We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Study Guides > Intermediate Algebra

Dividing Fractions

Learning Outcomes

  • 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 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 12\dfrac{1}{2} 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, 12\dfrac{1}{2}. Before we begin dividing fractions, let's cover some important terminology.
  • reciprocal: two fractions are reciprocals if their product is 11 (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 11 as a result, then the two numbers are reciprocals. Here are some examples of reciprocals:
Original number Reciprocal Product
34\dfrac{3}{4} 43\dfrac{4}{3} 3443=3443=1212=1\dfrac{3}{4}\cdot\dfrac{4}{3}=\dfrac{3\cdot 4}{4\cdot 3}=\dfrac{12}{12}=1
12\dfrac{1}{2} 21\dfrac{2}{1} 1221=1221=22=1\dfrac{1}{2}\cdot\dfrac{2}{1}=\dfrac{1\cdot2}{2\cdot1}=\dfrac{2}{2}=1
3=31 3=\dfrac{3}{1} 13\dfrac{1}{3} 3113=3113=33=1\dfrac{3}{1}\cdot\dfrac{1}{3}=\dfrac{3\cdot 1}{1\cdot 3}=\dfrac{3}{3}=1
213=732\dfrac{1}{3}=\dfrac{7}{3} 37\dfrac{3}{7} 7337=7337=2121=1\dfrac{7}{3}\cdot\dfrac{3}{7}=\dfrac{7\cdot3}{3\cdot7}=\dfrac{21}{21}=\normalsize 1
You can think of it as switching the numerator and denominator: swap the 22 with the 55 in 25\dfrac{2}{5} to get the reciprocal 52\dfrac{5}{2}. Make sure that if it's a negative fraction, the reciprocal is also negative. This is because the product of two negative numbers will give you the positive one that you are looking for.

Division by Zero

You know what it means to divide by 22 or divide by 1010, but what does it mean to divide a quantity by 00? Is this even possible? On the flip side, can you divide 00 by a number? Consider the fraction

08\dfrac{0}{8}

We can read it as, “zero divided by eight.” Since multiplication is the inverse of division, we could rewrite this as a multiplication problem. What number times 88 equals 00?

?8=0\text{?}\cdot{8}=0

We can infer that the unknown must be 00 since that is the only number that will give a result of 00 when it is multiplied by 88.

Now let’s consider the reciprocal of 08\dfrac{0}{8} which would be 80\dfrac{8}{0}. If we rewrite this as a multiplication problem, we will have "what times 00 equals 88?"

?0=8\text{?}\cdot{0}=8

This doesn't make any sense. There are no numbers that you can multiply by zero to get a result of 8. In fact, any number divided by 00 is impossible, or better defined, all division by zero is undefined.
CautionCaution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number a, a0\dfrac{a}{0} is undefined. Additionally, the reciprocal of 0a\dfrac{0}{a} will always be undefined.

Divide a Fraction by a Whole Number

When you divide by a whole number, you are also multiplying by the reciprocal. 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\div3=2. You can also multiply 66 by the reciprocal of 33, which is 13\dfrac{1}{3}, so the multiplication problem becomes

6113=63=2\dfrac{6}{1}\cdot\dfrac{1}{3}=\dfrac{6}{3}=\normalsize2

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 a recipe that needs to be divided in half, you can divide each ingredient by 22, or you can multiply each ingredient by 12\dfrac{1}{2} to find the new amount. If you have 34\dfrac{3}{4} of a candy bar and need to divide it among 55 people, each person gets 15\dfrac{1}{5} of the available candy:

15 of 34=1534=320\dfrac{1}{5}\normalsize\text{ of }\dfrac{3}{4}=\dfrac{1}{5}\cdot\dfrac{3}{4}=\dfrac{3}{20}

Each person gets 320\dfrac{3}{20} of a whole candy bar.

  For example, dividing by 66 is the same as multiplying by the reciprocal of 66, which is 16\dfrac{1}{6}. Look at the diagram of two pizzas below.  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 14\dfrac{1}{4} 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÷4\dfrac{2}{3}\div \normalsize 4

Answer: Write your answer in lowest terms. Dividing by 44 or 41\dfrac{4}{1} is the same as multiplying by the reciprocal of 44, which is 14\dfrac{1}{4}.

23÷4=2314\dfrac{2}{3}\normalsize\div 4=\dfrac{2}{3}\cdot\dfrac{1}{4}

Multiply numerators and multiply denominators.

2134=212\dfrac{2\cdot 1}{3\cdot 4}=\dfrac{2}{12}

Simplify to lowest terms by dividing numerator and denominator by the common factor 44.

16\dfrac{1}{6}

Answer

23÷4=16\dfrac{2}{3}\div4=\dfrac{1}{6}

Example

Divide. 9÷12 9\div\dfrac{1}{2}

Answer: Write your answer in lowest terms. Dividing by 12\dfrac{1}{2} is the same as multiplying by the reciprocal of 12\dfrac{1}{2}, which is 21\dfrac{2}{1}.

9÷12=91219\div\dfrac{1}{2}=\dfrac{9}{1}\cdot\dfrac{2}{1}

Multiply numerators and multiply denominators.

9211=181=18\dfrac{9\cdot 2}{1\cdot 1}=\dfrac{18}{1}=\normalsize 18

This answer is already simplified to lowest terms.

Answer

9÷12=189\div\dfrac{1}{2}=\normalsize 18

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 44 slices. How many 12\dfrac{1}{2} 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 12\dfrac{1}{2} 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.

Dividing with Fractions

  1. Find the reciprocal of the number that follows the division symbol.
  2. 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.

Example

Divide 23÷16\dfrac{2}{3}\div\dfrac{1}{6}

Answer: Multiply by the reciprocal. KEEP 23\dfrac{2}{3} CHANGE  ÷ \div to  \cdot FLIP  16\dfrac{1}{6}

2361\dfrac{2}{3}\cdot\dfrac{6}{1}

Multiply numerators and multiply denominators.

2631=123\dfrac{2\cdot6}{3\cdot1}=\dfrac{12}{3}

Simplify.

123=4\dfrac{12}{3}=\normalsize 4

Answer

23÷16=4\dfrac{2}{3}\div\dfrac{1}{6}=4

Example

Divide 35÷23\dfrac{3}{5}\div\dfrac{2}{3}

Answer: Multiply by the reciprocal. Keep 35\dfrac{3}{5}, change ÷ \div to \cdot, and flip 23\dfrac{2}{3}.

3532\dfrac{3}{5}\cdot\dfrac{3}{2}

Multiply numerators and multiply denominators.

3352=910\dfrac{3\cdot 3}{5\cdot 2}=\dfrac{9}{10}

Answer

35÷23=910\dfrac{3}{5}\div\dfrac{2}{3}=\dfrac{9}{10}

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(\text{i.e. } 5=\dfrac{5}{1}  and  134=74)1\dfrac{3}{4}=\dfrac{7}{4}). The final answer should always be simplified and written as a mixed number if larger than 11. In the following video you will see an example of how to divide an integer by a fraction, as well as an example of how to divide a fraction by another fraction. https://youtu.be/F5YSNLel3n8

Licenses & Attributions