Question 68·Medium·Linear Equations in Two Variables
A juice bar sells two sizes of fruit smoothies. A small smoothie costs $4.50, and a large smoothie costs $6.00. On a busy afternoon, the bar sold a total of 120 smoothies and collected $615 in sales.
Based on this information, how many small smoothies were sold?
For this type of SAT word problem, first define clear variables for the unknown quantities, then translate each sentence into an equation (often one for a total count and one for a total cost). Once you have a system of two linear equations, pick the simpler one to solve for a variable and substitute into the other; this substitution method is usually fastest under time pressure. Finally, check your result quickly by plugging it back into the original context (both the total number and the total money) to avoid arithmetic errors before selecting your answer.
Hints
Define what you are solving for
Let one variable represent the number of small smoothies and another variable represent the number of large smoothies. What two facts does the problem give you about these numbers?
Use the total smoothies
Write an equation that shows the total number of smoothies sold is 120 in terms of your two variables.
Use the total money
Write a second equation that shows how the prices ($4.50 and $6.00) and the numbers of each size relate to the total $615 collected.
Solve the system
Use substitution: solve the simpler equation for one variable, plug it into the other equation, and solve for the number of small smoothies.
Desmos Guide
Enter the system of equations
In Desmos, type the two equations using x and y: x + y = 120 and 4.5x + 6y = 615. Each equation should be on its own line so that two lines appear on the graph.
Find the intersection point
Zoom or pan the graph (if needed) until you can see where the two lines cross, then tap or click on the intersection point. Desmos will display the coordinates of this intersection.
Interpret the intersection
Use the x-coordinate of the intersection point as the number of small smoothies and the y-coordinate as the number of large smoothies. Read the x-value carefully to answer the question.
Step-by-step Explanation
Define variables
Let be the number of small smoothies sold and be the number of large smoothies sold.
From the problem:
- Total number of smoothies:
- Total money from sales:
Express one variable in terms of the other
Use the simpler equation to solve for one variable.
Solve for :
Substitute into the money equation
Now substitute into the money equation .
Distribute the 6:
Solve for s
Combine like terms on the left:
Subtract 720 from both sides:
Divide both sides by to isolate and get the number of small smoothies:
So, the juice bar sold 70 small smoothies.