What Is 7 Times 1/2

Lesson 4: Multiplying and Dividing Fractions

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Multiplying fractions

A fraction is a office of a whole. In the last lesson, y'all learned how to add and decrease fractions. But that's not the only kind of math y'all can do with fractions. There are times when information technology volition be useful to multiply fractions too.

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

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  Permit's set a multiplication example with fractions. Suppose you beverage 2/4 of a pot of coffee every morning.

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  But your doctor just told you that you demand to cut down your coffee drinking by half.

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

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  This may non await like a multiplication problem. Only when yous see the discussion of with fractions, it means you need to multiply.

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  To set up the example, we'll just replace the word of with a multiplication sign.

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  Now our example is set up to be solved.

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  Different regular multiplication, which gives you lot a larger number...

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  Unlike regular multiplication, which gives you a larger number...multiplying fractions will unremarkably give you a smaller number.

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  And then when nosotros multiply i/ii times ii/4...

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  So when we multiply 1/ii times 2/4...our answer volition be smaller than 2/4.

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  Here'southward another example. Let'south say you have 3/5 of a cup of chocolate filling.

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  Y'all want to put an equal amount of filling in each of these 4 cupcakes.

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  You could say that y'all want to put one/four of 3/five of a cup of filling in each cupcake.

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

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  And now our fractions are fix to be multiplied.

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Attempt This!

Try setting up the multiplication problem below. Don't worry almost 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 instance says the correct answer is 2/three x ane/2, recollect, with multiplying lodge does not matter. 1/2 ten 2/3 would also be correct.

Solving multiplication problems with fractions

Now that we know how to set up multiplication problems with fractions, let's practice solving a few. If yous feel comfy multiplying whole numbers, you're set to multiply fractions.

Click through slideshow to learn how to multiply two fractions.

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  Permit'due south multiply to find i/2 of 7/10.

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  Merely like we did before, we'll replace the word of with a multiplication sign. At present we're set to multiply.

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  First, we'll multiply the numerators: 1 and 7.

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  1 times seven equals 7, and so we'll write seven to the correct of the numerators.

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  When nosotros added fractions, the denominators stayed the same. But when we multiply, the denominators become multiplied besides.

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  ii times ten equals 20, so nosotros'll write 20 to the correct of the denominators.

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  Now we know i/2 times 7/10 equals 7/twenty.

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  We could besides say one/2 of 7/10 is 7/20.

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  Permit'due south try some other case: iii/v times ii/3.

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  First, we'll multiply our numerators. 3 times 2 equals 6.

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  Next, nosotros'll multiply our denominators. 5 times three equals 15.

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  So three/5 times 2/3 equals 6/fifteen.

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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. At that place's just one actress step: Before you lot tin can multiply, you'll need to turn the whole number into a fraction. This slideshow will show you how to do information technology.

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

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  Let'south multiply ii times ane/3. Remember, this is but another way of request, "What'south ane/3 of 2?"

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  Before we starting time, nosotros need to make sure these numbers are ready to exist multiplied.

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  Nosotros can't multiply a whole number and a fraction, so we're going to have to write ii as a fraction.

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  As you learned in Introduction to Fractions, nosotros can too write ii as two/1.That'south because 2 tin can be divided by 1 twice.

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  Now we're set to multiply!

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  Beginning, we'll multiply the numerators: 2 and 1.

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

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  Next, nosotros'll multiply the denominators: ane and three.

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

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  So 2/i times ane/iii equals 2/three. We could also say i/three of 2 is ii/3.

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  Let's try another instance: iv times 1/5.

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  We'll have to write four every bit a fraction before we start.

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  We'll rewrite four as four/1. Now nosotros're ready to multiply.

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

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  4 times 1 equals 4, then the numerator of our answer is 4.

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  Next, we'll multiply the denominators: i and 5.

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  1 times 5 equals 5, so v is the denominator of our reply.

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  So iv/ane times 1/5 equals 4/five.

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Try This!

Try solving the multiplication bug beneath.

Dividing fractions

Over the last few pages, yous've learned how to multiply fractions. You might have guessed that y'all tin can dissever fractions too. You divide fractions to see 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 four inches, you could divide 4 by 1/4.

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

Nosotros'll need to discover out how many thirds of a cup are in iii cups. In other words, we'll need to carve up three by i-third.

Nosotros'd write the trouble like this:

3 ÷ i/3

Try This!

Try setting up these partitioning issues with fractions. Don't worry about solving them still!

A recipe calls for 3/4 of a cup of water. You only have a ane/8 measuring loving cup.

Solving sectionalisation problems with fractions

Now that we know how to write sectionalisation problems, let'due south practice by solving a few. Dividing fractions is a lot like multiplying. It just requires one extra step. If you can multiply fractions, yous tin can separate them also!

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

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  Allow's divide 3 by 1/3. Remember, this is but another mode to inquire, "How many thirds are in 3?"

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

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  When dividing fractions, information technology will help to use the other symbol for division ( ÷ ) so we don't fault information technology for a fraction.

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  Just like multiplication, we'll showtime by looking for any whole numbers in our problem. At that place's ane: three.

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  Recollect, iii is the same matter as 3/1.

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  Before we can separate, nosotros need to make ane more alter.

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  We'll switch the numerator and the denominator of the fraction we're dividing by: ane/3 in this example.

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  So one/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 sectionalization sign ( ÷ ) to a multiplication sign ( x ).

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  That's because multiplication is the inverse of partition.

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  At present nosotros can treat this like a regular multiplication problem.

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  Start, we'll multiply the numerators: 3 and 3.

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  3 times iii equals 9, so we'll write that adjacent to the numerators.

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  Next, we'll multiply the denominators: one and ane.

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

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  As y'all can see, 3/one x i/3 = nine/1.

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  Remember, any fraction over one can too be expressed as a whole number. So 9/ane = 9.

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  3 ÷ 1/3 = 9. In other words, at that place are ix thirds in 3.

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  Permit'due south try another case: 5 divided by iv/seven .

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  Every bit always, we'll rewrite any whole numbers, so 5 becomes five/ane.

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  Next, nosotros'll find the reciprocal of four/7. That'south the fraction we're dividing by.

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

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  And so we'll modify the division sign ( ÷ ) to a multiplication sign ( x ).

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  Now we tin multiply every bit nosotros ordinarily would. Starting time, we'll multiply the numerators: 5 and vii.

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  5 times vii equals 35, so we'll write that next to the numerators.

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  Next, we'll multiply the denominators: 1 and 4.

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  1 times 4 equals 4, and then we'll write that next to the denominators.
  

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  Then 5/one ten four/seven = 35/4 .
  

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  As you learned earlier, nosotros could convert our improper fraction into a mixed number to make our respond easier to read.

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  35/4 = 8 iii/4. So 5 ÷ 4/7 = 8 3/4.
  

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Attempt This!

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

Dividing ii fractions

We merely learned how to split up a whole number past a fraction. You can use the same method to divide two fractions.

Click through the slideshow to learn how to dissever with 2 fractions.

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  Permit'south try a trouble with two fractions: 2/3 ÷ three/4. Hither, we want to know how many three/4 are in ii/three.
  

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  Showtime, we'll find the reciprocal of the fraction nosotros're dividing by: 3/iv.

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

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  Next, we'll alter the sectionalisation sign ( ÷ ) to a multiplication sign ( ten ).

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  Now we'll multiply the numerators. ii x 4 = 8, so we'll write 8 next to the height numbers.

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  Next, we'll multiply the denominators. 3 x 3 = ix, so nosotros'll write nine next to the bottom numbers.
  

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  Then two/three x 4/3 = viii/9.
  

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  Nosotros could also write this as 2/three ÷ 3/4 = viii/9.
  

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  Let's try some other example: 4/7 divided by 2/nine.

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  There are no whole numbers, and so we'll discover the reciprocal of the fraction we're dividing past. That'southward 2/ix.

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  To do that, we'll switch the numerator and denominator. So ii/ix becomes nine/2.

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

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  Offset, we'll multiply the numerators. 4 ten nine = 36.

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  Side by side, we'll multiply the denominators. 7 ten 2 = 14.

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  So iv/seven 10 9/ii = 36/xiv. Just like before, you lot could catechumen this improper fraction into a mixed number.

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  So 4/7 ÷ 2/nine = ii viii/14.

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Try This!

Try solving these segmentation problems. Don't worry about reducing the respond for now.

Multiplying and dividing mixed numbers

How would you solve a problem similar this?

As you lot learned in the previous lesson, whenever y'all're solving a problem with a mixed number you'll demand to convert information technology into an improper fraction showtime. And then you can multiply or divide as usual.

Using canceling to simplify bug

Sometimes yous might have to solve bug similar this:

Both of these fractions include large numbers. Yous could multiply these fractions the same way every bit any other fractions. However, large numbers like this can be difficult to understand. Tin you picture 21/fifty, or twenty-one fiftieths, in your head?

21/50 x 25/14 = 525/700

Even the answer looks complicated. It'due south 525/700, or five hundred twenty-5 vii-hundredths. What a mouthful!

If y'all don't similar working with large numbers, you can simplify a problem like this by using a method chosen canceling. When you lot cancel the fractions in a problem, you're reducing them both at the same fourth dimension.

Canceling may seem complicated at offset, but we'll bear witness yous how to do it step by step. Let's accept another wait at the instance we simply saw.

Step i

First, look at the numerator of the beginning fraction and the denominator of the second. We desire to meet if they tin be divided by the same number.

In our example, it looks like both 21 and 14 tin can be divided past 7.

Pace 2

Next, nosotros'll divide 21 and 14 by 7. Starting time, we'll divide our summit number on the left: 21.

21 ÷ 7 = 3

Then nosotros'll split the bottom number on the right: fourteen.

xiv ÷ 7 = two

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

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

Step 3

Let's 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 be divided by the same number?

Find they can both exist divided by 25! You might have also noticed they tin both be divided past 5. We could utilize 5 likewise, but generally when you are canceling, you want to expect for the biggest number both numbers can be divided past. This way you won't take to reduce the fraction again at the end.

Footstep 4

Next, we'll cancel just like we did in step 2.
We'll divide our bottom number on the left: 50.

50 ÷ 25 = 2

Then we'll divide the top number on the correct: 25.

25 ÷ 25 = ane

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

Step 5

Now that we've canceled the original fractions, we can multiply our new fractions like nosotros ordinarily would. Every bit e'er, multiply the numerators start:

3 x 1 = 3

Then multiply the denominators:

2 ten two = 4

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

Stride 6

Finally, allow's 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 see that 525/700 is equal to 3/4.

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

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What Is 7 Times 1/2,

Source: https://edu.gcfglobal.org/en/fractions/multiplying-and-dividing-fractions/1/

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