Question

A container contains only milk. 2/3 of the mixture is removed and replaced with water. This process is done once and then repeated another 3 times. What is the final ratio of milk and water?

Hint:

The formula \(V_{final} = V_{initial} \times (1 - x)^n\) is a powerful shortcut for all repeated replacement problems. Identify the initial amount, the fraction replaced (x), and the number of repetitions (n) to solve quickly.

Answer 1

Step 1: Understanding the Question:
The problem describes a repeated dilution process. We start with a container full of pure milk. In each step, a fraction (2/3) of the mixture is removed and replaced with water. We need to find the final ratio of milk to water after this process is performed a total of four times. The phrase "done another 3 times" means the total number of operations is 1 + 3 = 4.
Step 2: Key Formula or Approach:
For repeated dilutions where a fraction 'x' of a mixture is removed and replaced, the amount of the original substance remaining (\(V_{final}\)) after 'n' operations is given by the formula:
\[ V_{final} = V_{initial} \times (1 - x)^n \] where \(V_{initial}\) is the initial volume of the substance.
Step 3: Detailed Explanation:
Let the initial volume of the container be 'V'. Initially, the container is full of milk.
So, Initial volume of milk = V.
Initial volume of water = 0.
The fraction of the mixture replaced in each operation is \(x = 2/3\).
The process is performed a total of \(n = 1 + 3 = 4\) times.
Now, we can calculate the final volume of milk using the formula from Step 2:
\[ \text{Final Milk} = V \times \left(1 - \frac{2}{3}\right)^4 \] \[ \text{Final Milk} = V \times \left(\frac{1}{3}\right)^4 \] \[ \text{Final Milk} = V \times \frac{1}{81} = \frac{V}{81} \] The total volume of the mixture in the container always remains V. The final volume of water will be the total volume minus the final volume of milk.
\[ \text{Final Water} = \text{Total Volume} - \text{Final Milk} \] \[ \text{Final Water} = V - \frac{V}{81} = \frac{81V - V}{81} = \frac{80V}{81} \] Finally, we find the ratio of the final volume of milk to the final volume of water.
\[ \text{Ratio (Milk : Water)} = \frac{\text{Final Milk}}{\text{Final Water}} = \frac{V/81}{80V/81} \] The terms 'V' and '81' cancel out.
\[ \text{Ratio (Milk : Water)} = \frac{1}{80} \quad \text{or} \quad 1:80 \] Step 4: Final Answer:
The final ratio of milk to water is 1:80.