Amit can complete a piece of work in 45 days, while Bharath can complete the same work in 5 days less than Amit, that is, in 40 days. Amit and Bharath start the work together at their full efficiencies, but after working together for some days Bharath leaves. Amit then finishes the remaining work alone in 56 days, working at half of his usual efficiency during this period. For how many days did Amit and Bharath work together at full efficiency?

Introduction / Context:
This is a multi-step time and work problem that combines full and reduced efficiencies. Two workers, Amit and Bharath, start working together at full efficiency. After some time, Bharath leaves and Amit continues alone but at half his usual efficiency. You must use the given times to determine the number of days they worked together initially. This problem tests your understanding of variable work rates and how to translate them into equations.

Given Data / Assumptions:
Amit alone can complete the work in 45 days. Bharath alone can complete the same work in 40 days (5 days less). Both work together at full efficiency for an unknown number of days. After that, Bharath leaves and Amit continues alone for 56 days but at half of his usual efficiency. The total work is one complete job, and all efficiency changes are clearly specified and constant during the periods described.

Concept / Approach:
We first find the usual daily work rates of Amit and Bharath. Let Amit’s full-efficiency rate be a and Bharath’s rate be b. During the period they work together, the combined rate is a + b. During the period when Amit works alone at half efficiency, his rate is a / 2. If t is the number of days they work together, then the total work equation is t(a + b) + 56(a / 2) = 1. We substitute the known values of a and b from 45 and 40 days, solve for t, and interpret the result as the number of days of joint work.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Step 2: Amit alone completes the work in 45 days, so his full-efficiency rate is a = 1 / 45 of the work per day. Step 3: Bharath alone completes the work in 40 days, so his rate is b = 1 / 40 of the work per day. Step 4: Let t be the number of days during which Amit and Bharath work together at full efficiency. Step 5: In those t days, total work done by Amit and Bharath together is t(a + b) = t(1 / 45 + 1 / 40). Step 6: After t days, Bharath leaves. Amit continues alone for 56 days but at half of his usual efficiency, so his rate in this period is a / 2 = (1 / 45) / 2 = 1 / 90 of the work per day. Step 7: Work done by Amit during these 56 days is 56 * (1 / 90) = 56 / 90 = 28 / 45 of the total work. Step 8: Total work done is the sum of the work done during the joint period and Amit’s half-efficiency period: t(a + b) + 28 / 45 = 1. Step 9: Compute a + b: 1 / 45 + 1 / 40 = (40 + 45) / (45 * 40) = 85 / 1800. This simplifies to 17 / 360. Step 10: Substitute into the work equation: t * (17 / 360) + 28 / 45 = 1. Step 11: Convert 28 / 45 to a denominator of 360: 28 / 45 = (28 * 8) / (45 * 8) = 224 / 360. Step 12: The equation becomes t * (17 / 360) + 224 / 360 = 360 / 360. Step 13: Multiply both sides by 360: 17t + 224 = 360. Step 14: Solve for t: 17t = 360 - 224 = 136, so t = 136 / 17 = 8. Step 15: Thus, Amit and Bharath worked together at full efficiency for 8 days.

Verification / Alternative check:
We can confirm by computing the total work done using t = 8 days. Combined rate a + b is 17 / 360 per day. In 8 days, they complete 8 * (17 / 360) = 136 / 360 = 34 / 90 of the work. Amit then works 56 days at rate 1 / 90 per day, completing 56 / 90 = 56 / 90 of the work. Total work done is 34 / 90 + 56 / 90 = 90 / 90 = 1 job. This matches the requirement that the entire work is completed, confirming that t = 8 days is correct.

Why Other Options Are Wrong:
Values such as 6, 9, 12, or 15 days would change the total work done during the joint period and would not balance the equation t(a + b) + 28 / 45 = 1. For example, if t were 6 days, the combined work during the joint period would be 6 * (17 / 360) = 102 / 360, and adding Amit’s 28 / 45 would not produce exactly 1. Only t = 8 days satisfies the exact equality required for the complete job.

Common Pitfalls:
A major pitfall is forgetting that Amit works at half efficiency during the 56 days, leading to an incorrect rate for that period. Some learners also misinterpret the statement “5 days less than Amit” and incorrectly assign Bharath’s time. Another frequent error is to average the times of 45 and 40 days rather than building the correct algebraic equation for the total work. Always convert times into rates and construct a precise equation reflecting each phase of the work.

Final Answer:
Amit and Bharath worked together at full efficiency for 8 days before Bharath left.