Question

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?

Answer 1

Let the time taken by John, Jack, and Jim to complete the work individually be \( a \), \( b \), and \( c \) days, respectively.

Given Conditions:

1. \( a = 2b \)     (John's time is twice Jack's time)
2. \( \frac{bc}{b + c} = \frac{1}{3}a \)     (Jack and Jim working together take 1/3 of John's individual time)
3. \( a - \left( \frac{abc}{ab + bc + ac} \right) = 3 \)     (The difference between John's time and the time taken when all three work together is 3 days)

Step-by-Step Solution:

Step 1: From the first given equation:

\[ a = 2b \]

Step 2: Substitute \( a = 2b \) into the second equation:

\[ \frac{bc}{b + c} = \frac{1}{3}(2b) = \frac{2b}{3} \]

Cross-multiply to solve for \( c \):

\[ bc = \frac{2b}{3}(b + c) \]

\[ 3bc = 2b(b + c) \]

Since \( b eq 0 \), we can divide by \( b \):

\[ 3c = 2(b + c) \]

\[ 3c = 2b + 2c \]

\[ c = 2b \]

Step 3: Now, use the third equation:

\[ a - \left( \frac{abc}{ab + bc + ac} \right) = 3 \]

Substitute \( a = 2b \) and \( c = 2b \) into this equation:

\[ 2b - \left( \frac{2b \cdot b \cdot 2b}{2b \cdot b + b \cdot 2b + 2b \cdot 2b} \right) = 3 \]

\[ 2b - \left( \frac{4b^3}{2b^2 + 2b^2 + 4b^2} \right) = 3 \]

\[ 2b - \left( \frac{4b^3}{8b^2} \right) = 3 \]

\[ 2b - \frac{b}{2} = 3 \]

Combine the terms on the left side:

\[ \frac{4b - b}{2} = 3 \]

\[ \frac{3b}{2} = 3 \]

Solve for \( b \):

\[ b = 2 \]

Final Values:

• \( b = 2 \) days (Time taken by Jack)
• \( a = 2b = 2(2) = 4 \) days (Time taken by John)
• \( c = 2b = 2(2) = 4 \) days (Time taken by Jim)

✅ Answer:

Jack takes 2 days, and Jim takes 4 days.