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## Doubling and halving

These activities are designed to teach doubling and halving strategies.

Multiplication and division, AM (Stage 7)

## Prior knowledge

This activty involves using proportional adjustment to solve multiplication problems. Doubling and halving, and trebling and thirding can be used to make multiplication problems easier to solve. For example, 3 x 16 is the same as 6 x 8. The activity asks students to solve problems, fill in missing numbers in equations using proportional adjustment and solve word problems.

Algebraic notation The essential notation of 2 x (and 2) for doubling and /2 or ÷ 2 or ½ as notation for halving. This is then repeated with letters. This not only bridges the gap between “fill in the box” type problems and the x as an unknown number but also introduces students to such notational forms before they are expected to use them. This is essential introductory algebra to build understanding of the language of mathematics. Good discussion is warranted as a follow-up.

Students are also introduced to the concept of proof. It is likely that when students are asked to “explain why doubling and halving always gives the same answer as the original problem” many are likely to write a story. However, the concept of proof and the power of algebra can be followed up in discussions. A teacher led explanation of “what is going on when we play with the symbols using the rules of mathematics we know” should help decode the answer sheet for the problem. An explanation along the lines of “as we don’t know what numbers we actually started with – and just ended up with the same numbers, what we have shown must work for every pair of numbers we can think of…regardless of whether or not the process is actually useful!” should help explain what manipulating the symbols has shown (or proved).

Doubling and halving to find factors Doubling and halving (tripling and thirding etc) is a very useful strategy for finding a full set of factors. However, it does require some idea of prime numbers and how these operate. Start with 1 x n, and double and halve from there. For example 1 x 60 2 x 30 4 x 15 ← look for what goes into 15 20 x 3 ← 3 is a prime so this thread stops, work on the 20 10 x 6 5 x 12 ← other side is now a prime – so stop

## Comments on the Exercises

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## Objective 1

Dividing fractions, learning objectives.

- Find the reciprocal of a number
- Divide a fraction by a whole number
- Divide a fraction by a fraction

## Divide Fractions

There are times when you need to use division to solve a problem. For example, if painting one coat of paint on the walls of a room requires 3 quarts of paint and you have a bucket that contains 6 quarts of paint, how many coats of paint can you paint on the walls? You divide 6 by 3 for an answer of 2 coats. There will also be times when you need to divide by a fraction. Suppose painting a closet with one coat only required [latex] \frac{1}{2}[/latex] quart of paint. How many coats could be painted with the 6 quarts of paint? To find the answer, you need to divide 6 by the fraction, [latex] \frac{1}{2}[/latex].

Before we begin dividing fractions, let’s cover some important terminology.

- reciprocal: two fractions are reciprocals if their product is 1 (Don’t worry; we will show you examples of what this means.)
- quotient: the result of division

Dividing fractions requires using the reciprocal of a number or fraction. If you multiply two numbers together and get 1 as a result, then the two numbers are reciprocals. Here are some examples of reciprocals:

Sometimes we call the reciprocal the “flip” of the other number: flip [latex] \frac{2}{5}[/latex] to get the reciprocal [latex]\frac{5}{2}[/latex].

## Division by Zero

You know what it means to divide by 2 or divide by 10, but what does it mean to divide a quantity by 0? Is this even possible? Can you divide 0 by a number? Consider the fraction

[latex]\frac{0}{8}[/latex]

We can read it as, “zero divided by eight.” Since multiplication is the inverse of division, we could rewrite this as a multiplication problem.

[latex]\text{?}\cdot{8}=0[/latex].

We can infer that the unknown must be 0 since that is the only number that will give a result of 0 when it is multiplied by 8.

Now let’s consider the reciprocal of [latex]\frac{0}{8}[/latex] which would be [latex]\frac{8}{0}[/latex]. If we rewrite this as a multiplication problem, we will have

[latex]\text{?}\cdot{0}=8[/latex].

This doesn’t make any sense. There are no numbers that you can multiply by zero to get a result of 8. The reciprocal of [latex]\frac{8}{0}[/latex] is undefined, and in fact, all division by zero is undefined.

## Divide a Fraction by a Whole Number

When you divide by a whole number, you are multiplying by the reciprocal. In the painting example where you need 3 quarts of paint for a coat and have 6 quarts of paint, you can find the total number of coats that can be painted by dividing 6 by 3, [latex]6\div3=2[/latex]. You can also multiply 6 by the reciprocal of 3, which is [latex] \frac{1}{3}[/latex], so the multiplication problem becomes

[latex] \frac{6}{1}\cdot \frac{1}{3}=\frac{6}{3}=2[/latex].

## Dividing is Multiplying by the Reciprocal

For all division, you can turn the operation into multiplication by using the reciprocal. Dividing is the same as multiplying by the reciprocal.

The same idea will work when the divisor (the thing being divided) is a fraction. If you have [latex] \frac{3}{4}[/latex] of a candy bar and need to divide it among 5 people, each person gets [latex] \frac{1}{5}[/latex] of the available candy:

[latex] \frac{1}{5}\text{ of }\frac{3}{4}=\frac{1}{5}\cdot \frac{3}{4}=\frac{3}{20}[/latex]

Each person gets [latex]\frac{3}{20}[/latex] of a whole candy bar.

If you have a recipe that needs to be divided in half, you can divide each ingredient by 2, or you can multiply each ingredient by [latex]\frac{1}{2}[/latex] to find the new amount.

For example, dividing by 6 is the same as multiplying by the reciprocal of 6, which is [latex]\frac{1}{6}[/latex]. Look at the diagram of two pizzas below. How can you divide what is left (the red shaded region) among 6 people fairly?

Each person gets one piece, so each person gets [latex] \frac{1}{4}[/latex] of a pizza.

Dividing a fraction by a whole number is the same as multiplying by the reciprocal, so you can always use multiplication of fractions to solve division problems.

Find [latex] \frac{2}{3}\div 4[/latex].

Dividing by 4 or [latex] \frac{4}{1}[/latex] is the same as multiplying by the reciprocal of 4, which is [latex] \frac{1}{4}[/latex].

[latex] \frac{2}{3}\div 4=\frac{2}{3}\cdot \frac{1}{4}[/latex]

Multiply numerators and multiply denominators.

[latex] \frac{2\cdot 1}{3\cdot 4}=\frac{2}{12}[/latex]

Simplify to lowest terms by dividing numerator and denominator by the common factor 4.

[latex] \frac{1}{6}[/latex]

[latex]\frac{2}{3}\div4=\frac{1}{6}[/latex]

Divide. [latex] 9\div\frac{1}{2}[/latex].

Dividing by [latex]\frac{1}{2}[/latex] is the same as multiplying by the reciprocal of [latex]\frac{1}{2}[/latex], which is [latex] \frac{2}{1}[/latex].

[latex]9\div\frac{1}{2}=\frac{9}{1}\cdot\frac{2}{1}[/latex]

[latex] \frac{9\cdot 2}{1\cdot 1}=\frac{18}{1}=18[/latex]

This answer is already simplified to lowest terms.

[latex]9\div\frac{1}{2}=18[/latex]

## Divide a Fraction by a Fraction

Sometimes you need to solve a problem that requires dividing by a fraction. Suppose you have a pizza that is already cut into 4 slices. How many [latex]\frac{1}{2}[/latex] slices are there?

There are 8 slices. You can see that dividing 4 by [latex] \frac{1}{2}[/latex] gives the same result as multiplying 4 by 2.

What would happen if you needed to divide each slice into thirds?

You would have 12 slices, which is the same as multiplying 4 by 3.

## Dividing with Fractions

- Find the reciprocal of the number that follows the division symbol.
- Multiply the first number (the one before the division symbol) by the reciprocal of the second number (the one after the division symbol).

Any easy way to remember how to divide fractions is the phrase “keep, change, flip.” This means to KEEP the first number, CHANGE the division sign to multiplication, and then FLIP (use the reciprocal) of the second number.

Divide [latex] \frac{2}{3}\div \frac{1}{6}[/latex].

KEEP [latex] \frac{2}{3}[/latex]

CHANGE [latex] \div [/latex] to [latex]\cdot[/latex]

FLIP [latex]\frac{1}{6}[/latex]

[latex] \frac{2}{3}\cdot \frac{6}{1}[/latex]

[latex]\frac{2\cdot6}{3\cdot1}=\frac{12}{3}[/latex]

[latex] \frac{12}{3}=4[/latex]

[latex] \frac{2}{3}\div \frac{1}{6}=4[/latex]

Divide [latex] \frac{3}{5}\div \frac{2}{3}[/latex].

[latex] \frac{3}{5}\cdot \frac{3}{2}[/latex]

[latex] \frac{3\cdot 3}{5\cdot 2}=\frac{9}{10}[/latex]

- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
- Ex 1: Divide Fractions (Basic). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/F5YSNLel3n8 . License : CC BY: Attribution
- College Algebra. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:s7ku6WX5@2/Multiply-and-Divide-Fractions . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
- Unit 2: Fractions and Mixed Numbers, from Developmental Math: An Open Program. Provided by : Monterey Institute of Technology. Located at : http://nrocnetwork.org/dm-opentext . License : CC BY: Attribution

## Dividing Fractions

Turn the second fraction upside down, then multiply.

## There are 3 Simple Steps to Divide Fractions:

Example: 1 2 ÷ 1 6.

Step 1. Turn the second fraction upside down (it becomes a reciprocal ):

1 6 becomes 6 1

Step 2. Multiply the first fraction by that reciprocal :

(multiply tops ...)

1 2 × 6 1 = 1 × 6 2 × 1 = 6 2

(... multiply bottoms)

Step 3. Simplify the fraction:

6 2 = 3

## With Pen and Paper

And here is how to do it with a pen and paper (press the play button):

To help you remember:

♫ "Dividing fractions, as easy as pie, Flip the second fraction, then multiply. And don't forget to simplify, Before it's time to say goodbye" ♫

20 divided by 5 is asking "how many 5s in 20?" (=4) and so:

1 2 ÷ 1 6 is really asking:

how many 1 6 s in 1 2 ?

Now look at the pizzas below ... how many "1/6th slices" fit into a "1/2 slice"?

So now you can see why 1 2 ÷ 1 6 = 3

In other words "I have half a pizza, if I divide it into one-sixth slices, how many slices is that?"

## Another Example: 1 8 ÷ 1 4

Step 1. Turn the second fraction upside down (the reciprocal ):

1 4 becomes 4 1

1 8 × 4 1 = 1 × 4 8 × 1 = 4 8

4 8 = 1 2

## Fractions and Whole Numbers

What about division with fractions and whole numbers?

Make the whole number a fraction, by putting it over 1.

Example: 5 is also 5 1

Then continue as before.

## Example: 2 3 ÷ 5

Make 5 into 5 1 :

2 3 ÷ 5 1

5 1 becomes 1 5

2 3 × 1 5 = 2 × 1 3 × 5 = 2 15

The fraction is already as simple as it can be.

Answer = 2 15

## Example: 3 ÷ 1 4

Make 3 into 3 1 :

3 1 ÷ 1 4

1 4 becomes 4 1

3 1 × 4 1 = 3 × 4 1 × 1 = 12 1

## And Remember ...

You can rewrite a question like "20 divided by 5" into "how many 5s in 20"

So you can also rewrite "3 divided by ¼" into "how many ¼s in 3" (=12)

## Why Turn the Fraction Upside Down?

Because dividing is the opposite of multiplying!

But for DIVISION we:

- divide by the top number
- multiply by the bottom number

## Example: dividing by 5 / 2 is the same as multiplying by 2 / 5

So instead of dividing by a fraction, it is easier to turn that fraction upside down, then do a multiply.

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## Fractions - Multiplying and Dividing Fractions

Fractions -, multiplying and dividing fractions, fractions multiplying and dividing fractions.

## Fractions: Multiplying and Dividing Fractions

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 and subtract fractions. But that’s not the only 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 learn how to write a multiplication problem with fractions.

Let's set up a multiplication example with fractions. Suppose you drink 2/4 of a pot of coffee every morning.

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

Now you need to figure out how much 1/2 of 2/4 of a pot of coffee is.

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

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

Now our example is ready to be solved.

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

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

So when we multiply 1/2 times 2/4 ...

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

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

You want to put an equal amount of filling in each of these 4 cupcakes.

You could say that you want to put 1/4 of 3/5 of a cup of filling in each cupcake.

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

And now our fractions are ready to be multiplied.

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

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

Note : Although our example says the correct answer is 2/3 x 1/2, remember, with multiplying order does not matter. 1/2 x 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 you feel comfortable multiplying whole numbers , you're ready to multiply fractions.

Click through slideshow to learn how to multiply two fractions.

Let's multiply to find 1/2 of 7/10 .

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

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

1 times 7 equals 7 , so we'll write 7 to the right of the numerators.

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

2 times 10 equals 20 , so we'll write 20 to the right of the denominators.

Now we know 1/2 times 7/10 equals 7/20 .

We could also say 1/2 of 7/10 is 7/20 .

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

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

Next, we'll multiply our denominators. 5 times 3 equals 15 .

So 3/5 times 2/3 equals 6/15 .

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's just one extra step: Before you can multiply, you'll need to turn 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 1/3 . Remember, this is just another way of asking, "What's 1/3 of 2 ?"

Before we start, we need 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 2/1 .That's because 2 can be divided by 1 twice.

Now we're ready 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 3 .

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

So 2/1 times 1/3 equals 2/3 . We could also say 1/3 of 2 is 2/3 .

Let's try another example: 4 times 1/5 .

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

We'll rewrite 4 as 4/1 . Now we're ready to multiply.

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

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

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

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

So 4/1 times 1/5 equals 4/5 .

## 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 see 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, you could divide 4 by 1/4 .

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

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

We'd write the problem like this:

3 ÷ 1/3

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

A recipe calls for 3/4 of a cup of water. You only have a 1/8 measuring cup.

## Solving division problems with fractions

Now that we know how to write division problems, let's practice by solving a few. Dividing fractions is a lot like multiplying. It just requires one extra step. If you can multiply fractions, you can divide them too!

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

Let's divide 3 by 1/3 . Remember, this is just another way to ask, "How many thirds are in 3 ?"

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

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

Just like multiplication, we'll start by looking for any whole numbers in our problem. There's one: 3 .

Remember, 3 is the same thing as 3/1 .

Before we can divide, we need to make one more change.

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

So 1/3 becomes 3/1 .

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

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

That's because multiplication is the inverse of division.

Now we can treat this like a regular multiplication problem.

First, we'll multiply the numerators: 3 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 .

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

As you can see, 3/1 x 1/3 = 9/1 .

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

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

Let's try another example: 5 divided by 4/7 .

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

Next, we'll find the reciprocal of 4/7 . That's the fraction we're dividing by.

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

Then we'll change the division sign ( ÷ ) to a multiplication sign ( x ).

Now we can multiply as we normally would. First, we'll multiply the numerators: 5 and 7 .

5 times 7 equals 35 , so we'll write that next to the numerators.

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

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

So 5/1 x 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 . So 5 ÷ 4/7 = 8 3/4 .

Try solving these division problems. Don't worry about 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 divide two fractions .

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

Let's try a problem with two fractions: 2/3 ÷ 3/4 . Here, we want to know how many 3/4 are in 2/3 .

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

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

Next, we'll change the division 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.

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

So 2/3 x 4/3 = 8/9 .

We could also write this as 2/3 ÷ 3/4 = 8/9 .

Let's try another example: 4/7 divided by 2/9 .

There are no whole numbers, so we'll find the reciprocal of the fraction we're dividing by. That's 2/9 .

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

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

First, we'll multiply the numerators. 4 x 9 = 36 .

Next, we'll multiply the denominators. 7 x 2 = 14 .

So 4/7 x 9/2 = 36/14 . Just like before, you could convert this improper fraction into a mixed number.

So 4/7 ÷ 2/9 = 2 8/14.

## 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. Then you can multiply or divide as usual.

## Using canceling to simplify problems

Sometimes you might have to solve problems like this:

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

21/50 x 25/14 = 525/700

Even the answer looks complicated. It's 525/700 , or five 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 first, but we'll show you how to do it step by step. Let's take another look at the example we just saw.

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

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

Next, we'll divide 21 and 14 by 7 . First, we'll divide our top number on the left: 21 .

21 ÷ 7 = 3

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

14 ÷ 7 = 2

We'll write the answers to each problem next to the numbers we divided. Since 21 ÷ 7 equals 3 , we'll write 3 where the 21 was. 14 ÷ 7 equals 2 , so we'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?

Let's look at the other numbers in the fraction. This time we'll look at the denominator of the first fraction and the numerator of the second. Can they be divided by the same number?

Notice they can both be divided by 25 ! You might have also noticed they can both be divided by 5 . We could use 5 too, but generally when you are canceling, you want to look for the biggest number both numbers can be divided by. This way you won't have to reduce the fraction again at the end.

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 right: 25 .

25 ÷ 25 = 1

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

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

Then multiply the denominators:

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

Finally, let'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 , we can see that 525/700 is equal to 3/4 .

We 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 get the same answer, no matter when you reduce them.

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## Course: 5th grade > Unit 7

- Fraction division in context
- Fraction and whole number division in contexts
- Dividing fractions and whole number word problems

## Divide fractions and whole numbers word problems

- Divide fractions: FAQ
- Your answer should be
- an integer, like 6
- a simplified proper fraction, like 3 / 5
- a simplified improper fraction, like 7 / 4
- a mixed number, like 1 3 / 4
- an exact decimal, like 0.75
- a multiple of pi, like 12 pi or 2 / 3 pi

## Exploring Fractions

Age 5 to 11.

Published 2013 Revised 2019

- The first group gives you some starting points to explore with your class, which are applicable to a wide range of ages. The tasks in this first group will build on children's current understanding of fractions and will help them get to grips with the concept of the part-whole relationship.
- The second group of tasks focuses on the progression of ideas associated with fractions, through a problem-solving lens. So, the tasks in this second group are curriculum-linked but crucially also offer opportunities for learners to develop their problem-solving and reasoning skills.

- are applicable to a range of ages;
- provide contexts in which to explore the part-whole relationship in depth;
- offer opportunities to develop conceptual understanding through talk.

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## Problems on Division of Fractional Numbers | Division of Fractions Word Problems with Answers

Problems on Divison of Fractional Numbers are provided with the various types of problems. Follow this complete concept and learn more about the division of the Fractional Numbers topic. Here on this page, we will explain different methods which are given to solve a single problem. So, let us check out different problems with explanations for the division of Fractional Number Problems.

Also, Check:

- Problems on Multiplication of Fractional Numbers
- Properties of Multiplication of Fractional Numbers

## Fraction – Definition

A fractional number is nothing but a section, portion, or part of any given quantity. Fractions are usually represented in the form of \(\frac { m }{ n } \), where m is called a numerator, and n is known as a denominator.

## How to Divide Fractional Numbers?

We have to convert the given second fraction into its reciprocal and then multiply it with the given first fraction. Next, we just need to simplify the fraction to its lowest terms.

## Problems on Division of Fractional Numbers

All the below-mentioned solved problems on dividing fractions numbers will help you to get every piece of detailed information and also helps you to score better marks in the exam. So let’s see few problems.

## Division of a Fraction with a Whole Number

Solve the equation dividing a faction number \(\frac { 6 }{ 5 } \) with a whole number 10

First, we need to convert our given whole number 10 into a fractional number by simply just adding 1 as its denominator. which gives \(\frac { 10 }{ 1 } \)

Now we need to find the reciprocal of \(\frac { 10 }{ 1 } \) which gives \(\frac { 1 }{ 10 } \)

Now we have to multiply both fractions \(\frac { 6 }{ 5 } * \frac { 1 }{ 10 } \)

As we already know we can simplify this by multiplying numerators and denominators with each other \(\frac { 6 * 1 }{ 5 * 10 } \)

Which gives \(\frac { 30 }{ 10 } \).

The result of dividing a facrtion \(\frac { 6 }{ 5 } \) with a whole number 10 is \(\frac { 30 }{ 10 } \).

Fractional number \(\frac { 30 }{ 10 } \) can be simplifed into lowest terms as \(\frac { 3 }{ 1 } \) since both these integers can be divided by 2.

Solve the equation dividing a faction number \(\frac { 2 }{ 4 } \) with a whole number 6

First, we need to convert our given whole number 6 into a fractional number by simply just adding 1 as its denominator. which gives \(\frac { 6 }{ 1 } \)

Now we need to find the reciprocal of \(\frac { 6 }{ 1 } \) which gives \(\frac { 1 }{ 6 } \)

Now we have to multiply both fractions \(\frac { 2 }{ 4 } * \frac { 1 }{ 6 } \)

As we already know we can simplify this by multiplying numerators and denominators with each other \(\frac { 2 * 1 }{ 4 * 6 } \)

Which gives \(\frac { 2 }{ 24 } \).

The result of dividing a facrtion \(\frac { 2 }{ 4 } \) with a whole number 6 is \(\frac { 2 }{ 24 } \).

Fractional number \(\frac { 2 }{ 24 } \) can be simplifed into lowest terms as \(\frac { 1 }{ 12 } \) since both these integers can be divided by 2.

Answer: \(\frac { 1 }{ 12 } \)

## Division of a Whole Number with a Fractional Number.

Solve the equation dividing a whole number 5 with a factional number \(\frac { 3 }{ 15 } \)

First, we need to convert our given whole number 5 into a fractional number by simply just adding 1 as its denominator. which gives \(\frac { 5 }{ 1 } \)

Now we need to find the reciprocal of \(\frac { 5 }{ 1 } \) which gives \(\frac { 1 }{ 5 } \)

Now we have to multiply both fractions \(\frac { 1 }{ 5 } * \frac { 3 }{ 15 } \)

As we already know we can simplify this by multiplying numerators and denominators with each other \(\frac { 1 * 3 }{ 5 * 15 } \)

Which gives \(\frac { 3 }{ 75 } \)

The result of dividing a whole number 5 with a fractional number \(\frac { 3 }{ 15 } \) is \(\frac { 3 }{ 75 } \).

Fractional number \(\frac { 3 }{ 75 } \) can be simplifed into lowest terms as \(\frac { 1 }{ 25 } \) since both these integers can be divided by 3.

Answer: \(\frac { 1 }{ 25 } \)

Solve the equation dividing a whole number 2 with a factional number \(\frac { 5 }{ 4 } \)

First, we need to convert our given whole number 2 into a fractional number by simply just adding 1 as its denominator. which gives \(\frac { 2 }{ 1 } \)

Now we need to find the reciprocal of \(\frac { 2 }{ 1 } \) which gives \(\frac { 1 }{ 2 } \)

Now we have to multiply both fractions \(\frac { 1 }{ 2 } * \frac { 5 }{ 4 } \)

As we already know we can simplify this by multiplying numerators and denominators with each other \(\frac { 1 * 5 }{ 2 * 4 } \)

Which gives \(\frac { 5 }{ 6 } \)

The result of dividing a whole number 2 with a fractional number \(\frac { 5 }{ 4 } \) is \(\frac { 5 }{ 6 } \)

The answer remains the same since 5 and 6 do not have common factorials so it can be simplified further.

Answer: \(\frac { 5 }{ 6 } \).

## Dividing a Fractional Number with another Fractional Number

Solve the equation dividing these factional number \(\frac { 5 }{ 4 } \) and \(\frac { 2 }{ 3 } \).

First, we need to find the reciprocal of the second fractional number \(\frac { 2 }{ 3 } \) which gives \(\frac { 3 }{ 2 } \)

Now we have to multiply both fractions \(\frac { 5 }{ 4 } * \frac { 3 }{ 2 } \)

As we already know we can simplify this by multiplying numerators and denominators with each other \(\frac { 5 * 3 }{ 4 * 2 } \)

Which gives \(\frac { 15 }{ 8 } \)

The result of dividing a fractional number \(\frac { 5 }{ 4 } \)with another fractional number \(\frac { 2 }{ 3 } \) is \(\frac { 15 }{ 8 } \)

The answer remains the same since 15 and 8 do not have common factorials so it can be simplified further.

Answer: \(\frac { 15 }{ 8 } \).

Solve the equation dividing these factional number \(\frac { 9 }{ 4 } \) and \(\frac { 2 }{ 3 } \).

Now we have to multiply both fractions \(\frac { 9 }{ 4 } * \frac { 3 }{ 2 } \)

As we already know we can simplify this by multiplying numerators and denominators with each other \(\frac { 9 * 3 }{ 4 * 2 } \)

Which gives \(\frac { 27 }{ 8 } \)

The result of dividing a fractional number \(\frac { 9 }{ 4 } \)with another fractional number \(\frac { 2 }{ 3 } \) is \(\frac { 27 }{ 8 } \)

The answer remains the same since 27 and 8 do not have common factorials so it can be simplified further.

Answer: \(\frac { 27 }{ 8 } \)

## Division of a Whole Number with a Mixed Fractional Number

Solve the equation dividing a whole number 4 with a mixed factional number 2\(\frac { 9 }{ 13 } \)

First, we need to convert our given whole number 4 into a fractional number by simply just adding 1 as its denominator. which gives \(\frac { 4 }{ 1 } \)

Now we need to find the reciprocal of \(\frac { 4 }{ 1 } \) which gives \(\frac { 1 }{ 4 } \)

We have to convert the given mixed fractional number into the simple fractional number 2\(\frac { 9 }{ 13 } \) becomes \(\frac { 35 }{ 13 } \)

Now we have to multiply both fractions \(\frac { 1 }{ 4 } * \frac { 35 }{ 13 } \)

As we already know we can simplify this by multiplying numerators and denominators with each other \(\frac { 1 * 35 }{ 4 * 13 } \)

Which gives \(\frac { 35 }{ 52 } \)

The result of dividing a whole number 2 with a fractional number 2\(\frac { 9 }{ 13 } \) is \(\frac { 35 }{ 52 } \)

The answer remains the same since 35 and 52 do not have common factorials so it can be simplified further.

Answer: \(\frac { 35 }{ 52 } \).

Solve the equation dividing a whole number 6 with a mixed factional number 2\(\frac { 2 }{ 3 } \)

We have to convert the given mixed fractional number into the simple fractional number 2\(\frac { 2 }{ 3 } \) becomes \(\frac { 8 }{ 3 } \)

Now we have to multiply both fractions \(\frac { 1 }{ 6 } * \frac { 8 }{ 3 } \)

As we already know we can simplify this by multiplying numerators and denominators with each other \(\frac { 1 * 8 }{ 6 * 3 } \)

Which gives \(\frac { 8 }{ 18 } \)

The result of dividing a whole number 6 with a fractional number 2\(\frac { 2 }{ 3 } \) is \(\frac { 8 }{ 18 } \)

Fractional number \(\frac { 8 }{ 18 } \) can be simplifed into lowest terms as \(\frac { 4 }{ 9 } \) since both these integers can be divided by 2.

Answer: \(\frac { 4 }{ 9 } \).

## Division of a Mixed Fractional Number with a Whole Number

Solve the equation dividing mixed factional number 3\(\frac { 1 }{ 4 } \) with a whole number 3.

We have to convert the given mixed fractional number into the simple fractional number 3\(\frac { 1 }{ 4 } \) becomes \(\frac { 13 }{ 4 } \)

Now to convert our given whole number 3 into a fractional number by simply just adding 1 as its denominator. which gives \(\frac { 3 }{ 1 } \)

Let us find the reciprocal of \(\frac { 3 }{ 1 } \) which gives \(\frac { 1 }{ 3 } \)

Now we have to multiply both fractions \(\frac { 13 }{ 4 } * \frac { 1 }{ 3 } \)

As we already know we can simplify this by multiplying numerators and denominators with each other \(\frac { 13 * 1 }{ 4 * 3 } \)

Which gives \(\frac { 13 }{ 3 } \)

The result of dividing a mixed fractional number 3\(\frac { 1 }{ 4 } \) with a whole number 3 is \(\frac { 13 }{ 3 } \)

The answer remains the same since 13 and 3 do not have common factorials so it can be simplified further.

Answer: \(\frac { 13 }{ 3 } \).

Problem 10:

Solve the equation dividing mixed factional number 2\(\frac { 2 }{ 3 } \) with a whole number 8.

Now to convert our given whole number 8 into a fractional number by simply just adding 1 as its denominator. which gives \(\frac { 8 }{ 1 } \)

Let us find the reciprocal of \(\frac { 8 }{ 1 } \) which gives \(\frac { 1 }{ 8 } \)

Now we have to multiply both fractions \(\frac { 8 }{ 3 } * \frac { 1 }{ 8 } \)

As we already know we can simplify this by multiplying numerators and denominators with each other \(\frac { 8 * 1 }{ 3 * 8 } \)

Which gives \(\frac { 8 }{ 24 } \)

The result of dividing a mixed fractional number 2\(\frac { 2 }{ 3 } \) with a whole number 8 is \(\frac { 8 }{ 24 } \)

Fractional number \(\frac { 8 }{ 24 } \) can be simplifed into lowest terms as \(\frac { 1 }{ 3 } \) since both these integers can be divided by 2.

Answer: \(\frac { 1 }{ 3 } \).

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Home / United States / Math Classes / 5th Grade Math / Problem Solving using Fractions

## Problem Solving using Fractions

Fractions are numbers that exist between whole numbers. We get fractions when we divide whole numbers into equal parts. Here we will learn to solve some real-life problems using fractions. ...Read More Read Less

## Table of Contents

## What are Fractions?

Types of fractions.

- Fractions with like and unlike denominators
- Operations on fractions
- Fractions can be multiplied by using
- Let’s take a look at a few examples

## Solved Examples

- Frequently Asked Questions

Equal parts of a whole or a collection of things are represented by fractions . In other words a fraction is a part or a portion of the whole. When we divide something into equal pieces, each part becomes a fraction of the whole.

For example in the given figure, one pizza represents a whole. When cut into 2 equal parts, each part is half of the whole, that can be represented by the fraction \(\frac{1}{2}\) .

Similarly, if it is divided into 4 equal parts, then each part is one fourth of the whole, that can be represented by the fraction \(\frac{1}{4}\) .

Proper fractions

A fraction in which the numerator is less than the denominator value is called a proper fraction.

For example , \(\frac{3}{4}\) , \(\frac{5}{7}\) , \(\frac{3}{8}\) are proper fractions.

Improper fractions

A fraction with the numerator higher than or equal to the denominator is called an improper fraction .

Eg \(\frac{9}{4}\) , \(\frac{8}{8}\) , \(\frac{9}{4}\) are examples of improper fractions.

Mixed fractions

A mixed number or a mixed fraction is a type of fraction which is a combination of both a whole number and a proper fraction.

We express improper fractions as mixed numbers.

For example , 5\(\frac{1}{3}\) , 1\(\frac{4}{9}\) , 13\(\frac{7}{8}\) are mixed fractions.

Unit fraction

A unit fraction is a fraction with a numerator equal to one. If a whole or a collection is divided into equal parts, then exactly 1 part of the total parts represents a unit fraction .

## Fractions with Like and Unlike Denominators

Like fractions are those in which two or more fractions have the same denominator, whereas unlike fractions are those in which the denominators of two or more fractions are different.

For example,

\(\frac{1}{4}\) and \(\frac{3}{4}\) are like fractions as they both have the same denominator, that is, 4.

\(\frac{1}{3}\) and \(\frac{1}{4}\) are unlike fractions as they both have a different denominator.

## Operations on Fractions

We can perform addition, subtraction, multiplication and division operations on fractions.

Fractions with unlike denominators can be added or subtracted using equivalent fractions. Equivalent fractions can be obtained by finding a common denominator. And a common denominator is obtained either by determining a common multiple of the denominators or by calculating the product of the denominators.

There is another method to add or subtract mixed numbers, that is, solve the fractional and whole number parts separately, and then, find their sum to get the final answer.

## Fractions can be Multiplied by Using:

Division operations on fractions can be performed using a tape diagram and area model. Also, when a fraction is divided by another fraction then we can solve it by multiplying the dividend with the reciprocal of the divisor.

## Let’s Take a Look at a Few Examples

Addition and subtraction using common denominator

( \(\frac{1}{6} ~+ ~\frac{2}{5}\) )

We apply the method of equivalent fractions. For this we need a common denominator, or a common multiple of the two denominators 6 and 5, that is, 30.

\(\frac{1}{6} ~+ ~\frac{2}{5}\)

= \(\frac{5~+~12}{30}\)

= \(\frac{17}{30}\)

( \(\frac{5}{2}~-~\frac{1}{6}\) )

= \(\frac{12~-~5}{30}\)

= \(\frac{7}{30}\)

Examples of Multiplication and Division

Multiplication:

(\(\frac{1}{6}~\times~\frac{2}{5}\))

= (\(\frac{1~\times~2}{6~\times~5}\)) [Multiplying numerator of fractions and multiplying denominator of fractions]

= \(\frac{2}{30}\)

(\(\frac{2}{5}~÷~\frac{1}{6}\))

= (\(\frac{2 ~\times~ 5}{6~\times~ 1}\)) [Multiplying dividend with the reciprocal of divisor]

= (\(\frac{2 ~\times~ 6}{5 ~\times~ 1}\))

= \(\frac{12}{5}\)

Example 1: Solve \(\frac{7}{8}\) + \(\frac{2}{3}\)

Let’s add \(\frac{7}{8}\) and \(\frac{2}{3}\) using equivalent fractions. For this we need to find a common denominator or a common multiple of the two denominators 8 and 3, which is, 24.

\(\frac{7}{8}\) + \(\frac{2}{3}\)

= \(\frac{21~+~16}{24}\)

= \(\frac{37}{24}\)

Example 2: Solve \(\frac{11}{13}\) – \(\frac{12}{17}\)

Solution:

Let’s subtract \(\frac{12}{17}\) from \(\frac{11}{13}\) using equivalent fractions. For this we need a common denominator or a common multiple of the two denominators 13 and 17, that is, 221.

\(\frac{11}{13}\) – \(\frac{12}{17}\)

= \(\frac{187~-~156}{221}\)

= \(\frac{31}{221}\)

Example 3: Solve \(\frac{15}{13} ~\times~\frac{18}{17}\)

Multiply the numerators and multiply the denominators of the 2 fractions.

\(\frac{15}{13}~\times~\frac{18}{17}\)

= \(\frac{15~~\times~18}{13~~\times~~17}\)

= \(\frac{270}{221}\)

Example 4: Solve \(\frac{25}{33}~\div~\frac{41}{45}\)

Divide by multiplying the dividend with the reciprocal of the divisor.

\(\frac{25}{33}~\div~\frac{41}{45}\)

= \(\frac{25}{33}~\times~\frac{41}{45}\) [Multiply with reciprocal of the divisor \(\frac{41}{45}\) , that is, \(\frac{45}{41}\) ]

= \(\frac{25~\times~45}{33~\times~41}\)

= \(\frac{1125}{1353}\)

Example 5:

Sam was left with \(\frac{7}{8}\) slices of chocolate cake and \(\frac{3}{7}\) slices of vanilla cake after he shared the rest with his friends. Find out the total number of slices of cake he had with him. Sam shared \(\frac{10}{11}\) slices from the total number he had with his parents. What is the number of slices he has remaining?

To find the total number of slices of cake he had after sharing we need to add the slices of each cake he had,

= \(\frac{7}{8}\) + \(\frac{3}{7}\)

= \(\frac{49~+~24}{56}\)

= \(\frac{73}{56}\)

To find out the remaining number of slices Sam has \(\frac{10}{11}\) slices need to be deducted from the total number,

= \(\frac{73}{56}~-~\frac{10}{11}\)

= \(\frac{803~-~560}{616}\)

= \(\frac{243}{616}\)

Hence, after sharing the cake with his friends, Sam has \(\frac{73}{56}\) slices of cake, and after sharing with his parents he had \(\frac{243}{616}\) slices of cake left with him.

Example 6: Tiffany squeezed oranges to make orange juice for her juice stand. She was able to get 25 ml from one orange. How many oranges does she need to squeeze to fill a jar of \(\frac{15}{8}\) liters? Each cup that she sells carries 200 ml and she sells each cup for 64 cents. How much money does she make at her juice stand?

First \(\frac{15}{8}\) l needs to be converted to milliliters.

\(\frac{15}{8}\)l into milliliters = \(\frac{15}{8}\) x 1000 = 1875 ml

To find the number of oranges, divide the total required quantity by the quantity of juice that one orange can give.

The number of oranges required for 1875 m l of juice = \(\frac{1875}{25}\) ml = 75 oranges

To find the number of cups she sells, the total quantity of juice is to be divided by the quantity of juice that 1 cup has

= \(\frac{1875}{200}~=~9\frac{3}{8}\) cups

We know that, the number of cups cannot be a fraction, it has to be a whole number. Also each cup must have 200ml. Hence with the quantity of juice she has she can sell 9 cups, \(\frac{3}{8}\) th of a cup cannot be sold alone.

Money made on selling 9 cups = 9 x 64 = 576 cents

Hence she makes 576 cents from her juice stand.

## What is a mixed fraction?

A mixed fraction is a number that has a whole number and a fractional part. It is used to represent values between whole numbers.

## How will you add fractions with unlike denominators?

When adding fractions with unlike denominators, take the common multiple of the denominators of both the fractions and then convert them into equivalent fractions.

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## Dividing Algebraic Fractions

Related Topics: More Lessons for GCSE Maths Math Worksheets

Examples, solutions, and videos to help GCSE Maths students learn how to divide algebraic fractions.

How to divide Algebraic Fractions?

- Invert the second fraction.
- Change the ÷ to × .
- Factorize the numerators and denominators.
- Cancel the factors common to both the numerator and denominator.
- Apply the multiplication to obtain the answer.

Algebraic Fractions Lesson 4 (Part 1 of 2) Dividing fractions

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## Halving and Doubling Strategy with Easier Questions (A)

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Course: 6th grade > Unit 2. Lesson 5: Dividing fractions by fractions. Understanding division of fractions. Dividing fractions: 2/5 ÷ 7/3. Dividing fractions: 3/5 ÷ 1/2. Dividing fractions. Dividing mixed numbers. Divide mixed numbers. Writing fraction division story problems.

Get Started WALT - We are learning to... 7.3 - Use Doubling and Halving or Thirding and Trebling to solve multiplication and division problems Examples: 18 x 4 = 9 (half 18) x 8 (double 4) so 18 x 4 = 9 x...

To Turn or Not to Turn. Level Four. Number and Algebra. Numeracy activities. Solve division problems that have fraction answers.

The essential notation of 2 x (and 2 ) for doubling and /2 or ÷ 2 or ½ as notation for halving. This is then repeated with letters. This not only bridges the gap between "fill in the box" type problems and the x as an unknown number but also introduces students to such notational forms before they are expected to use them.

First write five different division word problems that use whole numbers. (Try to write at least a couple each of partitive and quotative division problems.) Then change the problems so that they are fraction division problems instead. You might need to rewrite the problem a bit so that it makes sense.

The first step to dividing fractions is to find the reciprocal (reverse the numerator and denominator) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. Finally, simplify the fractions if needed. Created by Sal Khan and Monterey Institute for Technology and Education. Questions Tips & Thanks

To find the answer, you need to divide 6 by the fraction, 1 2 1 2. Before we begin dividing fractions, let's cover some important terminology. reciprocal: two fractions are reciprocals if their product is 1 (Don't worry; we will show you examples of what this means.) Dividing fractions requires using the reciprocal of a number or fraction.

4 - 0 = 4 Bring down the next number of the dividend and insert it after the 4 so you have 48. Divide 48 by the divisor, 32. The answer is 1. You can ignore the remainder for now.

There are 3 Simple Steps to Divide Fractions: Step 1. Turn the second fraction (the one you want to divide by)upside down (this is now a reciprocal). Step 2. Multiplythe first fraction by that reciprocal Step 3. Simplifythe fraction (if needed) Example: Example: 12 ÷ 16 Step 1. Turn the second fraction upside down (it becomes a reciprocal):

Try setting up the multiplication problem below. Don't worry about solving it yet! A recipe calls for 2/3 of a cup of milk. You want to cut the recipe in half. Note: Although our example says the correct answer is 2/3 x 1/2, remember, with multiplying order does not matter. 1/2 x 2/3 would also be correct.

To divide a fraction like 8/3 by another fraction like 1/3, count the jumps of 1/3 needed to reach 8/3. Alternatively, multiply 8/3 by the reciprocal of the divisor (3/1) to get the same result. This concept applies to other fractions, such as dividing 8/3 by 2/3. Created by Sal Khan. Questions Tips & Thanks Sort by: Top Voted Chris Williams

Problem solving with fractions. Problem solving is a fantastic way to put your knowledge into practise. Word problems can sometimes seem scary and more like a comprehension question in English ...

Want to know how to use arrays to divide when the numbers are larger? Check out this POST ! Step 4: Number Line to Solve Division Word Problems. Number lines have become an import

Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

Step 1: Take the reciprocal of the dividing fraction. To take the reciprocal of a fraction, we simply have to flip the fraction. That is, we swap the numerator and the denominator. Step 2: Change the division sign to multiplication. Step 3: Multiply the numerators. Step 4: Multiply the denominators. Step 5: Simplify the final fraction if possible.

Both tasks have multiple solutions, so children are encouraged to use a variety of problem-solving skills, such as trial and improvement, and working systematically. If you would like to focus on supporting children to consider how to go about solving a problem, Andy's Marbles may well fit the bill. The task involves using fractions to ...

All the below-mentioned solved problems on dividing fractions numbers will help you to get every piece of detailed information and also helps you to score better marks in the exam. So let's see few problems. Division of a Fraction with a Whole Number. Problem 1: Solve the equation dividing a faction number \(\frac { 6 }{ 5 } \) with a whole ...

When we divide something into equal pieces, each part becomes a fraction of the whole. For example in the given figure, one pizza represents a whole. When cut into 2 equal parts, each part is half of the whole, that can be represented by the fraction \ (\frac {1} {2}\). Similarly, if it is divided into 4 equal parts, then each part is one ...

Wyzant is IXL's tutoring network and features thousands of tutors who can help with math, writing, science, languages, music, hobbies, and almost anything else you can imagine. For all ages, children to adults. Improve your math knowledge with free questions in "Divide fractions and mixed numbers: word problems" and thousands of other math skills.

Invert the second fraction. Change the ÷ to × . Factorize the numerators and denominators. Cancel the factors common to both the numerator and denominator. Apply the multiplication to obtain the answer. The following diagram shows how to divide algebraic fractions. Scroll down the page for more examples and solutions.

Students can use math worksheets to master a math skill through practice, in a study group or for peer tutoring. Use the buttons below to print, open, or download the PDF version of the Halving and Doubling Strategy with Easier Questions (A) math worksheet. The size of the PDF file is 27063 bytes. Preview images of the first and second (if ...