What Times 7 Equals 1

Lesson four: Multiplying and Dividing Fractions

/en/fractions/adding-and-subtracting-fractions/content/

Multiplying fractions

A fraction is a part of a whole. In the last lesson, y'all learned how to add together and subtract fractions. But that's not the only kind of math yous can do with fractions. There are times when information technology will be useful to multiply fractions as well.

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

Let'due south prepare up a multiplication example with fractions. Suppose you drink 2/4 of a pot of coffee every morning.
But your doctor but told you that yous need to cut downwards your coffee drinking past half.
Now you lot demand to figure out how much 1/2 of 2/4 of a pot of coffee is.
This may not look similar a multiplication problem. Merely when you run into the discussion of with fractions, it means you need to multiply.
To set up the example, nosotros'll only supplant the give-and-take of with a multiplication sign.
Now our instance is ready to be solved.
Unlike regular multiplication, which gives you a larger number...
Unlike regular multiplication, which gives you lot a larger number...multiplying fractions will normally give y'all a smaller number.
So when we multiply 1/2 times 2/4...
So when nosotros multiply 1/two times ii/4...our answer volition be smaller than two/4.
Hither'southward another example. Let'southward say you have iii/5 of a cup of chocolate filling.
You desire to put an equal amount of filling in each of these 4 cupcakes.
Yous could say that you want to put 1/4 of 3/5 of a cup of filling in each cupcake.
Simply like we did before, we'll change the give-and-take of into a multiplication sign.
And now our fractions are prepare to be multiplied.

Attempt This!

Try setting up the multiplication problem beneath. Don't worry about solving it nonetheless!

A recipe calls for 2/3 of a cup of milk. You want to cut the recipe in one-half.

Notation: Although our example says the right answer is 2/3 x i/2, remember, with multiplying society does not thing. 1/ii x 2/iii would also be correct.

Solving multiplication issues with fractions

Now that we know how to ready up multiplication problems with fractions, permit'due south practice solving a few. If you experience comfortable multiplying whole numbers, you lot're prepare to multiply fractions.

Click through slideshow to learn how to multiply 2 fractions.

Let'southward multiply to observe 1/ii of 7/10.
Only like we did earlier, we'll replace the discussion of with a multiplication sign. Now we're ready to multiply.
First, we'll multiply the numerators: one and 7.
1 times seven equals vii, so nosotros'll write 7 to the right of the numerators.
When we added fractions, the denominators stayed the same. Only when we multiply, the denominators get multiplied as well.
2 times 10 equals twenty, and then nosotros'll write 20 to the right of the denominators.
Now we know ane/2 times 7/10 equals 7/20.
We could also say 1/2 of 7/x is seven/20.
Let'southward try another example: 3/v times ii/3.
First, we'll multiply our numerators. 3 times ii equals 6.
Next, we'll multiply our denominators. 5 times three equals 15.
Then iii/5 times 2/3 equals half dozen/fifteen.

Attempt This!

Try solving the multiplication problems below.

Multiplying a fraction and a whole number

Multiplying a fraction and a whole number is similar to multiplying two fractions. There'due south just one actress step: Before you can multiply, you lot'll need to turn the whole number into a fraction. This slideshow will bear witness y'all how to do information technology.

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

Allow's multiply 2 times 1/3. Remember, this is just another way of request, "What's 1/3 of 2?"
Before we commencement, we need to make sure these numbers are fix to be multiplied.
We tin can't multiply a whole number and a fraction, and then we're going to have to write two every bit a fraction.
As yous learned in Introduction to Fractions, we can also write 2 as 2/1.That's considering 2 can be divided by 1 twice.
At present we're fix to multiply!
First, we'll multiply the numerators: 2 and 1.
2 times 1 equals 2. We'll line the 2 up with the numerators.
Next, we'll multiply the denominators: 1 and iii.
1 times 3 equals three. We'll line the three upwardly with the denominators.
So 2/1 times 1/3 equals 2/three. Nosotros could also say 1/3 of 2 is 2/3.
Permit'south try another example: 4 times 1/v.
We'll have to write 4 every bit a fraction before we offset.
Nosotros'll rewrite 4 as 4/1. At present we're ready to multiply.
First, we'll multiply the numerators: 4 and one.
iv times 1 equals 4, then the numerator of our answer is 4.
Next, we'll multiply the denominators: 1 and 5.
1 times v equals five, and so 5 is the denominator of our answer.
Then 4/i times 1/5 equals 4/five.

Attempt This!

Attempt solving the multiplication bug below.

Dividing fractions

Over the final few pages, you've learned how to multiply fractions. You might have guessed that you lot tin can divide fractions besides. Yous divide fractions to see how many parts of something are in something else. For example, if yous wanted to know how many fourths of an inch are in iv inches, you lot could divide four by 1/4.

Permit's try some other example. Imagine a recipe calls for 3 cups of flour, but your measuring cup merely holds 1/iii, or one-3rd, of a cup. How many thirds of a cup should you lot add together?

We'll need to find out how many thirds of a cup are in three cups. In other words, we'll need to split iii by ane-third.

We'd write the problem like this:

three ÷ 1/three

Try This!

Attempt setting up these division problems with fractions. Don't worry nearly solving them yet!

A recipe calls for 3/4 of a loving cup of h2o. You simply have a 1/eight measuring cup.

Solving sectionalization issues with fractions

Now that we know how to write division bug, let's practice by solving a few. Dividing fractions is a lot like multiplying. It just requires 1 actress stride. If you can multiply fractions, yous can dissever them besides!

Click through the slideshow to learn how to divide a whole number past a fraction.

Permit's divide 3 by 1/3. Remember, this is just some other mode to ask, "How many thirds are in 3?"
In our lesson on segmentation, you learned how to write the division sign like this (/).
When dividing fractions, it will aid to use the other symbol for partitioning ( ÷ ) then we don't mistake it for a fraction.
Just like multiplication, nosotros'll start by looking for any whole numbers in our trouble. There's one: 3.
Retrieve, iii is the same thing equally 3/i.
Before we tin divide, we need to brand one more change.
We'll switch the numerator and the denominator of the fraction we're dividing by: 1/3 in this example.
And so i/3 becomes 3/1.
This is called finding the reciprocal, or multiplicative changed, of the fraction.
Since we're switching our original fraction, we'll as well switch the sectionalisation sign ( ÷ ) to a multiplication sign ( x ).
That'south considering multiplication is the inverse of division.
Now we can treat this like a regular multiplication problem.
First, nosotros'll multiply the numerators: 3 and 3.
3 times iii equals 9, and then we'll write that next to the numerators.
Next, we'll multiply the denominators: 1 and 1.
1 times i equals i, then we'll write 1 adjacent to the denominator.
Equally you can run across, 3/1 ten 1/iii = 9/1.
Remember, whatever fraction over 1 can also be expressed as a whole number. And then 9/ane = 9.
3 ÷ 1/3 = 9. In other words, in that location are 9 thirds in 3.
Let'southward try some other example: 5 divided by 4/7 .
Every bit always, we'll rewrite whatever whole numbers, so five becomes five/ane.
Next, nosotros'll find the reciprocal of 4/7. That'southward the fraction we're dividing by.
To exercise that, nosotros'll switch the numerator and denominator, then 4/7 becomes seven/4.
Then we'll alter the partition sign ( ÷ ) to a multiplication sign ( x ).
Now we can multiply every bit nosotros normally would. First, we'll multiply the numerators: 5 and vii.
5 times 7 equals 35, and then we'll write that next to the numerators.
Adjacent, we'll multiply the denominators: 1 and iv.
one times 4 equals four, so we'll write that next to the denominators.
So 5/1 ten 4/7 = 35/4 .
As you learned before, we could convert our improper fraction into a mixed number to make our answer easier to read.
35/4 = 8 3/4. Then 5 ÷ 4/seven = eight 3/4.

Try This!

Endeavor solving these partitioning problems. Don't worry nearly reducing the answer for now.

Dividing two fractions

We just learned how to divide a whole number past a fraction. Y'all can use the same method to dissever ii fractions.

Click through the slideshow to learn how to dissever with ii fractions.

Let'southward effort a problem with 2 fractions: 2/iii ÷ 3/4. Here, we want to know how many 3/iv are in ii/three.
First, we'll find the reciprocal of the fraction we're dividing past: 3/4.
To practise that, we'll switch the numerator and denominator. So 3/4 becomes iv/3.
Side by side, we'll change the division sign ( ÷ ) to a multiplication sign ( x ).
Now nosotros'll multiply the numerators. two ten 4 = 8, so nosotros'll write 8 next to the top numbers.
Next, we'll multiply the denominators. three ten 3 = ix, and so nosotros'll write nine adjacent to the bottom numbers.
So 2/3 x 4/3 = eight/9.
We could also write this as two/iii ÷ three/4 = eight/nine.
Let's endeavor another example: 4/vii divided past 2/ix.
In that location are no whole numbers, then we'll detect the reciprocal of the fraction we're dividing by. That'southward ii/9.
To do that, we'll switch the numerator and denominator. So 2/9 becomes 9/two.
Now we'll alter the partitioning sign ( ÷ ) to a multiplication sign ( x ) and multiply as normal.
First, we'll multiply the numerators. 4 10 nine = 36.
Next, nosotros'll multiply the denominators. seven x 2 = 14.
And so 4/7 x ix/2 = 36/xiv. Just like before, you could convert this improper fraction into a mixed number.
Then iv/7 ÷ 2/nine = 2 8/xiv.

Try This!

Try solving these division bug. Don't worry almost reducing the respond for at present.

Multiplying and dividing mixed numbers

How would you solve a problem similar this?

As y'all learned in the previous lesson, whenever you're solving a problem with a mixed number you'll need to convert it into an improper fraction first. And then you can multiply or separate as usual.

Using canceling to simplify problems

Sometimes you lot might have to solve issues similar this:

Both of these fractions include large numbers. You could multiply these fractions the same way as whatever other fractions. Even so, large numbers like this can be hard to understand. Can you picture 21/50, or twenty-ane fiftieths, in your caput?

21/50 x 25/xiv = 525/700

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

If you don't similar working with big numbers, you can simplify a trouble like this by using a method called canceling. When you lot cancel the fractions in a problem, yous're reducing them both at the aforementioned fourth dimension.

Canceling may seem complicated at commencement, just we'll prove you how to do it step past stride. Let's take another look at the example we but saw.

Step i

First, look at the numerator of the offset fraction and the denominator of the second. We desire to see if they can be divided by the same number.

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

Footstep 2

Next, we'll separate 21 and xiv by 7. Beginning, we'll divide our top number on the left: 21.

21 ÷ seven = three

Then we'll divide the lesser number on the right: 14.

14 ÷ seven = 2

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

Our problem looks a lot simpler now, doesn't it?

Step 3

Allow'due south look at the other numbers in the fraction. This time nosotros'll look at the denominator of the first fraction and the numerator of the second. Tin can they exist divided by the same number?

Observe they can both be divided by 25! You might accept also noticed they can both be divided by v. We could apply 5 as well, but by and large when you are canceling, you desire to look for the biggest number both numbers tin be divided past. This style you won't have to reduce the fraction over again at the stop.

Step iv

Side by side, we'll abolish just similar we did in step ii.
We'll divide our bottom number on the left: l.

50 ÷ 25 = 2

So we'll split up the top number on the correct: 25.

25 ÷ 25 = 1

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

Step 5

Now that we've canceled the original fractions, nosotros can multiply our new fractions like we normally would. As ever, multiply the numerators first:

3 x one = three

Then multiply the denominators:

ii x ii = 4

And then 3/2 x 1/2 = 3/four, or three-fourths.

Step 6

Finally, permit's double bank check our work. 525/700 would have been our answer if we had solved the problem without canceling. If we divide both 525 and 700 by 175, we tin run into that 525/700 is equal to 3/4.

Nosotros could also say that we're reducing 525/700 to 3/4. Remember, canceling is just another way of reducing fractions before solving a problem. You'll become the aforementioned answer, no matter when you reduce them.

/en/fractions/converting-percentages-decimals-and-fractions/content/