Ayush and Bilal can complete a job together in D days. Ayush alone can…

Concept

If one worker finishes a job in x days, that worker’s rate is 1/x job per day.

For people working together, their rates add. Therefore, the reciprocal of the joint time equals the sum of the reciprocal individual times.

Application

1. Let the joint completion time be D days. Then the joint rate is 1/D job per day.

2. Ayush’s rate is 1/(D + 3), and Bilal’s rate is 1/(D + 12).

3. Using the combined-rate rule: 1/(D + 3) + 1/(D + 12) = 1/D.

4. Combining the fractions gives (2D + 15)/[(D + 3)(D + 12)] = 1/D.

5. Cross-multiplication gives D(2D + 15) = (D + 3)(D + 12).

6. Expanding both sides gives 2D2 + 15D = D2 + 15D + 36, so D2 = 36.

7. Because time must be positive, D = 6 days.

Cross-check

1. At D = 6, Ayush takes 9 days and Bilal takes 18 days.

2. Their combined rate is 1/9 + 1/18 = 1/6 job per day, which corresponds to 6 days.

Therefore, D = 6 days.