A drum of 20 litres is filled with milk. A milkman has only two measuring vessels of 3 litres and 5 litres without any calibration. He has to measure four litres of milk for a customer without using any other vessel. Minimum how many operations are required for this work, where an operation is counted if the milk is transferred from one vessel to another vessel?

Options

A. 5
B. 6
C. 8
D. 11

Answer

To solve the problem of measuring 4 liters of milk using a 3-liter and a 5-liter vessel, we can follow these steps:

### Step-by-Step Solution:

1. **Fill the 3-liter vessel from the drum.**
- **Operation Count:** 1
- **Vessels Status:** 3L vessel has 3 liters, 5L vessel is empty.

2. **Pour the milk from the 3-liter vessel into the 5-liter vessel.**
- **Operation Count:** 2
- **Vessels Status:** 3L vessel is empty, 5L vessel has 3 liters.

3. **Fill the 3-liter vessel again from the drum.**
- **Operation Count:** 3
- **Vessels Status:** 3L vessel has 3 liters, 5L vessel has 3 liters.

4. **Pour from the 3-liter vessel into the 5-liter vessel until the 5-liter vessel is full.**
- Since the 5-liter vessel already has 3 liters, it can only take 2 more liters from the 3-liter vessel.
- **Operation Count:** 4
- **Vessels Status:** 3L vessel has 1 liter (3 - 2 = 1), 5L vessel is full (5 liters).

5. **Empty the 5-liter vessel back into the drum.**
- **Operation Count:** 5
- **Vessels Status:** 3L vessel has 1 liter, 5L vessel is empty.

6. **Pour the remaining 1 liter from the 3-liter vessel into the 5-liter vessel.**
- **Operation Count:** 6
- **Vessels Status:** 3L vessel is empty, 5L vessel has 1 liter.

7. **Fill the 3-liter vessel again from the drum.**
- **Operation Count:** 7
- **Vessels Status:** 3L vessel has 3 liters, 5L vessel has 1 liter.

8. **Pour from the 3-liter vessel into the 5-liter vessel until the 5-liter vessel is full.**
- The 5-liter vessel currently has 1 liter, so it can take 4 more liters from the 3-liter vessel.
- **Operation Count:** 8
- **Vessels Status:** 3L vessel has 0 liters (3 - 3 = 0), 5L vessel has 5 liters (1 + 4 = 5).

9. **Finally, empty the 5-liter vessel back into the drum.**
- **Operation Count:** 9
- **Vessels Status:** 3L vessel is empty, 5L vessel is empty.

10. **Fill the 3-liter vessel again from the drum.**
- **Operation Count:** 10
- **Vessels Status:** 3L vessel has 3 liters, 5L vessel is empty.

11. **Pour from the 3-liter vessel into the 5-liter vessel.**
- **Operation Count:** 11
- **Vessels Status:** 3L vessel has 0 liters, 5L vessel has 3 liters.

12. **Fill the 3-liter vessel again from the drum.**
- **Operation Count:** 12
- **Vessels Status:** 3L vessel has 3 liters, 5L vessel has 3 liters.

13. **Pour from the 3-liter vessel into the 5-liter vessel until the 5-liter vessel is full.**
- The 5-liter vessel can take 2 more liters, so we pour 2 liters from the 3-liter vessel.
- **Operation Count:** 13
- **Vessels Status:** 3L vessel has 1 liter, 5L vessel has 5 liters.

14. **Empty the 5-liter vessel back into the drum.**
- **Operation Count:** 14
- **Vessels Status:** 3L vessel has 1 liter, 5L vessel is empty.

15. **Pour the remaining 1 liter from the 3-liter vessel into the 5-liter vessel.**
- **Operation Count:** 15
- **Vessels Status:** 3L vessel is empty, 5L vessel has 1 liter.

16. **Fill the 3-liter vessel again from the drum.**
- **Operation Count:** 16
- **Vessels Status:** 3L vessel has 3 liters, 5L vessel has 1 liter.

17. **Pour from the 3-liter vessel into the 5-liter vessel until the 5-liter vessel is full.**
- The 5-liter vessel can take 4 more liters, so we pour 3 liters from the 3-liter vessel.
- **Operation Count:** 17
- **Vessels Status:** 3L vessel has 0 liters, 5L vessel has 4 liters.

At the end of these operations, we have successfully measured 4 liters of milk in the 5-liter vessel.

### Total Operations Required: 17