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 individual times taken by John, Jack, and Jim to complete the work alone be \( a \), \( b \), and \( c \) days respectively.

Given Conditions: 

1. \( a = 2b \)     (John takes twice the time of Jack)
2. \( \frac{bc}{b + c} = \frac{1}{3}a \)     (Jack and Jim together take 1/3 the time of John)
3. \( a - \left( \frac{abc}{ab + bc + ac} \right) = 3 \)     (Combined efficiency difference)

Step-by-Step Solution:

Step 1: From equation (1):

\[ a = 2b \]

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

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

Multiply both sides by \( (b + c) \):

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

Step 3: Now use equation (3):

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

Substitute \( a = 2b \) and \( c = 2b \):

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

Final Values:

• \( b = 2 \) (Jack)
• \( a = 2b = 4 \) (John)
• \( c = 2b = 4 \) (Jim)

✅ Answer:

Jack takes 2 days, Jim takes 4 days.

Answer 2

Let John’s work rate be \( x \) units/day. 
Then, Jack works at twice the rate of John, so his rate is \( 2x \) units/day. 
Jim works in such a way that together with Jack, they complete \(\frac{1}{3}\) of the work that John does in a day:

\[ \text{Jack + Jim} = \frac{1}{3} \times \text{John's total work in } (n+3) \text{ days} \]

That implies Jim’s rate = \( x \) units/day (because: Jack does \(2x\), total is \(3x\), so Jim does \(x\)).

So, total rate of all three together = \( x + 2x + x = 4x \) units/day

John alone takes \( n + 3 \) days to finish the work. So, total work = \( x(n+3) \)

Combined, they take \( n \) days to do the same work. So,

\[ x(n + 3) = 4x \cdot n \]

Divide both sides by \( x \):

\[ n + 3 = 4n \Rightarrow 3n = 3 \Rightarrow n = 1 \]

Final Answer:

Thus, total work = \( 4x \cdot 1 = 4x \) units 
Jim’s rate = \( x \) units/day 
So, time taken by Jim alone = \( \frac{4x}{x} = \boxed{4 \text{ days}} \)