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Lesson 4: Multiplying and Dividing Fractions

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Multiplying fractions

A fraction is a part of a whole. In the last lesson, you learned how to add together and subtract fractions. But that'south not the only kind of math yous can do with fractions. There are times when it will be useful to multiply fractions too.

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

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  Let's prepare upward a multiplication instance with fractions. Suppose you beverage 2/4 of a pot of coffee every morning.

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  Simply your dr. simply told you that you demand to cut downwardly your coffee drinking by one-half.

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  At present you need to figure out how much 1/2 of 2/iv of a pot of coffee is.

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  This may not await like a multiplication problem. Just when you see the word of with fractions, it means you need to multiply.

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  To set up the instance, nosotros'll just supervene upon the word of with a multiplication sign.

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  At present our instance is ready to be solved.

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  Unlike regular multiplication, which gives yous a larger number...

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  Different regular multiplication, which gives you a larger number...multiplying fractions volition usually give you a smaller number.

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  So when we multiply 1/two times ii/4...

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  So when nosotros multiply i/two times 2/4...our answer will exist smaller than 2/4.

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  Here's some other example. Let'due south say you lot have three/5 of a cup of chocolate filling.

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  You lot desire to put an equal amount of filling in each of these iv cupcakes.

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  Yous could say that y'all want to put one/4 of iii/5 of a cup of filling in each cupcake.

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  Just like we did before, we'll change the word of into a multiplication sign.

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  And now our fractions are prepare to be multiplied.

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Attempt This!

Effort setting upward the multiplication problem below. Don't worry nearly solving it yet!

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

Annotation: Although our case says the correct answer is 2/3 x one/ii, remember, with multiplying order does not matter. i/2 x two/iii would too be correct.

Solving multiplication bug with fractions

Now that we know how to set upward multiplication bug with fractions, let'south practice solving a few. If you experience comfortable multiplying whole numbers, you're ready to multiply fractions.

Click through slideshow to learn how to multiply two fractions.

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  Let's multiply to find i/2 of 7/10.

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  Only like nosotros did earlier, nosotros'll supercede the word of with a multiplication sign. Now we're ready to multiply.

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  Outset, we'll multiply the numerators: ane and 7.

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  i times 7 equals 7, so we'll write 7 to the correct of the numerators.

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  When we added fractions, the denominators stayed the same. But when we multiply, the denominators get multiplied as well.

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  two times x equals 20, so we'll write 20 to the right of the denominators.

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  Now we know one/2 times vii/10 equals 7/twenty.

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  We could likewise say i/two of vii/10 is 7/xx.

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  Let's endeavor another example: 3/5 times 2/3.

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  Get-go, nosotros'll multiply our numerators. 3 times 2 equals half-dozen.

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  Side by side, nosotros'll multiply our denominators. v times 3 equals xv.

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  And then 3/five times 2/3 equals half-dozen/15.

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Try 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 2 fractions. In that location'due south simply one extra footstep: Before you can multiply, y'all'll need to turn the whole number into a fraction. This slideshow will show yous how to practice it.

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

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  Let's multiply two times one/3. Think, this is simply some other way of asking, "What's 1/three of 2?"

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  Before we start, we need to brand certain these numbers are prepare to be multiplied.

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  We can't multiply a whole number and a fraction, so we're going to take to write 2 as a fraction.

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  Every bit you learned in Introduction to Fractions, nosotros tin can as well write 2 as 2/ane.That'south considering 2 can be divided by 1 twice.

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  Now we're ready to multiply!

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  First, we'll multiply the numerators: ii and 1.

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  2 times 1 equals 2. We'll line the 2 upwards with the numerators.

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  Next, nosotros'll multiply the denominators: 1 and iii.

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  1 times 3 equals 3. We'll line the 3 upwardly with the denominators.

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  And then two/1 times 1/three equals two/three. We could as well say 1/3 of 2 is 2/iii.

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  Let'south endeavor another instance: four times i/five.

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  Nosotros'll have to write 4 as a fraction before we start.

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  We'll rewrite 4 equally 4/1. Now we're set up to multiply.

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  First, we'll multiply the numerators: 4 and 1.

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  4 times i equals 4, so the numerator of our answer is 4.

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  Next, nosotros'll multiply the denominators: 1 and 5.

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  i times 5 equals 5, and then five is the denominator of our answer.

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  And so four/1 times 1/5 equals 4/5.

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Try This!

Try solving the multiplication problems below.

Dividing fractions

Over the last few pages, you've learned how to multiply fractions. Y'all might accept guessed that you lot tin divide fractions also. Yous split fractions to come across how many parts of something are in something else. For instance, if you wanted to know how many fourths of an inch are in four inches, you could split up 4 by 1/iv.

Allow's try another case. Imagine a recipe calls for 3 cups of flour, but your measuring loving cup simply holds 1/3, or ane-third, of a loving cup. How many thirds of a cup should you add?

We'll need to find out how many thirds of a loving cup are in three cups. In other words, we'll demand to separate three by one-tertiary.

We'd write the problem like this:

3 ÷ 1/iii

Try This!

Try setting up these partition problems with fractions. Don't worry almost solving them yet!

A recipe calls for 3/4 of a cup of water. You only take a i/8 measuring cup.

Solving division issues with fractions

Now that nosotros know how to write sectionalisation problems, let's practice by solving a few. Dividing fractions is a lot like multiplying. Information technology merely requires one actress step. If you tin multiply fractions, you can divide them too!

Click through the slideshow to acquire how to divide a whole number by a fraction.

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  Allow's divide 3 past 1/3. Remember, this is just another style to ask, "How many thirds are in three?"

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  In our lesson on division, you learned how to write the segmentation sign like this (/).

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  When dividing fractions, it will help to employ the other symbol for sectionalization ( ÷ ) so we don't mistake it for a fraction.

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  Just similar multiplication, we'll start past looking for whatever whole numbers in our problem. There's i: 3.

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  Remember, three is the aforementioned thing equally 3/ane.

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  Earlier we can divide, we need to brand one more change.

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  We'll switch the numerator and the denominator of the fraction nosotros're dividing by: 1/3 in this case.

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  Then 1/3 becomes three/1.

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  This is called finding the reciprocal, or multiplicative inverse, of the fraction.

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  Since nosotros're switching our original fraction, we'll also switch the division sign ( ÷ ) to a multiplication sign ( 10 ).

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  That'due south because multiplication is the inverse of division.

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  Now nosotros tin can care for this similar a regular multiplication problem.

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  First, we'll multiply the numerators: three and 3.

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  iii times 3 equals 9, and then we'll write that side by side to the numerators.

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  Side by side, we'll multiply the denominators: ane and one.

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  1 times i equals 1, so we'll write i next to the denominator.
  

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  Equally you can see, 3/one x 1/three = 9/1.

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  Remember, any fraction over 1 can also exist expressed as a whole number. So ix/ane = 9.

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  3 ÷ 1/3 = ix. In other words, there are 9 thirds in iii.

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  Let's try another instance: five divided by four/seven .

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  As ever, we'll rewrite any whole numbers, and so 5 becomes 5/1.

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  Next, nosotros'll discover the reciprocal of 4/seven. That'southward the fraction we're dividing past.

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  To do that, nosotros'll switch the numerator and denominator, so 4/7 becomes seven/4.

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  Then we'll change the sectionalization sign ( ÷ ) to a multiplication sign ( 10 ).

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  Now nosotros can multiply every bit we usually would. First, we'll multiply the numerators: 5 and 7.

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  5 times seven equals 35, and then we'll write that adjacent to the numerators.

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  Next, we'll multiply the denominators: i and 4.

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  1 times four equals four, so we'll write that side by side to the denominators.
  

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  So 5/1 10 4/7 = 35/4 .
  

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  As you learned before, we could catechumen our improper fraction into a mixed number to make our answer easier to read.

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  35/4 = viii 3/4. So five ÷ four/7 = 8 3/four.
  

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Try This!

Endeavor solving these partition problems. Don't worry about reducing the reply for at present.

Dividing two fractions

We just learned how to divide a whole number by a fraction. Yous can use the same method to separate two fractions.

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

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  Let's try a trouble with two fractions: 2/3 ÷ 3/iv. Here, we want to know how many 3/four are in 2/3.
  

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  First, we'll notice the reciprocal of the fraction we're dividing by: 3/4.

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  To practice that, we'll switch the numerator and denominator. So 3/iv becomes 4/3.

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  Side by side, we'll change the partitioning sign ( ÷ ) to a multiplication sign ( ten ).

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  Now we'll multiply the numerators. 2 x 4 = 8, then we'll write 8 side by side to the top numbers.

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  Next, we'll multiply the denominators. 3 x three = 9, so we'll write 9 next to the bottom numbers.
  

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  So two/3 ten 4/3 = 8/9.
  

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  We could as well write this equally 2/3 ÷ 3/four = viii/9.
  

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  Let'due south attempt another example: four/7 divided by 2/9.

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  At that place are no whole numbers, so we'll notice the reciprocal of the fraction nosotros're dividing by. That's 2/ix.

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  To do that, we'll switch the numerator and denominator. Then 2/ix becomes 9/2.

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  Now we'll change the division sign ( ÷ ) to a multiplication sign ( x ) and multiply every bit normal.

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  First, we'll multiply the numerators. four x 9 = 36.

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  Next, nosotros'll multiply the denominators. 7 x two = 14.

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  Then 4/7 x 9/2 = 36/fourteen. Just like before, y'all could convert this improper fraction into a mixed number.

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  And then iv/seven ÷ 2/9 = 2 viii/14.

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Try This!

Effort solving these segmentation problems. Don't worry about reducing the answer for at present.

Multiplying and dividing mixed numbers

How would you solve a problem like this?

As you 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. So you can multiply or divide as usual.

Using canceling to simplify problems

Sometimes you might take to solve issues like this:

Both of these fractions include large numbers. You could multiply these fractions the same fashion as whatever other fractions. However, large numbers like this can be hard to understand. Can you film 21/50, or twenty-one fiftieths, in your head?

21/50 x 25/14 = 525/700

Even the reply looks complicated. It's 525/700, or 5 hundred twenty-5 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 y'all cancel the fractions in a problem, y'all're reducing them both at the same time.

Canceling may seem complicated at outset, simply we'll show you how to practice information technology step past step. Let'southward take another look at the example we just saw.

Step 1

Starting time, look at the numerator of the start fraction and the denominator of the 2nd. Nosotros want to see if they can be divided past the same number.

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

Step 2

Side by side, we'll carve up 21 and 14 by 7. Beginning, we'll divide our height number on the left: 21.

21 ÷ 7 = 3

So we'll divide the bottom number on the correct: fourteen.

14 ÷ 7 = two

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

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

Step 3

Permit's look at the other numbers in the fraction. This time we'll expect at the denominator of the offset fraction and the numerator of the 2nd. Can they be divided by the same number?

Find they tin can both exist divided by 25! You might have as well noticed they can both exist divided past 5. We could use 5 too, but generally when you are canceling, you want to wait for the biggest number both numbers tin be divided by. This style you won't have to reduce the fraction again at the end.

Footstep iv

Adjacent, we'll cancel merely similar we did in step 2.
We'll separate our bottom number on the left: 50.

50 ÷ 25 = 2

Then we'll carve up the top number on the right: 25.

25 ÷ 25 = i

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

Step 5

At present that we've canceled the original fractions, we tin can multiply our new fractions like we normally would. Every bit e'er, multiply the numerators first:

three x 1 = iii

And so multiply the denominators:

2 x 2 = 4

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

Step 6

Finally, let'southward double cheque our work. 525/700 would have been our respond if we had solved the problem without canceling. If we divide both 525 and 700 by 175, we can see that 525/700 is equal to three/4.

We could likewise say that nosotros're reducing 525/700 to 3/4. Remember, canceling is just another manner of reducing fractions before solving a trouble. You'll get the same respond, no matter when you reduce them.

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3 1 2 Times 3,

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