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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?
(CAT 2017)
Correct answer is '15'. Can you explain this answer?
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A person can complete a job in 120 days. He works alone on Day 1. On D...
Let the rate of work of a person by Y units/day.
Hence, the total work = 120Y .
Given : On first day, only one person works, on the second day two people work and so on.
Hence, the work done on day 1, day 2,… will be Y ,2Y ,3Y …respectively.
Let the total number of days required to complete the work is n.
Thus sum should be equal to 120Y.
So, Y + 2Y + 3Y +…+ nY = 120Y
or, 120Y = Y x n (n - 1) / 2
or, n2 + n - 240 = 0 or (n - 15)(n + 16) = 0
So, n = 15 or n = -16
But, n = 15 is the only positive value.
Hence, it will take 15 days to complete the work.
Hence, the total work = 120Y .
Given : On first day, only one person works, on the second day two people work and so on.
Hence, the work done on day 1, day 2,… will be Y ,2Y ,3Y …respectively.
Let the total number of days required to complete the work is n.
Thus sum should be equal to 120Y.
So, Y + 2Y + 3Y +…+ nY = 120Y
or, 120Y = Y x n (n - 1) / 2
or, n2 + n - 240 = 0 or (n - 15)(n + 16) = 0
So, n = 15 or n = -16
But, n = 15 is the only positive value.
Hence, it will take 15 days to complete the work.
Most Upvoted Answer
A person can complete a job in 120 days. He works alone on Day 1. On D...
Problem Analysis:
- We are given that a person can complete the job in 120 days.
- On each day, a new person joins the work, and all the people work together.
- We need to find out how many days are required to complete the job.
Solution:
Let's analyze the problem step by step:
Day 1:
- The first person works alone, and therefore, completes 1/120th of the job.
Day 2:
- The second person joins on Day 2.
- Now, two people are working, and their combined efficiency is 2/120 or 1/60.
- Hence, they complete 1/60th of the remaining job, which means they complete 1/60 x (1 - 1/120) of the entire job.
Day 3:
- On Day 3, the third person joins.
- Now, three people are working, and their combined efficiency is 3/120 or 1/40.
- Hence, they complete 1/40th of the remaining job, which means they complete 1/40 x (1 - 1/120 - 1/60) of the entire job.
Generalizing the Pattern:
We can observe a pattern in the above calculations. On each day, the combined efficiency of the people working is equal to the sum of their individual efficiencies. Hence, on Day N, the combined efficiency is N/120.
Using this pattern, we can write the general formula for the amount of work completed on Day N as follows:
Work completed on Day N = (1 - 1/120 - 1/60 - 1/40 - ... - 1/(120/N)) = 1 - (1/120 + 1/60 + 1/40 + ... + 1/(120/N))
Calculating the Number of Days:
We need to find the value of N for which the work completed on Day N is equal to 1 (i.e., the entire job is completed).
1 - (1/120 + 1/60 + 1/40 + ... + 1/(120/N)) = 1
1/(120/1) + 1/(120/2) + 1/(120/3) + ... + 1/(120/N) = 1
(1 + 2 + 3 + ... + N)/120 = 1
(N(N + 1))/2 = 120
N^2 + N - 240 = 0
(N - 15)(N + 16) = 0
Since N cannot be negative, the only valid solution is N = 15.
Conclusion:
Therefore, it will take 15 days to complete the job when a new person joins every day, starting from Day 1.
- We are given that a person can complete the job in 120 days.
- On each day, a new person joins the work, and all the people work together.
- We need to find out how many days are required to complete the job.
Solution:
Let's analyze the problem step by step:
Day 1:
- The first person works alone, and therefore, completes 1/120th of the job.
Day 2:
- The second person joins on Day 2.
- Now, two people are working, and their combined efficiency is 2/120 or 1/60.
- Hence, they complete 1/60th of the remaining job, which means they complete 1/60 x (1 - 1/120) of the entire job.
Day 3:
- On Day 3, the third person joins.
- Now, three people are working, and their combined efficiency is 3/120 or 1/40.
- Hence, they complete 1/40th of the remaining job, which means they complete 1/40 x (1 - 1/120 - 1/60) of the entire job.
Generalizing the Pattern:
We can observe a pattern in the above calculations. On each day, the combined efficiency of the people working is equal to the sum of their individual efficiencies. Hence, on Day N, the combined efficiency is N/120.
Using this pattern, we can write the general formula for the amount of work completed on Day N as follows:
Work completed on Day N = (1 - 1/120 - 1/60 - 1/40 - ... - 1/(120/N)) = 1 - (1/120 + 1/60 + 1/40 + ... + 1/(120/N))
Calculating the Number of Days:
We need to find the value of N for which the work completed on Day N is equal to 1 (i.e., the entire job is completed).
1 - (1/120 + 1/60 + 1/40 + ... + 1/(120/N)) = 1
1/(120/1) + 1/(120/2) + 1/(120/3) + ... + 1/(120/N) = 1
(1 + 2 + 3 + ... + N)/120 = 1
(N(N + 1))/2 = 120
N^2 + N - 240 = 0
(N - 15)(N + 16) = 0
Since N cannot be negative, the only valid solution is N = 15.
Conclusion:
Therefore, it will take 15 days to complete the job when a new person joins every day, starting from Day 1.
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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?(CAT 2017)Correct answer is '15'. Can you explain this answer?
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