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أدلة الدراسة > Prealgebra

Multiplying and Dividing Mixed Numbers

Learning Outcomes

  • Multiply mixed numbers and fractions
  • Divide an integer by a fraction
  • Divide a mixed number by an integer
In the previous section, you learned how to multiply and divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a mixed number to an improper fraction. And you learned how to do that in Visualize Fractions.

Example

Multiply: 313⋅583\frac{1}{3}\cdot \frac{5}{8} Solution:
313⋅583\frac{1}{3}\cdot \frac{5}{8}
Convert 3133\frac{1}{3} to an improper fraction. 103⋅58\frac{10}{3}\cdot \frac{5}{8}
Multiply. 10⋅53⋅8\frac{10\cdot 5}{3\cdot 8}
Look for common factors. 2⋅5⋅53⋅2⋅4\frac{\color{red}{2}\cdot 5\cdot 5}{3\cdot \color{red}{2} \cdot 4}
Remove common factors. 5⋅53⋅4\frac{5\cdot 5}{3\cdot 4}
Simplify. 2512\frac{25}{12}
Notice that we left the answer as an improper fraction, 2512\frac{25}{12}, and did not convert it to a mixed number. In algebra, it is preferable to write answers as improper fractions instead of mixed numbers. This avoids any possible confusion between 21122\frac{1}{12} and 2⋅1122\cdot \frac{1}{12}.

Try it

#146092 [ohm_question height="270"]146092[/ohm_question]
Watch the following video for another example of how to multiply a mixed number by a fraction. https://youtu.be/SvTd_ZxvqCM

Multiply or divide mixed numbers

  1. Convert the mixed numbers to improper fractions.
  2. Follow the rules for fraction multiplication or division.
  3. Simplify if possible.

Example

Multiply, and write your answer in simplified form: 245(−178)2\frac{4}{5}\left(-1\frac{7}{8}\right)

Answer: Solution:

245(−178)2\frac{4}{5}\left(-1\frac{7}{8}\right)
Convert mixed numbers to improper fractions. 145(−158)\frac{14}{5}\left(-\frac{15}{8}\right)
Multiply. −14⋅155⋅8-\frac{14\cdot 15}{5\cdot 8}
Look for common factors. −2⋅7⋅5⋅35⋅2⋅4-\frac{\color{red}{2} \cdot 7\cdot \color{red}{5} \cdot 3}{\color{red}{5} \cdot \color{red}{2}\cdot 4}
Remove common factors. −7⋅34-\frac{7\cdot 3}{4}
Simplify. −214-\frac{21}{4}

Try It

#146160 [ohm_question height="270"]146160[/ohm_question]
In the following video we show more examples of how to multiply mixed numbers that are negative. https://youtu.be/ahTOIf0fkOc

Example

Divide, and write your answer in simplified form: 347÷53\frac{4}{7}\div 5

Answer: Solution:

347÷53\frac{4}{7}\div 5
Convert mixed numbers to improper fractions. 257÷51\frac{25}{7}\div \frac{5}{1}
Multiply the first fraction by the reciprocal of the second. 257⋅15\frac{25}{7}\cdot \frac{1}{5}
Multiply. 25⋅17⋅5\frac{25\cdot 1}{7\cdot 5}
Look for common factors. 5⋅5⋅17⋅5\frac{\color{red}{5} \cdot 5\cdot 1}{7\cdot \color{red}{5} }
Remove common factors. 5⋅17\frac{5\cdot 1}{7}
Simplify. 57\frac{5}{7}

Try It

#146099 [ohm_question height="270"]146099[/ohm_question]

Example

Divide: 212÷1142\frac{1}{2}\div 1\frac{1}{4}

Answer: Solution:

212÷1142\frac{1}{2}\div 1\frac{1}{4}
Convert mixed numbers to improper fractions. 52÷54\frac{5}{2}\div \frac{5}{4}
Multiply the first fraction by the reciprocal of the second. 52⋅45\frac{5}{2}\cdot \frac{4}{5}
Multiply. 5⋅42⋅5\frac{5\cdot 4}{2\cdot 5}
Look for common factors. 5⋅2⋅22⋅1⋅5\frac{\color{red}{5} \cdot \color{red}{2} \cdot 2}{\color{red}{2} \cdot 1\cdot \color{red}{5}}
Remove common factors. 21\frac{2}{1}
Simplify. 22

Try It

#146100 [ohm_question height="270"]146100[/ohm_question]
The next video provides several more examples of dividing mixed numbers, whole numbers and fractions. https://youtu.be/zw7WdhQnXHw

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