# Multiplication of Fractions

## Primarily proper fractions

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Practice Multiplication of Fractions

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Multiply and Divide Fractions and Mixed Numbers

Objective:  To multiply and divide fractions and mixed numbers.

### Guidance

Multiplying and dividing fractions is a lot less work then adding and subtracting them.

To multiply two fractions, simply multiply the numerators to get the numerator of the product, and multiply the denominators to get the denominator of the product.

To divide two fractions, you first need to find the reciprocal of the divisor. That means that you need to flip the second fraction upside down. Then multiply the numerators and multiply the denominators.

Multiply: \begin{align*}\frac{2}{7} \times \frac{3}{5}\end{align*}

Multiply the numerators and multiply the denominators.

\begin{align*}\frac{2}{7}\times\frac{3}{5}=\frac{2\times3}{7\times5}=\frac{6}{35}\end{align*}

Now let’s look at dividing fractions.

Divide: \begin{align*}4\frac{3}{10}\div \frac{1}{2}\end{align*}

Wow! This one has a mixed number and a fraction. Don’t let that throw you! You can work with mixed numbers quite easily. Just remember to convert them to improper fractions first.

First change the mixed number to an improper fraction.

\begin{align*}4\frac{3}{10}=\frac{4\times10+3}{10}=\frac{43}{10}\end{align*}

Then flip the second fraction and multiply.

\begin{align*}\frac{43}{10}\div\frac{1}{2}=\frac{43}{10}\times\frac{2}{1}=\frac{86}{10}\end{align*}

Finally, simplify the fraction.

\begin{align*}\frac{86}{10}=8\frac{6}{10}=8\frac{3}{5}\end{align*}

This is our answer.

#### Example A

\begin{align*}9\frac{1}{4} \div \frac{1}{3}\end{align*}

Solution:  \begin{align*}27 \frac{3}{4}\end{align*}

#### Example B

\begin{align*}\frac{1}{4}\times\frac{5}{6}\end{align*}

Solution:  \begin{align*}\frac{5}{24}\end{align*}

#### Example C

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

Solution:  7 \begin{align*}\frac{1}{2}\end{align*}

### Vocabulary

Greatest Common Factor
a number that will divide evenly into both the numerator and the denominator of a fraction.
Product
the answer in a multiplication problem.
Quotient
the answer in a division problem.
Fraction
a part of a whole
Mixed Number
a number with a whole number and a fraction.
Improper Fraction
a number that is greater than a whole with a larger top number and a smaller bottom number.

### Guided Practice

Here is one for you to try on your own.

\begin{align*}\frac{2}{3}\times\frac{4}{6}\end{align*}

Solution

Multiply the numerators and the denominators.

\begin{align*}\frac{2}{3}\times\frac{4}{6}=\frac{8}{18}\end{align*}

Now we need to simplify the product.

That’s alright. We can review it here.

When we simplify a fraction, we rewrite it as an equal fraction that is smaller than the fraction we have as our answer. We look for the greatest common factor that will divide into both the numerator and the denominator. The greatest common factor is the largest number that will divide into both the numerator and the denominator. This is how we will rewrite the fraction in simplest form.

The greatest common factor of 8 and 18 is 2. We divide both the numerator and denominator by 2.

\begin{align*}\frac{8}{18}=\frac{8\div2}{18\div2}=\frac{4}{9}\end{align*}

This is our answer.

### Practice

Directions: Multiply the following fractions. Be sure to simplify your answer when necessary.

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

Directions: Divide the following fractions. Be sure to convert any improper fractions to mixed numbers.

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

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