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Difficulty: MediumSolving Linear Equations

A local coffee shop sells two types of coffee blends: House Blend and organic Reserve Blend. A bag of Reserve Blend costs 4.504.50 dollars less than three times the cost of a bag of House Blend. A customer purchases 33 bags of House Blend and 22 bags of Reserve Blend for a total of 54.0054.00 dollars, excluding tax. What is the cost, in dollars, of a bag of Reserve Blend?

  1. A
    7.00
  2. B
    10.50
  3. C
    15.00
  4. 16.50Answer
  5. E
    19.50

Answer

The cost of a bag of Reserve Blend is 16.50 dollars.
The cost of a bag of Reserve Blend is 16.50 dollars. Let xx represent the cost of a bag of House Blend. A bag of Reserve Blend costs 3x−4.503x - 4.50. Since the total cost for 33 bags of House Blend and 22 bags of Reserve Blend is 54.0054.00, we write the linear equation 3x+2(3x−4.50)=543x + 2(3x - 4.50) = 54. Expanding the terms gives 3x+6x−9=543x + 6x - 9 = 54, which simplifies to 9x−9=549x - 9 = 54. Adding 99 to both sides yields 9x=639x = 63. Dividing by 99 gives x=7.00x = 7.00. Substituting 7.007.00 back into the Reserve Blend expression yields 3(7.00)−4.50=16.503(7.00) - 4.50 = 16.50 dollars.

Step-by-Step Solution

1
Define variables for the costs of each blend.
Let xx represent the cost of a bag of House Blend in dollars. The cost of a bag of Reserve Blend is represented by the expression 3x−4.503x - 4.50.
Establishing algebraic expressions for the unknowns based on the problem description allows us to set up a linear equation.
2
Set up the linear equation based on the total cost of the purchase.
The total cost of 33 bags of House Blend and 22 bags of Reserve Blend is 54.0054.00 dollars: 3(x)+2(3x−4.50)=543(x) + 2(3x - 4.50) = 54.
The sum of the individual total costs of the two blends must equal the overall purchase total.
3
Distribute and combine like terms to simplify the equation.
3x+6x−9=543x + 6x - 9 = 54 simplifies to 9x−9=549x - 9 = 54.
Applying the distributive property removes the parentheses, allowing like terms to be combined.
4
Isolate the variable term by adding 99 to both sides.
9x=639x = 63.
To solve for xx, we must first isolate the term containing the variable by performing the inverse operation.
5
Solve for xx by dividing both sides by 99.
x=7.00x = 7.00.
Dividing isolates xx, giving the cost of a bag of House Blend.
6
Calculate the cost of a bag of Reserve Blend using the expression from Step 1.
3(7.00)−4.50=21.00−4.50=16.503(7.00) - 4.50 = 21.00 - 4.50 = 16.50 dollars.
The question asks for the cost of a bag of Reserve Blend, not the House Blend, so we evaluate the expression 3x−4.503x - 4.50 at x=7.00x = 7.00.

Key Concept

Setting up and solving a single-variable linear equation to solve a real-world word problem with multiple unknown quantities.
Estimated Time:1m 30s
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Topics

Solving Linear Equations

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