2.2.1: Multiplying Fractions And Mixed Numbers

Multiplying Mixed Numbers

If you want to multiply two mixed numbers, or a fraction and a mixed number, you can again rewrite any mixed number as an improper fraction.

So, to multiply two mixed numbers, rewrite each as an improper fraction and then multiply as usual. Multiply numerators and multiply denominators and simplify. And, as before, when simplifying, if the answer comes out as an improper fraction, then convert the answer to a mixed number.

Example

\(\ 2 \frac{1}{5} \cdot 4 \frac{1}{2}\)

Multiply. Simplify the answer and write as a mixed number.

Solution

\(\ 2 \frac{1}{5}=\frac{11}{5}\)	Change 2 \(\ \frac{1}{5}\) to an improper fraction. \(\ 5 \cdot 2+1=11\), and the denominator is 5.
\(\ 4 \frac{1}{2}=\frac{9}{2}\)	Change \(\ 4 \frac{1}{2}\) to an improper fraction. \(\ 2 \cdot 4+1=9\), and the denominator is 2.
\(\ \frac{11}{5} \cdot \frac{9}{2}\)	Rewrite the multiplication problem, using the improper fractions.
\(\ \frac{11 \cdot 9}{5 \cdot 2}=\frac{99}{10}\)	Multiply numerators and multiply denominators.
\(\ \frac{99}{10}=9 \frac{9}{10}\)	Write as a mixed number. \(\ 99 \div 10=9\) with a remainder of 9.

\(\ 2 \frac{1}{5} \cdot 4 \frac{1}{2}=9 \frac{9}{10}\)

Example

\(\ \frac{1}{2} \cdot 3 \frac{1}{3}\)

Multiply. Simplify the answer and write as a mixed number.

Solution

\(\ 3 \frac{1}{3}=\frac{10}{3}\)	Change \(\ 3 \frac{1}{3}\) to an improper fraction. \(\ 3 \cdot 3+1=10\), and the denominator is 3.
\(\ \frac{1}{2} \cdot \frac{10}{3}\)	Rewrite the multiplication problem, using the improper fraction in place of the mixed number.
\(\ \frac{1 \cdot 10}{2 \cdot 3}=\frac{10}{6}\)	Multiply numerators and multiply denominators.
\(\ \frac{10}{6}=1 \frac{4}{6}\)	Rewrite as a mixed number. \(\ 10 \div 6=1\) with a remainder of 4.
\(\ 1 \frac{2}{3}\)	Simplify the fractional part to lowest terms by dividing the numerator and denominator by the common factor 2.

\(\ \frac{1}{2} \cdot 3 \frac{1}{3}=1 \frac{2}{3}\)

As you saw earlier, sometimes it’s helpful to look for common factors in the numerator and denominator before you simplify the products.

Example

\(\ 1 \frac{3}{5} \cdot 2 \frac{1}{4}\)

Multiply. Simplify the answer and write as a mixed number.

Solution

\(\ 1 \frac{3}{5}=\frac{8}{5}\)	Change \(\ 1 \frac{3}{5}\) to an improper fraction. \(\ 5 \cdot 1+3=8\), and the denominator is 5.
\(\ 2 \frac{1}{4}=\frac{9}{4}\)	Change \(\ 2 \frac{1}{4}\) to an improper fraction. \(\ 4 \cdot 2+1=9\), and the denominator is 4.
\(\ \frac{8}{5} \cdot \frac{9}{4}\)	Rewrite the multiplication problem using the improper fractions.
\(\ \frac{8 \cdot 9}{5 \cdot 4}=\frac{9 \cdot 8}{5 \cdot 4}\)	Reorder the numerators so that you can see a fraction that has a common factor.
\(\ \frac{9 \cdot 8}{5 \cdot 4}=\frac{9 \cdot 2}{5 \cdot 1}\)	Simplify. \(\ \frac{8}{4}=\frac{8 \div 4}{4 \div 4}=\frac{2}{1}\)
\(\ \frac{18}{5}\)	Multiply.
\(\ \frac{18}{5}=3 \frac{3}{5}\)	Write as a mixed fraction.

\(\ 1 \frac{3}{5} \cdot 2 \frac{1}{4}=3 \frac{3}{5}\)

In the last example, the same answer would be found if you multiplied numerators and multiplied denominators without removing the common factor. However, you would get \(\ \frac{72}{20}\), and then you would need to simplify more to get your final answer.

Exercise

\(\ 1 \frac{3}{5} \cdot 3 \frac{1}{3}\)

1. \(\ \frac{80}{15}\)
2. \(\ 5 \frac{5}{15}\)
3. \(\ 4 \frac{14}{15}\)
4. \(\ 5 \frac{1}{3}\)

Answer

1. Incorrect. You probably wrote both mixed numbers as improper fractions correctly. You probably also correctly multiplied numerators and denominators. However, this improper fraction still needs to be rewritten as a mixed number and simplified. Dividing \(\ 80 \div 15=5\) with a remainder of 5 or \(\ 5 \frac{5}{15}\), then simplifying the fractional part, the correct answer is \(\ 5 \frac{1}{3}\).
2. Incorrect. You probably wrote both mixed numbers as improper fractions correctly. You probably also correctly multiplied numerators and denominators, and wrote the answer as a mixed number. However, the mixed number is not in lowest terms. \(\ \frac{5}{15}\) can be simplified to \(\ \frac{1}{3}\) by dividing numerator and denominator by the common factor 5. The correct answer is \(\ 5 \frac{1}{3}\).
3. Incorrect. This is the result of adding the two numbers. To multiply, rewrite each mixed number as an improper fraction: \(\ 1 \frac{3}{5}=\frac{8}{5}\) and \(\ 3 \frac{1}{3}=\frac{10}{3}\). Next, multiply numerators and multiply denominators: \(\ \frac{8}{5} \cdot \frac{10}{3}=\frac{80}{15}\). Then, write the resulting improper fraction as a mixed number: \(\ \frac{80}{15}=5 \frac{5}{15}\). Finally, simplify the fractional part by dividing both numerator and denominator by the common factor, 5. The correct answer is \(\ 5 \frac{1}{3}\).
4. Correct. First, rewrite each mixed number as an improper fraction: \(\ 1 \frac{3}{5}=\frac{8}{5}\) and \(\ 3 \frac{1}{3}=\frac{10}{3}\). Next, multiply numerators and multiply denominators: \(\ \frac{8}{5} \cdot \frac{10}{3}=\frac{80}{15}\). Then write as a mixed fraction \(\ \frac{80}{15}=5 \frac{5}{15}\). Finally, simplify the fractional part by dividing both numerator and denominator by the common factor 5.