nosotros'll use the largest number that 6 and 9 tin can be divided by.
That's iii. This is called the greatest mutual divisor, or GCD. (You tin can besides call information technology the greatest mutual gene, or GCF.)
3 is the GCD of 6 and ix considering information technology'southward the largest number they can be divided by.
And so we'll divide the numerator by 3. 6 divided by 3 is 2.
Then we'll carve up the denominator by three. nine divided by 3 is 3.
Now we've reduced vi/9 to two/3, which is its simplest class. Nosotros could also say that half dozen/ix is equal to 2/iii.

Irreducible fractions

Not all fractions can be reduced. Some are already as simple equally they can exist. For case, you lot can't reduce 1/2 because at that place's no number other than 1 that both 1 and two tin can be divided by. (For that reason, yous can't reduce any fraction that has a numerator of one.)

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

Endeavour This!

Reduce each fraction to its simplest form.

Mixed numbers and improper fractions

In the previous lesson, yous learned well-nigh mixed numbers. A mixed number has both a fraction and a whole number. An instance is i ii/iii. Y'all'd read ane 2/three like this: one and two-thirds.

Another way to write this would exist v/3, or v-thirds. These two numbers await different, but they're really the aforementioned. 5/3 is an improper fraction. This simply means the numerator is larger than the denominator.

In that location are times when you may prefer to apply an improper fraction instead of a mixed number. Information technology's easy to change a mixed number into an improper fraction. Let'due south learn how:

Let'due south convert ane 1/4 into an improper fraction.
Kickoff, nosotros'll need to find out how many parts make up the whole number: 1 in this example.
To do this, we'll multiply the whole number, 1, by the denominator, 4.
1 times 4 equals 4.
Now, let'south add together that number, 4, to the numerator, 1.
4 plus 1 equals 5.
The denominator stays the same.
Our improper fraction is 5/4, or five-fourths. So we could say that one ane/four is equal to five/4.
This means there are v i/4s in 1 1/4.
Let's convert another mixed number: 2 2/5.
First, we'll multiply the whole number by the denominator. 2 times v equals 10.
Adjacent, we'll add 10 to the numerator. x plus ii equals 12.
Equally ever, the denominator will stay the same.
So ii 2/five is equal to 12/five.

Attempt This!

Try converting these mixed numbers into improper fractions.

Converting improper fractions into mixed numbers

Improper fractions are useful for math issues that use fractions, as you'll acquire later. However, they're besides more hard to read and understand than mixed numbers. For instance, it's a lot easier to picture ii 4/7 in your head than eighteen/7.

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

Allow's turn 10/4 into a mixed number.
You can think of any fraction as a sectionalization problem. Just treat the line between the numbers like a partitioning sign (/).
So we'll carve up the numerator, 10, by the denominator, four.
10 divided by iv equals 2...
10 divided past four equals ii... with a remainder of two.
The respond, 2, will go our whole number because 10 can be divided by 4 twice.
And the rest, two, will become the numerator of the fraction because we have 2 parts left over.
The denominator remains the same.
And then 10/4 equals 2 2/four.
Permit's try another example: 33/3.
We'll split up the numerator, 33, by the denominator, 3.
33 divided past three...
33 divided past 3... equals 11, with no remainder.
The answer, 11, volition go our whole number.
There is no remainder, so we can encounter that our improper fraction was actually a whole number. 33/iii equals 11.