The ratio of water to milk in a container is 2 : 3. If 40 litres of the mixture are removed from the container and then 40 litres of pure milk are added in their place, the ratio of water to milk becomes 1 : 4. What was the initial quantity of the mixture in the container (in litres)?

Introduction:
This question is a mixture and replacement problem involving changes in the ratio of water to milk when some mixture is removed and replaced by pure milk. It tests your ability to use ratios and algebra to track how the quantities of components change in a container.

Given Data / Assumptions:

• Initially, the ratio of water to milk in a container is 2 : 3.
• A total of 40 litres of the mixture are removed.
• Then 40 litres of pure milk are added to the container.
• After these operations, the new ratio of water to milk becomes 1 : 4.
• The container volume remains constant throughout.
• We need to find the initial quantity of the mixture.

Concept / Approach:
We represent the initial amounts of water and milk using a common variable, based on the given ratio. Since the initial ratio is 2 : 3, we take water as 2k litres and milk as 3k litres. The total initial volume is 5k litres. When 40 litres of mixture are removed, water and milk are removed in the same ratio 2 : 3. Then 40 litres of pure milk are added, altering only the milk quantity. Using the final ratio, we set up an equation in k and solve for the initial total volume.

Step-by-Step Solution:
Step 1: Let initial water = 2k litres and initial milk = 3k litres. Step 2: Initial total mixture = 2k + 3k = 5k litres. Step 3: From 5k litres, 40 litres of mixture are removed. Step 4: Water removed = 40 * (2k / 5k) = 16 litres. Step 5: Milk removed = 40 * (3k / 5k) = 24 litres. Step 6: Water remaining = 2k - 16 litres. Step 7: Milk remaining before adding new milk = 3k - 24 litres. Step 8: Now add 40 litres of pure milk: new milk = 3k - 24 + 40 = 3k + 16 litres. Step 9: Final water = 2k - 16 litres, final milk = 3k + 16 litres. Step 10: The final ratio of water to milk is 1 : 4, so (2k - 16) / (3k + 16) = 1 / 4. Step 11: Cross multiply: 4 * (2k - 16) = 3k + 16. Step 12: This gives 8k - 64 = 3k + 16. Step 13: Rearrange: 8k - 3k = 16 + 64, so 5k = 80 and k = 16. Step 14: Initial total mixture = 5k = 5 * 16 = 80 litres.

Verification / Alternative check:
With k = 16, initial water = 32 litres, initial milk = 48 litres. Removing 40 litres in ratio 2 : 3 removes 16 litres of water and 24 litres of milk. Remaining water = 16 litres and remaining milk = 24 litres. After adding 40 litres of milk, final milk = 64 litres, final water = 16 litres. Thus final ratio of water to milk is 16 : 64 which simplifies to 1 : 4, confirming the result.

Why Other Options Are Wrong:
75, 85, 90 and 100 litres do not satisfy the equation for the final ratio 1 : 4 when the same process is applied. Substituting these values produces different ratios that do not simplify to 1 : 4.

Common Pitfalls:
Students sometimes mistakenly subtract 40 litres only from milk or only from water instead of from the mixture as a whole. Another common error is forgetting that the removed mixture has the same composition as the original mixture. Also, some learners misread the final ratio as milk to water instead of water to milk, which leads to the wrong equation.

Final Answer:
The initial quantity of the mixture in the container is 80 litres.