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

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Accepted Answer

The problem involves calculating the final ratio of milk to water in a glass after repeatedly replacing two-thirds of its content with water. Here's the step-by-step solution:

**Initial Setup:**

- Let the initial quantity of milk in the glass be $$y$$ liters.
   - After each iteration, two-thirds of the current content is replaced with water, leaving one-third of the milk.

**Formula for Final Milk Concentration:**

- The concentration of milk after $$n$$ iterations is given by:
     $$
     	ext{Final concentration of milk} = 	ext{Initial concentration} 	imes \left(1 - \frac{x}{v}ight)^n
     $$
     Here:
     - $$x$$ = volume replaced = $$\frac{2}{3}$$.
     - $$v$$ = total volume = $$1$$ unit.
     - $$n$$ = number of iterations = $$4$$.

Substituting values:
   $$
   	ext{Final concentration} = y 	imes \left(1 - \frac{2}{3}ight)^4 = y 	imes \left(\frac{1}{3}ight)^4 = y 	imes \frac{1}{81}.
   $$

**Final Volumes:**

- Volume of milk:
     $$
     	ext{Milk volume} = \frac{1}{81}y.
     $$
   - Volume of water:
     $$
     	ext{Water volume} = y - 	ext{Milk volume} = y - \frac{1}{81}y = \frac{80}{81}y.
     $$

**Final Ratio:**

The ratio of milk to water is:
   $$
   	ext{Milk : Water} = \frac{	ext{Milk volume}}{	ext{Water volume}} = \frac{\frac{1}{81}y}{\frac{80}{81}y} = 1 : 80.
   $$

### Final Answer:
The final ratio of milk to water in the glass is **1 : 80**.  
Correct option: **D**

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