Ben and Joel are raising money for their class by selling wrapping paper. Ben raised $43.50 by selling 12 rolls of solid paper and 9 of printed paper. Joel raised $51.50 by selling 8 rolls of solid and 15 rolls of printed paper. Find the cost of each type of wrapping paper.
The Correct Answer and Explanation is:
Let’s analyze the problem step by step.
Problem Summary:
- Ben sold 12 rolls of solid wrapping paper and 9 rolls of printed wrapping paper, raising $43.50.
- Joel sold 8 rolls of solid wrapping paper and 15 rolls of printed wrapping paper, raising $51.50.
We want to find the cost per roll of solid wrapping paper and printed wrapping paper.
Step 1: Define variables
Let:
- xx = cost per roll of solid wrapping paper (in dollars)
- yy = cost per roll of printed wrapping paper (in dollars)
Step 2: Write the system of equations
From Ben’s sales: 12x+9y=43.5012x + 9y = 43.50
From Joel’s sales: 8x+15y=51.508x + 15y = 51.50
Step 3: Solve the system of linear equations
We have: 12x+9y=43.50(1)12x + 9y = 43.50 \quad (1) 8x+15y=51.50(2)8x + 15y = 51.50 \quad (2)
Let’s use the elimination method to eliminate one variable.
Multiply equation (1) by 5 and equation (2) by 3, so the coefficients of yy are the same: 5×(12x+9y)=5×43.50 ⟹ 60x+45y=217.50(3)5 \times (12x + 9y) = 5 \times 43.50 \implies 60x + 45y = 217.50 \quad (3) 3×(8x+15y)=3×51.50 ⟹ 24x+45y=154.50(4)3 \times (8x + 15y) = 3 \times 51.50 \implies 24x + 45y = 154.50 \quad (4)
Step 4: Subtract equation (4) from (3)
(60x+45y)−(24x+45y)=217.50−154.50(60x + 45y) – (24x + 45y) = 217.50 – 154.50 60x−24x+45y−45y=6360x – 24x + 45y – 45y = 63 36x=6336x = 63 x=6336=1.75x = \frac{63}{36} = 1.75
So, the cost per roll of solid wrapping paper is $1.75.
Step 5: Substitute x=1.75x = 1.75 into one of the original equations to find yy
Use equation (1): 12(1.75)+9y=43.5012(1.75) + 9y = 43.50 21+9y=43.5021 + 9y = 43.50 9y=43.50−21=22.509y = 43.50 – 21 = 22.50 y=22.509=2.50y = \frac{22.50}{9} = 2.50
So, the cost per roll of printed wrapping paper is $2.50.
Final answer:
- Solid wrapping paper: $1.75 per roll
- Printed wrapping paper: $2.50 per roll
Explanation (300 words):
This problem involves finding the cost of two types of wrapping paper, given the total sales amounts from two people selling different quantities of each type. We use algebraic methods, particularly solving a system of linear equations, to find the prices.
First, we assign variables to unknowns — the cost per roll of solid paper as xx and the cost per roll of printed paper as yy. Then, using the information about how many rolls were sold by Ben and Joel and how much money they raised, we create two equations representing total money raised by each seller.
Ben’s total sales lead to the equation 12x+9y=43.5012x + 9y = 43.50, meaning 12 rolls of solid at xx dollars each plus 9 rolls of printed at yy dollars each equals $43.50. Joel’s sales give us 8x+15y=51.508x + 15y = 51.50.
To solve the system, the elimination method is effective. By multiplying the equations to make the coefficients of one variable the same, we can subtract one from the other to eliminate that variable. This leaves a single-variable equation, which is easy to solve. After solving for xx, we substitute back into one of the original equations to find yy.
This stepwise approach ensures accuracy. The solution shows that the solid paper costs $1.75 per roll, while the printed paper is $2.50 per roll. This makes sense logically because printed paper, typically more decorative, is expected to cost more than solid paper.
The key to solving these word problems is translating the scenario into mathematical equations and then systematically solving them, checking back for reasonableness. This problem is a classic example of applying linear algebra to everyday practical problems like pricing.