Question

A container has 40 liters of milk. Then, 4 liters are removed from the container and replaced with 4 liters of water. This process of replacing 4 liters of the liquid in the container with an equal volume of water is continued repeatedly. The smallest number of times of doing this process, after which the volume of milk in the container becomes less than that of water, is

• A. 5
• B. 7
• C. 4
• D. None of Above

Answer 1

The Correct Option is B

Step 1: Understanding the Substitution Process

We are told that each time \(10\%\) of the mixture is substituted with pure adulterant (water), the concentration of the mixture decreases to \(90\%\) of its initial concentration.

In general, when a proportion \(p\) of a mixture is substituted with pure adulterant, the concentration of the resulting mixture becomes \((1 - p)\) times the previous concentration. Here, \(p = \frac{4}{40} = 0.1\).

Initially, the mixture contains pure milk, so the concentration or strength is \(100\%\) or \(1\) (in terms of proportion). After undergoing the substitution process \(n\) times, the concentration of milk in the mixture will be represented by the expression:

\(1 \times (0.9)^n\).

Step 2: Finding the Smallest \( n \) for Milk Concentration Less Than 50%

We are required to find the smallest number of substitutions, \( n \), for which the concentration of milk becomes less than 50%. This condition is represented as:

\(1 \times (0.9)^n < 0.5\).

Step 3: Solving the Inequality

To solve this, take the logarithm of both sides:

\(\log \left( (0.9)^n \right) < \log (0.5)\)

Using the properties of logarithms, this simplifies to:

\(n \log (0.9) < \log (0.5)\)

Step 4: Calculating the Logarithms

The logarithms are approximately:

\(\log (0.9) \approx -0.045757\) and \(\log (0.5) \approx -0.3010\)

Now, substitute these values into the inequality:

\(n \times (-0.045757) < -0.3010\)

Solving for n

\(n > \frac{-0.3010}{-0.045757} \approx 6.58\)

Step 5: Conclusion

The smallest integer greater than 6.58 is \(n = 7\).

Final Answer:

The smallest number of substitutions, \(n\), that satisfies this condition is 7.

The correct option is (B): 7.