A drum contains 20 litres of a paint. From this, 2 litres of paint is taken out and replaced by 2 litres of oil. Again 2 litres of this mixture is taken out and replaced by 2 litres of oil. If this operation is performed once again, then what would be the final ratio of paint and oil in the drum ?

Answer

To solve the problem step by step, we will analyze the operations performed on the drum containing paint and oil.

### Step 1: Initial Setup
Initially, the drum contains 20 litres of paint and 0 litres of oil.
- Paint = 20 litres
- Oil = 0 litres

### Step 2: First Operation
We take out 2 litres of paint and replace it with 2 litres of oil.
- Paint after removal: \(20 - 2 = 18\) litres
- Oil after addition: \(0 + 2 = 2\) litres
- After the first operation:
- Paint = 18 litres
- Oil = 2 litres

### Step 3: Second Operation
We take out 2 litres of the mixture (which now contains both paint and oil) and replace it with 2 litres of oil.
- The total volume of the mixture is still 20 litres.
- The ratio of paint to the total mixture is \( \frac{18}{20} \) and the ratio of oil to the total mixture is \( \frac{2}{20} \).
- Amount of paint removed in the second operation:
\[ 2 \times \frac{18}{20} = 1.8 \text{ litres} \]
- Amount of oil removed in the second operation:
\[ 2 \times \frac{2}{20} = 0.2 \text{ litres} \]
- Paint after removal:
\[ 18 - 1.8 = 16.2 \text{ litres} \]
- Oil after removal:
\[ 2 - 0.2 = 1.8 \text{ litres} \]
- Adding 2 litres of oil:
\[ 1.8 + 2 = 3.8 \text{ litres} \]
- After the second operation:
- Paint = 16.2 litres
- Oil = 3.8 litres

### Step 4: Third Operation
We again take out 2 litres of the new mixture and replace it with 2 litres of oil.
- The total volume of the mixture is still 20 litres.
- The ratio of paint to the total mixture is \( \frac{16.2}{20} \) and the ratio of oil to the total mixture is \( \frac{3.8}{20} \).
- Amount of paint removed in the third operation:
\[ 2 \times \frac{16.2}{20} = 1.62 \text{ litres} \]
- Amount of oil removed in the third operation:
\[ 2 \times \frac{3.8}{20} = 0.38 \text{ litres} \]
- Paint after removal:
\[ 16.2 - 1.62 = 14.58 \text{ litres} \]
- Oil after removal:
\[ 3.8 - 0.38 = 3.42 \text{ litres} \]
- Adding 2 litres of oil:
\[ 3.42 + 2 = 5.42 \text{ litres} \]
- After the third operation:
- Paint = 14.58 litres
- Oil = 5.42 litres

### Step 5: Final Ratio
To find the final ratio of paint to oil:
\[ \text{Ratio} = \frac{\text{Paint}}{\text{Oil}} = \frac{14.58}{5.42} \]
To simplify:
- Divide both numbers by 2:
\[ \frac{14.58 \div 2}{5.42 \div 2} = \frac{7.29}{2.71} \]
This can be approximated further if needed, but the final ratio is approximately \( \frac{14.58}{5.42} \).

### Final Answer
The final ratio of paint to oil in the drum is approximately \( 14.58 : 5.42 \).

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