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?

The correct answer is 4 days. Let's build the solution step-by-step from the given conditions.

Step 1: Assign a variable and establish John's time

Let Jack take x days to finish the job alone.

Since John takes twice as much time as Jack:

John's time = 2x days

Step 2: Find Jack and Jim's combined time

Jack and Jim together take one-third of the time John takes alone:

$$\text{Time for (Jack + Jim) together}=\frac{1}{3}\times 2x=\frac{2x}{3}\text{days}$$

Step 3: Find Jim's individual rate (and time)

Their combined rate (work done per day) is:

$$\text{Rate of (Jack + Jim)}=\frac{3}{2x}$$

Jack's individual rate:

$$\text{Rate of Jack}=\frac{1}{x}$$

Jim's rate = (Jack + Jim)'s combined rate − Jack's rate:

$$\text{Rate of Jim}=\frac{3}{2x}-\frac{1}{x}=\frac{3}{2x}-\frac{2}{2x}=\frac{1}{2x}$$

So Jim alone takes 2x days to finish the job.

Step 4: Find all three working together

Combined rate of John + Jack + Jim:

$$\text{Combined Rate}=\frac{1}{2x}+\frac{1}{x}+\frac{1}{2x}=\frac{1+2+1}{2x}=\frac{4}{2x}=\frac{2}{x}$$

Time for all three together:

$$\text{Time (all three)}=\frac{x}{2}\text{days}$$

Step 5: Apply the final condition — "John takes 3 more days than all three together"

$$\text{John's time}-\text{Time (all three)}=3$$

$$2x-\frac{x}{2}=3$$

$$\frac{4x-x}{2}=3$$

$$\frac{3x}{2}=3$$

$$x=2$$

Step 6: Find Jim's time

We established that Jim takes 2x days alone. Substituting x = 2:

$$\text{Jim's time}=2\times 2=4\text{days}$$

Verification Summary:

• Jack = 2 days  |  John = 4 days  |  Jim = 4 days
• Jack + Jim together = 1 day (rate = ½ + ¼ = ¾ → time = 4/3 days) — wait, let's verify: rate of Jack(1/2) + Jim(1/4) = 3/4 → time = 4/3 days = (1/3) × 4 ✔
• All three rate = 1/4 + 1/2 + 1/4 = 1 → time = 1 day
• John (4 days) − All three (1 day) = 3 days ✔

✅ Jim will finish the job working alone in 4 days.