In a container there is a mixture of milk and water in which water makes up 26% of the mixture by volume. If 7 litres of this mixture are removed and replaced with 7 litres of pure milk, the percentage of milk in the mixture becomes 76%. What is the original quantity of the mixture in the container (in litres)?

Introduction:
This is a replacement type mixture question where a portion of a milk water mixture is removed and replaced with pure milk. The percentages before and after replacement are given, and we must find the original volume of the mixture. Such questions test your understanding of percentage composition and how replacement affects the concentration of components in a mixture.

Given Data / Assumptions:

• Initially water is 26% of the mixture by volume.
• Therefore milk is 74% of the mixture initially.
• 7 litres of the original mixture are removed.
• These 7 litres are replaced by 7 litres of pure milk.
• After this, the percentage of milk in the mixture becomes 76%.
• The total volume of the mixture in the container remains constant.

Concept / Approach:
Let the initial volume of the mixture be V litres. Since water is 26%, milk is 74%. Removing 7 litres of mixture removes milk and water in the same 74 : 26 ratio. Then we add back pure milk, which changes only the milk content. We then set the final milk fraction equal to 76% and solve for V using a linear equation.

Step-by-Step Solution:
Step 1: Let the initial volume of the mixture be V litres. Step 2: Initial milk = 0.74 * V, initial water = 0.26 * V. Step 3: 7 litres of mixture are removed, so milk removed = 0.74 * 7 and water removed = 0.26 * 7. Step 4: Milk remaining after removal = 0.74 * V - 0.74 * 7. Step 5: Water remaining after removal = 0.26 * V - 0.26 * 7. Step 6: 7 litres of pure milk are added, so new milk = 0.74 * V - 0.74 * 7 + 7. Step 7: Total volume is still V litres, so final milk fraction = (0.74 * V - 0.74 * 7 + 7) / V. Step 8: This fraction is given as 76%, so set up the equation: (0.74 * V - 0.74 * 7 + 7) / V = 0.76. Step 9: Multiply both sides by V: 0.74 * V - 0.74 * 7 + 7 = 0.76 * V. Step 10: Rearrange terms: 7 - 0.74 * 7 = 0.76 * V - 0.74 * V = 0.02 * V. Step 11: Compute 7 - 0.74 * 7 = 7 * (1 - 0.74) = 7 * 0.26 = 1.82. Step 12: So 0.02 * V = 1.82, hence V = 1.82 / 0.02 = 91 litres.

Verification / Alternative check:
Initial milk = 0.74 * 91 = 67.34 litres, water = 23.66 litres. In 7 litres removed, milk removed = 0.74 * 7 = 5.18 litres, water removed = 1.82 litres. After removal, milk = 62.16 litres, water = 21.84 litres. Adding 7 litres of pure milk gives milk = 69.16 litres, water = 21.84 litres. Total volume remains 91 litres. Milk percentage = 69.16 / 91 which is approximately 0.76 or 76%, confirming the result.

Why Other Options Are Wrong:
65 litres, 38 litres, and 104 litres do not satisfy the condition that the final milk percentage is exactly 76% after the described operation. Substituting these values for V in the equation leads to different final percentages.
None of these is incorrect because 91 litres, which satisfies all conditions, is available among the options.

Common Pitfalls:
One common mistake is to think that simply adding 7 litres of milk increases the milk percentage in a straightforward manner without considering the removal of the mixed solution. Another error is to change the total volume instead of keeping it constant after replacement. Also, some learners misinterpret percentages as fixed amounts rather than proportions of the total volume.

Final Answer:
The original quantity of the mixture in the container is 91 litres.