Question

A glass is initially filled entirely with milk. In each step, \(\frac{2}{3}\) of the milk is replaced with water. This process is repeated 3 times. What is the final ratio of water to milk in the glass?

• A. \(1 : 27\)
• B. \(1 : 20\)
• C. \(1 : 26\)
• D. \(26 : 1\)

Answer 1

The Correct Option is D

Initially, the glass is filled entirely with milk. We denote the initial amount of milk as \(M=1\). In the first step, \(\frac{2}{3}\) of the milk is replaced with water.

The remaining milk after step 1 is:

\(M_1=M-\frac{2}{3}M=\frac{1}{3}M\) 

Water added = \(\frac{2}{3}\)

In the second step, \(\frac{2}{3}\) of the remaining milk is replaced with water:

Remaining milk after step 2:

\(M_2=\frac{1}{3}M_1=\frac{1}{3}\times\frac{1}{3}M=\frac{1}{9}M\)

Water added = \(\frac{2}{3}+\frac{2}{3}\times\frac{1}{3}=\frac{2}{3}+\frac{2}{9}=\frac{8}{9}\)

In the third step, \(\frac{2}{3}\) of the remaining milk is again replaced with water:

Remaining milk after step 3:

\(M_3=\frac{1}{3}M_2=\frac{1}{3}\times\frac{1}{9}M=\frac{1}{27}M\)

Water added = \(1-\frac{1}{27}=\frac{26}{27}\)

The final water to milk ratio is:

\(\frac{\frac{26}{27}}{\frac{1}{27}}=26:1\)