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# Mathematical Expressions

## Write variable expressions to represent word problems

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Practice Mathematical Expressions
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Translate Verbal Phrases into Variable Expressions

Have you ever had to figure out a math problem that was described in words? Look at this dilemma.

Kelly and her brother sold lemonade and cookies at the school fair. They sold the lemonade for two dollars per glass and the cookies for one dollar and fifty cents a piece. When finished, Kelly realized that they had sold fifty glasses of lemonade and twenty cookies. She said this to her brother.

"We sold fifty times two dollars and twenty times one dollar and fifty cents."

Kelly's brother isn't sure how to write this expression. Pay attention and you will be able to help him at the end of the Concept.

### Guidance

Do you know how to take a verbal phrase and write it as a variable expression?

To accomplish this task, you will need to think about what different words mean. A verbal expression is a mathematical statement that is expressed in words.

You will have to work as a detective to figure out what different words mean. Once you know what those words mean, you will be able to write different variable expressions.

Let’s start by looking at some mathematical operations written as words.

Addition

Sum

Plus

Increased by

More

Subtraction

Difference

Less than

Take away

Multiplication

Product

Times

Division

Quotient

Split up

This list does not include ALL of the ways to write the operations, but it will give you a good place to start.

Take a few minutes and write these words down in your notebook.

Now we can look at the following chart which starts with a verbal phrase and writes it as a variable expression.

Verbal Phrase Variable Expression
Three minus a number \begin{align*}3 - x\end{align*}
A number increased by seven \begin{align*}n + 7\end{align*}
The difference between an unknown quantity and twenty-six \begin{align*}s - 26\end{align*}
A number decreased by nine \begin{align*}w - 9\end{align*}
Ten times a number plus four \begin{align*}10f + 4\end{align*}

Notice that words like “a number” and “an unknown quantity” let us know that we need to use a variable.

#### Example A

Write a variable expression that reads “The product of a number and six plus four.”

Solution: \begin{align*}6x+4\end{align*}

#### Example B

Write a variable expression that reads “Ninety divided by a number minus eight.”

Solution: \begin{align*}\frac{90}{b}-8\end{align*}

#### Example C

Write a variable expression that reads “Two less than a number, multiplied by thirty-six.”

Solution: \begin{align*}36(n - 2)\end{align*}

Now let's go back to the dilemma from the beginning of the Concept. Kelly explained the sales to her brother in this way.

"We sold fifty times two dollars and twenty times one dollar and fifty cents."

First, use the information in the statement to write an expression.

\begin{align*}50(2.00) + 20(1.50)\end{align*}

Notice that we have fifty times two dollars plus twenty times one dollar and fifty cents. This shows the number of glasses of lemonade and cookies times each price.

Next, we can figure out how much money they made.

\begin{align*}130.00\end{align*}

This is our final answer.

### Vocabulary

Variable Expression
a group of numbers, operations and variables without an equal sign.
Variable
a letter used to represent an unknown number.
Constant
a number in an expression that does not have a variable.
Verbal Expression
using language to write a mathematical expression instead of numbers, symbols and variables.

### Guided Practice

Here is one for you to try on your own.

Write a variable expression that reads “Eighty-five divided by a number minus thirteen.”

Solution

We could do this in several different ways. We could use a symbol, \begin{align*}\div\end{align*}, to show division or we could use a fraction bar to show division.

Because you are moving toward Algebra, let’s use a fraction bar.

The answer is \begin{align*}\frac{85}{a}-13\end{align*}.

### Practice

Directions: Write a variable expression for each verbal expression.

1. The sum of a number and twelve.

2. The difference between a number and eight.

3. Three times a number

4. A number squared plus five

5. A number divided by two plus seven

6. Four times the quantity of a number plus six

7. A number times two divided by four

8. A number times six plus the same number times two

9. A number squared plus seven take a way four

10. A number divided by three plus twelve

11. A number times five and another number times six

12. Sixteen less than a number times negative four

13. A number times eight divided by two

14. A number divided by six and another number times negative five

15. A number divided by four plus another number divided by sixteen

### Vocabulary Language: English

Evaluate

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.
Variable

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.
Verbal Expression

Verbal Expression

A verbal expression (or verbal model) uses words to decipher the mathematical information in a problem. An equation can often be written from a verbal model.

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