Question

A vessel contains a liquid in which there is 5 part milk and 3 part water. What part of the mixture should be taken out and replaced with water so that the ratio of a milk and water may become 1 : 1?

• A. 2/5
• B. 1/3
• C. 1/4
• D. 1/5

Answer 1

Answer:

D. 1/5

Initial State of the Mixture

• The original mixture contains Milk and Water in the ratio 5 : 3.

• This implies that out of a total of 8 parts (5+3), Milk constitutes 5 parts and Water constitutes 3 parts.

• Therefore, the initial proportion of Milk is 5/8 and Water is 3/8.

The Replacement Process Explained

• Step 1: Taking out a part of the mixture.

  • When a certain fraction (let's say 'x') of the mixture is taken out, both Milk and Water are removed in their existing ratio (5:3).

  • If 'V' is the total volume, 'xV' volume of mixture is removed. This means (5/8) * xV of Milk and (3/8) * xV of Water are removed.

  • The ratio of Milk to Water in the remaining mixture still remains 5:3.

• Step 2: Replacing with Water.

  • The removed 'xV' volume is replaced entirely by water.

  • This action increases the water content but does not affect the remaining milk content (which only decreased in the first step).

  • The total volume of the mixture is restored to its original amount 'V'.

Focusing on the Constant Component (Milk)

• For competitive exams, the most efficient way to solve these problems is to focus on the component whose absolute quantity is not replenished. In this case, Milk is only removed, never added back. Water is removed and then added back.

• The final ratio of Milk to Water is 1 : 1. This means that after the process, Milk will constitute 1/2 of the total mixture, and Water will also constitute 1/2.

• Let 'x' be the fraction of the mixture that was taken out and replaced.

• If 'x' fraction is removed, the remaining fraction of the mixture is (1 - x).

• The initial proportion of Milk was 5/8. After removing 'x' of the mixture, the amount of milk remaining will be (5/8) * (1 - x) of the original total volume.

• Since the volume is restored by adding water, this remaining milk now represents the final proportion of milk in the mixture.

• Equating the final proportion of Milk:Initial Milk Proportion * (1 - Fraction Removed) = Final Milk Proportion(5/8) * (1 - x) = 1/2

• Solving the equation:

  • Multiply both sides by 8: 5 * (1 - x) = 4

  • Distribute: 5 - 5x = 4

  • Rearrange: 5 - 4 = 5x

  • Simplify: 1 = 5x

  • Solve for x: x = 1/5