From 90 liters of 4: 5 milk-water mixtures, 15 liters are taken out and replaced by same amount of water. This process is repeated once more. Now, 9 liters of the mixture is taken out and replaced by same amount of milk. The amount of water at the end of third cycle is how much percent of the amount of water at the end of second cycle?

Sure, let's break this down step by step:

1. Initially, the milk-water mixture is in the ratio 4:5 in 90 liters. This means there are 36 liters of milk (4/9 * 90) and 54 liters of water (5/9 * 90).

2. In the first cycle, 15 liters of the mixture is taken out. This means 6 liters of milk (4/9 * 15) and 9 liters of water (5/9 * 15) are taken out. They are replaced by 15 liters of water. So, the new amounts are 30 liters of milk (36-6) and 60 liters of water (54-9+15).

3. In the second cycle, again 15 liters of the mixture is taken out. This means 5 liters of milk (1/3 * 15) and 10 liters of water (2/3 * 15) are taken out. They are replaced by 15 liters of water. So, the new amounts are 25 liters of milk (30-5) and 65 liters of water (60-10+15).

4. In the third cycle, 9 liters of the mixture is taken out. This means 2.25 liters of milk (1/4 * 9) and 6.75 liters of water (3/4 * 9) are taken out. They are replaced by 9 liters of milk. So, the new amounts are 31.75 liters of milk (25-2.25+9) and 58.25 liters of water (65-6.75).

5. Now, we need to find the amount of water at the end of the third cycle as a percentage of the amount of water at the end of the second cycle. This is (58.25/65) * 100 = 89.62%.

So, the amount of water at the end of the third cycle is 89.62% of the amount of water at the end of the second cycle.