8.6: Equal Costs
Equal Costs – Write and Solve Equations for Equal Cost Problems
Teacher Notes
In Equal Costs, students are given information about the purchases of two people who spend the same amount of money for groups of items. In each group some items are the same. Students write an expression for the purchases for each person, and set the expressions equal to one another to represent the equal costs. Students then solve the equation for the value of the unknown. This is the first problem set in which students encounter equations with unknowns and constants on both sides of the equal sign. They add or subtract equal amounts from each side of the equation to figure out the value of the unknown.
Solutions:

\begin{align*}2b = b + 4; 2b  b = b + 4  b; b = 4\end{align*}
2b=b+4;2b−b=b+4−b;b=4 ; one sandwich costs $4. 
\begin{align*}4c + 2 = 3c + 5; 4c + 2  3c = 3c + 5  3c; c + 2 = 5; c + 2  2 = 5  2; c = 3;\end{align*}
4c+2=3c+5;4c+2−3c=3c+5−3c;c+2=5;c+2−2=5−2;c=3; one small soda costs $3. 
\begin{align*}6d = 4d + 3; 6d  4d = 4d + 3  4d; 2d = 3; d = 1.5\end{align*}
6d=4d+3;6d−4d=4d+3−4d;2d=3;d=1.5 ; one muffin costs $1.50. 
\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*}
5f+5=3f+29;5f+5−3f=3f+29−3f;2f+5=29;2f+5−5=29−5;2f=24;f=12; one CD costs $12. 
\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*}
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; one movie ticket costs $5. 
\begin{align*}10j = 5j + 10; 10j  5j = 5j + 10  5j; 5j = 10; j = 2\end{align*}
10j=5j+10;10j−5j=5j+10−5j;5j=10;j=2 ; one pound of apples costs $2.
Equal Costs – Write and Solve Equations for Equal Cost Problems
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.
Use \begin{align*}a\end{align*}
Write an equation to represent the costs of the two people.
Solve for the value of \begin{align*}a\end{align*}.
Show your work.
What is the cost of one notebook?
\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+22a=2a+82a\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ a+2=8\\ &&& \text{Subtract} \ 2 \ \text{from each side.} \qquad \qquad \ \ a+22=82\\ &&& \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*}
1. 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.
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.
What is the cost of one sandwich?
2. 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.
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.
What is the cost of one small soda?
3. Erin bought 6 muffins
Fred bought 4 muffins and a $3 drink.
All muffins cost the same.
They spent the same amount of money.
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.
What is the cost of one muffin?
4. 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.
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.
What is the cost of one CD?
5. 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.
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.
What is the cost of one movie ticket?
6. 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.
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.
What is the cost of one pound of apples?
Extra for Experts: Equal Costs – Write and Solve Equations for Equal Cost Problems
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.
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.
What is the cost of one bagel?
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.
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.
What is the cost of one pad of paper?
3. Mary bought 4 decks of cards and a $2.00 crossword puzzle book.
Nina bought 2 decks of cards and 2 boxes of dominos for $3.50 each.
All card decks cost the same.
They spent the same amount of money.
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.
What is the cost of one deck of cards?
Solutions:
 \begin{align*}2a + 2.50 = 4a + 1.00; 2a + 2.50  2a = 4a + 1.00  2a; 2.50 = 2a + 1.00; 2.50  1.00 = 2a + 1.00  1.00; 1.50 = 2a; 0.75 = a;\end{align*} one bagel costs $0.75.
 \begin{align*}5b + 1.50 = 2b + 6.00; 5b + 1.50  2b = 2b + 6.00  2b; 3b + 1.50 = 6.00; 3b + 1.50  1.50 = 6.00  1.50; 3b = 4.50; b = 1.50;\end{align*} one pad of paper costs $1.50.
 \begin{align*}4c + 2.00 = 2c + 2 \times 3.50; 4c + 2.00 = 2c + 7.00; 4c + 2.00  2c = 2c + 7.00  2c; 2c + 2.00 = 7.00; 2c + 2.00  2.00 = 7.00  2.00; 2c = 5.00; c = 2.50;\end{align*} one deck of cards costs $2.50.