Question

A glass is filled with milk. Two-thirds of its content is poured out and replaced with water. If this process of pouring out two-thirds the content and replacing with water is repeated three more times, then the final ratio of milk to water in the glass, is

• A. 1:80
• B. 1:27
• C. 1:26
• D. 1:81

Answer 1

The Correct Option is A

To solve this problem, we can use the method known as "successive dilution". Initially, let's assume the glass contains 1 unit of milk.

First Iteration: Two-thirds of the milk is poured out, leaving one-third inside. The remaining amount of milk is:

Remaining Milk = Initial Milk × (1/3) = 1 × (1/3) = 1/3 

Then, the glass is filled with water up to 1 unit, so the total is still 1 unit, now with a new milk-to-water ratio.

Second Iteration: Repeat the process:

Remaining Milk = (1/3) × (1/3) = 1/9

Third Iteration:

Remaining Milk = (1/9) × (1/3) = 1/27

Fourth Iteration:

Remaining Milk = (1/27) × (1/3) = 1/81

After four iterations, the milk fraction left is 1/81. The remaining portion in the glass is water, making the water fraction:

Water Fraction = 1 - Milk Fraction = 1 - (1/81) = 80/81

Thus, the final ratio of milk to water is 1:80, confirming the correct answer is 1:80.

Answer 2

Let the initial amount of milk be 1. After the first step, the amount of milk remaining is $\frac{1}{3}$. In each subsequent step, two-thirds of the content is replaced, so the amount of milk remaining after each step follows the pattern:

Milk after step 1 $= \frac{1}{3}$,

Milk after step 2 $= \frac{1}{9}$, 

Milk after step 3 $= \frac{1}{27}$, 

Milk after step 4 $= \frac{1}{81}$

Thus, the final amount of milk is $\frac{1}{81}$ and the remaining content is water. The ratio of milk to water is 1 : 80.