Time and Work Question 4 - CAT 2023
The amount of job that Amal, Sunil and Kamal can individually do in a day, are in harmonic progression. Kamal takes twice as much time as Amal to do the same amount of job. If Amal and Sunil work for 4 days and 9 days, respectively, Kamal needs to work for 16 days to finish the remaining job. Then the number of days Sunil will take to finish the job working alone, is:
Step-by-Step Explanation
Let the daily work efficiencies of Amal, Sunil, and Kamal be a, s, and k, respectively.
- Apply the Harmonic Progression (HP) property:
The amount of work they can do in a day represents their efficiencies. We are told these efficiencies form an HP:
2/s = 1/a + 1/k --- (Equation 1)
- Incorporate the time constraint:
Kamal takes twice as long as Amal to finish the same job, which means Kamal's daily efficiency is half of Amal's:
a = 2k
- Express efficiencies in terms of s:
Substitute a = 2k into the HP equation:
2/s = 1/(2k) + 1/k
= 3/(2k)
âźą 4k = 3s
âźą k = 3s/4
a = 2k
= 2(3s/4)
= 3s/2
- Set up the total work equation:
Amal works for 4 days, Sunil works for 9 days, and Kamal works for 16 days to complete the task:
4a + 9s + 16k = W
Substitute our expressions for a and k in terms of s:
4(3s/2) + 9s + 16(3s/4) = W
6s + 9s + 12s = W
âźą 27s = W
Time taken by Sunil alone = W/s
= 27 days
Correct Answer: 27
đź’ˇ FundaMakers Tip:
"Amount of job done in a day" is just another way of saying efficiency. Since the question asks for Sunil's standalone time, express all other efficiencies relative to Sunil (s). This converts the total work equation into a single-variable sum, leading directly to the answer.