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Question:

A person can complete a job in 120 days. He works alone on Day 1. On Day 2, he is joined by another person who also can complete the job in exactly 120 days. On Day 3, they are joined by another person of equal efficiency. Like this, everyday a new person with the same efficiency joins the work. How many days are required to complete the job?

Explanation:

Let the number of days required to complete the job be n.

1 person works on day 1, 2 on day 2, 3 on day 3, …. n on day n.

Using Unitary Method:
Each person has the same efficiency = $\frac{1}{120}$

Work done on
Day 1 = $1 \times \frac{1}{120}$
Day 2 = $2 \times \frac{1}{120}$
...
Day n = $n \times \frac{1}{120}$

∴ Total Work = $1 \times \frac{1}{120}$ + $2 \times \frac{1}{120}$ + ... + $n \times \frac{1}{120}$

This is also equal to 1.

∴ $\frac{1}{120} + \frac{2}{120} + \frac{3}{120} + . . + \frac{n}{120}$ = 1

∑n = 120
⇒ $\frac{n (n + 1)}{2}$ = 120
⇒ n = 15

Hence, 15.

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