Questions: You are choosing between two plans at a discount warehouse. Plan A offers an annual membership fee of 120 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?

Solution Steps

To find the amount of merchandise you need to purchase for both plans to cost the same, set up an equation where the total cost of Plan A equals the total cost of Plan B. The total cost for each plan is the sum of the annual membership fee and the percentage of the manufacturer's recommended list price. Solve for the amount of merchandise, \( x \), that makes these costs equal. Then, calculate the total cost for each plan using this value of \( x \).

Let \( x \) be the amount of merchandise purchased in dollars. The total cost for Plan A can be expressed as: \[ \text{Cost}_A = 120 + 0.8x \] The total cost for Plan B can be expressed as: \[ \text{Cost}_B = 40 + 0.9x \]

To find the value of \( x \) where both plans cost the same, set the two cost equations equal to each other: \[ 120 + 0.8x = 40 + 0.9x \] Rearranging gives: \[ 120 - 40 = 0.9x - 0.8x \] \[ 80 = 0.1x \] Thus, solving for \( x \) yields: \[ x = \frac{80}{0.1} = 800 \]

Now, substitute \( x = 800 \) back into the cost equations to find the total costs for each plan: \[ \text{Cost}_A = 120 + 0.8 \times 800 = 120 + 640 = 760 \] \[ \text{Cost}_B = 40 + 0.9 \times 800 = 40 + 720 = 760 \]

Final Answer