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John takes twice as much time as Jack to finish a job. Jack and Jim together take one-thirds of the time to finish the job than John takes working alone. Moreover, in order to finish the job, John takes three days more than that taken by three of them working together. In how many days will Jim finish the job working alone?
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Updated On: Jan 15, 2026
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The Correct Option is B

Solution and Explanation

Let Jack's work completion time be \(x\) days. Consequently, John's time is \(2x\) days. Let Jim's time be \(y\) days.

According to the problem statement: The average time for Jack and Jim working together equals the time for John and Jim working together.

Their combined work rates per day are:
Jack and Jim: \(\frac{1}{x} + \frac{1}{y}\)
John and Jim: \(\frac{1}{2x} + \frac{1}{y}\)

The problem states that the average time of Jack and Jim is equivalent to the time taken by Jack, John, and Jim collectively.

The average of Jack and Jim's time is calculated as the harmonic mean:
Equating the given conditions:
\(\frac{2xy}{x + y} = \frac{2x}{3}\)

Cross-multiplication yields:
\(\frac{2x}{3} = \frac{xy}{x + y}\)
Multiplying both sides by \(x + y\):
\(\frac{2x(x + y)}{3} = xy\)

Simplifying the equation:
\(2x^2 + 2xy = 3xy\)
\(2x^2 = xy\)
Dividing both sides by \(x\):
\(2x = y\)

Thus, we have determined that \( y = 2x \).

Now, consider the condition: John requires 3 days more than the combined working time of all three individuals.

The combined work rate of all three per day is:
\(\frac{1}{x} + \frac{1}{2x} + \frac{1}{y}\)
Substituting \(y = 2x\):
\(\frac{1}{x} + \frac{1}{2x} + \frac{1}{2x} = \frac{1}{x} + \frac{1}{x} = \frac{2}{x}\)

Therefore, the time taken by all three working together is \(\frac{x}{2}\) days.
John's time is \(2x\) days.
Given that John takes 3 days longer than the combined time of all three:

\(2x - \frac{x}{2} = 3\)
\(\frac{4x - x}{2} = 3 \Rightarrow \frac{3x}{2} = 3\)
Multiplying both sides by 2:
\(3x = 6 \Rightarrow x = 2\)

Substituting the value of \(x\) back:
Jack's time = \(x = 2\) days
Jim's time = \(y = 2x = 4\) days

Therefore, the time taken by Jim to complete the work is \(\boxed{4}\) days.

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