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Difficulty: MediumAlgebraic Word Problems and Modeling

A hotel renovated its guest rooms and purchased a total of 100 lighting fixtures, consisting of wall sconces costing $45\$45 each and ceiling pendants costing $70\$70 each. If the average (arithmetic mean) cost per fixture for the entire purchase was $52\$52, how many wall sconces were purchased?

Answer: 72

Answer

72
Setting up the weighted total cost equation 45x+70(100−x)=520045x + 70(100 - x) = 5200 simplifies to −25x=−1800-25x = -1800, giving x=72x = 72 wall sconces.

Step-by-Step Solution

1
Define the variable for the unknown quantity
Let xx represent the number of wall sconces purchased. The number of ceiling pendants is 100−x100 - x.
Defining the target quantity as xx allows the problem to be modeled using a single-variable linear equation.
2
Set up the algebraic equation for total cost
45x+70(100−x)=52×10045x + 70(100 - x) = 52 \times 100
The total cost of all fixtures is the sum of the total sconce cost and total pendant cost, which equals the overall average cost multiplied by the total number of fixtures.
3
Expand and solve the linear equation
45x+7000−70x=5200  ⟹  −25x=−1800  ⟹  x=7245x + 7000 - 70x = 5200 \implies -25x = -1800 \implies x = 72
Simplifying algebraic terms isolates xx to find the exact number of wall sconces.

Key Concept

Linear Algebraic Modeling and Weighted Averages
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Topics

Algebraic Word Problems and Modeling

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