Akshay is 3x as effective as Kartik. Akshay completes the work in 40 days…

Concept: Time-and-work problems link a worker's efficiency (work done per day) to the time needed to finish a fixed amount of work: efficiency and time are inversely proportional for the same work. If one worker is k times as effective as another, the more effective worker needs 1/k times the time the other needs. When two people work together, their combined efficiency is the sum of their individual efficiencies, and the time to finish the work together is the reciprocal of that combined efficiency.

Application:

1. Let Kartik's time to complete the work alone be T days. Since Akshay is 3 times as effective as Kartik, Akshay's time to complete the same work is T/3 days.

2. Akshay completes the work 40 days lesser than Kartik, so T - T/3 = 40.

3. Simplifying the left side: 2T/3 = 40.

4. Solving for T: T = 60.

5. So Kartik's time = 60 days and Akshay's time = 60/3 = 20 days.

6. Kartik's work rate = 1/60 of the work per day; Akshay's work rate = 1/20 of the work per day.

7. Combined work rate = 1/60 + 1/20 = 1/60 + 3/60 = 4/60 = 1/15 of the work per day, so together they finish the full work in 15 days.

8. Time to finish half the work together = 15/2 = 7.5 days.

Cross-check: Take the total work as 60 units (a convenient common multiple). Kartik's efficiency is then 60 ÷ 60 = 1 unit per day, and Akshay's efficiency is 3 times that, 3 units per day, consistent with '3x as effective'. Their combined efficiency is 1 + 3 = 4 units per day, so the full work (60 units) takes 60 ÷ 4 = 15 days, and half the work (30 units) takes 30 ÷ 4 = 7.5 days, matching the rate-based calculation above.