Question 418·Hard·Algebra

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At a campus vending machine, each protein bar costs $p$ cents and each granola bar costs $g$ 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?

Your answer

Two sales totals involving the same two unknown prices signal a system of linear equations: two equations that must both hold. Multiply each bar count by its own price to build one equation per day. Graph both equations in Desmos and click their intersection to find the prices. Don’t stop at the intersection if the question asks how much more one item costs.

Hints

Hint 1
Each day’s total is a sum of two amounts: the number of protein bars times their price, plus the number of granola bars times their price. What equation does Monday’s sale give in cents?
Hint 2
A system is two equations that share the same unknown prices. Desmos graphs equations using $x$ and $y$ , so decide which price each coordinate will represent before you click the intersection.
Hint 3
The intersection gives two individual prices, not their difference. To find how much more a granola bar costs, subtract the protein price from the granola price. Which coordinate should come first?

Step-by-step

Approach 1: Graph the two sales totals
Step 1
Write Monday’s cost equation
Monday’s total adds the cost of both kinds of bars. Each of the $15$ protein bars costs $p$ cents, so they bring in $15p$ cents. The $10$ granola bars bring in $10g$ cents. Together, they brought in $2{,}450$ cents:
$15p+10g=2450.$
Step 2
Write Tuesday’s cost equation
The prices are the same on Tuesday. Multiply each bar count by its own price, then add to match the $2{,}160$ -cent total:
$8p+12g=2160.$
Step 3
Find both prices in Desmos
Use $x$ for the protein price $p$ and $y$ for the granola price $g$ . Type both equations on separate Desmos lines: $15x+10y=2450$ and $8x+12y=2160$ . Click their intersection, where both totals hold. Desmos shows $(78,128)$ , so a protein bar costs $78$ cents and a granola bar costs $128$ cents.
Step 4
Find how much more granola costs
“Granola costs more than protein” means granola price minus protein price. Type $128-78$ on a new Desmos line; it displays $50$ . A granola bar costs $50$ more than a protein bar. Grid in 50.

Approach 2: Build the price difference directly
Step 1
Reduce Monday’s equation
The question asks for $g-p$ , so look for equations whose coefficients will subtract to that difference. Divide every term of Monday’s equation by $5$ :
$\begin{aligned}\frac{15p+10g}{5}&=\frac{2450}{5}\\[1.4em]3p+2g&=490.\end{aligned}$
Step 2
Reduce Tuesday’s equation
Divide every term of Tuesday’s equation by $4$ to make its price coefficients small too:
$\begin{aligned}\frac{8p+12g}{4}&=\frac{2160}{4}\\[1.4em]2p+3g&=540.\end{aligned}$
Step 3
Subtract to get the difference
Subtract the reduced Monday equation from the reduced Tuesday equation. Their coefficients leave one granola price minus one protein price:
$\begin{aligned}(2p+3g)-(3p+2g)&=540-490\\[1.4em]2p-3p+3g-2g&=540-490\g-p&=50.\end{aligned}$
So a granola bar costs $50$ more than a protein bar. Grid in 50.

Question 418·Hard·Algebra

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