Madhur works two times faster than Sagar. If Sagar can complete a certain job alone in 18 days at his usual rate, then in how many days can Madhur and Sagar together finish the entire job?

Introduction / Context:
This time and work question describes two workers, Madhur and Sagar, with a known relationship between their efficiencies. Sagar's individual time to complete the job is known, and Madhur is said to be twice as fast as Sagar. The problem asks for the total time when both work together from the start.

Given Data / Assumptions:

Sagar alone can complete the job in 18 days.
Madhur works two times faster than Sagar, that is, Madhur's rate is twice Sagar's rate.
Both work together at constant rates from the beginning until the job is finished.
The total work is taken as one complete unit of job.

Concept / Approach:
First, compute Sagar's daily work rate from his given completion time. Then use the information that Madhur is twice as fast to get Madhur's rate. The combined daily rate is the sum of their individual rates. Finally, find the time taken by taking the reciprocal of this combined rate, since Time = Work / Rate and we consider Work = 1 unit.

Step-by-Step Solution:
Let the total work be 1 unit. Given that Sagar alone completes the job in 18 days. So, Sagar's daily rate = 1/18 of the job per day. Madhur works two times faster, so Madhur's rate = 2 * (1/18) = 1/9 of the job per day. Combined daily rate of Madhur and Sagar = 1/18 + 1/9. Convert 1/9 to 2/18 for easy addition. Thus, combined rate = 1/18 + 2/18 = 3/18 = 1/6 of the job per day. Time taken together = total work / combined rate = 1 / (1/6) = 6 days.

Verification / Alternative check:
To verify, in 6 days at a combined rate of 1/6 per day, they complete exactly 1 job. Also, note that Sagar alone requires 18 days, while Madhur alone would require 9 days. Working together, it is reasonable that they take less time than 9 days and significantly less than 18 days. The answer of 6 days fits these expectations and confirms the correctness of the calculation.

Why Other Options Are Wrong:
Option 5 days would correspond to a combined rate of 1/5 per day, which is higher than the computed 1/6 and does not match the given efficiency relationship.
Option 2 days is far too small and would imply an unrealistically high combined rate that contradicts the individual times of 9 and 18 days.
Option 4 days also leads to a combined rate of 1/4 per day, which is not equal to the sum of 1/18 and 1/9.
Option 3 days would assume an even faster rate of 1/3 per day, clearly impossible based on the problem data.

Common Pitfalls:
A common mistake is misinterpreting “two times faster” as “twice as many days,” which is the opposite of what it means. Faster speed corresponds to fewer days and higher daily work rate. Another pitfall is adding or averaging days rather than rates. Always convert times into rates, apply the speed relationship correctly and then add the rates to find the combined efficiency.

Final Answer:
Thus, Madhur and Sagar working together can complete the job in 6 days, so the correct option is 6 days.