# 8.5: Equal Costs

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

Carla and Dan went to the store. Carla bought 3 notebooks and a $2 pen. Dan bought 2 notebooks and an $8 stapler. All notebooks cost the same. They spent the same amount of money. Can you figure out the cost of one notebook? In this concept, we will learn how to use equations to solve problems having to do with equal costs.

### Guidance

In order to answer questions about equal costs like the one above, we can use the problem solving steps to help.

- First,
**describe**what you know. What did each person buy? - Second, identify what
**your job**is. In these problems, your job will be to solve for the price of an item. - Third, make a
**plan**. In these problems, your plan should be to write an expression for what each person spent. Then, set those expressions equal to each other since each person spent the same amount. Finally, solve the equation. - Fourth,
**solve**. Implement your plan. - Fifth,
**check**. Substitute your answer into the original problem and make sure it works.

#### Example A

Al bought 2 sandwiches. Bob bought one sandwich and a $4 large soda. All sandwiches cost the same. They spent the same amount of money. What is the cost of one sandwich?

- Use \begin{align*}b\end{align*} to stand for the cost of one sandwich.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}b\end{align*}.

- Show your work.

**Solution:**

We can use problem solving steps to help us with this problem.

\begin{align*}& \mathbf{Describe:} && \text{Al:} \ 2 \ \text{sandwiches}\\ &&& \text{Bob:} \ 1 \ \text{sandwich and a} \ \$4 \ \text{large soda}\\ &&& 2 \ \text{sandwiches} \ \text{cost the same as} \ 1 \ \text{sandwich and} \ \$4\\ \\ & \mathbf{My \ job:} && \text{Figure out the cost of one sandwich.}\\ \\ & \mathbf{Plan:} && \text{Use} \ b \ \text{to stand for the cost of one sandwich.}\\ &&& \text{Write an expression showing the money spent by each person.}\\ &&& \text{Since they spent the same amount of money, the two expressions are equal.}\\ &&& \text{Write the equation.}\\ &&& \text{Solve for} \ b.\\ \\ & \mathbf{Solve:} && \text{Al's cost:} \ 2b\\ &&& \text{Bob's cost:} \ b+4.\\ &&& \text{The costs are the same so} \ 2b=b+4. \ \text{Solve the equation.}\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 2b=b+4\\ &&& \text{Subtract} \ b \ \text{from each side.} \qquad \qquad 2b-b=b+4-b\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ b=4\\ &&& \text{One sandwich is} \ \$4\\ \\ & \mathbf{Check:} && \text{Al:} \ 2 \ \text{sandwiches is} \ 2 \times \$4, \ \text{or} \ \$8.\\ &&& \text{Bob:} \ 1 \ \text{sandwich and a} \ \$4 \ \text{large soda is} \ 1 \times \$4+\$4, \ \text{or} \ \$8.\\ &&& \$8= \$8\end{align*}

#### Example B

Camilla bought 4 small sodas and a $2 cookie. Darla bought 3 small sodas and a $5 dessert. All small sodas cost the same. They spent the same amount of money. What is the cost of one small soda?

- Use \begin{align*}c\end{align*} to stand for the cost of one small soda.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}c\end{align*}.

- Show your work.

**Solution:**

We can use problem solving steps to help us with this problem.

\begin{align*}& \mathbf{Describe:} && \text{Camilla:} \ 4 \ \text{small sodas and a } \ \$2 \ \text{cookie}\\ &&& \text{Dana:} \ 3 \ \text{small sodas and a } \ \$5 \ \text{dessert}\\ &&& 4 \ \text{small sodas and } \ \$2 \ \text{costs the same as} \ 3 \ \text{notebooks and} \ \$5\\ \\ & \mathbf{My \ job:} && \text{Figure out the cost of one small soda.}\\ \\ & \mathbf{Plan:} && \text{Use} \ c \ \text{to stand for the cost of one small soda.}\\ &&& \text{Write an expression showing the money spent by each person.}\\ &&& \text{Since they spent the same amount of money, the two expressions are equal.}\\ &&& \text{Write the equation.}\\ &&& \text{Solve for} \ c.\\ \\ & \mathbf{Solve:} && \text{Camilla's cost:} \ 4c+2\\ &&& \text{Dana's cost:} \ 3c+5.\\ &&& \text{The costs are the same so} \ 4c+2=3c+5. \ \text{Solve the equation.}\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 4c+2=3c+5\\ &&& \text{Subtract} \ 3c \ \text{from each side.} \qquad \qquad 4c+2-3c=3c+5-3c\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ c+2=5\\ &&& \text{Subtract} \ 2 \ \text{from each side.} \qquad \qquad \ \ c=3\\ &&& \text{One small soda is} \ \$3\\ \\ & \mathbf{Check:} && \text{Camilla:} \ 4 \ \text{small sodas and a} \ \$2 \ \text{cookie is} \ 4 \times \$3+\$2, \ \text{or} \ \$14.\\ &&& \text{Dana:} \ 3 \ \text{small sodas and a} \ \$5 \ \text{dessert is} \ 3 \times \$3+\$5, \ \text{or} \ \$14.\\ &&& \$14= \$14\end{align*}

#### Example C

Erin bought 6 muffins Fred bought 4 muffins and a $3 drink. All muffins cost the same. They spent the same amount of money. What is the cost of one muffin?

- Use \begin{align*}d\end{align*} to stand for the cost of one muffin.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}d\end{align*}.

- Show your work.

**Solution:**

We can use problem solving steps to help us with this problem.

\begin{align*}& \mathbf{Describe:} && \text{Erin:} \ 6 \ \text{muffins}\\ &&& \text{Fred:} \ 4 \ \text{muffins and a} \ \$3 \ \text{drink}\\ &&& 6 \ \text{muffins} \ \text{costs the same as} \ 4 \ \text{notebooks and} \ \$3\\ \\ & \mathbf{My \ job:} && \text{Figure out the cost of one muffin.}\\ \\ & \mathbf{Plan:} && \text{Use} \ d \ \text{to stand for the cost of one muffin.}\\ &&& \text{Write an expression showing the money spent by each person.}\\ &&& \text{Since they spent the same amount of money, the two expressions are equal.}\\ &&& \text{Write the equation.}\\ &&& \text{Solve for} \ d.\\ \\ & \mathbf{Solve:} && \text{Erin's cost:} \ 6d\\ &&& \text{Fred's cost:} \ 4d+3.\\ &&& \text{The costs are the same so} \ 6d=4d+3. \ \text{Solve the equation.}\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 6d=4d+3\\ &&& \text{Subtract} \ 4d \ \text{from each side.} \qquad \qquad 6d-4d=4d+3-4d\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ 2d=3\\ &&& \text{Divide each side by} \ 2. \qquad \qquad \ \ d=1.5\\ &&& \text{One muffin is} \ \$1.50\\ \\ & \mathbf{Check:} && \text{Erin:} \ 6 \ \text{muffins is} \ 6 \times \$1.50, \ \text{or} \ \$9.\\ &&& \text{Fred:} \ 4 \ \text{muffins and a} \ \$3 \ \text{drink is} \ 4 \times \$1.50+\$3, \ \text{or} \ \$9.\\ &&& \$9= \$9\end{align*}

#### Concept Problem Revisited

Remember the problem about Carla and Dan? Carla bought 3 notebooks and a $2 pen. Dan bought 2 notebooks and an $8 stapler. All notebooks cost the same. They spent the same amount of money. What is the cost of one notebook?

We can use problem solving steps to help us with this problem.

\begin{align*}& \mathbf{Describe:} && \text{Carla:} \ 3 \ \text{notebooks and a} \ \$2 \ \text{pen}\\ &&& \text{Dan:} \ 2 \ \text{notebooks and an} \ \$8 \ \text{stapler}\\ &&& 3 \ \text{notebooks and} \ \$2 \ \text{costs the same as} \ 2 \ \text{notebooks and} \ \$8\\ \\ & \mathbf{My \ job:} && \text{Figure out the cost of one notebook.}\\ \\ & \mathbf{Plan:} && \text{Use} \ a \ \text{to stand for the cost of one notebook.}\\ &&& \text{Write an expression showing the money spent by each person.}\\ &&& \text{Since they spent the same amount of money, the two expressions are equal.}\\ &&& \text{Write the equation.}\\ &&& \text{Solve for} \ a.\\ \\ & \mathbf{Solve:} && \text{Carla's cost:} \ 3a+2\\ &&& \text{Dan's cost:} \ 2a+8.\\ &&& \text{The costs are the same so} \ 3a+2=2a+8. \ \text{Solve the equation.}\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 3a+2=2a+8\\ &&& \text{Subtract} \ 2a \ \text{from each side.} \qquad \qquad 3a+2-2a=2a+8-2a\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ a+2=8\\ &&& \text{Subtract} \ 2 \ \text{from each side.} \qquad \qquad \ \ a+2-2=8-2\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ a=6\\ &&& \text{One notebook is} \ \$6.\\ \\ & \mathbf{Check:} && \text{Carla:} \ 3 \ \text{notebooks and} \ \$2 \ \text{pen is} \ 3 \times \$6+\$2, \ \text{or} \ \$20.\\ &&& \text{Dan:} \ 2 \ \text{notebooks and} \ \$8 \ \text{stapler is} \ 2 \times \$6+\$8, \ \text{or} \ \$20.\\ &&& \$20 = \$20\end{align*}

### Vocabulary

In math, an ** expression** is a phrase that can contain numbers, operations, and variables without an equal's sign. An

**is a statement that two**

*equation***are equal. An**

*expressions***is two**

*equation***combined with an equals sign. To**

*expressions***an equation means to figure out the value(s) for the variable(s) that make the equation true.**

*solve*### Guided Practice

1. Gary bought 5 CDs and a $5 CD case. Helen bought 3 CDs and a set of $29 earphones. All CDs cost the same. They spent the same amount of money. What is the cost of one CD?

- Use \begin{align*}f\end{align*} to stand for the cost of one CD.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}f\end{align*}.

- Show your work.

2. Ina bought 4 movie tickets and a $2 soda. Jen bought 2 movie tickets, a $4 bag of popcorn, a $3 drinks, and a $5 bag of candy. All movie tickets cost the same. They spent the same amount of money. What is the cost of one movie ticket?

- Use \begin{align*}g\end{align*} to stand for the cost of one movie ticket.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}g\end{align*}.

- Show your work.

3. Ken bought 10 pounds of apples. Larry bought 5 pounds of apples and a $10 jar of honey. Each pound of apples cost the same. They spent the same amount of money. What is the cost of one pound of apples?

- Use \begin{align*}j\end{align*} to stand for the cost of one pound of apples.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}j\end{align*}.

- Show your work.

**Answers:**

1. One CD costs $12. Here is the equation you should have written and the steps to solve:

\begin{align*}5f + 5 &= 3f + 29\\ 5f + 5 - 3f &= 3f + 29 - 3f\\ 2f + 5 &= 29\\ 2f + 5 - 5 &= 29 - 5\\ 2f &= 24\\ f &= 12\end{align*}

2. One movie ticket costs $5. Here is the equation you should have written and the steps to solve:

\begin{align*}4g + 2 &= 2g + 4 + 3 + 5\\ 4g + 2 &= 2g + 12\\ 4g + 2 - 2g &= 2g + 12 - 2g\\ 2g + 2 &= 12\\ 2g + 2 - 2 &= 12 - 2\\ 2g &= 10\\ g &= 5\end{align*}

3. One pound of apples costs $2. Here is the equation you should have written and the steps to solve:

\begin{align*}10j &= 5j + 10\\ 10j - 5j &= 5j + 10 - 5j\\ 5j &= 10\\ j &= 2\end{align*}

### Explore More

1. Hal bought 2 bagels and a large hot chocolate for $2.50. Jon bought 4 bagels and a $1.00 cream cheese. All bagels cost the same. They spent the same amount of money. What is the cost of one bagel?

- Use \begin{align*}a\end{align*} to stand for the cost of one bagel.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}a\end{align*}.

- Show your work.

2. Kaelyn bought 5 pads of paper and a $1.50 box of binder clips. Lexa bought 2 pads of paper and a $6.00 box of pens. All pads of paper cost the same. They spent the same amount of money. What is the cost of one pad of paper?

- Use \begin{align*}b\end{align*} to stand for the cost of one pad of paper.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}b\end{align*}.

- Show your work.

3. Mary bought 4 decks of cards and a $2.00 crossword puzzle book. Nina bought 2 decks of cards and 2 boxes of dominoes for $3.50 each. All card decks cost the same. They spent the same amount of money. What is the cost of one deck of cards?

- Use \begin{align*}c\end{align*} to stand for the cost of one deck of cards.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}c\end{align*}.

- Show your work.

4. Mark bought 5 sandwiches and a $2 bag of chips. Dave bought 3 sandwiches and a $10 pie. All sandwiches cost the same and they spent the same amount of money. What is the cost of one sandwich?

- Use \begin{align*}d\end{align*} to stand for the cost of one sandwich.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}d\end{align*}.

- Show your work.

5. Jess bought 2 boxes of paper clips and a $1 notepad. John bought one box of paper clips and a $1.75 pen. They spent the same amount of money. What is the cost of one box of paper clips?

- Use \begin{align*}e\end{align*} to stand for the cost of one box of paper clips.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}e\end{align*}.

- Show your work.

6. Sarah bought 7 binders and a $4.50 pack of pens. Ben bought 8 binders. They spent the same amount of money. What is the cost of one binder?

- Use \begin{align*}f\end{align*} to stand for the cost of one binder.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}f\end{align*}.

- Show your work.

7. Bob bought 2 t-shirts and a $22.75 pair of pants. Jeff bought 3 t-shirts and a $12.50 hat. All t-shirts cost the same. They spent the same amount of money. What is the cost of one t-shirt?

- Use \begin{align*}g\end{align*} to stand for the cost of one t-shirt.

- Write an equation to represent the costs of the two people.

- Solve for the value of \begin{align*}g\end{align*}.

- Show your work.

### Image Attributions

Students are given information about what two people bought and told that they spent the same amount. Students set up equations to solve the price of an item. Students use problem solving steps to help.

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