2.7: Mixed Numbers in Applications
Suppose you're skateboarding at an average rate of \begin{align*}15 \frac{1}{2}\end{align*}
Guidance
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*}
For any real numbers \begin{align*}a, b, c,\end{align*}
\begin{align*}\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}\end{align*}
Example A
The original cookie recipe calls for 8 cups flour. How much is needed for the reduced recipe?
Solution: Begin by writing the multiplication situation. \begin{align*}8 \cdot \frac{1}{3}\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*}
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*}
\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*}
\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.
Example B
Ayinde is making a dog house that is \begin{align*}3\frac{1}{2}\end{align*}
Solution:
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*}
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.
Example C
Doris’s truck gets \begin{align*}10 \frac{2}{3}\end{align*}
How far can she travel?
Solution: 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*}
Guided Practice
Anne has a bar of chocolate and she offers Bill a piece. Bill quickly breaks off \begin{align*}\frac{1}{4}\end{align*}
Solution: Think of the bar as one whole.
\begin{align*}1 \frac{1}{4} = \frac{3}{4}\end{align*}
\begin{align*}\frac{1}{3} \times \frac{3}{4} = \frac{1}{4}\end{align*}
\begin{align*}\frac{3}{4}  \frac{1}{4} = \frac{2}{4} = \frac{1}{2}\end{align*}
Anne divides the remaining bar into two equal pieces. Every person receives \begin{align*}\frac{1}{4}\end{align*}
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Multiplication of Rational Numbers (8:56)
Multiply the following rational numbers.

\begin{align*}\frac{3}{4} \times \frac{1}{3}\end{align*}
34×13 
\begin{align*}\frac{15}{11} \times \frac{9}{7}\end{align*}
1511×97 
\begin{align*}\frac{2}{7} \cdot 3.5\end{align*}
27⋅−3.5 
\begin{align*}\frac{1}{13} \times \frac{1}{11}\end{align*}
113×111 
\begin{align*}\frac{7}{27} \times \frac{9}{14}\end{align*}
727×914  \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 5andahalf inches by 15andahalf 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*}.
Associative Property of Multiplication
For any real numbers and .Commutative Property of Multiplication
For any real numbers and .Different Sign Multiplication Rule
The product of a positive number and a negative number is a negative number.Same Sign Multiplication Rule
The product of two positive or two negative numbers is positive.Image Attributions
Here you'll multiply mixed numbers and fractions to solve realworld multiplication problems.
Concept Nodes:
Associative Property of Multiplication
For any real numbers and .Commutative Property of Multiplication
For any real numbers and .Different Sign Multiplication Rule
The product of a positive number and a negative number is a negative number.Same Sign Multiplication Rule
The product of two positive or two negative numbers is positive.