A thief steals four gallons of liquid soap kept in a train compartment’s…

2024

A thief steals four gallons of liquid soap kept in a train compartment’s bathroom from a container that is full of liquid soap. He then fills it with water to avoid detection. Unable to resist the temptation he steals 4 gallons of the mixture again, and fills it with water. When the liquid soap is checked at a station it is found that the ratio of the liquid soap now left in the container to that of the water in it is 36 : 13. What was the initial amount of the liquid soap in the container if it is known that the liquid soap is neither used nor augmented by anybody else during the entire period?

  1. A.

    13

  2. B.

    23

  3. C.

    28

  4. D.

    31

Attempted by 6 students.

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Correct answer: C

Answer: 28 gallons.

Let the container's initial volume be V gallons. Follow the quantities step by step:

  1. After the first theft of 4 gallons (pure soap) and refilling with 4 gallons of water, the amount of soap is V − 4 and the total volume is V.

  2. Before the second theft the fraction of soap in the mixture is (V − 4)/V. When 4 gallons of this mixture are removed, the amount of soap removed is 4 × (V − 4)/V.

  3. So the soap left after the second theft is (V − 4) − 4 × (V − 4)/V = (V − 4) × (1 − 4/V) = (V − 4)^2 / V.

  4. After refilling with 4 gallons of water the total volume is again V, so the final amount of soap is (V − 4)^2 / V. The final ratio of soap to water is given as 36:13, so soap = 36/49 of the total volume V.

  5. Set (V − 4)^2 / V = (36/49) V. Multiply both sides by V to get (V − 4)^2 = (36/49) V^2. Taking the positive square root gives (V − 4)/V = 6/7.

  6. Solve 1 − 4/V = 6/7 ⇒ 4/V = 1/7 ⇒ V = 28.

Therefore the initial amount of liquid soap in the container was 28 gallons.

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