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?

• A. 15
• B. 13
• C. 16
• D. 14

Answer 1

The Correct Option is A

Step 1: Understanding the problem

We are told that:

• On Day 1 → 1 person works
• On Day 2 → 2 people work
• On Day 3 → 3 people work
• … and so on until Day \(n\) → \(n\) people work.

All workers have the same efficiency, and **1 person alone can complete the job in 120 days**.

 

Step 2: Work rate of one person

If 1 person can finish the whole job in 120 days, then the work done by **one person in one day** is: \[ \frac{1}{120} \ \text{of the job} \]

Step 3: Work done over \( n \) days

Day 1: \( 1 \times \frac{1}{120} \) of the job Day 2: \( 2 \times \frac{1}{120} \) of the job Day 3: \( 3 \times \frac{1}{120} \) of the job … Day \(n\): \( n \times \frac{1}{120} \) of the job

Total work done in \(n\) days: \[ \frac{1}{120} + \frac{2}{120} + \frac{3}{120} + \dots + \frac{n}{120} \]

Step 4: Simplify the sum

Factor out \(\frac{1}{120}\): \[ \frac{1}{120} \left( 1 + 2 + 3 + \dots + n \right) \] We know the sum of the first \(n\) natural numbers is: \[ 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} \] So: \[ \frac{1}{120} \cdot \frac{n(n+1)}{2} = 1 \] (because the total work equals 1 job)

Step 5: Solve for \(n\)

Multiply both sides by 120: \[ \frac{n(n+1)}{2} = 120 \] Multiply through by 2: \[ n(n+1) = 240 \]

Step 6: Find \(n\)

We need two consecutive integers whose product is 240: \[ n^2 + n - 240 = 0 \] Using the quadratic formula: \[ n = \frac{-1 \pm \sqrt{1 + 4 \cdot 240}}{2} \] \[ n = \frac{-1 \pm \sqrt{961}}{2} \] \[ n = \frac{-1 \pm 31}{2} \] Only the positive root is valid: \[ n = \frac{-1 + 31}{2} = \frac{30}{2} = 15 \]

Final Answer:

\[ \boxed{\text{The job will be completed in 15 days.}} \]