7.2: Multiplication and Division Phrases as Equations
Have you ever been to a major league baseball game?
Kara and her twin brother Marc are going to be spending one month in Boston with their grandparents. They are very excited! Not only is it summer in Boston and the weather will be terrific, but Boston has a fantastic subway system called the T and they will be able to ride it to get all around.
Kara is very excited about visiting the museums, but Marc who is a huge baseball fan is hoping for a trip to Fenway Park before the month is up. Seeing the Red Sox play would be a huge bonus!
On the first day, the twins’ Grandpa told them that they were going to spend the day learning how to ride the subway. Given that the twins are 14 and that they will be traveling together, Grandpa feels comfortable that they will be fine. After a quick discussion on safety, they were ready to go.
Grandpa said that each ride on the subway would cost .85 for a teenager. They would be riding to the Boston Common and back, and they may take another ride too.
Kara goes upstairs to get her money and tries to figure out how much they will need in all. She is having a tough time because she doesn’t know how many train rides they will take in all. She tells Marc what she is trying to figure out.
“Well until we know the number of rides, you can’t figure out the total,” he says. “But you can write an expression.”
An expression? How can Kara do this? What does an expression mean? Why would you want one?
This Concept is all about expressions and equations. Learning how to write expressions can be very helpful when you have a variable that can change such as the number of train rides. Pay attention and at the end of the Concept you will be able to help Kara write an expression to show the amount of money needed for multiple train rides.
Guidance
Just as you can write addition and subtraction expressions from words or phrases, you can also write multiplication and division expressions. Once again, you can use key words to help you with this. The more familiar you become with the key words that identify a multiplication or division expression, the better you will become at writing expressions.
Here are some statements of how words or phrases can be translated into multiplication or division expressions.
Multiplication Expressions | Division Expressions | ||
---|---|---|---|
9 times \begin{align*}k\end{align*} | \begin{align*}9 \times k \ \ \text{or} \ \ 9k\end{align*} | 8 divided into \begin{align*}n\end{align*} groups | \begin{align*}8 \div n \ \ \text{or} \ \ \frac{8}{n}\end{align*} |
twice as much as \begin{align*}m\end{align*} | \begin{align*}2 \times m \ \ \text{or} \ \ 2m\end{align*} | \begin{align*}q\end{align*} shared equally by 3 people | \begin{align*}q \div 3 \ \ \text{or} \ \ \frac{q}{3}\end{align*} |
half of \begin{align*}r\end{align*} | \begin{align*}r \div 2 \ \ \text{or} \ \ \frac{r}{2}\end{align*} | ||
one-third of \begin{align*}p\end{align*} | \begin{align*}p \div 3 \ \ \text{or} \ \ \frac{p}{3}\end{align*} |
The bolded key words in the phrases above provide clues about whether or not you should write a multiplication or a division expression. Remember, key words can be a helpful guide, but you should always think about which operation makes sense for a particular situation.
Write these key words in your notebook and then continue with the Concept.
Write an algebraic expression to represent this phrase: 3 times a number, \begin{align*}t\end{align*}.
The phrase is “3 times a number, \begin{align*}t\end{align*}.” Use a number, an operation sign, or a variable to represent each part of that phrase.
\begin{align*}& \underline{3} \ \underline{times} \ a \ \underline{number, \ t}\\ & \downarrow \quad \ \downarrow \qquad \quad \downarrow\\ & \ 3 \quad \times \qquad \ \ t\end{align*}
Notice that in this phrase, the key word “times” means you should write a multiplication expression.
So, the expression \begin{align*}3 \times t\end{align*} or \begin{align*}3t\end{align*} represents the phrase. We also could have written this as \begin{align*}t \times 3\end{align*} because multiplication is commutative. That means, the order in which numbers are multiplied does not matter.
Here is another one.
Mr. Warren separated 30 students into \begin{align*}n\end{align*} equal groups. Write an algebraic expression to represent the number of students in each group.
The phrase is “separated 30 students into \begin{align*}n\end{align*} equal groups.”
Think about what makes sense. Separating 30 students into \begin{align*}n\end{align*} equal groups means dividing 30 students into \begin{align*}n\end{align*} equal groups. So, write a division expression.
\begin{align*}& separated \ \underline{30 \ students} \ \underline{into} \ \underline{n \ equal \ groups}\\ & \qquad \qquad \qquad \downarrow \qquad \qquad \downarrow \qquad \qquad \downarrow\\ & \qquad \qquad \qquad 30 \qquad \quad \ \div \qquad \quad \ \ n\end{align*}
Division is not commutative. The order in which we divide numbers matters. So, while \begin{align*}30 \div n\end{align*} represents the number of students in each group, \begin{align*}n \div 30\end{align*} does not.
The answer is \begin{align*}30 \div n\end{align*}.
Sometimes an expression becomes an equation when there is a quantity that it equals.
\begin{align*}30 \div n = 5\end{align*}
This is an equation that we can solve by using the inverse operation. But this Concept is about writing equations not solving them, so we can stop here.
Write a multiplication or division equation for each phrase.
Example A
Four times a number is eight
Solution:\begin{align*}4x = 8\end{align*}
Example B
Sixteen divided into a number of groups is two.
Solution: \begin{align*}\frac{16}{x} = 2\end{align*}
Example C
The product of five and a number is fifteen.
Solution: \begin{align*}5x = 15\end{align*}
Here is the original problem once again.
On the first day visiting their grandparents, the twins were told them that they were going to spend the day learning how to ride the subway. Given that the Kara and Marc are 14 and that they will be traveling together, Grandpa feels comfortable that they will be fine. After a quick discussion on safety, they were ready to go.
Grandpa said that each ride on the subway would cost .85 for a teenager. They would be riding to the Boston Common and back, and they may take another ride too.
Kara goes upstairs to get her money and tries to figure out how much they will need in all. She is having a tough time because she doesn’t know how many train rides they will take in all. She tells Marc what she is trying to figure out.
“Well until we know the number of rides, you can’t figure out the total,” he says. “But you can write an expression.”
An expression? How can Kara do this? What does an expression mean? Why would you want one?
Now that you know what an expression is, you can write one to help Kara figure out the amount of money depending on the number of train rides.
The amount of money per ride does not vary. It costs .85 per ride for a teenager.
The number of train rides does vary. This is where a variable is very useful. It can change according to the number of train rides. Let’s use \begin{align*}x\end{align*}.
The expression is \begin{align*}.85x\end{align*}.
If Kara changes the variable \begin{align*}x\end{align*} according to the number of train rides that she and Marc take, then she can figure out the cost per day of riding the train.
If they go on four train rides for example, here is the expression.
.85(4)
Evaluating this expression, the cost would be $3.40.
Expressions are very helpful for figuring out different problems with changeable parts like this one.
Vocabulary
Here are the vocabulary words in this Concept.
- Expression
- a number sentence with variables, numbers and operations.
- Variable
- a letter used to represent an unknown quantity.
- Algebraic Expression
- a combination of multiple variables, numbers and operations.
- Equation
- a number sentence with an equal sign where the quantity on one side of the equals is the same as the quantity on the other side.
Guided Practice
Here is one for you to try on your own.
Write a single variable equation for the following phrase.
Keith bought tickets to the movies. The tickets were $8.50 each. Keith spent a total of $34.00. Write an equation to show this dilemma.
Answer
To begin, we can identify our unknown as the number of tickets Keith bought. Let's use \begin{align*}t\end{align*} to represent this unknown number.
The cost of each ticket was $8.50.
Now we can write an equation.
\begin{align*}8.50t = 34.00\end{align*}
This is our answer.
Video Review
Here is a video for review.
Practice
Directions: Write an equation for each phrase.
1. The product of four and a number is twelve.
2. Six times a number is thirty.
3. Twelve times a number is forty - eight.
4. Fourteen divided by a number is twenty -eight.
5. The product of five and a number is thirty.
6. Eight times a number is sixty-four.
7. Twenty divided by a number is four.
8. Eighty divided by a number is four.
9. Nineteen times an unknown number is ninety- five.
10. Thirteen times an unknown number is thirty -nine.
11. Twelve divided into groups is six.
12. An unknown number divided by two is eight.
13. An unknown number divided by seven is fourteen.
14. An unknown number times five is thirty- five.
15. An unknown number divided by twelve is twelve.
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Image Attributions
Here you'll learn to write multiplication and division phrases as single variable equations.