Question

There is a tank with some capacity, 9 liters are taken out and replaced with water 2 times. Now milk:water = 16:9, find the capacity of the tank.

Hint:

In problems involving mixtures, the remaining amount of the first substance after each replacement can be calculated using proportional reduction.

Answer 1

Step 1: Understanding the problem.
Let the capacity of the tank be \( C \) liters. Initially, the tank is full of milk, so it contains \( C \) liters of milk. Each time, 9 liters of the mixture are taken out and replaced by 9 liters of water. After two such replacements, the milk-to-water ratio is 16:9.
Step 2: First replacement.
After the first removal of 9 liters, the remaining milk in the tank is: \[ C - 9 \] Then, 9 liters of water are added. Therefore, the milk-to-water ratio immediately after the first replacement is: \[ \frac{C - 9}{9} \]
Step 3: Second replacement.
After the second removal of 9 liters, the remaining milk in the tank is: \[ (C - 9) \times \left(\frac{C - 9}{C}\right) = C \times \left(\frac{C - 9}{C}\right) \] Now the milk-to-water ratio is 16:9. Solving for the capacity \( C \), we find that: \[ C = 45 \]