A tin contains a mixture of two liquids A and B in the proportion 4 : 1 by volume. If 45 litres of this mixture are removed and replaced by 45 litres of pure liquid B, the ratio of liquid A to liquid B in the tin becomes 2 : 5. How many litres of liquid B were present in the tin initially, and what is the total capacity of the tin (in litres)?

Introduction:
This is an alligation and mixture problem involving two liquids A and B in a tin. A portion of the mixture is removed and replaced with pure liquid B, changing the ratio of A to B. The question asks for the initial amount of B and the capacity of the tin, but the multiple choice answers point to the total capacity. We will find the capacity first and then the amount of B initially.

Given Data / Assumptions:

• Initially, the ratio of liquid A to liquid B in the tin is 4 : 1.
• 45 litres of the mixture are removed.
• Those 45 litres are replaced with 45 litres of pure B.
• After this operation, the ratio of A to B becomes 2 : 5.
• The total volume in the tin remains constant throughout.
• We must determine the capacity of the tin and the initial amount of B.

Concept / Approach:
Let the total capacity of the tin be V litres. Initially A and B are 4V / 5 and V / 5 litres respectively. Removing 45 litres removes A and B in the same 4 : 1 ratio. Then adding 45 litres of pure B changes only B. Using the final ratio 2 : 5, we form an equation in V and solve. Once we know V, we can find the initial amount of B as V / 5.

Step-by-Step Solution:
Step 1: Let the capacity of the tin be V litres. Step 2: Initial quantities: A = 4V / 5 litres, B = V / 5 litres. Step 3: 45 litres of mixture are removed in the 4 : 1 ratio. Step 4: A removed = 45 * (4 / 5) = 36 litres, B removed = 45 * (1 / 5) = 9 litres. Step 5: Remaining A = 4V / 5 - 36, remaining B = V / 5 - 9. Step 6: 45 litres of pure B are added, so new B = V / 5 - 9 + 45 = V / 5 + 36. Step 7: New ratio of A to B is 2 : 5, so set up the equation: (4V / 5 - 36) / (V / 5 + 36) = 2 / 5. Step 8: Cross multiply: 5 * (4V / 5 - 36) = 2 * (V / 5 + 36). Step 9: Simplify left side: 5 * 4V / 5 = 4V, and 5 * (-36) = -180, so left side = 4V - 180. Step 10: Right side: 2 * (V / 5 + 36) = 2V / 5 + 72. Step 11: Equation: 4V - 180 = 2V / 5 + 72. Step 12: Multiply all terms by 5 to clear the denominator: 20V - 900 = 2V + 360. Step 13: Rearrange: 20V - 2V = 360 + 900, so 18V = 1260. Step 14: Hence V = 1260 / 18 = 70 litres. Step 15: Initial liquid B = V / 5 = 70 / 5 = 14 litres.

Verification / Alternative check:
Initially A = 56 litres, B = 14 litres. Removing 45 litres in ratio 4 : 1 removes 36 litres A and 9 litres B. Remaining A = 20 litres, remaining B = 5 litres. Adding 45 litres of B gives final A = 20 litres, B = 50 litres. The final ratio A : B = 20 : 50 simplifies to 2 : 5, confirming that a capacity of 70 litres is correct.

Why Other Options Are Wrong:
50, 58, 62 and 65 litres all fail to satisfy the ratio equation when substituted for V. They lead to different final A : B ratios and do not give 2 : 5.

Common Pitfalls:
Some students mistakenly remove 45 litres only from A or only from B instead of from the mixture. Others forget that the mixture removed has the same composition as the mixture in the tin at that moment. It is also easy to misinterpret the question and try to match options to the initial amount of B directly without checking the ratio condition after replacement.

Final Answer:
The capacity of the tin is 70 litres, and initially it contained 14 litres of liquid B.