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# Expression Evaluation with Products of Mixed Numbers

## Evaluating expressions multiplying fractions > 1.

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Expression Evaluation with Products of Mixed Numbers

As Julie works on her project she learns that there are many problems facing today’s rainforest. The rainforest is an important resource for our environment and much of it is being destroyed. This is mainly due to development where companies such as logging companies only see the rainforest as a valuable commercial resource. Julie is amazed that these companies don’t seem to understand that many rare animals and plants live in the rainforest, or that so much of the world’s water is in the rainforest and that many medicines are found because of the resources there. As she reads, Julie finds herself getting more and more irritated. “Are you alright Julie,” Mr. Gibbons asks, as he pauses in his walk around the room checking on students.

“No, I’m not,” Julie says, and proceeds to tell Mr. Gibbons all about what she has learned about the rainforest. “Look here,” she says pointing to her book. “It says that we lose 112\begin{align*}1 \frac{1}{2}\end{align*} acres of land every second!”

The amount of land lost every second could change if the harvesting of trees continues. This could be a changeable number. Because of this, we can call this a variable x.

Can you write an expression that shows how many acres could be lost in 212\begin{align*}2 \frac{1}{2}\end{align*} acres per second? How about 312\begin{align*}3 \frac{1}{2}\end{align*} acres per second?

This Concept is about evaluating expressions involving products of mixed numbers. It is just what you need to complete this task.

### Guidance

To begin, let’s review the difference between a numerical expression and an algebraic expression.

A Numerical Expression has numbers and operations, but does not have an equals sign. We evaluate a numerical expression.

An Algebraic Expression has numbers, operations and variables in it. It also does not have an equals sign. We evaluate an algebraic expression as well.

How can we evaluate a numerical expression that has mixed number in it?

We can work through a problem like this just as we would if we were solving an equation. Here we will be evaluating an expression, but our work will be the same. Sometimes an expression will also use different signs to show multiplication, like a dot ()\begin{align*}(\cdot)\end{align*} or two sets of parentheses next to each other ( )( ).

Evaluate \begin{align*}\left ( 3\frac{1}{3} \right ) \left ( 1 \frac{1}{2} \right )\end{align*}

When evaluating this expression, follow the same steps as we did when we were multiplying mixed numbers. First, convert each to an improper fraction.

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

Next, we can rewrite the expression and finish our work.

\begin{align*}\frac{10}{3} \cdot \frac{3}{2} = \frac{5}{1} \cdot \frac{1}{1} = 5\end{align*}

What about algebraic expressions? How do we evaluate an algebraic expression?

An algebraic expression uses variables, numbers and operations. Often you will be given a value for the one or more variables in the expression.

Evaluate \begin{align*}\frac{1}{2} x\end{align*} when \begin{align*}x\end{align*} is \begin{align*}4\frac{2}{3}\end{align*}

To evaluate this expression, we substitute four and two-thirds in for \begin{align*}x\end{align*}. Notice that the \begin{align*}x\end{align*} is next to the one-half which means we are going to multiply to evaluate this expression.

\begin{align*}\frac{1}{2} \cdot 4\frac{2}{3}\end{align*}

Next, we change four and two-thirds to an improper fraction, simplify, and multiply.

\begin{align*}4\frac{2}{3} & = \frac{14}{3}\\ \frac{1}{2} \cdot \frac{14}{3} & = \frac{1}{1} \cdot \frac{7}{3} = \frac{7}{3} = 2\frac{1}{3}\end{align*}

Our final answer is \begin{align*}2\frac{1}{3}\end{align*}.

#### Example A

Evaluate \begin{align*}2\frac{1}{3} x\end{align*} when \begin{align*}x\end{align*} is \begin{align*}\frac{4}{5}\end{align*}.

Solution:\begin{align*}1 \frac{13}{15}\end{align*}

#### Example B

Evaluate \begin{align*}\left ( 2\frac{1}{7} \right ) \left ( 1\frac{1}{2} \right )\end{align*}

Solution: \begin{align*}3 \frac{3}{14}\end{align*}

#### Example C

Evaluate \begin{align*}\left ( 8\frac{1}{2} \right ) (12)\end{align*}

Solution: 102

Now back to the rainforest dilemma. Here is the original problem once again.

As Julie works on her project she learns that there are many problems facing today’s rainforest. The rainforest is an important resource for our environment and much of it is being destroyed. This is mainly due to development where companies such as logging companies only see the rainforest as a valuable commercial resource. Julie is amazed that these companies don’t seem to understand that many rare animals and plants live in the rainforest, or that so much of the world’s water is in the rainforest and that many medicines are found because of the resources there. As she reads, Julie finds herself getting more and more irritated. “Are you alright Julie,” Mr. Gibbons asks, as he pauses in his walk around the room checking on students.

“No, I’m not,” Julie says, and proceeds to tell Mr. Gibbons all about what she has learned about the rainforest. “Look here,” she says pointing to her book. “It says that we lose \begin{align*}1 \frac{1}{2}\end{align*} acres of land every second!”

The amount of land lost every second could change if the harvesting of trees continues. This could be a changeable number. Because of this, we can call this a variable x.

Can you write an expression that shows how many acres could be lost if \begin{align*}2 \frac{1}{2}\end{align*} acres per second were lost in one minute? How about \begin{align*}3 \frac{1}{2}\end{align*} acres per second in one minute?

To write an expression, we use x to represent the changing acreage and 60 seconds to one minute. Here is the expression.

\begin{align*}x(60)\end{align*}

Since x is a variable, we can substitute the two given values in for it and multiply.

\begin{align*}2 \frac{1}{2} \times 60\end{align*}

\begin{align*}\frac{5}{2} \times 60 = 150\end{align*}

At that rate, 150 acres are lost in one minute.

\begin{align*}3 \frac{1}{2} \times 60\end{align*}

\begin{align*}\frac{7}{2} \times 30 = 210\end{align*}

At that rate, 210 acres are lost in one minute.

### Vocabulary

Mixed Number
a number that has both wholes and parts.
Improper Fraction
a number where the numerator is greater than the denominator.
Numerical Expression
has numbers and operations but no equals sign.
Algebraic Expression
has numbers, operations and variables but no equals sign.

### Guided Practice

Here is one for you to try on your own.

Evaluate \begin{align*}\left ( 3\frac{1}{9} \right ) \left ( 2\frac{1}{3} \right )\end{align*}

To begin with this problem, we have to convert each mixed number to an improper fraction.

\begin{align*}3 \frac{1}{9}\end{align*} becomes \begin{align*}\frac{28}{9}\end{align*}

\begin{align*}2 \frac{1}{3}\end{align*} becomes \begin{align*}\frac{7}{3}\end{align*}

Now we can multiply the two fractions together.

\begin{align*}\frac{28}{9} \times \frac{7}{3} = \frac{196}{27} = 7 \frac{7}{27}\end{align*}

### Practice

Directions:Evaluate each expression if x = \begin{align*}1 \frac{1}{2}\end{align*}. Be sure your answer is in simplest form.

1. \begin{align*}\left ( 3\frac{1}{9} \right )x\end{align*}

2. \begin{align*}\left ( 2\frac{1}{2} \right )x\end{align*}

3. \begin{align*}\left ( 1\frac{1}{3} \right )x\end{align*}

4. \begin{align*}\left ( 4\frac{1}{3} \right )x\end{align*}

5. \begin{align*}\left ( 5\frac{1}{2} \right )x\end{align*}

6. \begin{align*}\left ( 6\frac{1}{9} \right )x\end{align*}

7. \begin{align*}\left ( 4\frac{2}{3} \right )x\end{align*}

8. \begin{align*}\left ( 3\frac{1}{5} \right )x\end{align*}

9. \begin{align*}\left ( 4\frac{2}{3} \right )x\end{align*}

10. \begin{align*}\left ( 3\frac{1}{6} \right )x\end{align*}

11. \begin{align*}\left ( 4\frac{1}{2} \right )x\end{align*}

12. \begin{align*}\left ( 1\frac{6}{7} \right )x\end{align*}

13. \begin{align*}\left ( 1\frac{5}{9} \right )x\end{align*}

14. \begin{align*}\left ( 2\frac{2}{9} \right )x\end{align*}

15. \begin{align*}\left ( 4\frac{3}{7} \right )x\end{align*}

16. \begin{align*}\left ( 6\frac{7}{8} \right )x\end{align*}

### Vocabulary Language: English

Algebraic Equation

An algebraic equation contains numbers, variables, operations, and an equals sign.

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.

Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.