A group of workers having equal efficiency can complete a job in 4 days. But it so happened that every alternate day starting from the second day, 3 workers are withdrawn from the job and every alternate day starting from the third day, 2 workers are added to the group. If it now takes 7 days to complete the job, find the number of workers who started the job.

### Concept & Formula This problem is about tracking fluctuating work capacity day by day. We use the concept of **man-days** where Total Work is the sum of the number of workers present on each individual day. $$ \text{Total Work} = W_1 + W_2 + W_3 + \dots + W_n $$ where $W_i$ is the number of workers on day $i$. ### Step-by-Step Solution * **Initial Scenario:** Let the initial number of workers be $x$. Total Work = $4 \times x = 4x$ man-days. * **Track Daily Workers:** The job took 7 days with changing workforce. Day 1: $x$ Day 2 (withdraw 3): $x - 3$ Day 3 (add 2): $(x - 3) + 2 = x - 1$ Day 4 (withdraw 3): $(x - 1) - 3 = x - 4$ Day 5 (add 2): $(x - 4) + 2 = x - 2$ Day 6 (withdraw 3): $(x - 2) - 3 = x - 5$ Day 7 (add 2): $(x - 5) + 2 = x - 3$ * **Equate Total Work:** The sum of work done each day equals the total work. $x + (x - 3) + (x - 1) + (x - 4) + (x - 2) + (x - 5) + (x - 3) = 4x$ Combine like terms: $7x - (3 + 1 + 4 + 2 + 5 + 3) = 4x$ $7x - 18 = 4x$ * **Solve for $x$:** $7x - 4x = 18$ $3x = 18$ $x = 6$ ### Exam Strategy & Shortcut Pattern recognition helps speed this up. Notice the pattern of worker changes: $0, -3, -1, -4, -2, -5, -3$. Sum of the changes over 7 days = $-18$. Total work over 7 days = $7x - 18$. Since total work is also $4x$, we immediately get $7x - 18 = 4x \Rightarrow 3x = 18 \Rightarrow x = 6$. ### Common Pitfall The phrasing "every alternate day starting from..." can be confusing. It implies alternating operations: withdraw, add, withdraw, add. Misinterpreting this sequence (e.g., withdrawing on day 2 and day 4, but adding on day 3 and day 5 simultaneously) will lead to an incorrect sum. ### Final Answer Therefore, the correct answer is **6**.