The Empire Carpet Company orders merchandise for $17,700, including $550 in shipping charges, from Mohawk Carpet Mills on May 4. Carpets valued at $1,390 will be returned because they are damaged. The terms of sale are 2/10, n30 ROG. The shipment arrives on May 26 and Empire wants to take advantage of the cash discount. By what date must Empire pay the invoice?

To determine the date by which Empire Carpet Company must pay the invoice to take advantage of the cash discount, we need to consider the terms of sale and the date of shipment arrival.

The terms of sale 2/10, n30 ROG can be broken down as follows:
- 2/10: This indicates a 2% cash discount if the invoice is paid within 10 days.
- n30: This means that the full payment is due within 30 days.
- ROG: This stands for "Receipt of Goods," indicating that the payment terms start from the date of shipment arrival.

Given that the shipment arrives on May 26, we need to calculate the payment due date by considering both the cash discount period and the full payment due date.

1. Cash Discount Period:
The cash discount period is 10 days from the date of the shipment arrival, which is May 26. So, the last day to take advantage of the cash discount is June 4 (May 26 + 10 days).

2. Full Payment Due Date:
The full payment is due within 30 days from the date of the shipment arrival. Therefore, the payment due date is June 25 (May 26 + 30 days).

So, Empire Carpet Company must pay the invoice by June 4 in order to take advantage of the cash discount.