12.2: Multiplication and Division Phrases as Expressions
Mrs. Hawk's class hasn't been able to decide which trip to take yet. But they are leaning towards the amusement park.
The amusement park has an cost of $6.50 per student for admission.
A couple of students aren't sure that they will be able to go on the trip.
The exact number of students attending the trip is unknown.
Do you have enough information to write an expression to describe this situation?
This is a tricky question that will be answered in this Concept.
Guidance
In the last Concept, you worked on writing addition and subtraction phrases as variable expressions. In this Concept, you will write multiplication and division phrases and single-variable expressions.
Let’s start by looking at some of the key words that mean multiplication and division.
Multiplication
Product, times, groups
Division
Split up, quotient, divided
When deciphering phrases, you will be looking for the same things as the last Concept.
- Identify any numbers
- Identify the operation involved
- Identify the variable
The product of eight and a number
First, identify any numbers. The number in this phrase is eight.
8
Next, identify the operation. The word “product” means multiply.
\begin{align*}\times\end{align*}
Finally, identify the variable. “A number” means use a variable.
\begin{align*}y\end{align*}
Next, we write the expression.
\begin{align*}8y\end{align*}
Remember that a number next to a variable means multiply. We could also have used parentheses to show multiplication.
Now let’s look at a phrase with division.
Twenty-four divided by a number
First, identify any numbers.
24
Divided \begin{align*}\div\end{align*}
A number means use a variable \begin{align*}y\end{align*}
Now put it all together to write an expression
\begin{align*}24 \div y\end{align*}
We could also use a fraction bar to show division. \begin{align*}\frac{24}{y}\end{align*}
Practice writing expressions from the following phrases.
Example A
Six times an unknown number
Solution: \begin{align*}6x\end{align*}
Example B
An unknown number divided by two
Solution: \begin{align*}\frac{x}{2}\end{align*}
Example C
The product of seven and a number
Solution: \begin{align*}7y\end{align*}
Here is the original problem once again.
Mrs. Hawk's class hasn't been able to decide which trip to take yet. But they are leaning towards the amusement park.
The amusement park has an cost of $6.50 per student for admission.
A couple of students aren't sure that they will be able to go on the trip.
The exact number of students attending the trip is unknown.
Do you have enough information to write an expression to describe this situation?
We do have enough information to solve this problem, but we can't solve it for the total cost until we know how many students are going on the trip.
The number of students is the variable. We can call that \begin{align*}y\end{align*}.
We multiply the number of students times the cost.
\begin{align*}6.50y\end{align*}
This is our answer.
Vocabulary
Here are the vocabulary words in this Concept.
- Expression
- a variable expression has variables or unknown quantities, numbers and operations without an equal sign.
- Variable
- a letter used to represent an unknown quantity in an expression or equation
Guided Practice
Here is one for you to try on your own.
Write an expression for the quantity six times an unknown number divided by 2.
Answer
This is tricky because there are two operations in this problem. Let's break it down.
The quantity six times an unknown number becomes \begin{align*}6x\end{align*}.
Divided by two is the next part. We can use a fraction bar for division.
\begin{align*}\frac{}{2}\end{align*}
Because it is divided by the quantity, we put the \begin{align*}6x\end{align*} on top of the fraction bar.
\begin{align*}\frac{6x}{2}\end{align*}
This is our answer.
Video Review
Here are videos for review.
James Sousa, Introduction to Variables and Variable Expressions
James Sousa, Example of Writing Variable Expressions
James Sousa, Example of Writing Basic Algebraic Expressions
James Sousa, Another Example of Writing Basic Algebraic Expressions
Practice
Directions: Write multiplication and division phrases as single-variable expressions.
1. The product of six and a number
2. A number divided by two
3. Fifteen divided by an unknown number
4. A number times seven
5. The product of ten and a number
6. Eighteen divided by a number
7. Twenty times a number
8. A number divided by three
9. An unknown number divided by twelve
10. An unknown number times sixteen
11. The product of five and an unknown quantity
12. The quantity six times an unknown number divided by three
13. The quantity four times an unknown number divided by seven
14. The quantity six times an unknown number divided by ten
15. The product of three and an unknown times four
Variable
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.Variable Expression
A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.Image Attributions
Here you'll learn to write multiplication and division phrases as single variable expressions.