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Division of Fractions by Whole Numbers

Understand how to divide a fraction by a whole number.

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Division of Fractions by Whole Numbers
Source: https://pixabay.com/en/diary-pen-calculator-case-work-582976/
License: CC BY-NC 3.0

Ray spends \begin{align*}\frac{1}{3}\end{align*} of his day at work. He needs to divide his time equally between 3 different projects. How much time should Ray spend on each project? Find the answer in terms of hours. 

In this concept, you will learn how to divide fractions by whole numbers.

Dividing Fractions by Whole Numbers

Think about what is happening when you divide a fraction by a whole number. You are taking a part of something and splitting it up into more parts. Here is a division problem.

\begin{align*}\frac{1}{2} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

This problem is asking you to take one-half and divide it into three parts. Here is a picture of one-half.

Divide each half into three parts.

Each section is \begin{align*}\frac{1}{6}\end{align*} of the whole. One-half divided by 3 is \begin{align*}\frac{1}{6}\end{align*}

There are two things to remember when dividing fractions. The first is that you can solve the problem by using the inverse operation. The inverse or opposite of division is multiplication. The second is that you will multiply by the reciprocal of the divisor. Remember that the reciprocal of a fraction is a fraction with the numerator and denominator change places. 

To divide a fraction, multiply by the reciprocal of the divisor. Here is the division problem again. 

\begin{align*}\frac{1}{2} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

First, change the operation to multiplication and change 3 to its reciprocal. 3 can be written as the fraction \begin{align*}\frac{3}{1}\end{align*}. The reciprocal of \begin{align*}\frac{3}{1}\end{align*} is \begin{align*}\frac{1}{3}\end{align*}

\begin{align*}\frac{1}{2} \div 3 = \frac{1}{2} \times \frac{1}{3}\end{align*}

Then, multiply the fractions to solve. 

\begin{align*}\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\end{align*}

The answer is the same as the diagram above. Dividing a fraction by whole number is the same as multiplying a fraction by the reciprocal of the divisor.  

Examples

Example 1

Earlier, you were given a problem about Ray's day at work.

Ray needs to evenly divide \begin{align*}\frac{1}{3}\end{align*} of his day between 3 projects. Divide \begin{align*}\frac{1}{3}\end{align*} by 3 to find how much time he should spend on each project.

First, set up a division problem.

\begin{align*}\frac{1}{3} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}

Then, change the division problem. Multiply by the reciprocal of the divisor.

\begin{align*}\frac{1}{3} \div 3 = \frac{1}{3} \times \frac{1}{3}\end{align*}

Next, multiply the fractions.

\begin{align*} \frac{1}{3} \times \frac{1}{3}=\frac{1}{9}\end{align*}

Now find \begin{align*}\frac{1}{9}\end{align*} of a day in terms of hours. One day is 24 hours. The word "of" tell you to multiply \begin{align*}\frac{1}{9}\end{align*} by 24.

\begin{align*}\frac{1}{9}\times 24=\frac{1}{\cancel{9}^3}\times\frac{\cancel{24}^8}{1}=\frac{8}{3}=2 \frac{2}{3}\end{align*}

Ray should spend \begin{align*}2\frac{2}{3}\end{align*} hours on each project.

Example 2

Divide the fraction: \begin{align*}\frac{6}{8} \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, change the operation to multiplication and change 4 to its reciprocal. 

\begin{align*}\frac{6}{8} \div 4 =\frac{6}{8} \times \frac{1}{4} \end{align*}

Then, multiply the fractions.

\begin{align*} \frac{6}{8} \times \frac{1}{4} = \frac{6}{32}\end{align*}

Next, simplify the fraction. The greatest common factor of 6 and 32 is 2.

\begin{align*} \frac{6}{32}= \frac{3}{16}\end{align*}

The answer is \begin{align*} \frac{3}{16}\end{align*}.

Example 3

Divide the fraction: \begin{align*}\frac{1}{4} \div 2 = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, change the expression. Multiply by the inverse of the divisor. 

\begin{align*}\frac{1}{4}\div 2=\frac{1}{4}\times \frac{1}{2}\end{align*}

Then, multiply the fractions.

\begin{align*}\frac{1}{4}\times \frac{1}{2}=\frac{1}{8}\end{align*}

The fraction is in simplest form.

The answer is \begin{align*} \frac{1}{8}\end{align*}.

Example 4

Divide the fraction: \begin{align*}\frac{3}{4} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, change the expression. Multiply by the inverse of the divisor. 

\begin{align*}\frac{3}{4} \div 3 =\frac{3}{4}\times\frac{1}{3}\end{align*}

Then, multiply the fractions. You can simplify the 3s before multiplying.

\begin{align*}\frac{\cancel{3}}{4}\times\frac{1}{\cancel{3}}=\frac {1}{4}\end{align*}

The fraction is in simplest form.

The answer is \begin{align*} \frac{1}{4}\end{align*}.

Example 5

Divide the fraction: \begin{align*}\frac{4}{5} \div 2 = \underline{\;\;\;\;\;\;\;}\end{align*}. Answer in simplest form.

First, change the expression. Multiply by the inverse of the divisor. 

\begin{align*}\frac{4}{5} \div 2 =\frac{4}{5} \times \frac{1}{2}\end{align*}

Then, multiply the fractions.

\begin{align*}\frac{4}{5} \times \frac{1}{2} = \frac{4}{10}\end{align*}

Next, simplify the fraction by the greatest common factor of 2.

\begin{align*}\frac{4}{10}=\frac{2}{5}\end{align*}

The answer is \begin{align*} \frac{2}{5}\end{align*}.

Review

Divide each fraction and whole number.

  1. \begin{align*}\frac{1}{2} \div 6= \underline{\;\;\;\;\;\;\;}\end{align*} 
  2. \begin{align*}\frac{1}{4} \div 8= \underline{\;\;\;\;\;\;\;}\end{align*} 
  3. \begin{align*}\frac{1}{3} \div 5= \underline{\;\;\;\;\;\;\;}\end{align*} 
  4. \begin{align*}\frac{1}{3} \div 4= \underline{\;\;\;\;\;\;\;}\end{align*} 
  5. \begin{align*}\frac{1}{2} \div 7= \underline{\;\;\;\;\;\;\;}\end{align*} 
  6. \begin{align*}\frac{1}{5} \div 11= \underline{\;\;\;\;\;\;\;}\end{align*} 
  7. \begin{align*}\frac{1}{2} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}
  8. \begin{align*}\frac{1}{4} \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}
  9. \begin{align*}\frac{1}{9} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}
  10. \begin{align*}\frac{2}{3} \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}
  11. \begin{align*}\frac{4}{7} \div 3 = \underline{\;\;\;\;\;\;\;}\end{align*}
  12. \begin{align*}\frac{2}{5} \div 2 = \underline{\;\;\;\;\;\;\;}\end{align*}
  13. \begin{align*}\frac{3}{7} \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}
  14. \begin{align*}\frac{1}{5} \div 6 = \underline{\;\;\;\;\;\;\;}\end{align*}
  15. \begin{align*}\frac{8}{9} \div 2 = \underline{\;\;\;\;\;\;\;}\end{align*}
  16. \begin{align*}\frac{6}{7} \div 4 = \underline{\;\;\;\;\;\;\;}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.8. 

Resources

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Vocabulary

Inverse Operation

Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.

reciprocal

The reciprocal of a number is the number you can multiply it by to get one. The reciprocal of 2 is 1/2. It is also called the multiplicative inverse, or just inverse.

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