Suppose you're skateboarding at an average rate of \begin{align*}15 \frac{1}{2}\end{align*} miles per hour, and you skateboard for \begin{align*}\frac{3}{5}\end{align*} of an hour. How far did you travel? In this Concept, you'll learn how to multiply mixed numbers and fractions so that you can answer real-world questions such as this.

### Mixed Numbers in Applications

You’ve decided to make cookies for a party. The recipe you’ve chosen makes 6 dozen cookies, but you only need 2 dozen. How do you reduce the recipe?

In this case, you should not use subtraction to find the new values. Subtraction means to make less by taking away. You haven’t made any cookies; therefore, you cannot take any away. Instead, you need to make \begin{align*}\frac{2}{6}\end{align*} or \begin{align*}\frac{1}{3}\end{align*} of the original recipe. This process involves multiplying fractions.

For any real numbers \begin{align*}a, b, c,\end{align*} and \begin{align*}d\end{align*}, where \begin{align*}\ b \neq 0\end{align*} and \begin{align*}\ d \neq 0\end{align*}:

\begin{align*}\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}\end{align*}

#### The original cookie recipe calls for 8 cups flour. How much is needed for the reduced recipe?

Begin by writing the multiplication situation. \begin{align*}8 \cdot \frac{1}{3}\end{align*}. You need to rewrite this product in the form of the property above. In order to perform this multiplication, you need to rewrite 8 as the fraction \begin{align*}\frac{8}{1}\end{align*}.

\begin{align*}8 \times \frac{1}{3} = \frac{8}{1} \times \frac{1}{3} = \frac{8 \cdot 1}{1 \cdot 3} = \frac{8}{3} = 2 \frac{2}{3}\end{align*}

You will need \begin{align*}2 \ \frac{2}{3}\end{align*} cups of flour.

**Multiplication Properties**

Properties that hold true for addition such as the Associative Property and Commutative Property also hold true for multiplication. They are summarized below.

The **Associative Property of Multiplication:** For any real numbers \begin{align*}a, \ b,\end{align*} and \begin{align*}c,\end{align*}

\begin{align*}(a \cdot b)\cdot c = a \cdot (b \cdot c)\end{align*}

The **Commutative Property of Multiplication:** For any real numbers \begin{align*}a\end{align*} and \begin{align*}b,\end{align*}

\begin{align*}a(b) = b(a)\end{align*}

The **Same Sign Multiplication Rule:** The product of two positive or two negative numbers is positive.

The **Different Sign Multiplication Rule:** The product of a positive number and a negative number is a negative number.

#### Ayinde is making a dog house that is \begin{align*}3\frac{1}{2}\end{align*} feet long and \begin{align*}2\frac{2}{3}\end{align*} feet wide. How many square feet of area will the dog house be?

Since the formula for area is

\begin{align*} area=length \times width\end{align*},

we plug in the values for length and width:

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

We first need to turn the mixed fractions into improper fractions:

\begin{align*} area=3\frac{1}{2} \times 2\frac{2}{3} =\frac{7}{2} \times \frac{8}{3}= \frac{7\times 8}{2\times 3} = \frac{56}{6}. \end{align*}

Now we turn the improper fraction back into a mixed fraction. Since 56 divided by 6 is 9 with a remainder of 2, we get:

\begin{align*} \frac{56}{6}=9\frac{2}{6}=9\frac{1}{3}\end{align*}

The dog house will have an area of \begin{align*}9\frac{1}{3}\end{align*} square feet.

**Note:** The units of the area are square feet, or feet squared, because we multiplied two numbers each with units of feet:

\begin{align*}feet \times feet=feet^2\end{align*},

which we call square feet.

#### Doris’s truck gets \begin{align*}10 \frac{2}{3}\end{align*} miles per gallon. Her tank is empty so she puts in \begin{align*}5 \frac{1}{2}\end{align*} gallons of gas. How far can she travel?

Begin by writing each mixed number as an improper fraction.

\begin{align*}10 \frac{2}{3} = \frac{32}{3} && 5 \frac{1}{2} = \frac{11}{2}\end{align*}

Now multiply the two values together.

\begin{align*}\frac{32}{3} \cdot \frac{11}{2} = \frac{352}{6} = 58\frac{4}{6} = 58\frac{2}{3}\end{align*}

Doris can travel \begin{align*}58 \ \frac{2}{3}\end{align*} miles on 5.5 gallons of gas.

### Example

Anne has a bar of chocolate and she offers Bill a piece. Bill quickly breaks off \begin{align*}\frac{1}{4}\end{align*} of the bar and eats it. Another friend, Cindy, takes \begin{align*}\frac{1}{3}\end{align*} of what was left. Anne splits the remaining candy bar into two equal pieces, which she shares with a third friend, Dora. How much of the candy bar does each person get?

Think of the bar as one whole.

\begin{align*}1- \frac{1}{4} = \frac{3}{4}\end{align*}. This is the amount remaining after Bill takes his piece.

\begin{align*}\frac{1}{3} \times \frac{3}{4} = \frac{1}{4}\end{align*}. This is the fraction Cindy receives.

\begin{align*}\frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}\end{align*}. This is the amount remaining after Cindy takes her piece.

Anne divides the remaining bar into two equal pieces. Every person receives \begin{align*}\frac{1}{4}\end{align*} of the bar.

### Review

Multiply the following rational numbers.

- \begin{align*}\frac{3}{4} \times \frac{1}{3}\end{align*}
- \begin{align*}\frac{15}{11} \times \frac{9}{7}\end{align*}
- \begin{align*}\frac{2}{7} \cdot -3.5\end{align*}
- \begin{align*}\frac{1}{13} \times \frac{1}{11}\end{align*}
- \begin{align*}\frac{7}{27} \times \frac{9}{14}\end{align*}
- \begin{align*}\left (\frac{3}{5} \right )^2\end{align*}
- \begin{align*}\frac{1}{11} \times \frac{22}{21} \times \frac{7}{10}\end{align*}
- \begin{align*}5.75 \cdot 0\end{align*}

In 9 – 11, state the property that applies to each of the following situations.

- A gardener is planting vegetables for the coming growing season. He wishes to plant potatoes and has a choice of a single 8 by 7 meter plot, or two smaller plots of 3 by 7 meters and 5 by 7 meters. Which option gives him the largest area for his potatoes?
- Andrew is counting his money. He puts all his money into $10 piles. He has one pile. How much money does Andrew have?
- Nadia and Peter are raising money by washing cars. Nadia is charging $3 per car, and she washes five cars in the first morning. Peter charges $5 per car (including a wax). In the first morning, he washes and waxes three cars. Who has raised the most money?

- Teo is making a flower box that is 5-and-a-half inches by 15-and-a-half inches. How many square inches will he have in which to plant flowers?

**Mixed Review**

- Compare these rational numbers: \begin{align*}\frac{16}{27}\end{align*}
*and*\begin{align*}\frac{2}{3}\end{align*}. - Define
*rational numbers.* - Give an example of a proper fraction. How is this different from an improper fraction?
- Which property is being applied? \begin{align*}16 - (-14) = 16 + 14 = 30\end{align*}
- Simplify \begin{align*}11 \frac{1}{2} + \frac{2}{9}\end{align*}.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 2.7.