Problem 37 - College Algebra

Question

You are choosing between two plans at a discount warehouse. Plan A offers an annual membership fee of \(\$ 100\) and you pay \(80 \%\) of the manufacturer's recommended list price. Plan B offers an annual membership fee of \(\$ 40\) and you pay \(90 \%\) of the manufacturer's recommended list price. How many dollars of merchandise would you have to purchase in a year to pay the same amount under both plans? What will be the cost for each plan?

Step-by-Step Solution

Answer

You would need to purchase \$600 worth of merchandise in a year to pay the same amount under both plans. The cost for each plan would then be \$100 + 0.8(\$600) = \$580 for Plan A, and \$40 + 0.9(\$600) = \$580 for Plan B.

Step 1: Determine the Cost Structure for Each Plan

For Plan A, the total cost if you purchase \( X \) dollars worth of goods would be \( \$100 + 0.8X \) where \$100 is the membership fee and 80% (0.8) of \( X \) is the price paid for the goods. Similarly for Plan B, the total cost if you purchase \( X \) dollars worth of goods would be \( \$40 + 0.9X \) where \$40 is the membership fee and 90% (0.9) of \( X \) is the price paid for the goods.

Step 2: Set Up an Equation to Make the Costs of Both Plans Equal

In order to find the amount of merchandise one would need to purchase in a year to pay the same amount under both plans, the total cost under Plan A should equal the total cost under Plan B. This gives us the following equation: \( \$100 + 0.8X = \$40 + 0.9X \).

Step 3: Simplify and Solve the Equation

Simplifying the equation by collecting like terms yields: \( (0.8X - 0.9X) = \$40 - \$100 \). This simplifies further to give \( -0.1X = -\$60 \). Solving for \( X \) gives: \( X = (-\$60 / -0.1) = \$600 \).