A vessel contains 100 litres of milk. If 20 litres of milk is replaced with water, and the process is repeated two more times, how much milk is left in the vessel after the third replacement?

This problem involves the concept of successive dilution. Let's walk through it step-by-step to determine how much milk remains after three cycles of replacement. 

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### Step 1: Basic Concept of Replacement
When \( x \) litres of liquid is removed from a solution of volume \( V \), and replaced by another liquid, the concentration of the original liquid decreases exponentially. The remaining amount of the original liquid after \( n \) replacements is given by the formula:

\[
\text{Remaining quantity} = V \left( 1 - \frac{x}{V} \right)^n
\]

In our case:
- Total volume \( V = 100 \) litres
- Replaced volume \( x = 20 \) litres
- Number of replacements \( n = 3 \)

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### Step 2: Calculate the Remaining Milk After 3 Replacements
Using the formula:

\[
\text{Remaining milk} = 100 \left( 1 - \frac{20}{100} \right)^3
\]

\[
\text{Remaining milk} = 100 \left( \frac{80}{100} \right)^3 = 100 \times 0.8^3
\]

\[
0.8^3 = 0.512
\]

\[
\text{Remaining milk} = 100 \times 0.512 = 51.2 \, \text{litres}
\]

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### Step 3: Final Answer
After three successive replacements, **51.2 litres of milk** remains in the vessel.

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Do you want further details or have any questions? 

### Related Questions:
1. What if the replacement volume was different each time?
2. How does the formula change if we don't refill to full capacity after each replacement?
3. What is the amount of water in the vessel after the three replacements?
4. How does increasing the number of replacements affect the remaining milk quantity?
5. Can we generalize this to infinite replacement steps?

**Tip:** Successive dilution problems become easier if you recognize the pattern in geometric progression for the concentration changes.

Explore how to solve a successive dilution problem involving a vessel containing 100 litres of milk. Learn how replacing 20 litres of milk with water, repeated three times, affects the remaining milk in the vessel. The solution applies exponential decay and geometric progression concepts. Suitable for students in Grades 9-12.

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