Gloria’s mom, Mrs. Thomas, has given Gloria permission to have a sleepover. Gloria has invited 8 of her friends to the sleepover. Mrs. Thomas is going to pick up pizzas and hot wings for the 9 girls to feast on while they watch a movie. Ken, the guy at the pizzeria, tells Mrs. Thomas that a large pizza contains 12 slices. Mrs. Thomas wants each of the girls to have at least 4 slices, so she needs a total of 36 slices. How can she use this information to write a variable expression that will help her decide how many pizzas she should pick up?

In this concept, you will learn to write and simplify products and quotients of variable expressions.

### Writing Products and Quotients of Variable Expressions

To evaluate an expression you must have a given value for the variable. If you have not been given a value for the variable, then you simplify the expression.

Here is an expression.

\begin{align*}3(5x)\end{align*}

This is a multiplication problem. The number next to the parentheses means that you are going to multiply. However, you haven’t been given a value for \begin{align*}x\end{align*} so you are going to simplify this expression. You can’t do anything with the variable, but you can multiply the numerical part of each term.

\begin{align*}\begin{array}{rcl} && 3(5x) \\ && 3 \times 5 = 15 \end{array}\end{align*}

Keeping the ‘\begin{align*}x\end{align*}’ variable, the expression is simplified as \begin{align*}15x\end{align*}.

The answer is \begin{align*}15x\end{align*}.

Let’s look at another example.

\begin{align*}4 \cdot (12 \cdot y)\end{align*}

These dots mean to multiply.

First, complete the operation in parentheses.

\begin{align*}12 \ \text{times}\ y = 12y\end{align*}

Now, the expression becomes \begin{align*}4 \cdot (12y)\end{align*}.

Next, multiply the number parts of the two terms.

\begin{align*}4 \times 12 = 48\end{align*}

Then, keeping the ‘\begin{align*}y\end{align*}’ variable, the expression is simplified as \begin{align*}48y\end{align*}.

The answer is \begin{align*}48y\end{align*}.

You can also simplify quotients. Remember that a quotient means division.

Here is another expression.

\begin{align*}\frac{4x}{2x}\end{align*}

Whether it is a number or a variable, a fraction over itself is equal to one.

\begin{align*}\frac{6}{6} =1\end{align*}

This is called simplifying a fraction. Variables can also be simplified when you divide. When simplifying a quotient with a variable expression, simplify the variables first.

For example:

\begin{align*}\frac{4x}{2x} = \frac{4}{2}\end{align*}

Remember, \begin{align*}x\end{align*} divided by \begin{align*}x\end{align*} is equal to one. So, the \begin{align*}x\end{align*}’s cancel each other out.

Then, you are left with four divided by two.

\begin{align*}4 \div 2 = 2\end{align*}

The answer is 2.

### Examples

#### Example 1

Earlier, you were given a problem about the slumber party feast.

Each pizza has 12 slices and Gloria’s mom wants each of the nine girls to get at least four slices, which means she needs a total of at least 36 slices. How can she use this information to write a variable expression that will help her decide how many pizzas she should pick up?

Begin by expressing the situation in a phrase.

She needs ‘\begin{align*}x\end{align*}’ number of pizzas each with 12 slices to be divided by 9 girls.

First, identify the key words, variables, and numbers.

- 12 is the number of slices in each pizza
- \begin{align*}x\end{align*} is the unknown number of pizzas
- “each with” suggests multiplication
- “divided by” suggests division
- 9 is the number of girls

Next, write the phrase as an expression.

‘\begin{align*}x\end{align*}’ number of pizzas each with 12 slices becomes \begin{align*}12(x)\end{align*}

\begin{align*}12x\end{align*}

divided by 9 girls becomes

\begin{align*}\frac{\Box}{9}\end{align*}

Then, put it together as an expression.

\begin{align*}\frac{12x}{9}\end{align*}

Note that ‘\begin{align*}x\end{align*}’ is the only variable in the expression, so you can’t do anything with the \begin{align*}x\end{align*}.

Now, simplify the number values by dividing 12 and 9 by their greatest common factor, 3.

\begin{align*}\begin{array}{rcl} 12 \div 3 &=& 4 \\ 9 \div 3 &=& 3 \end{array}\end{align*}

This becomes \begin{align*}\frac{4}{3}\end{align*}.

Finally, add the \begin{align*}x\end{align*} in.

\begin{align*}\frac{4x}{3}\end{align*}

The answer is \begin{align*}\frac{4x}{3}\end{align*}.

#### Example 2

Simplify the following expression.

\begin{align*}\frac{8xy}{4x}\end{align*}

There are like \begin{align*}x\end{align*}’s in both the numerator and denominator.

First, simplify the \begin{align*}x\end{align*}’s to 1 because any number or variable divided by itself is equal to 1. Here is what you are left with.

\begin{align*}\begin{array}{rcl} & \frac{8 (1) y}{4(1)} \\ & \frac{8y}{4} \end{array}\end{align*}

Next, simplify the number values by dividing four into eight.

\begin{align*}8 \div 4 = 2\end{align*}

Then, replace the \begin{align*}y\end{align*}.

The answer is \begin{align*}2y\end{align*}.

#### Example 3

Simplify the following expression.

\begin{align*}5 \cdot (4 \cdot x)\end{align*}

Remember, the dots mean to multiply.

First, complete the operation in parentheses.

\begin{align*}4 \ \text{times}\ x = 4x\end{align*}

The expression becomes \begin{align*}5 \cdot (4x)\end{align*}.

Next, multiply the number parts of the two terms.

\begin{align*}5 \times 4 = 20\end{align*}

Then, keeping the ‘\begin{align*}x\end{align*}’ variable, the expression is simplified as \begin{align*}20x\end{align*}.

The answer is \begin{align*}20x\end{align*}.

#### Example 4

Simplify the following expression.

\begin{align*}\frac{6y}{3}\end{align*}

First, simplify the number values by dividing 6 by 3.

\begin{align*}6 \div 3 = 2\end{align*}

The answer is \begin{align*}2y\end{align*}.

#### Example 5

Simplify the following expression.

\begin{align*}\frac{14xy}{7y}\end{align*}

First, simplify the like \begin{align*}y\end{align*}’s in the numerator and denominator to 1 and cancel them out. Here is what you are left with.

\begin{align*}\begin{array}{rcl} & \frac{14 x(1)}{7(1)} \\ & \frac{14 x}{7} \end{array}\end{align*}

Next, simplify the number values by dividing 7 into 14.

\begin{align*}14 \div 7 = 2\end{align*}

The answer is \begin{align*}2x\end{align*}.

### Review

Simplify the following expressions.

- \begin{align*}5x(4)\end{align*}
- \begin{align*}3(5x)\end{align*}
- \begin{align*}4y(2)\end{align*}
- \begin{align*}2 \cdot ( 8 \cdot a)\end{align*}
- \begin{align*}4 \cdot (7x)\end{align*}
- \begin{align*}\frac{9y}{3y}\end{align*}
- \begin{align*}\frac{64x}{8x}\end{align*}
- \begin{align*}\frac{12xy}{4}\end{align*}
- \begin{align*}\frac{10 ab}{5 a}\end{align*}
- \begin{align*}\frac{18xy}{6y}\end{align*}
- \begin{align*}15x (4y)\end{align*}
- \begin{align*}7x(2)\end{align*}
- \begin{align*}12x (4y)\end{align*}
- \begin{align*}13x (4y)\end{align*}
- \begin{align*}12 x (7)\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.7.