6 Times What Equals 8

Lesson 4: Multiplying and Dividing Fractions

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Multiplying fractions

A fraction is a part of a whole. In the final lesson, you learned how to add and subtract fractions. But that's not the only kind of math you can practice with fractions. In that location are times when information technology will be useful to multiply fractions too.

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

Permit's gear up upward a multiplication example with fractions. Suppose you beverage two/4 of a pot of coffee every morning.
Only your medico merely told you that you lot demand to cut down your coffee drinking past one-half.
Now you need to figure out how much ane/2 of ii/4 of a pot of coffee is.
This may not expect similar a multiplication problem. Only when you see the word of with fractions, it means you need to multiply.
To set up upwards the instance, we'll merely supersede the discussion of with a multiplication sign.
At present our instance is ready to be solved.
Unlike regular multiplication, which gives y'all a larger number...
Dissimilar regular multiplication, which gives yous a larger number...multiplying fractions will normally give you a smaller number.
So when we multiply ane/ii times 2/4...
And then when nosotros multiply 1/two times 2/4...our respond will be smaller than 2/4.
Here's another example. Let's say yous have three/5 of a cup of chocolate filling.
Yous desire to put an equal amount of filling in each of these 4 cupcakes.
You could say that yous want to put 1/4 of 3/5 of a loving cup of filling in each cupcake.
Only similar nosotros did before, we'll alter the give-and-take of into a multiplication sign.
And at present our fractions are set up to exist multiplied.

Effort This!

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

A recipe calls for two/3 of a cup of milk. Y'all want to cut the recipe in half.

Note: Although our example says the correct respond is 2/3 x ane/ii, recollect, with multiplying gild does not matter. i/ii x 2/3 would also exist correct.

Solving multiplication issues with fractions

Now that nosotros know how to set up multiplication bug with fractions, permit's do solving a few. If you feel comfortable multiplying whole numbers, y'all're ready to multiply fractions.

Click through slideshow to learn how to multiply two fractions.

Let's multiply to find 1/ii of 7/ten.
Just like we did before, we'll replace the give-and-take of with a multiplication sign. Now we're gear up to multiply.
First, we'll multiply the numerators: i and 7.
ane times 7 equals 7, so we'll write 7 to the right of the numerators.
When we added fractions, the denominators stayed the aforementioned. But when we multiply, the denominators become multiplied too.
2 times 10 equals 20, and so nosotros'll write xx to the right of the denominators.
Now we know 1/2 times 7/10 equals 7/20.
We could also say i/ii of 7/ten is 7/twenty.
Let's attempt another example: 3/5 times two/3.
Commencement, nosotros'll multiply our numerators. 3 times 2 equals half dozen.
Next, we'll multiply our denominators. v times 3 equals 15.
So 3/5 times ii/three equals 6/fifteen.

Attempt This!

Try solving the multiplication problems below.

Multiplying a fraction and a whole number

Multiplying a fraction and a whole number is like to multiplying two fractions. At that place'due south but i extra step: Before yous tin can multiply, you lot'll need to plow the whole number into a fraction. This slideshow will show you how to do it.

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

Let's multiply 2 times i/3. Remember, this is simply some other way of asking, "What's 1/3 of 2?"
Before we offset, we need to make sure these numbers are ready to be multiplied.
We tin't multiply a whole number and a fraction, and then we're going to have to write 2 as a fraction.
As you learned in Introduction to Fractions, nosotros can also write 2 as 2/1.That'south because 2 tin can be divided by one twice.
Now we're fix to multiply!
Commencement, we'll multiply the numerators: ii and 1.
2 times 1 equals two. We'll line the 2 upward with the numerators.
Next, nosotros'll multiply the denominators: 1 and 3.
one times 3 equals 3. Nosotros'll line the three upwards with the denominators.
And then ii/ane times i/3 equals 2/iii. We could too say one/iii of 2 is 2/iii.
Let's try another example: 4 times ane/5.
We'll take to write 4 equally a fraction earlier we beginning.
We'll rewrite 4 as 4/1. Now we're gear up to multiply.
First, nosotros'll multiply the numerators: four and ane.
4 times 1 equals 4, so the numerator of our reply is four.
Next, we'll multiply the denominators: 1 and 5.
1 times 5 equals v, so five is the denominator of our answer.
So iv/i times 1/5 equals 4/5.

Try This!

Try solving the multiplication problems below.

Dividing fractions

Over the last few pages, you've learned how to multiply fractions. You might have guessed that you can divide fractions too. You divide fractions to come across how many parts of something are in something else. For example, if you wanted to know how many fourths of an inch are in four inches, yous could separate four past 1/iv.

Let's try some other example. Imagine a recipe calls for 3 cups of flour, but your measuring loving cup only holds 1/3, or i-third, of a cup. How many thirds of a loving cup should you lot add?

We'll need to observe out how many thirds of a cup are in iii cups. In other words, we'll need to carve up three past one-3rd.

We'd write the problem similar this:

3 ÷ 1/3

Try This!

Try setting up these division issues with fractions. Don't worry nigh solving them all the same!

A recipe calls for three/4 of a cup of water. Y'all simply have a 1/8 measuring cup.

Solving sectionalisation bug with fractions

Now that nosotros know how to write division issues, let'south exercise past solving a few. Dividing fractions is a lot similar multiplying. Information technology just requires one extra step. If you can multiply fractions, you tin divide them too!

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

Permit's dissever 3 by ane/3. Remember, this is just some other style to inquire, "How many thirds are in 3?"
In our lesson on division, you learned how to write the division sign like this (/).
When dividing fractions, information technology will help to use the other symbol for division ( ÷ ) so we don't mistake information technology for a fraction.
Just like multiplication, nosotros'll start past looking for whatsoever whole numbers in our problem. In that location'south i: 3.
Call up, 3 is the same thing as 3/i.
Before nosotros tin divide, we need to brand 1 more than alter.
We'll switch the numerator and the denominator of the fraction we're dividing by: 1/three in this example.
So 1/3 becomes 3/1.
This is called finding the reciprocal, or multiplicative inverse, of the fraction.
Since nosotros're switching our original fraction, nosotros'll also switch the sectionalisation sign ( ÷ ) to a multiplication sign ( x ).
That'southward because multiplication is the inverse of division.
Now we can care for this like a regular multiplication problem.
First, we'll multiply the numerators: iii and 3.
3 times 3 equals 9, so we'll write that next to the numerators.
Next, we'll multiply the denominators: 1 and 1.
one times 1 equals 1, so we'll write 1 side by side to the denominator.
Equally you lot can see, 3/1 x 1/3 = 9/1.
Think, any fraction over 1 can also be expressed every bit a whole number. So 9/1 = 9.
3 ÷ ane/3 = 9. In other words, in that location are 9 thirds in 3.
Permit's try another example: five divided past 4/7 .
As e'er, we'll rewrite whatever whole numbers, and then 5 becomes five/1.
Next, we'll find the reciprocal of four/7. That's the fraction we're dividing past.
To do that, we'll switch the numerator and denominator, and so 4/seven becomes 7/4.
Then we'll modify the sectionalisation sign ( ÷ ) to a multiplication sign ( x ).
Now we tin multiply equally we normally would. Starting time, we'll multiply the numerators: 5 and 7.
5 times vii equals 35, and so nosotros'll write that adjacent to the numerators.
Next, nosotros'll multiply the denominators: 1 and iv.
1 times iv equals iv, so we'll write that adjacent to the denominators.
So 5/1 x 4/vii = 35/4 .
Every bit you lot learned before, we could convert our improper fraction into a mixed number to brand our respond easier to read.
35/four = viii 3/iv. So five ÷ 4/7 = 8 3/4.

Try This!

Try solving these segmentation problems. Don't worry well-nigh reducing the answer for now.

Dividing two fractions

We just learned how to divide a whole number by a fraction. You can use the same method to split 2 fractions.

Click through the slideshow to larn how to carve up with two fractions.

Allow's try a trouble with ii fractions: 2/3 ÷ 3/4. Here, nosotros want to know how many 3/4 are in 2/3.
First, we'll find the reciprocal of the fraction we're dividing by: iii/four.
To do that, we'll switch the numerator and denominator. So 3/4 becomes 4/three.
Next, nosotros'll modify the partition sign ( ÷ ) to a multiplication sign ( x ).
Now we'll multiply the numerators. 2 x 4 = 8, so we'll write 8 next to the top numbers.
Adjacent, we'll multiply the denominators. 3 10 iii = 9, so we'll write 9 next to the bottom numbers.
And then 2/3 10 4/3 = viii/9.
We could likewise write this as 2/3 ÷ three/four = 8/9.
Let'south attempt another case: 4/7 divided by 2/9.
At that place are no whole numbers, so we'll find the reciprocal of the fraction we're dividing by. That's ii/ix.
To do that, we'll switch the numerator and denominator. And then two/9 becomes 9/ii.
Now nosotros'll change the partitioning sign ( ÷ ) to a multiplication sign ( 10 ) and multiply as normal.
First, we'll multiply the numerators. 4 x 9 = 36.
Next, we'll multiply the denominators. 7 ten two = 14.
So 4/vii x 9/2 = 36/14. Only like before, you could convert this improper fraction into a mixed number.
So iv/7 ÷ ii/ix = two 8/14.

Endeavour This!

Try solving these division issues. Don't worry almost reducing the answer for now.

Multiplying and dividing mixed numbers

How would you solve a problem similar this?

As you learned in the previous lesson, whenever yous're solving a trouble with a mixed number you'll demand to catechumen it into an improper fraction offset. Then you can multiply or carve up every bit usual.

Using canceling to simplify problems

Sometimes you might take to solve issues similar this:

Both of these fractions include large numbers. You could multiply these fractions the same way as any other fractions. However, big numbers like this can be hard to understand. Can you picture 21/l, or twenty-one fiftieths, in your head?

21/l ten 25/14 = 525/700

Even the answer looks complicated. Information technology's 525/700, or v hundred twenty-five 7-hundredths. What a mouthful!

If you don't like working with large numbers, you can simplify a trouble similar this past using a method called canceling. When yous abolish the fractions in a problem, y'all're reducing them both at the same time.

Canceling may seem complicated at first, but we'll evidence y'all how to practice it step past step. Let's take another await at the example we simply saw.

Step 1

First, look at the numerator of the get-go fraction and the denominator of the second. We want to come across if they can exist divided by the same number.

In our example, it looks similar both 21 and 14 can be divided by 7.

Step 2

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

21 ÷ 7 = 3

Then we'll divide the bottom number on the right: xiv.

fourteen ÷ seven = 2

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

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

Pace iii

Let's expect at the other numbers in the fraction. This time we'll expect at the denominator of the first fraction and the numerator of the second. Tin can they be divided by the aforementioned number?

Find they can both be divided by 25! Y'all might have also noticed they can both exist divided by v. We could employ v too, but mostly when yous are canceling, you want to look for the biggest number both numbers can exist divided by. This way you won't have to reduce the fraction once again at the end.

Footstep iv

Adjacent, we'll abolish merely like we did in step ii.
Nosotros'll dissever our lesser number on the left: l.

50 ÷ 25 = 2

And so we'll divide the superlative number on the right: 25.

25 ÷ 25 = i

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

Step five

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

three x i = iii

Then multiply the denominators:

2 ten 2 = 4

So iii/2 x 1/ii = 3/4, or three-fourths.

Pace half dozen

Finally, let'southward double 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, nosotros can encounter that 525/700 is equal to three/4.

We could as well say that we're reducing 525/700 to iii/4. Remember, canceling is just another way of reducing fractions before solving a problem. You'll go the same respond, no affair when you reduce them.

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