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

**At Grade**Created by: CK-12

**Practice**Expressions for Real-Life Situations

For her birthday, Sara received a ticket to go to the ballet to see The Nutcracker. Sara needs to figure out the cost for the tickets for her friends, but she isn’t sure exactly how many of her friends will be going, it could be three or four. The cost for one ticket is $35.00, and there is a $2.00 one-time fee for the ticket purchase. Sara needs to know the price if she buys three tickets and also if she buys four tickets. If Sara could write a variable expression to model her problem, how could she use it to figure out the cost of the tickets?

In this concept, you will learn how to write a real world problem as a variable expression and how to evaluate the expression you have written using the order of operations.

### Variable Expressions and Order of Operations

When you have an unknown quantity in a problem, you can use a **variable** to represent the unknown quantity and then use it to evaluate the problem. In the real world there are many scenarios to which you can apply what you know about variable expressions and evaluating them.

Let’s look at an example.

Many events require that you purchase a ticket for admission. Tickets can have various prices if tickets are sold for adults, seniors or children. If you know the number and type of tickets being purchased and the price for each ticket, then you can figure out the total cost of the tickets.

An amusement park charges each person eight dollars admission and one dollar and fifty-cents per ride. Write a variable expression to model this situation and use it to find the cost of admission and five ride tickets.

First, notice there is a cost of eight dollars for admission. You can start with the number $8.00

Next, notice there is a charge of $1.50 per ride. Therefore you would have to add \begin{align*}1.50 \ x\end{align*}, where \begin{align*}x\end{align*} is the number of rides, to the admission cost.

Then, write a variable expression you can use to determine the cost of going to the amusement park and enjoying the rides.

\begin{align*}\$ 8.00 + \$ 1.50 \ x\end{align*}

Remember the variable ‘\begin{align*}x\end{align*}’ * *represents the number of rides. You can now evaluate the variable expression to find the cost of going to the amusement park and going on five rides.

First substitute \begin{align*}x=5 \end{align*} into \begin{align*}8.00+1.50 \ x\end{align*} and write the new expression.

\begin{align*}8.00+1.50(5)\end{align*}

Use the order of operations to evaluate the expression.

Multiply:\begin{align*}1.50\times 5=7.50\end{align*} to clear parenthesis and write the new expression.

\begin{align*}8.00+7.50\end{align*}

Next, add: \begin{align*}8.00+7.50=\$ 15.50\end{align*}

The answer is $15.50.

Let’s look at one more problem.

An ice cream cone costs $3.50 plus an additional $1.25 for chocolate dip. Write an expression to model the cost of two chocolate dip ice cream cones.

You must write a numerical expression to model the real-world problem.

\begin{align*}2(3.50+1.25)\end{align*}

Use the order of operations to evaluate the expression.

First, perform the indicated operation inside the parenthesis.

Next, add:\begin{align*}3.50+1.25=4.75\end{align*} and write the new expression.

\begin{align*}2(4.75)\end{align*}

Then, multiply: \begin{align*}2 \times 4.75=9.50\end{align*}

The answer is $9.50.

### Examples

#### Example 1

Earlier, you were given a problem about Sara and her ballet tickets. She needs to know the prices to purchase three tickets and four tickets. The price of one ticket is $35.00 and there is a $2.00 processing fee.

To determine the cost of the tickets Sara has to write a variable expression.

First, determine the variable. The variable will be the number of tickets purchased since that is the changeable amount. Let ’\begin{align*}x\end{align*}’ be the variable.

Remember the price of $35.00 of must be included with the variable.

Next, write this part of the expression\begin{align*}35x\end{align*}.

Then, include the onetime fee of $2.00 and write the variable expression.

The answer is \begin{align*}35x+2\end{align*}.

Sara needs to use the variable expression and the order of operations to calculate the cost of 3 tickets and of 4 tickets.

First, substitute \begin{align*}x=3\end{align*} into \begin{align*}35x+2\end{align*} and write the new expression.

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

Next, multiply: \begin{align*}35 \times 3 = 105\end{align*} to clear parenthesis and write the new expression.

\begin{align*}105+2\end{align*}

Then, add:\begin{align*}105+2= \$107.00\end{align*}

The answer is $107.00

First, substitute \begin{align*}x=4\end{align*} into \begin{align*}35x+2\end{align*} and write the new expression.

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

Next, multiply:\begin{align*}35 \times 3 = 140\end{align*} to clear parenthesis and write the new expression.

\begin{align*}140+2\end{align*}

Then, add:\begin{align*}140+2= \$ 142.00\end{align*}

The answer is $142.00

Sara will have to pay $107.00 for three tickets or $142.00 for four tickets.

#### Example 2

Harriet is making an almond cake for her sister and needs some ingredients. When she arrives at the store, Harriet sees that a can of evaporated milk costs $1.99 and almonds are $3.99 per pound.

Write a variable expression to model this problem so that Harriet can calculate her total cost based on how many pounds of almonds she purchases. Use your variable expression to calculate the cost of her purchase if Harriet buys 3 pounds of almonds.

First, determine the variable. The variable is going to be the number of pounds of almonds purchased since that is the changeable amount. Let ‘\begin{align*}x\end{align*}’ be the variable.

Remember the price of \begin{align*}$3.99\end{align*} must be included with the variable.

Next, write this part of the expression \begin{align*}3.99 \ x\end{align*} .

Harriet isn’t only purchasing almonds; she is also buying a can of evaporated milk. You must include this cost in your variable expression.

Then, add the cost of the can of milk to the first part of your expression:

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

The answer is \begin{align*}3.99 \ x + 1.99\end{align*}

First, substitute \begin{align*}x = 3 \ \text{into} \ 3.99 \ x + 1.99\end{align*}and write the new expression.

\begin{align*}3.99(3)+1.99\end{align*}

Use the order of operations to evaluate the expression.

Next, multiply:\begin{align*}3.99 \times 3 = 11.97\end{align*} to clear parentheissi and write the new expression.

\begin{align*}11.97+1.99\end{align*}

Then, add: \begin{align*}11.97+1.99=13.96\end{align*}

The answer is $13.96.

#### Example 3

The cost for three people to fly from Sydney, Nova Scotia to Calgary, Alberta is $1535.75. The airline also applies an additional charge of $25.00 for each piece of checked luggage. Write an expression to model the problem.

You must write a variable expression to model the problem.

\begin{align*}1535.75 + 25x\end{align*}where ‘\begin{align*}x\end{align*}*’ *represents the number of pieces of checked luggage.

Use your variable expression to figure out the total cost of the trip if 5 pieces of luggage are checked.

First, substitute \begin{align*}x=5\end{align*} into \begin{align*}1535.75+25x\end{align*} and write the new expression.

\begin{align*}1535.75+25(5)\end{align*}

Use the order of operations to evaluate the expression.

Next, multiply: \begin{align*}25 \times 5 = 125\end{align*}to clear parenthesis and write the new expression.

\begin{align*}1535.75+125\end{align*}

Then, add: \begin{align*}1535.75+125=1660.75\end{align*}

The answer is $1660.75.

#### Example 4

Maria wants to sew a new valence for her living room window. The window used to be one pane of glass 5 feet wide but the new window has a center section that is 5 feet wide and two smaller sections on either side. The new sections are each 3 feet long. The length of the valence must be double the width of the window. Write a variable expression to model the problem.

\begin{align*}3(5+2x) \end{align*}where ‘\begin{align*}x\end{align*}’ represents the width of two smaller sections of the window.

Since the width of each of the smaller window sections is 3 three feet, evaluate the variable expression when \begin{align*}x=3\end{align*}.

First, substitute the value \begin{align*}x=3\end{align*} into the expression.

\begin{align*}3(5+2(3))\end{align*}

Use the order of operations to evaluate the variable expression.

Next, perform the operations inside the parenthesis.

Multiply:\begin{align*}2(3)=6\end{align*} and write the new expression.

\begin{align*}3(5+6)\end{align*}

Next, add: \begin{align*}5+6=11\end{align*} and write the new expression.

\begin{align*}3(11)\end{align*}

Then, multiply \begin{align*}3(11)=33\end{align*} to clear the parenthesis.

The answer is 33.

### Review

Write a variable expression for each situation described below.

- A pound of apples costs $4.50. Kelly isn’t sure how many pounds she is going to purchase.
- Each touch down at a football game is worth 7 points. Write a variable expression where the number of touchdowns scored can change.
- Alex bought several bunches of bananas. Each bunch costs $.89. Write a variable expression to describe this situation.
- 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.
- 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.
- A turkey costs $6.75 per pound. Write a variable expression where the weight of the turkey is the changeable amount.
- 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.
- 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.

Now go back to each of the variable expressions that you have written as the answers for the above questions and evaluate each expression using 4 as the given value for the variable. These are your answers for numbers 9–16.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.5.

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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.### Image Attributions

In this concept, you will learn how to write a real world problem as a variable expression and how to evaluate the expression you have written using the order of operations.