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1.5: Connect Variable Expressions and the Order of Operations with Real-World Problems

Difficulty Level: At Grade Created by: CK-12
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Have you ever been to a ballet? Sara is going to see the "Nutcracker", but she needs to figure out the cost for tickets. She isn't sure if three or four of her friends are going to join her at the performance. The cost for one ticket is $35.00, and there is a $2.00 one time fee for the ticket purchase.

Can you write a variable expression to explain this situation? Can you figure out the total cost if three people attend the performance? Can you figure out the total cost if four people attend the performance?

Pay attention to this Concept, and you will be able to help Sara at the end of the Concept.


When we have an unknown quantity in a problem, we can use a variable to help us to find the value of our problem. You can find many real-world scenarios where variable expressions would be helpful in solving problems.

Let’s think about ticket pricing.

Many different places sell tickets. There are also prices for adults and prices for children. The number of tickets can vary, or they can be the same. If we know the number of tickets, then we can figure out the total amount of revenue earned based on the cost of the ticket for an adult and for a child.

Take a look at this situation.

An amusement park charges eight dollars admission and one dollar and fifty-cents per ride. Write an expression to find the cost of admission and five ride tickets.

First, notice that we have eight dollars admission. That is the first part of the expression.


Next, they charge $1.50 per ride.

\begin{align*}8 + 1.50x\end{align*}

We used the variable for the number of rides since this is the variable or changeable facet. The number of rides can change. In this example, we were asked to figure out the cost for 5 rides. That is the value that we can substitute into our expression for \begin{align*}x\end{align*}.

\begin{align*}8 + 1.50(5)\end{align*}

Now we can use the order of operations to solve this problem.

\begin{align*}& 8 + 7.50 \\ & 15.50\end{align*}

The cost of admission plus five ride tickets is $15.50.

Using variable expressions can help you solve for unknown quantities. Just remember to use the order of operations so that your work is accurate!!

Example A

An ice cream cone costs $3.50 plus an additional .35 for sprinkles. Write an expression to show how you would figure out the cost for two ice cream cones with sprinkles.

Solution: \begin{align*}2(3.50 + .35)\end{align*}

Example B

How much would it cost for two cones?

Solution: \begin{align*}$7.70\end{align*}

Example C

How much would it cost for five cones with sprinkles?

Solution: \begin{align*}$19.25\end{align*}

Now let's go back to the dilemma at the beginning of the Concept. Let's think about what we know.

Sara knows the cost for one ticket.


She isn't sure how many people are going. That is our variable. Let's use \begin{align*}x\end{align*}.


But there is also a \begin{align*}$2.00\end{align*} one time charge. Let's add that to our expression.

\begin{align*}35x + 2\end{align*}

Now we can figure out the cost for three people by substituting three in for \begin{align*}x\end{align*}.

\begin{align*}35(3) + 2 = $107\end{align*}

We can also figure out the cost for four people.

\begin{align*}35(4) + 2 = $142\end{align*}

Our work is complete.


to figure out the value of a numerical or variable expression.
a mathematical statement with an equals sign where one side of the equation has the same value as the other side.
a group of numbers, symbols and variables that represents a quantity.
Numerical Expression
a group of numbers and operations.
Variable Expression
a group of numbers, operations and at least one variable.
a letter used to represent an unknown quantity.
Grouping Symbols
parentheses or brackets used to group numbers and operations.

Guided Practice

Here is one for you to try on your own.

Harriet is making a cake. She needs to go to the store and purchase some almonds and a can of evaporated milk. When she arrives at the store, she picks up the can of evaporated milk first. It costs $1.99. Then she goes to get the almonds. Almonds are $3.99 per pound.

Can you write a variable expression so that Harriet can calculate her total cost based on how many pounds she purchases?


To accomplish this task, first let's figure out the variable. The variable is going to be the number of pounds purchased since that is the changeable amount. The price per pound is $3.99. Now let's write the first part of the variable expression.


Harriet isn't only purchasing almonds, she is also buying a can of evaporated milk. We can add that cost to our variable expression.

\begin{align*}3.99x + 1.99\end{align*}

This is our final answer.

Video Review

[ http://www.khanacademy.org/video?v=piIcRV2dx7E Khan Academy More Complicated Example with Order of Operations]


Directions:Write a variable expression for each situation described below.

1. A pound of apples costs $4.50. Kelly isn't sure how many pounds she is going to purchase.

2. Each touchdown at a football game is worth 7 points. Write a variable expression where the number of touchdowns scored can change.

3. Alex bought several bunches of bananas. Each bunch costs $.89. Write a variable expression to describe this situation.

4. If an ice cream cone costs $3.20 and hot fudge is an additional $.45, write a variable expression where the number of ice cream cones can change.

5. There are six children in the Smith family. If each child gets a haircut, write a variable expression to show the cost of the haircut so that the total cost can be calculated.

6. A turkey costs $6.75 per pound. Write a variable expression where the weight of the turkey is the changeable amount.

7. The car wash charges $15.00 per car. Write a variable expression that can be used to calculate the number of cars washed in one hour.

8. Kelly bought a pair of sneakers for $35.00. She also bought a pile of different laces. Each set of laces costs $3.00. Write a variable expression to show how Kelly could calculate her total cost.

Directions:Now go back to each variable expression that you wrote in numbers 1 - 8 and evaluate each expression as if the given value for the variable was 4. These are your answers for numbers 11 - 16.

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Algebraic Expression An expression that has numbers, operations and variables, but no equals sign.
Equation An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
Evaluate To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.
Expression An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
Grouping Symbols Grouping symbols are parentheses or brackets used to group numbers and operations.
Numerical expression A numerical expression is a group of numbers and operations used to represent a quantity.

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Date Created:
Dec 19, 2012
Last Modified:
Jan 26, 2017
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