Problem 324 - Elementary Algebra

Question

In the following exercises, solve. Kellen wants to rent a banquet room in a restaurant for her cousin's baby shower. The restaurant charges \(\$ 350\) for the banquet room plus \(\$ 32.50\) per person for lunch. How many people can Kellen have at the shower if she wants the maximum cost to be \(\$ 1,500 ?\)

Step-by-Step Solution

Answer

Kellen can have 35 people.

Step 1 - Setup the Equation

Let the number of people be represented by the variable, say, \(x\). The total cost includes a fixed charge of \(\$350\) plus an additional charge of \(\$32.50\) per person. Therefore, the equation representing the total cost \(C\) is given by \[C = 350 + 32.50x\].

Step 2 - Set the Maximum Cost

Kellen wants the maximum cost to be \(\$1500\). Therefore, set the equation from Step 1 equal to \$1500: \[350 + 32.50x = 1500\].

Step 3 - Isolate the Variable

To find the number of people, isolate \(x\) by subtracting \$350 from both sides of the equation: \[32.50x = 1500 - 350\]. This simplifies to \[32.50x = 1150\].

Step 4 - Solve for the Variable

Solve for \(x\) by dividing both sides of the equation by \$32.50: \[x = \frac{1150}{32.50}\].

Step 5 - Calculate the Solution

Divide \$1150 by \$32.50 to get the number of people: \[x = 35.38\]. Since the number of people must be a whole number, round down to 35.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that use variables to represent unknown values. These equations help us find unknown quantities by performing operations like addition, subtraction, multiplication, and division. In the context of our problem, we use an algebraic equation to determine the number of people Kellen can invite within the cost limit.
The equation set up in this exercise is \[C = 350 + 32.50x\], where:

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