Azhar is four times as good a workman as Balaraj and therefore Azhar can finish a job in 45 days less than Balaraj takes to finish the same job alone. If Azhar and Balaraj work together, in how many days will they complete the job?

Introduction / Context:

This time and work problem involves comparative efficiency. One worker is stated to be four times as good as another, meaning he can complete four times as much work in the same time. We are also told how much earlier the faster worker finishes compared with the slower worker. The challenge is to use these relationships to find their individual times and then the time taken when they work together.

Given Data / Assumptions:

• Azhar is four times as efficient as Balaraj.
• Azhar finishes the job 45 days earlier than Balaraj would.
• We assume constant work rates and treat the entire job as one unit of work.
• We are asked to find the time taken if both work together.

Concept / Approach:

If Azhar is four times as good as Balaraj, then Azhar's work rate is four times Balaraj's work rate. Completion time is the reciprocal of work rate, so Azhar's time is one fourth of Balaraj's time. Combining this with the information about the 45 day difference in their completion times, we can set up an equation for Balaraj's time, solve it, and then use both individual times to find the combined work rate and joint completion time.

Step-by-Step Solution:

Verification / Alternative Check:

We can recheck by verifying the 45 day difference. With Balaraj needing 60 days and Azhar needing 15 days, the difference is indeed 45 days. Also, if they work together for 12 days at a combined rate of 1/12, they complete 12 * 1/12 = 1 full job, confirming the joint completion time is correct.

Why Other Options Are Wrong:

A joint time of 6 days would mean a combined rate of 1/6 and suggest extremely high efficiency beyond what four times comparison justifies. A time of 18 or 24 days would mean slower combined work, conflicting with the fact that Azhar alone already finishes the work in 15 days. When a very fast worker is joined by a slower one, the joint time must be less than the faster worker's time, and 12 days is the only option matching that condition and the algebraic solution.

Common Pitfalls:

Students may misinterpret "four times as good" as subtracting or adding four days instead of treating it as a comparison of rates. Others may incorrectly handle the equation for the 45 day difference, leading to arithmetic mistakes. Remember that efficiency is inversely proportional to time taken, and that differences are applied to times, not rates, when setting up such equations.

Final Answer:

Working together, Azhar and Balaraj will complete the job in 12 days.