Question 84·Hard·Linear Equations in Two Variables
At a campus vending machine, each protein bar costs cents and each granola bar costs cents. On Monday, 15 protein bars and 10 granola bars were sold for a total of 2,450 cents. On Tuesday, 8 protein bars and 12 granola bars were sold for a total of 2,160 cents.
According to these data, how many more cents does a granola bar cost than a protein bar?
(Express the answer as an integer)
For SAT word problems that describe two different purchases with totals, quickly define variables for the unknown prices and write one equation for each situation. Simplify both equations if possible to keep numbers small, then use elimination (subtract one equation from the other) to cancel a variable and get a direct relationship between the unknowns. Finally, answer exactly what the question asks—often a difference like —without doing extra work to find each individual value unless necessary.
Hints
Translate the words into equations
Assign variables to the prices: one for a protein bar and one for a granola bar. Use the totals from Monday and Tuesday to write two equations involving these variables.
Make the equations easier to work with
Look at the coefficients in each equation. Can you divide both sides of each equation by a common number to make the coefficients smaller?
Use elimination to compare the prices
Once the equations are simplified, try subtracting one from the other so that you get an equation involving and with small coefficients. Use that result to find , since the question asks how many more cents the granola bar costs.
Desmos Guide
Enter the equations as lines
In Desmos, let represent the protein bar price (in cents) and represent the granola bar price (in cents). Type 15x+10y=2450 on one line and 8x+12y=2160 on another line to graph the two lines.
Find the intersection point
Adjust the view (zoom in or out) until you see where the two lines intersect. Tap or click the intersection point to see its coordinates , which represent the prices of the protein and granola bars.
Compute the price difference
In a new expression line, type the granola price minus the protein price using the intersection coordinates (for example, if the point is , enter b-a). The result shown is the number of cents that a granola bar costs more than a protein bar.
Step-by-step Explanation
Define variables and write equations
Let be the price (in cents) of a protein bar and be the price (in cents) of a granola bar.
From the problem:
- Monday:
- Tuesday:
These two equations describe the total sales in cents for each day.
Simplify the equations to smaller numbers
Simplify each equation by dividing by a common factor:
- Monday: divide everything by 5:
- Tuesday: divide everything by 4:
Now the system is:
Eliminate one variable to relate p and g
Subtract the second simplified equation from the first to eliminate one variable:
On the left:
So you get:
This tells you how the two prices compare, but in the order protein minus granola.
Answer the question: g minus p
The question asks how many more cents a granola bar costs than a protein bar, which is , not .
From , multiply both sides by :
So a granola bar costs 50 cents more than a protein bar.