Question

A bottle is full of pure alcohol. One third of it is taken out and then equal amount of water is poured into the bottle to fill it. This operation is done four times. Find the final ratio of alcohol and water in the bottle?

• A. 8 : 30
• B. 36 : 44
• C. 16 : 65
• D. 32:65

Answer 1

Answer:

C. 16 : 65

Key Formula for Repeated Dilution

• When a vessel contains V units of a liquid, and x units are drawn out and replaced with water, and this operation is repeated n times, the quantity of the original liquid remaining in the vessel after 'n' operations is given by the formula:

  Quantity of Original Liquid Remaining = V * (1 - x/V)ⁿ

  • Here:

  • V = Initial total volume of the liquid in the vessel.

  • x = Quantity of the mixture removed in each operation.

  • n = Number of times the operation is performed.

Step-by-Step Application to the Problem

• Identify the Given Values:

  • Initial volume (let's assume total volume) = V (e.g., 1 unit or an arbitrary quantity)

  • Quantity taken out in each operation (x) = One third of the total volume = V/3

  • Number of operations (n) = 4

• Calculate the Fraction Removed (x/V):

  • x/V = (V/3) / V = 1/3

• Calculate the Quantity of Alcohol Remaining:

  • Using the formula: Quantity of Alcohol Remaining = V * (1 - 1/3)⁴

  • Quantity of Alcohol Remaining = V * (2/3)⁴

  • Quantity of Alcohol Remaining = V * (16/81)

• Determine the Quantity of Water:

  • Since the total volume in the bottle remains constant at V (because the removed liquid is replaced by an equal amount of water), the quantity of water present will be the total volume minus the remaining alcohol.

  • Quantity of Water = Total Volume - Quantity of Alcohol Remaining

  • Quantity of Water = V - V * (16/81)

  • Quantity of Water = V * (1 - 16/81)

  • Quantity of Water = V * ((81 - 16)/81)

  • Quantity of Water = V * (65/81)

• Find the Final Ratio of Alcohol to Water:

  • Ratio = Quantity of Alcohol : Quantity of Water

  • Ratio = (V * 16/81) : (V * 65/81)

  • Ratio = 16 : 65