1. **Problem statement:** Jack has a 5-gallon pail and Jill has a 3-gallon pail. They want to measure exactly 1 gallon of water. 2. **Key idea:** Use the pails to transfer water back and forth, filling and emptying them to isolate exactly 1 gallon. 3. **Step-by-step solution:** - Fill the 3-gallon pail completely (3 gallons). - Pour the 3 gallons from the 3-gallon pail into the 5-gallon pail. - Fill the 3-gallon pail again (3 gallons). - Pour water from the 3-gallon pail into the 5-gallon pail until the 5-gallon pail is full. Since the 5-gallon pail already has 3 gallons, it can only take 2 more gallons. - This leaves exactly 1 gallon in the 3-gallon pail (because 3 - 2 = 1). 4. **Result:** The 3-gallon pail now contains exactly 1 gallon of water. This method uses the principle of filling, transferring, and subtracting volumes to isolate the desired amount.