# How are fractions and division related?

One way to think of a fraction is as a division that hasn't been done yet. Why do we even use fractions? Why don't we just divide the two numbers and use the decimal instead? In this day of cheap calculators, that's a very good question.

Fractions were invented long before decimal numbers, as a way of showing portions less than 1, and they're still hanging around.

They're used in cooking, in building, in sewing, in the stock market – they're everywhere, and we need to understand them.

• Just to review, the number above the bar is called the numerator, and the number below the bar is called the denominator.
• We can read this fraction as three-fourths, three over four, or three divided by four.
• Every fraction can be converted to a decimal by dividing. If you use the calculator to divide 3 by 4, you'll find that it is equal to 0.75.

Here are some other fractions and their decimal equivalents. Remember, you can find the decimal equivalent of any fraction by dividing. Here are some terms that are very important when working with fractions.

Proper fraction When the numerator is less than the denominator, we call the expression a proper fraction. These are some examples of proper fractions.

1. 2. Improper fraction An improper fraction occurs when the numerator is greater than or equal to the denominator. These are some examples of improper fractions:
3. 4. Mixed number When an expression consists of a whole number and a proper fraction, we call it a mixed number. Here are some examples of mixed numbers:
5. We can convert a mixed number to an improper fraction. First, multiply the whole number by the denominator of the fraction. Then, add the numerator of the fraction to the product. Finally, write the sum over the original denominator. In this example, since three thirds is a whole, the whole number 1 is three thirds plus one more third, which equals four thirds.

Convert 1-1/3 to an improper fraction: Equivalent fractions There are many ways to write a fraction of a whole. Fractions that represent the same number are called equivalent fractions.

This is basically the same thing as equal ratios. For example, ½, 2/4, and 4/8 are all equivalent fractions. To find out if two fractions are equivalent, use a calculator and divide.

If the answer is the same, then they are equivalent.

Reciprocal When the product of two fractions equals 1, the fractions are reciprocals. Every nonzero fraction has a reciprocal. It's easy to determine the reciprocal of a fraction since all you have to do is switch the numerator and denominator–just turn the fraction over. Here's how to find the reciprocal of three-fourths. To find the reciprocal of a whole number, just put 1 over the whole number. For example, the reciprocal of 2 is ½.

## Dividing Fractions – Definition with Examples

A fraction is part of a whole number. It has two parts – a numerator and a denominator. Dividing a fraction

Dividing a fraction by another fraction is the same as multiplying the fraction by the reciprocal (inverse) of the other. We get the reciprocal of a fraction by interchanging its numerator and denominator.

For example, the reciprocal of  25 is52 • Consider the following example:
• 1 2 ÷ 1 3
• Step 1:
• Find the reciprocal of the second fraction (divisor).
• Reciprocal of   1 3  is  3 1 or 3
• Step 2:
• Multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor).
• 13 x 31
• Step 3:
• Multiply the numerators and denominators of the fractions.
• 12 x 31 = 1x32x1 = 32
• Dividing Fractions Song
• Take the fractions for the division to apply
• Flip the second one and then multiply
• In the end, you need to simplify.
• So next time you divide the fractions don’t forget to apply
• The simple rule is to flip and multiply. 1. Dividing a Fraction by a Whole Number
2. Follow a simple rule that a number divided by 1 is the number itself.
3. Steps to divide a fraction by a whole number:
• Convert the whole number into a fraction by using a denominator 1.
• Flip this number.
• Multiply by the fraction.
• Simplify the result, if needed.

Take an example, Divide 35 by 5

• Convert 5 into a fraction  =51
• Flip 51by  get 15
• Multiply the fractions: 35 x 15 = 325
 Fun-facts  Why do we turn the fraction upside down ?  This is because the inverse operation of division is multiplication.

## Dividing Fractions

Turn the second fraction upside down, then multiply.

### There are 3 Simple Steps to Divide Fractions:

• 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

1. And here is how to do it with a pen and paper (press the play button):
3. ♫ “Dividing fractions, as easy as pie,
Flip the second fraction, then multiply.
And don't forget to simplify,

4. Before it's time to say goodbye”

 Another way to remember is: “leave me, change me, turn me over”
• 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”?

 How many in ? Answer: 3

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?”

1. Step 1. Turn the second fraction upside down (the reciprocal):
2. 1 4   becomes   4 1
3. Step 2. Multiply the first fraction by that reciprocal:
4. 1 8   ×   4 1 = 1 × 4 8 × 1 = 4 8
5. Step 3. Simplify the fraction:
6. 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.
• Then continue as before.
1. Make 5 into 5 1 :
2. 2 3   ÷   5 1
3. Then continue as before.
4. Step 1. Turn the second fraction upside down (the reciprocal):
5. 5 1 becomes 1 5
6. Step 2. Multiply the first fraction by that reciprocal:
7. 2 3 × 1 5 = 2 × 1 3 × 5 = 2 15
8. Step 3. Simplify the fraction:
9. The fraction is already as simple as it can be.
• Make 3 into 3 1 :
• 3 1   ÷   1 4
• Then continue as before.
• Step 1. Turn the second fraction upside down (the reciprocal):
• 1 4 becomes 4 1
• Step 2. Multiply the first fraction by that reciprocal:
• 3 1 × 4 1 = 3 × 4 1 × 1 = 12 1
• Step 3. Simplify the fraction:
• 12 1 = 12

## Explainer: Dividing Fractions

In this explainer, we will learn how to divide proper fractions by proper fractions.

For this purpose, we are going to start with dividing fractions by whole numbers and vice versa before going into dividing fractions by fractions. We are going to look at the two different meanings such divisions can have, and, with the help of diagrams, we will find how to compute them. This will lead us to a general method to compute divisions by fractions.

Let us first recall that a fraction compares a part to a whole and describes what we call a proportion. The denominator of the fraction is the number of equal shares (or “portions”) the whole is split into, while the numerator is the number of these shares that make the part we are considering.

Now we are going to start with a very simple example of a division of a fraction by a whole number.

We are considering the fraction 23 here; that is, a whole has been divided into three equal shares, and the part we are considering is made of two of these shares.

We now want to divide the part (the shaded area in the diagram) into three parts. By dividing each third in three parts, we have our whole divided into 9 equal shares, and we see that 23÷3 is two of them, that is, two-ninths: 23÷3=29.

Notice here how dividing a fraction is further splitting the whole; the denominator, which is the number of shares, is indeed multiplied! So, dividing a half in three is creating sixths of the whole.

Let us look at another example where we can use this.

Which of the following division expressions has a quotient of 12?

1. 78÷19
2. 23÷16
3. 12÷57
4. 38÷34
5. 56÷47

We are looking for a division whose result is one-half; in other words, we are looking for a division where the dividend is half the divisor. In all the options given, the dividends and divisors are fractions. So, we are looking for a fraction divided by another fraction that is double its value.

Let us take two examples to see how a fraction can be one-half of another. First, 25 is half of 45 as both fractions have the same denominator but the numerator of the first is half that of the second.

And, second, 16 is half of 13 since both have the same numerator but the denominator has been divided by two in the second fraction, meaning that the number of shares has been halved, so their size has been indeed doubled.

Therefore, we are looking for one of these situations here: either the denominators are the same and the numerator of the dividend is half that of the divisor ????????÷2????????, or the numerators are the same and the denominator of the dividend is double that of the divisor ????2????÷????????.

Hence, the answer is 38÷34 (option D).

In the next example, we are looking at the division of a whole number by a fraction.

Evaluate 10÷12.

We are asked here to find the result of dividing 10 by 12. This can be understood as finding how many times there is 12 in 10.

There are two halves in 1, so there will be 10 times as many in 10; that is, 10×2=20.

Hence, 10÷12=10×2=20.

Let us look now at dividing a fraction by a fraction. For instance, we want to find 1213÷313, that is, how many 313 are in 1213. We can use a diagram to help us visualize this.

We see that, in this case, because both fractions have the same denominator, this division is simply 12 divided by 3: there are 4 times 313 in 1213.

That is, we have 1213÷313=12÷3=4.

From this, we see that a strategy to divide a fraction by another fraction with a different denominator is to rewrite the fractions so that they have the same denominator and then divide the numerators. It is a good strategy, for example, to compute 78÷34 since 34 can be easily rewritten as 68, and so 78÷34=78÷68=76.

In the following, we are going to discover a general method equivalent to rewriting the fractions with the same denominator and dividing the numerators, but simpler.

So far, we have interpreted the division by a fraction as a measurement division: how many of this fraction are there in a given number?

There is another way to envision such a division. For instance, imagine that Oscar has written 12 pages for his assignment and he says that it is three-quarters of what is required. How can we find how many pages he is required to write for his assignment?

Let us use a diagram to help us.

We know that 12 is three-quarters 34 of what is required. So, if we start with a rectangle to represent what is required, we need to split it in 4 and the 12 pages already written are three of these shares.

Now, let us see how to find the number of pages required. First, we find the value of each quarter by dividing the 12 by 3. This gives 4. The number of pages required is 4 times this value (which is 4 quarters); that is, 4×4=16.

Now, we were told that 12 pages are three-quarters of what is required. We can write that as 12=34×.pagesrequired

Now, think of the complementarity between division and multiplication sentences. For instance, 2×5=10 (two groups of 5 make 10) means that 10÷2=5 (10 can be split in 2 groups of 5). Hence, we can rewrite the above equation as 12÷34=.pagesrequired

What we have found by reasoning on the diagram was nothing else as the result of 12÷34. And the stages were

• divide 12 by 3,
• multiply the result by 4.

We see that 12÷34=123×4. This equation can be rewritten as 12÷34=12×43.

We have found here a very important result: dividing by a fraction is mathematically the same as multiplying by the multiplicative inverse, or reciprocal, of the fraction (i.e., the numerator and denominator have been swapped over).

Let us look at a division of a fraction by a fraction. For instance, let us find a number knowing that four-fifths 45 of this unknown number equals 23. That is, 45×=23unknownnumber. For this, we are going to use three diagrams.

In the top diagram, 23 is represented in dark green, and the bigger rectangle represents our whole. We know that the dark green rectangle represents 45 of the unknown number. So, in as the first stage, we need to split our 23 in 4 shares of value ???? (second diagram): ????=23÷4=16.

Note that, without any diagram, we would have probably written ????=23÷4=212 and then simplified this fraction to 16. This is of course completely correct.

• The number we are looking for, called ????, is made of 5 of these shares (third diagram): ????=5×16=56.
• The unknown number is found to be five-sixths of our whole.
• Here, again, the two stages of the division by 45 were
• divide by 4,
• multiply the result by 5.

We have found that 23÷45=23×54=56.

Let us now interpret this division as a measurement division; that is, how many 45 are there in 23?

For this, we have already seen that it is helpful to rewrite the fractions with the same denominator. The lowest common multiple of 3 and 5 is 15. As 23=2×53×5=1015 and 45=4×35×3=1215, we have 23÷45=1015÷1215.

This step of rewriting the dividend and divisor with the same denominator can be visualized in the diagram.

Now that both fractions have the same denominator, finding how many 1215 there are in 1015 is simply finding how many 12s there are in 10: it is given by 10÷12=1012. This number is smaller than 1 because 12 is bigger than 10.

We have found that 23÷45=1012. Let us look back at how we got these numbers, 10 and 12, from 23÷45 when we renamed the fractions with the same denominator. The 10 was given by 2×5 and the 12 by 3×4. So, we did find that 23÷45=2×53×4;

that is, 23÷45=23×54.

We can write the general method we have found for dividing fractions by fractions.

Dividing by a fraction is equivalent to dividing by the numerator of the fraction and then multiplying by its denominator: ????÷????????=????÷????×????.

In other words, it is equivalent to multiplying by its multiplicative inverse or reciprocal. Hence, when a fraction is divided by another fraction, we can write ????????÷????????=????????×????????.

Let us look at an example where we are going to use our understanding of division by fractions.

Find 12÷23.

To find the result of dividing by a fraction, we use the fact that dividing by a fraction is equivalent to multiplying by its inverse (i.e., a fraction where the numerator and the denominator have been swapped). Hence, we have 12÷23=12⋅32.

Now, we just need to multiply the numerators together and the denominators together. We find 12÷23=34.

We have found out that one half is three-quarters of two-thirds and two-thirds of three-quarters!

Our last example is a word problem.

Natalie and Mason went out to get some ice cream. Natalie had 79 pt of chocolate chip ice cream, while Mason had 67 pt of strawberry-flavored ice cream. Determine how many times as much ice cream Mason had as Natalie.

Both Natalie and Mason have got a fraction of a pint of ice cream. We want to find how many times as much ice cream Mason had as Natalie. This is given mathematically by dividing the amount of ice cream Mason had by the amount of ice cream Natalie had, that is, 67÷79.

We know that when we divide by a fraction, this is equivalent to multiplying by its reciprocal, so 67÷79=67×97=5449.

This fraction is an improper fraction: the numerator is larger than the denominator. Hence, we will express it as a mixed number: 5449=49+549=1549.

1. A fraction compares a part to a whole. The denominator of the fraction is the number of equal shares (or “portions”) the whole is split into, while the numerator is the number of these shares that make the part we are considering.
2. If we know that a number is a fraction of another number, for instance, 12 is three quarters of an unknown number, we can write 12=34⋅unknownnumber. As we know, for instance, that 6=3⋅2 is equivalent to 6÷3=2, we can write that 12÷34=unknownnumber. Reasoning with the diagram, we find that, to find this unknown number, we need to divide 12 by 3 to find the value of one quarter and then multiply this by 4 to find the unknown number. So, we have 12÷34=12⋅43 since dividing by 3 and multiplying by 4 is the same as multiplying by 43.

## How to Convert a Division to a Fraction

••• Stockbyte/Stockbyte/Getty Images

Updated April 24, 2017

By Charlotte Johnson

Division is a mathematical process in which you calculate how many times a certain value will fit into another value. This process is the opposite of multiplication. The traditional way to write division problems is with a division bracket.

Another method of writing division calculations is to use fractions. In a fraction, the top number, or numerator, is divided by the bottom number, or denominator.

You may have to convert between traditional and fractional division forms in a high school or college math class.

Write down the dividend. This is the number that appears under the division bracket. This will be the numerator in the fraction.

Draw a dividing bar under the dividend or to the right of the dividend.

Write the divisor underneath the dividing bar or to the right of the bar. The divisor is the number to the left of the division bracket. This will be the denominator.

Write an equal sign, followed by the quotient if you are required to provide the solution to the problem. If you are looking at a completed division problem in traditional form, the quotient will appear on top of the division bracket. For example, if you have 50 divide by 5, you could write this as 50/5 = 10.

Charlotte Johnson is a musician, teacher and writer with a master's degree in education. She has contributed to a variety of websites, specializing in health, education, the arts, home and garden, animals and parenting.

## Dividing Fractions

Dividing fractions can be a little tricky. It’s the only operation that requires using the reciprocal. Using the reciprocal simply means you flip the fraction over, or invert it.

• For example, the reciprocal of 2/3 is 3/2.
• After we give you the division rule, we will show you WHY you have to use the reciprocal in the first place.
• But for now…

### Here’s the Rule for Division

To divide, convert the fraction division process to a multiplication process by using the following steps.

1. Change the “÷” (division sign) to “x” (multiplication sign) and invert the number to the right of the sign.
2. Multiply the numerators.
3. Multiply the denominators.
4. Re-write your answer in its simplified or reduced form, if needed

Once you complete Step #1 for dividing fractions, the problem actually changes from division to multiplication.

### Example 1: Dividing Fractions by Fractions

1. 1/2 ÷ 1/3 = 1/2 x 3/1
2. 1/2 x 3/1 = 3/2
3. Simplified Answer is 1 1/2

### Example 2: Dividing Fractions by Whole Numbers

• 1/2 ÷ 5 = 1/2 ÷ 5/1
• (Remember to convert
whole numbers to fractions, FIRST!)
• 1/2 ÷ 5/1 = 1/2 x 1/5
• 1/2 x 1/5 = 1/10

### Example 3: Dividing Whole Numbers by Fractions

1. 6 ÷ 1/3 = 6/1 ÷ 1/3
2. (Remember to convert
whole numbers to fractions, FIRST!)
3. 6/1 ÷ 1/3 = 6/1 x 3/1
4. 6/1 x 3/1 = 18/1 = 18

Now that’s all there is to it.

The main things you have to remember when you divide is to convert whole numbers to fractions first, then invert the fraction to the right of the division sign, and change the sign to multiplication.

The “divisor” has some other considerations you should keep in mind…

### Special Notes!

• Remember to only invert the divisor.
• The divisor’s numerator or denominator can not be “zero”.
• Convert the operation to multiplication BEFORE performing any cancellations.

I promised to try to explain why the rule requires inverting the divisor.

Here goes..

### Why Dividing Fractions Requires Inverting The Divisor

• Let’s use our simple example to actually validate this strange Rule for division.
• If you really think about it, we are dividing a fraction by a fraction, which forms what is called a “complex fraction”