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Multiplying and Dividing Mixed Numbers

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Multiplying and Dividing Mixed Numbers

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 a previous lesson.

Example

Multiply: \(3\frac{1}{3}·\frac{5}{8}\)

Solution

 \(3\frac{1}{3}·\frac{5}{8}\)
Convert \(3\frac{1}{3}\) to an improper fraction.\(\frac{10}{3}·\frac{5}{8}\)
Multiply.\(\frac{10·5}{3·8}\)
Look for common factors.\(\frac{2̸·5·5}{3·2̸·4}\)
Remove common factors.\(\frac{5·5}{3·4}\)
Simplify.\(\frac{25}{12}\)

Notice that we left the answer as an improper fraction, \(\frac{25}{12},\) 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 \(2\frac{1}{12}\) and \(2·\frac{1}{12}.\)

Multiplying or Dividing 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: \(2\frac{4}{5}\phantom{\rule{0.2em}{0ex}}\left(-1\frac{7}{8}\right).\)

Solution

 \(2\frac{4}{5}\phantom{\rule{0.2em}{0ex}}\left(-1\frac{7}{8}\right)\)
Convert mixed numbers to improper fractions.\(\frac{14}{5}\phantom{\rule{0.2em}{0ex}}\left(-\frac{15}{8}\right)\)
Multiply.\(-\phantom{\rule{0.2em}{0ex}}\frac{14·15}{5·8}\)
Look for common factors.\(-\phantom{\rule{0.2em}{0ex}}\frac{2̸·7·5̸·3}{5̸·2̸·4}\)
Remove common factors.\(-\phantom{\rule{0.2em}{0ex}}\frac{\phantom{\rule{0.2em}{0ex}}7·3}{4}\)
Simplify.\(-\phantom{\rule{0.2em}{0ex}}\frac{\phantom{\rule{0.2em}{0ex}}21}{4}\)

Example

Divide, and write your answer in simplified form: \(3\frac{4}{7}\phantom{\rule{0.2em}{0ex}}÷\phantom{\rule{0.2em}{0ex}}5.\)

Solution

 \(3\frac{4}{7}\phantom{\rule{0.2em}{0ex}}÷\phantom{\rule{0.2em}{0ex}}5\)
Convert mixed numbers to improper fractions.\(\frac{25}{7}\phantom{\rule{0.2em}{0ex}}÷\phantom{\rule{0.2em}{0ex}}\frac{5}{1}\)
Multiply the first fraction by the reciprocal of the second.\(\frac{25}{7}·\frac{1}{5}\)
Multiply.\(\frac{25·1}{7·5}\)
Look for common factors.\(\frac{5̸·5·1}{7·5̸}\)
Remove common factors.\(\frac{5·1}{7}\)
Simplify.\(\frac{5}{7}\)

Example

Divide: \(2\frac{1}{2}÷1\frac{1}{4}.\)

Solution

 \(2\frac{1}{2}÷1\frac{1}{4}\)
Convert mixed numbers to improper fractions.\(\frac{5}{2}÷\frac{5}{4}\)
Multiply the first fraction by the reciprocal of the second.\(\frac{5}{2}·\frac{4}{5}\)
Multiply.\(\frac{5·4}{2·5}\)
Look for common factors.\(\frac{5̸·2̸·2}{2̸·1·5̸}\)
Remove common factors.\(\frac{2}{1}\)
Simplify.\(2\)

Optional Video: Division Involving Mixed Numbers

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