Cash Discount versus Credit Terms: A merchant receives an invoice for a motor boat for $20,000 with credit terms 4/30, n/100 (a 4% discount if paid within 30 days, otherwise the full amount is due in 100 days). What is the highest simple annual interest rate (in %) at which the merchant can borrow money and still find it worthwhile to take advantage of the discount?

Introduction / Context:
This question compares cash discount terms with delayed payment terms and asks you to find the highest simple annual interest rate at which it is still beneficial to borrow money to pay early. It is a classic time value of money and trade credit problem, where you relate the discount saved to the time period over which payment is advanced.

Given Data / Assumptions:

• Invoice amount = $20,000.
• Terms 4/30, n/100: 4% discount if paid within 30 days; otherwise pay full in 100 days.
• If paying early, the merchant pays within 30 days instead of 100 days.
• Therefore, the merchant pays 70 days earlier (100 − 30 = 70 days).
• Interest rate sought is a simple annual rate.
• We use simple interest and a 360 day year (commonly used in such problems) to match the given options.

Concept / Approach:
If the merchant takes the discount, he pays 96% of the invoice now instead of 100% later. The discount saved is 4% of 20,000, which is $800. Effectively, he is earning $800 by paying $19,200 earlier than otherwise required. This is like earning interest on $19,200 for 70 days. The implied simple interest rate is (interest / principal) * (year length / days), converted to a percentage.

Step-by-Step Solution:
Step 1: Compute the discounted price: 96% of $20,000 = 0.96 * 20,000 = $19,200. Step 2: Discount saved = 20,000 − 19,200 = $800. Step 3: The merchant advances payment by 70 days (100 − 30). Step 4: Think of this as earning $800 interest on a "principal" of $19,200 over 70 days. Step 5: Simple rate r (per year) is given by r = (I / P) * (year length / time in days) * 100. Step 6: Using a 360 day year, r = (800 / 19,200) * (360 / 70) * 100. Step 7: First compute 800 / 19,200 ≈ 0.0416667. Step 8: Next compute 360 / 70 ≈ 5.142857. Step 9: Multiply: 0.0416667 * 5.142857 ≈ 0.2142857. Step 10: Convert to percent: r ≈ 0.2142857 * 100 ≈ 21.43% per annum.

Verification / Alternative check:
Because the discount is 4% for 70 days, a quick rule of thumb is to annualise the discount rate: approximate annual rate ≈ (discount / discounted price) * (360 / days). Here, 4 / 96 ≈ 4.17%, and 4.17% * (360 / 70) ≈ 21.43%. This matches the detailed calculation above. Taking any borrowing rate lower than this is beneficial, since the cost of borrowing would be less than the effective return from taking the discount.

Why Other Options Are Wrong:

• 18.00% and 15.00%: These are below the implied rate; if the merchant can borrow at these rates, he certainly should take the discount, but they are not the highest break even rate.
• 24.00% and 30.00%: These are higher than the calculated effective rate; borrowing at such rates may make taking the discount uneconomical.

Common Pitfalls:
Common mistakes include forgetting to use the discounted price as the principal, using 70 / 360 instead of 360 / 70, or computing the discount as a percentage of the full amount without adjusting for the shorter time period. Some students also confuse the 4% discount with a 4% annual rate, which is incorrect. Always annualise the effective short term return to compare with an annual borrowing rate.

Final Answer:
The highest simple annual interest rate at which taking the discount remains worthwhile is approximately 21.43% per annum.