X Divided By 4 Equals

Lesson 4: Multiplying and Dividing Fractions

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Multiplying fractions

A fraction is a part of a whole. In the final lesson, y'all learned how to add and subtract fractions. But that'southward non the but kind of math you can do with fractions. There are times when it will be useful to multiply fractions too.

Click through the slideshow to larn how to write a multiplication trouble with fractions.

Allow's fix a multiplication example with fractions. Suppose you drink ii/4 of a pot of java every morning.
But your dr. but told you that you need to cutting down your coffee drinking by one-half.
Now y'all need to figure out how much one/2 of 2/iv of a pot of coffee is.
This may not look similar a multiplication problem. But when you see the word of with fractions, it ways yous need to multiply.
To ready up the example, nosotros'll just replace the word of with a multiplication sign.
Now our case is ready to be solved.
Unlike regular multiplication, which gives you a larger number...
Different regular multiplication, which gives you a larger number...multiplying fractions will usually give you a smaller number.
So when we multiply 1/2 times 2/iv...
And then when nosotros multiply ane/2 times 2/4...our answer will exist smaller than two/4.
Hither'southward another case. Permit'south say you have 3/five of a loving cup of chocolate filling.
You lot desire to put an equal corporeality of filling in each of these four cupcakes.
You could say that you desire to put 1/4 of 3/5 of a loving cup of filling in each cupcake.
Just similar we did before, nosotros'll change the word of into a multiplication sign.
And now our fractions are ready to exist multiplied.

Effort This!

Try setting up the multiplication problem below. Don't worry almost solving it yet!

A recipe calls for ii/3 of a loving cup of milk. Yous desire to cut the recipe in one-half.

Note: Although our example says the correct answer is 2/three ten 1/ii, remember, with multiplying gild does not matter. 1/2 x 2/3 would too be correct.

Solving multiplication problems with fractions

At present that we know how to prepare up multiplication problems with fractions, let's practice solving a few. If y'all feel comfy multiplying whole numbers, y'all're ready to multiply fractions.

Click through slideshow to larn how to multiply two fractions.

Let's multiply to discover 1/2 of 7/10.
Simply similar nosotros did earlier, we'll replace the word of with a multiplication sign. Now we're ready to multiply.
First, we'll multiply the numerators: i and vii.
i times 7 equals 7, so we'll write 7 to the right of the numerators.
When we added fractions, the denominators stayed the same. Simply when we multiply, the denominators go multiplied also.
two times ten equals xx, so we'll write twenty to the right of the denominators.
Now nosotros know 1/two times seven/10 equals 7/xx.
We could likewise say ane/2 of 7/ten is 7/20.
Let's try another instance: iii/5 times 2/three.
Beginning, we'll multiply our numerators. 3 times 2 equals six.
Next, we'll multiply our denominators. v times iii equals 15.
So 3/5 times two/iii equals 6/fifteen.

Try This!

Attempt solving the multiplication issues below.

Multiplying a fraction and a whole number

Multiplying a fraction and a whole number is similar to multiplying 2 fractions. There's simply 1 extra footstep: Before you lot can multiply, you'll need to turn the whole number into a fraction. This slideshow will show yous how to exercise it.

Click through the slideshow to larn how to multiply a fraction and a whole number.

Let's multiply two times one/iii. Remember, this is just another way of request, "What's 1/iii of 2?"
Before nosotros offset, we demand to make sure these numbers are ready to be multiplied.
We can't multiply a whole number and a fraction, so we're going to have to write 2 as a fraction.
As you learned in Introduction to Fractions, we can also write 2 as 2/ane.That's because 2 tin can exist divided by 1 twice.
At present nosotros're gear up to multiply!
First, nosotros'll multiply the numerators: two and ane.
two times ane equals ii. We'll line the 2 upward with the numerators.
Next, we'll multiply the denominators: ane and three.
1 times iii equals 3. We'll line the 3 up with the denominators.
So 2/one times 1/3 equals ii/iii. We could besides say 1/iii of two is two/three.
Let's effort another example: 4 times 1/5.
We'll have to write four as a fraction before nosotros commencement.
We'll rewrite 4 as iv/1. Now we're ready to multiply.
Offset, nosotros'll multiply the numerators: 4 and 1.
four times 1 equals four, so the numerator of our answer is 4.
Next, we'll multiply the denominators: 1 and five.
1 times v equals five, so 5 is the denominator of our respond.
So iv/1 times ane/v equals 4/5.

Try This!

Endeavor solving the multiplication problems below.

Dividing fractions

Over the last few pages, you lot've learned how to multiply fractions. You might take guessed that you lot can separate fractions too. You divide fractions to run into how many parts of something are in something else. For case, if you lot wanted to know how many fourths of an inch are in 4 inches, you could divide 4 past 1/4.

Allow's try another example. Imagine a recipe calls for 3 cups of flour, but your measuring cup only holds i/3, or one-3rd, of a cup. How many thirds of a cup should you lot add?

Nosotros'll need to find out how many thirds of a cup are in 3 cups. In other words, we'll need to divide iii by one-third.

We'd write the trouble like this:

3 ÷ i/3

Try This!

Try setting upwards these partitioning problems with fractions. Don't worry nearly solving them yet!

A recipe calls for three/4 of a cup of water. You only have a one/8 measuring cup.

Solving sectionalization issues with fractions

At present that nosotros know how to write partitioning problems, let'southward practice past solving a few. Dividing fractions is a lot like multiplying. It just requires i extra step. If you can multiply fractions, you lot tin can carve up them too!

Click through the slideshow to larn how to separate a whole number by a fraction.

Allow's divide 3 past 1/3. Call up, this is just another way to enquire, "How many thirds are in 3?"
In our lesson on sectionalization, you learned how to write the division sign similar this (/).
When dividing fractions, information technology will assist to use the other symbol for division ( ÷ ) so we don't mistake it for a fraction.
Only similar multiplication, nosotros'll start by looking for any whole numbers in our trouble. In that location'south 1: 3.
Recollect, 3 is the aforementioned thing as 3/one.
Before we can separate, we need to make one more change.
We'll switch the numerator and the denominator of the fraction we're dividing past: one/iii in this example.
So 1/3 becomes 3/1.
This is called finding the reciprocal, or multiplicative changed, of the fraction.
Since nosotros're switching our original fraction, we'll also switch the partitioning sign ( ÷ ) to a multiplication sign ( x ).
That'southward because multiplication is the changed of division.
Now we can treat this like a regular multiplication trouble.
Outset, we'll multiply the numerators: 3 and 3.
3 times 3 equals ix, and then we'll write that next to the numerators.
Next, we'll multiply the denominators: 1 and 1.
ane times ane equals i, so we'll write 1 next to the denominator.
As you lot tin see, 3/one x 1/3 = 9/1.
Recollect, any fraction over 1 can also be expressed as a whole number. Then ix/1 = nine.
3 ÷ 1/3 = 9. In other words, there are ix thirds in 3.
Let'southward try some other example: 5 divided by 4/7 .
Equally ever, nosotros'll rewrite whatsoever whole numbers, and then 5 becomes 5/ane.
Next, nosotros'll find the reciprocal of 4/7. That'southward the fraction we're dividing by.
To do that, we'll switch the numerator and denominator, and so 4/7 becomes 7/4.
Then we'll alter the division sign ( ÷ ) to a multiplication sign ( 10 ).
Now we can multiply as we normally would. First, we'll multiply the numerators: 5 and seven.
5 times 7 equals 35, so we'll write that adjacent to the numerators.
Next, we'll multiply the denominators: 1 and 4.
one times 4 equals 4, so nosotros'll write that next to the denominators.
So 5/one x 4/7 = 35/4 .
As you learned before, we could catechumen our improper fraction into a mixed number to brand our answer easier to read.
35/four = viii 3/4. Then v ÷ 4/seven = eight 3/4.

Attempt This!

Try solving these division problems. Don't worry about reducing the answer for now.

Dividing two fractions

We just learned how to split up a whole number by a fraction. You can use the aforementioned method to divide two fractions.

Click through the slideshow to learn how to divide with two fractions.

Allow's try a problem with ii fractions: two/3 ÷ 3/four. Here, we want to know how many 3/4 are in ii/iii.
Beginning, nosotros'll find the reciprocal of the fraction we're dividing past: 3/4.
To practise that, we'll switch the numerator and denominator. So 3/iv becomes 4/3.
Adjacent, we'll change the division sign ( ÷ ) to a multiplication sign ( x ).
Now we'll multiply the numerators. two ten iv = viii, so we'll write 8 adjacent to the top numbers.
Side by side, we'll multiply the denominators. iii x 3 = ix, so nosotros'll write 9 side by side to the bottom numbers.
So ii/3 x 4/3 = viii/9.
Nosotros could also write this every bit ii/3 ÷ iii/iv = eight/9.
Let's endeavour another example: 4/7 divided by 2/9.
There are no whole numbers, so nosotros'll observe the reciprocal of the fraction nosotros're dividing by. That'southward 2/9.
To do that, nosotros'll switch the numerator and denominator. And so 2/9 becomes ix/2.
At present we'll change the partitioning sign ( ÷ ) to a multiplication sign ( x ) and multiply equally normal.
Start, we'll multiply the numerators. 4 x nine = 36.
Next, nosotros'll multiply the denominators. vii x 2 = 14.
So 4/seven x nine/two = 36/xiv. Just like before, you could convert this improper fraction into a mixed number.
So four/7 ÷ 2/nine = ii 8/fourteen.

Endeavour This!

Attempt solving these division problems. Don't worry most reducing the answer for now.

Multiplying and dividing mixed numbers

How would you lot solve a problem like this?

As you learned in the previous lesson, whenever you lot're solving a problem with a mixed number you'll need to catechumen it into an improper fraction first. Then you lot can multiply or divide as usual.

Using canceling to simplify problems

Sometimes you might have to solve bug similar this:

Both of these fractions include large numbers. Y'all could multiply these fractions the aforementioned mode equally any other fractions. However, large numbers like this can be hard to sympathise. Tin can you picture 21/fifty, or 20-ane fiftieths, in your caput?

21/fifty 10 25/14 = 525/700

Even the answer looks complicated. It's 525/700, or v hundred twenty-five seven-hundredths. What a mouthful!

If you don't like working with large numbers, you can simplify a problem like this by using a method called canceling. When you cancel the fractions in a problem, you're reducing them both at the same time.

Canceling may seem complicated at beginning, only nosotros'll show you how to do it step by stride. Allow'due south have another look at the instance we just saw.

Step one

Commencement, expect at the numerator of the showtime fraction and the denominator of the 2nd. We desire to run across if they can exist divided past the same number.

In our example, it looks like both 21 and fourteen can be divided by seven.

Step 2

Next, we'll divide 21 and 14 past seven. First, we'll divide our peak number on the left: 21.

21 ÷ 7 = 3

And then we'll divide the bottom number on the right: 14.

fourteen ÷ vii = 2

We'll write the answers to each trouble next to the numbers we divided. Since 21 ÷ vii equals 3, we'll write 3 where the 21 was. 14 ÷ 7 equals 2, so we'll write ii where the 14 was. We can cross out, or abolish, the numbers nosotros started with.

Our trouble looks a lot simpler now, doesn't information technology?

Step 3

Let'due south expect at the other numbers in the fraction. This fourth dimension we'll await at the denominator of the commencement fraction and the numerator of the second. Can they be divided by the same number?

Find they tin can both be divided by 25! You might take also noticed they tin both exist divided by five. We could apply 5 too, but generally when you are canceling, yous want to look for the biggest number both numbers tin exist divided by. This way y'all won't accept to reduce the fraction over again at the cease.

Pace 4

Side by side, we'll cancel just like we did in stride 2.
We'll carve up our bottom number on the left: 50.

fifty ÷ 25 = 2

And then nosotros'll divide the top number on the right: 25.

25 ÷ 25 = 1

We'll write the answers to each problem next to the numbers nosotros divided.

Step 5

Now that we've canceled the original fractions, nosotros tin multiply our new fractions similar we normally would. Every bit always, multiply the numerators offset:

3 x i = iii

Then multiply the denominators:

2 ten ii = 4

So 3/2 x 1/ii = 3/four, or three-fourths.

Pace 6

Finally, permit'southward double check our work. 525/700 would have been our answer if we had solved the trouble without canceling. If we divide both 525 and 700 past 175, nosotros tin can see that 525/700 is equal to 3/4.

We could too say that we're reducing 525/700 to 3/iv. Call up, canceling is just another way of reducing fractions before solving a problem. You'll get the same answer, no matter when you reduce them.

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