Dividing Fractions

Learning Outcomes

  • 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 33 quarts of paint and you have a bucket that contains 66 quarts of paint, how many coats of paint can you paint on the walls? You divide 66 by 33 for an answer of 22 coats. There will also be times when you need to divide by a fraction. Suppose painting a closet with one coat only required 1212 quart of paint. How many coats could be painted with the 6 quarts of paint? To find the answer, you need to divide 66 by the fraction, 1212.

Divide a Fraction by a Whole Number

When you divide by a whole number, you are also multiplying by the reciprocal.  Review how to find a reciprocal here. In the painting example where you need 33 quarts of paint for a coat and have 66 quarts of paint, you can find the total number of coats that can be painted by dividing 66 by 33, 6÷3=26÷3=2. You can also multiply 66 by the reciprocal of 33, which is 1313, so the multiplication problem becomes

6113=63=26113=63=2

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.

If you have a recipe that needs to be divided in half, you can divide each ingredient by 22, or you can multiply each ingredient by 1212 to find the new amount.

If you have 3434 of a candy bar and need to divide it among 55 people, each person gets 1515 of the available candy:

15 of 34=1534=32015 of 34=1534=320

Each person gets 320320 of a whole candy bar.

If you have 3232 of a pizza left over, how can you divide what is left (the red shaded region) among 66 people fairly?

Two pizzas divided into fourths. One pizza has all four pieces shaded, and the other pizza has two of the four slices shaded. 3/2 divided by 6 is equal to 3/2 times 1/6. This is 3/2 times 1/6 equals 1/4.

Each person gets one piece, so each person gets 1414 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.

Example

Find 23÷423÷4

The same idea will work when the divisor (the number being divided) is a fraction.

Divide a Whole Number by a Fraction

Let’s use money to model 2÷142÷14. We often read 1414 as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in the image below. So we can think of 2÷142÷14 as, “How many quarters are there in two dollars?” One dollar is 44 quarters, so 22 dollars would be 88 quarters. So again, 2÷14=2141=82÷14=2141=8.

The U.S. coin called a quarter is worth one-fourth of a dollar.

A picture of a United States quarter is shown.

Let’s look at another way to model 2÷142÷14.

Example

Divide: 2÷142÷14

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Divide: 2÷132÷13

Divide: 3÷123÷12

 

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The next video shows more examples of how to divide a whole number by a fraction.

Example

Divide. 9÷129÷12

Divide a Fraction by a Fraction

Sometimes you need to solve a problem that requires dividing a fraction by a fraction. Suppose we want to find the quotient: 12÷1612÷16. We need to figure out how many 16s16s there are in 1212. We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown below. Notice, there are three 1616 tiles in 1212, so 12÷16=312÷16=3.

A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.

Example

Model: 14÷1814÷18

Solution:
We want to determine how many 18s18s are in 1414. Start with one 1414 tile. Line up 1818 tiles underneath the 1414 tile.

A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth.
There are two 1818s in 1414.
So, 14÷18=214÷18=2.

Try It

Model: 13÷1613÷16

Model: 12÷1412÷14

The following video shows another way to model division of two fractions.

Using fraction tiles, we showed that 12÷16=312÷16=3. Notice that 1261=31261=3 also. How are 1616 and 6161 related? They are reciprocals. This leads us to the procedure for fraction division.  Suppose you have a pizza that is already cut into 44 slices. How many 1212 slices are there?

A pizza divided into four equal pieces. There are four slices. A pizza divided into four equal slices. Each slice is then divided in half. There are now 8 slices.

There are 88 slices. You can see that dividing 44 by 1212 gives the same result as multiplying 44 by 22.

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

A pizza divided into four equal slice. Each slice is divided into thirds. There are now 12 slices.

You would have 1212 slices, which is the same as multiplying 44 by 33.

Fraction Division

If a,b,c, and da,b,c, and d are numbers where b0,c0, and d0b0,c0, and d0, then

ab÷cd=abdcab÷cd=abdc

To divide fractions, multiply the first fraction by the reciprocal of the second.

We need to say b0,c0 and d0b0,c0 and d0 to be sure we don’t divide by zero.

Tip:  Here’s a rhyme to help you with dividing fractions.  When dividing fractions don’t ask why, just flip the second and multiply.

Dividing with Fractions

  1. Find the reciprocal of the divisor (the number that follows the division symbol).
  2. Multiply the dividend (the number before the division symbol) by the reciprocal of the divisor (the number 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.

Example

Divide 23÷1623÷16

Example

Divide 35÷2335÷23

Watch this video for more examples of dividing fractions using a reciprocal.

When solving a division problem by multiplying by the reciprocal, remember to write all whole numbers and mixed numbers as improper fractions before doing calculations  (i.e. 5=51(i.e. 5=51  and  134=74)134=74).  You can review how to convert mixed numbers to improper fractions here. The final answer should always be simplified and written as a mixed number if larger than 11.

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Example

Divide, and write the answer in simplified form: 25÷(37)25÷(37)

Example

Divide, and write the answer in simplified form: 34÷(78)34÷(78)

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The following video shows more examples of dividing fractions that are negative.

Example

Divide, and write the answer in simplified form: 23÷n523÷n5

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