# 12.7: Products and Quotients of Variable Expressions

**At Grade**Created by: CK-12

**Practice**Products and Quotients of Variable Expressions

Have you ever eaten lunch at an amusement park?

Well, Kelly and Keith are in line at the snack bar at amusement park. When they get to the counter, they order two of the hot dog specials which includes a hot dog, fries and a soda. There are other specials as well.

The cost of the special varies depending on which special you order.

Kelly and Keith both want to order the same special, but aren't sure which one to order.

If three pairs of people behind them also order this special, can you write an expression that includes all of this information?

**This Concept will teach you how to write variable expressions with multiplication as well as how to simplify them.**

### Guidance

In an earlier Concept, you learned how to simplify expressions. You simplified expressions that had addition and subtraction in them. Now it is time to simplify expressions with multiplication and division in them. Let’s start by reviewing what we mean by the word “simplify.”

**To** *simplify***means to make smaller**. When simplifying an expression, we don’t evaluate the expression, we just simplify it. To evaluate an expression we must have a given value for the variable. If you have not been given a value for the variable, then you will be simplifying the expression.

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

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

3 \begin{align*}\times\end{align*} 5 \begin{align*}=\end{align*} 15

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

Let’s look at another one.

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

These dots mean to multiply. First, we complete the operation in parentheses.

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

Next we multiply the number parts of the two terms.

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

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

**We can also simplify quotients. Remember that a quotient means division.**

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

**That is a great question! Do you remember back to fractions? You know that a fraction over itself is equal to one.**

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

**We call that simplifying a fraction. Well, we can simplify the variables too when we divide.**

**That is the first step. When simplifying a quotient with a variable expression, simplify the variables first.**

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

\begin{align*}x\end{align*} divided by \begin{align*}x\end{align*} is equal to one. Since 4 x 1 = 4, and 2 x 1 = 2, those x's really cancel each other out. Then we are left with four divided by two.

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

**The answer is 2.**

Now practice a few of these on your own.

#### Example A

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

**Solution: \begin{align*}20x\end{align*}**

#### Example B

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

**Solution: \begin{align*}2\end{align*}**

#### Example C

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

**Solution: \begin{align*}2x\end{align*}**

Here is the original problem once again.

Have you ever eaten lunch at an amusement park?

Well, Kelly and Keith are in line at the snack bar at amusement park. When they get to the counter, they order two of the hot dog specials which includes a hot dog, fries and a soda. There are other specials as well.

The cost of the special varies depending on which special you order.

Kelly and Keith both want to order the same special, but aren't sure which one to order.

If three pairs of people behind them also order this special, can you write an expression that includes all of this information?

First, let's write the part of the expression that only includes Kelly and Keith.

\begin{align*}2y\end{align*}

Why \begin{align*}y\end{align*}?

We use \begin{align*}y\end{align*} to represent the unknown cost of the special. Since the cost can change depending upon which special is chosen.

If three more pairs of people buy the same special, what would the expression look like?

\begin{align*}3(2y)\end{align*}

Now we can simplify this expression.

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

**This is our answer.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Product
- the answer to a multiplication problem

- Quotient
- the answer to a division problem

### Guided Practice

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

**Answer**

Here we have like \begin{align*}x\end{align*}’s in both the numerator and denominator. We can simplify those to one and cancel them out. Here is what we are left with.

\begin{align*}\frac{8\bcancel{x}y}{4\bcancel{x}} = \frac{8y}{4}\end{align*}

Now we can’t do anything with the \begin{align*}y\end{align*}, so we leave it alone.

We can divide four into eight.

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

Don’t forget to add the \begin{align*}y\end{align*} in.

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

### Video Review

Here is a video for review.

Khan Academy: CA Algebra I: Simplifying Expressions

### Practice

Directions: Simplify the products and quotients of the following single-variable expressions.

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

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

3. \begin{align*}4y(2)\end{align*}

4. \begin{align*}2 \cdot (8 \cdot a)\end{align*}

5. \begin{align*}4 \cdot (7x)\end{align*}

6. \begin{align*}\frac{9y}{3y}\end{align*}

7. \begin{align*}\frac{64x}{8x}\end{align*}

8. \begin{align*}\frac{12xy}{4}\end{align*}

9. \begin{align*}\frac{10ab}{5a}\end{align*}

10. \begin{align*}\frac{18xy}{6y}\end{align*}

11. \begin{align*}15x(4y)\end{align*}

12. \begin{align*}7x(2)\end{align*}

13. \begin{align*}12x(4y)\end{align*}

14. \begin{align*}13x(4y)\end{align*}

15. \begin{align*}12x(7)\end{align*}

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### Image Attributions

Here you'll learn to simplify products and quotients of variable expressions.