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Products and Quotients of Variable Expressions

Simplify expressions using multiplication and division.

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Products and Quotients of Variable Expressions
Credit: jeffreyw
Source: https://www.flickr.com/photos/jeffreyww/12509472093/in/photolist-k4qjek-osZXr8-bWKZtb-24yW9s-cHnTZ-soZAqp-5FcgB6-hM79qw-aef6PP-9cmbDu-5b1mUW-bV2Zgi-7tWkNz-4aMBad-81JJJ6-dVbYrV-abz33F-d8w44G-aiKifC-wcoQsb-9jWtQZ-6ixm63-csPz1w-nwGwBf-aZFaXi-9ohnMK-7dJNS5-f2sh5E-7YVJH4-MhLH-8mnQTF-Fm8Tq-4xdggs-piNLB-4Sjz1K-bPa4Jp-5x4bgb-deFGD4-hw5Hg7-4gCjEC-ozM26F-34uZs9-8vT7S5-bV2YHK-8yQWBQ-4zyZoA-kqXdxV-6981q-qrhbHX-bmt5FN
License: CC BY-NC 3.0

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*}3(5x)

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*}x 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*}3(5x)3×5=15

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

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

Let’s look at another example.

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

These dots mean to multiply.

First, complete the operation in parentheses.

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

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

Next, multiply the number parts of the two terms.

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

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

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

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

Here is another expression.

\begin{align*}\frac{4x}{2x}\end{align*}4x2x

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

\begin{align*}\frac{6}{6} =1\end{align*}66=1

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*}4x2x=42

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

Then, you are left with four divided by two.

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

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

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

divided by 9 girls becomes \begin{align*}\frac{\Box}{9}\end{align*}

\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.

  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{10 ab}{5 a}\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*}12 x (7)\end{align*} 

Review (Answers)

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

Resources

 

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Vocabulary

Product

The product is the result after two amounts have been multiplied.

Quotient

The quotient is the result after two amounts have been divided.

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