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 [latex]3[/latex] quarts of paint and you have a bucket that contains [latex]6[/latex] quarts of paint, how many coats of paint can you paint on the walls? You divide [latex]6[/latex] by [latex]3[/latex] for an answer of [latex]2[/latex] coats. There will also be times when you need to divide by a fraction. Suppose painting a closet with one coat only required [latex]\dfrac{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 [latex]6[/latex] by the fraction, [latex]\dfrac{1}{2}[/latex]. Before we begin dividing fractions, let's cover some important terminology.- reciprocal: two fractions are reciprocals if their product is [latex]1[/latex] (Don't worry; we will show you examples of what this means.)
- quotient: the result of division
Original number | Reciprocal | Product |
---|---|---|
[latex]\dfrac{3}{4}[/latex] | [latex]\dfrac{4}{3}[/latex] | [latex]\dfrac{3}{4}\cdot\dfrac{4}{3}=\dfrac{3\cdot 4}{4\cdot 3}=\dfrac{12}{12}=1[/latex] |
[latex]\dfrac{1}{2}[/latex] | [latex]\dfrac{2}{1}[/latex] | [latex]\dfrac{1}{2}\cdot\dfrac{2}{1}=\dfrac{1\cdot2}{2\cdot1}=\dfrac{2}{2}=1[/latex] |
[latex] 3=\dfrac{3}{1}[/latex] | [latex]\dfrac{1}{3}[/latex] | [latex]\dfrac{3}{1}\cdot\dfrac{1}{3}=\dfrac{3\cdot 1}{1\cdot 3}=\dfrac{3}{3}=1[/latex] |
[latex]2\dfrac{1}{3}=\dfrac{7}{3}[/latex] | [latex]\dfrac{3}{7}[/latex] | [latex]\dfrac{7}{3}\cdot\dfrac{3}{7}=\dfrac{7\cdot3}{3\cdot7}=\dfrac{21}{21}=\normalsize 1[/latex] |
Division by Zero
You know what it means to divide by [latex]2[/latex] or divide by [latex]10[/latex], but what does it mean to divide a quantity by [latex]0[/latex]? Is this even possible? On the flip side, can you divide [latex]0[/latex] by a number? Consider the fraction[latex]\dfrac{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. What number times [latex]8[/latex] equals [latex]0[/latex]?[latex]\text{?}\cdot{8}=0[/latex]
We can infer that the unknown must be [latex]0[/latex] since that is the only number that will give a result of [latex]0[/latex] when it is multiplied by [latex]8[/latex].
Now let’s consider the reciprocal of [latex]\dfrac{0}{8}[/latex] which would be [latex]\dfrac{8}{0}[/latex]. If we rewrite this as a multiplication problem, we will have "what times [latex]0[/latex] equals [latex]8[/latex]?"[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. In fact, any number divided by [latex]0[/latex] is impossible, or better defined, all division by zero is 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 [latex]3[/latex] quarts of paint for a coat and have [latex]6[/latex] quarts of paint, you can find the total number of coats that can be painted by dividing [latex]6[/latex] by [latex]3[/latex], [latex]6\div3=2[/latex]. You can also multiply [latex]6[/latex] by the reciprocal of [latex]3[/latex], which is [latex]\dfrac{1}{3}[/latex], so the multiplication problem becomes[latex]\dfrac{6}{1}\cdot\dfrac{1}{3}=\dfrac{6}{3}=\normalsize2[/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.[latex]\dfrac{1}{5}\normalsize\text{ of }\dfrac{3}{4}=\dfrac{1}{5}\cdot\dfrac{3}{4}=\dfrac{3}{20}[/latex]
Each person gets [latex]\dfrac{3}{20}[/latex] of a whole candy bar.
For example, dividing by [latex]6[/latex] is the same as multiplying by the reciprocal of [latex]6[/latex], which is [latex]\dfrac{1}{6}[/latex]. Look at the diagram of two pizzas below. How can you divide what is left (the red shaded region) among [latex]6[/latex] people fairly? Each person gets one piece, so each person gets [latex]\dfrac{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.Example
Find [latex]\dfrac{2}{3}\div \normalsize 4[/latex]Answer: Write your answer in lowest terms. Dividing by [latex]4[/latex] or [latex]\dfrac{4}{1}[/latex] is the same as multiplying by the reciprocal of [latex]4[/latex], which is [latex]\dfrac{1}{4}[/latex].
[latex]\dfrac{2}{3}\normalsize\div 4=\dfrac{2}{3}\cdot\dfrac{1}{4}[/latex]
Multiply numerators and multiply denominators.[latex]\dfrac{2\cdot 1}{3\cdot 4}=\dfrac{2}{12}[/latex]
Simplify to lowest terms by dividing numerator and denominator by the common factor [latex]4[/latex].[latex]\dfrac{1}{6}[/latex]
Answer
[latex-display]\dfrac{2}{3}\div4=\dfrac{1}{6}[/latex-display]Example
Divide. [latex] 9\div\dfrac{1}{2}[/latex]Answer: Write your answer in lowest terms. Dividing by [latex]\dfrac{1}{2}[/latex] is the same as multiplying by the reciprocal of [latex]\dfrac{1}{2}[/latex], which is [latex]\dfrac{2}{1}[/latex].
[latex]9\div\dfrac{1}{2}=\dfrac{9}{1}\cdot\dfrac{2}{1}[/latex]
Multiply numerators and multiply denominators.[latex]\dfrac{9\cdot 2}{1\cdot 1}=\dfrac{18}{1}=\normalsize 18[/latex]
This answer is already simplified to lowest terms.Answer
[latex-display]9\div\dfrac{1}{2}=\normalsize 18[/latex-display]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 [latex]4[/latex] slices. How many [latex]\dfrac{1}{2}[/latex] slices are there?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).
Example
Divide [latex]\dfrac{2}{3}\div\dfrac{1}{6}[/latex]Answer: Multiply by the reciprocal. KEEP [latex]\dfrac{2}{3}[/latex] CHANGE [latex] \div [/latex] to [latex]\cdot[/latex] FLIP [latex]\dfrac{1}{6}[/latex]
[latex]\dfrac{2}{3}\cdot\dfrac{6}{1}[/latex]
Multiply numerators and multiply denominators.[latex]\dfrac{2\cdot6}{3\cdot1}=\dfrac{12}{3}[/latex]
Simplify.[latex]\dfrac{12}{3}=\normalsize 4[/latex]
Answer
[latex-display]\dfrac{2}{3}\div\dfrac{1}{6}=4[/latex-display]Example
Divide [latex]\dfrac{3}{5}\div\dfrac{2}{3}[/latex]Answer: Multiply by the reciprocal. Keep [latex]\dfrac{3}{5}[/latex], change [latex] \div [/latex] to [latex]\cdot[/latex], and flip [latex]\dfrac{2}{3}[/latex].
[latex]\dfrac{3}{5}\cdot\dfrac{3}{2}[/latex]
Multiply numerators and multiply denominators.[latex]\dfrac{3\cdot 3}{5\cdot 2}=\dfrac{9}{10}[/latex]
Answer
[latex-display]\dfrac{3}{5}\div\dfrac{2}{3}=\dfrac{9}{10}[/latex-display]Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex 1: Divide Fractions (Basic). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- College Algebra. Provided by: OpenStax Located at: https://openstax.org/books/prealgebra/pages/4-2-multiply-and-divide-fractions. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].