Given only an 11 liter jug and an 8 liter jug, each with no markings, find the method of using the two jugs to measure out eactly 1 liter of water while using the smallest amount of water. Argue clearly why you think that your procedure uses the smallest amount!
To solve this problem, we can use a simple method known as the Water Jug Problem. Here's a step-by-step procedure to measure out exactly 1 liter of water using the 11-liter and 8-liter jugs:
1. Fill the 11-liter jug to its maximum capacity.
2. Pour water from the 11-liter jug into the 8-liter jug until the 8-liter jug is full. This leaves the 11-liter jug with 3 liters of water.
3. Empty the 8-liter jug entirely.
4. Pour the remaining 3 liters of water from the 11-liter jug into the empty 8-liter jug.
5. Fill the 11-liter jug to its maximum capacity again.
6. Slowly pour water from the 11-liter jug into the 8-liter jug until the 8-liter jug is full. At this point, the remaining water in the 11-liter jug is 6 liters.
7. Take the 8-liter jug and pour one liter of water into any other container.
8. Finally, pour the remaining 5 liters of water from the 11-liter jug into the container with 1 liter of water.
Following this procedure, we end up with exactly 1 liter of water separated from the rest. Now let's argue why this procedure uses the smallest amount of water!
To minimize water usage, we need to find a way to measure out 1 liter using the smallest possible amount of excess water, i.e., the amount of water left in either jug. In this procedure, we used a total of 8 liters of excess water, which is the smallest possible amount achievable given the constraints of the problem.
Here's the breakdown of water usage:
- Initially, the 11-liter jug has 11 liters of water. We pour 8 liters into the 8-liter jug, leaving 3 liters.
- We reset, pouring the 3 liters from the 11-liter jug into the 8-liter jug, leaving it empty.
- Again, we fill the 11-liter jug to its maximum, yielding 11 liters. Pouring 8 liters into the 8-liter jug leaves 3 liters in the 11-liter jug.
- Now, we pour 1 liter from the 8-liter jug into another container, leaving 2 liters.
- Finally, we pour the remaining 5 liters from the 11-liter jug into the container with the 1 liter we set aside.
Hence, the total amount of water used in this procedure is 8 liters, which is an optimal solution to measure out exactly 1 liter while minimizing water usage.