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For a certain retail company, a customer can purchase a membership and receive a percent discount applied to all purchases made from that company for one year, or the customer can make purchases from the company without buying a membership and pay full price for purchases. There are exactly four membership types:
Provided that a customer expected to spend exactly \(\$2{,}400\) on purchases made from the retail company during a one-year period (before any discounts are applied), select for Most advantageous and for Least advantageous the membership option that would be most financially advantageous and least financially advantageous, respectively, for the customer based on the information provided. Make only two selections, one in each column.
No membership
A 1% discount membership
A 2% discount membership
A 5% discount membership
A 10% discount membership
Let's organize this membership discount problem using a comparison table to evaluate each option:
| Membership Type | Membership Fee | Discount Rate | Expected Purchases | Discount Savings | Total Cost |
| No membership | \(\$0\) | \(0\%\) | \(\$2{,}400\) | \(\$0\) | ? |
| \(1\%\) discount | \(\$100\) | \(1\%\) | \(\$2{,}400\) | ? | ? |
| \(2\%\) discount | \(\$200\) | \(2\%\) | \(\$2{,}400\) | ? | ? |
| \(5\%\) discount | \(\$500\) | \(5\%\) | \(\$2{,}400\) | ? | ? |
| \(10\%\) discount | \(\$1{,}000\) | \(10\%\) | \(\$2{,}400\) | ? | ? |
We need to find:
Key formula: \(\mathrm{Total~Cost} = \mathrm{Membership~Fee} + (\mathrm{Expected~Purchases} \times (1 - \mathrm{Discount~Rate}))\)
Let's calculate the total cost for each option:
No membership:
Total Cost = \(\$0 + \$2{,}400 = \$2{,}400\)
\(1\%\) discount membership:
Discount Savings = \(\$2{,}400 \times 0.01 = \$24\)
Total Cost = \(\$100 + (\$2{,}400 - \$24) = \$100 + \$2{,}376 = \$2{,}476\)
\(2\%\) discount membership:
Discount Savings = \(\$2{,}400 \times 0.02 = \$48\)
Total Cost = \(\$200 + (\$2{,}400 - \$48) = \$200 + \$2{,}352 = \$2{,}552\)
\(5\%\) discount membership:
Discount Savings = \(\$2{,}400 \times 0.05 = \$120\)
Total Cost = \(\$500 + (\$2{,}400 - \$120) = \$500 + \$2{,}280 = \$2{,}780\)
\(10\%\) discount membership:
Discount Savings = \(\$2{,}400 \times 0.10 = \$240\)
Total Cost = \(\$1{,}000 + (\$2{,}400 - \$240) = \$1{,}000 + \$2{,}160 = \$3{,}160\)
Let's update our table with the results:
| Membership Type | Total Cost | Financial Impact |
| No membership | \(\$2{,}400\) | Baseline |
| \(1\%\) discount | \(\$2{,}476\) | Costs \(\$76\) more |
| \(2\%\) discount | \(\$2{,}552\) | Costs \(\$152\) more |
| \(5\%\) discount | \(\$2{,}780\) | Costs \(\$380\) more |
| \(10\%\) discount | \(\$3{,}160\) | Costs \(\$760\) more |
Key insight: All membership options actually cost MORE than no membership because the membership fees exceed the discount savings at this spending level.
Most advantageous: No membership (Total cost: \(\$2{,}400\))
Least advantageous: A \(10\%\) discount membership (Total cost: \(\$3{,}160\))