Rohan and Mohit together can build a wall in 8 days. Mohit and Vikas can build the same wall in 10 days, and Vikas and Rohan together can build it in 12 days. In how many days will Rohan, Mohit, and Vikas working all together be able to complete the wall?

Introduction / Context:
This problem is a classic example of a work and time question where we know how long different pairs of workers take to complete a job. The goal is to find how long all three workers, Rohan, Mohit, and Vikas, take when they work together. It tests understanding of combined work rates and solving simple linear equations with multiple unknowns.

Given Data / Assumptions:

• Rohan and Mohit together complete the wall in 8 days.
• Mohit and Vikas together complete the wall in 10 days.
• Vikas and Rohan together complete the wall in 12 days.
• The total work is considered as 1 unit (one complete wall).
• All work rates are constant and additive.

Concept / Approach:
Assign individual daily work rates to Rohan, Mohit, and Vikas. Use the information about their pairwise times to build equations in terms of these rates. Then solve for the sum of the three individual rates, which equals the combined rate when all three work together. Finally, take the reciprocal of that rate to find the required number of days for the combined team.

Step-by-Step Solution:
Step 1: Let R, M, and V be the daily work rates of Rohan, Mohit, and Vikas respectively. Step 2: Rohan and Mohit together finish in 8 days, so R + M = 1/8. Step 3: Mohit and Vikas together finish in 10 days, so M + V = 1/10. Step 4: Vikas and Rohan together finish in 12 days, so V + R = 1/12. Step 5: Add all three equations: (R + M) + (M + V) + (V + R) = 1/8 + 1/10 + 1/12. Step 6: The left side becomes 2(R + M + V). The right side is 1/8 + 1/10 + 1/12. Step 7: Compute 1/8 + 1/10 + 1/12. Using common denominator 120, we get 15/120 + 12/120 + 10/120 = 37/120. Step 8: Therefore 2(R + M + V) = 37/120, so R + M + V = 37/240. Step 9: When all three work together, their combined rate is 37/240 units per day. Hence time taken = 1 / (37/240) = 240/37 days.

Verification / Alternative check:
We can approximate the value 240/37. Since 37 * 6 = 222 and 37 * 7 = 259, the time is between 6 and 7 days, approximately 6.49 days. That is reasonable because each pair alone takes between 8 and 12 days, so three people together should take less time than any pair, which matches our result.

Why Other Options Are Wrong:

• 120/37 days: This is roughly 3.24 days, which is unrealistically fast compared with the pairwise times.
• 150/37 days: This is around 4.05 days, still too fast compared with the individual pairwise times of 8, 10, and 12 days.
• 180/37 days: This is close to 4.86 days, which does not match the derived combined rate and is an incorrect result of manipulating the equations.

Common Pitfalls:
One common mistake is to average the three given times directly, which does not work for rate problems. Another error is forgetting to divide the sum of the three pairwise equations by 2 to get R + M + V. Students may also mishandle fraction addition, so it is important to use a common denominator carefully and simplify step by step.

Final Answer:
All three, Rohan, Mohit, and Vikas, working together will complete the wall in 240/37 days.