Problem 38 - College Algebra

Question

You are choosing between two plans at a discount warehouse. Plan A offers an annual membership fee of \(\$ 300\) and you pay \(70 \%\) 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

\(x = 1300\), which means you have to buy $1300 worth of merchandise for both plans to cost the same. The annual cost for each plan is $1210.

Step 1: Define the Equations for Both Plans

Start by defining two equations that represent the annual cost for each plan. For Plan A: \(Cost_A = 300 + 0.70x\), where \(x\) is the amount of merchandise purchased. For Plan B: \(Cost_B = 40 + 0.90x\) .

Step 2: Set the Equations Equal to Each Other

To find the amount of merchandise that will cost the same under both plans, set \(Cost_A\) equal to \(Cost_B\). So, \(300 + 0.70x = 40 + 0.90x\).

Step 3: Solve the Equation

Solve the equation for \(x\) : subtract \(0.70x\) and 40 on both sides to get \(0.20x = 260\). Then divide both sides by `0.20` to solve for \(x\).

Step 4: Calculate the Costs for Each Plan

Substitute the \(x\) value into the equations for \(Cost_A\) and \(Cost_B\) to calculate the annual cost for each plan.