Question

A bottle contains 50 liters of milk. From this bottle 5 liters of milk was taken out and replaced with water. This process was repeated further for three times. How much milk is now contained in the bottle?

• A. 32.8 litres
• B. 34.4 litres
• C. 36.8 litres
• D. 46.5 litres

Hint:

Use the successive dilution formula: milk left = original volume times (1 minus removed fraction) raised to the number of times the process happens, here 4 times.

Answer 1

The Correct Option is A

To solve this problem, we need to understand how the replacement of milk with water affects the quantity of milk in the bottle after repeated procedures. The key here is the understanding of geometric progression in the context of a repeated dilution process.

Initially, the bottle contains 50 liters of milk. 

In the first operation, 5 liters of milk are taken out, and the same amount of water is added. The remaining milk in the bottle after this step is given by:

\(\frac{50 - 5}{50} \times 50 = 45 \text{ liters}\)

In the second operation, another 5 liters of the mixture is replaced with water. The amount of milk after this operation is:

\(\frac{45}{50} \times 45 = \frac{9}{10} \times 45 = 40.5 \text{ liters}\)

During the third operation, the process is repeated, and the calculation is:

\(\frac{45}{50} \times 40.5 = \frac{9}{10} \times 40.5 = 36.45 \text{ liters}\)

Finally, for the fourth operation:

\(\frac{45}{50} \times 36.45 = \frac{9}{10} \times 36.45 = 32.805 \approx 32.8 \text{ liters}\)

Thus, the amount of milk remaining in the bottle after repeating this process four times is approximately 32.8 liters.

Conclusion: The correct answer is 32.8 liters.

Answer 2

Step 1: Set up the replacement formula.
When you repeatedly take out \(y\) units from a mixture of total volume \(x\) and replace it with water, the milk left after \(n\) such steps is \(x\left(1-\frac{y}{x}\right)^n\). Here \(x = 50\) litres and \(y = 5\) litres taken out each time.

Step 2: Count how many times the removal happens.
The problem says 5 litres was taken out once, and then this process was repeated further for three times. So the removal and replacement happens 4 times in total, meaning \(n = 4\).

Step 3: Plug the values into the formula.
Milk left \(= 50\left(1-\frac{5}{50}\right)^4 = 50(0.9)^4\). Now \(0.9^2 = 0.81\) and \(0.9^4 = 0.81 \times 0.81 = 0.6561\).

Step 4: Work out the final volume.
Milk left \(= 50 \times 0.6561 = 32.805\) litres, which rounds to 32.8 litres.

Final Answer:
The bottle now has about 32.8 litres of milk. \[ \boxed{32.8 \text{ litres}} \]