Time and Work Question 4 - CAT 2021
Amar, Akbar and Anthony are working on a project. Working together Amar and Akbar can complete the project in 1 year, Akbar and Anthony can complete in 16 months, Anthony and Amar can complete in 2 years. If the person who is neither the fastest nor the slowest works alone, the time in months he will take to complete the project is:
Step-by-Step Explanation
- First, standardize all time parameters into months:
Amar + Akbar = 1 year = 12 months
Akbar + Anthony = 16 months
Anthony + Amar = 2 years = 24 months
Let the total work be the LCM of 12, 16, and 24, which is 48 units.
Let the monthly work outputs of Amar, Akbar, and Anthony be m, k, and n, respectively.
m + k = 48/12 = 4 units/month --- (Equation 1)
k + n = 48/16 = 3 units/month --- (Equation 2)
n + m = 48/24 = 2 units/month --- (Equation 3)
- Add all three equations together:
2(m + k + n) = 4 + 3 + 2 = 9
⟹ m + k + n = 4.5 units/month
- Isolate the individual monthly rates:
Anthony's rate (n) = 4.5 − (m + k)
= 4.5 − 4
= 0.5 units/month
(Slowest)
Akbar's rate (k) = 4.5 − (n + m)
= 4.5 − 2
= 2.5 units/month
(Fastest)
Amar's rate (m) = 4.5 − (k + n)
= 4.5 − 3
= 1.5 units/month
(Middle)
The person who is neither the fastest nor the slowest is Amar (1.5 units/month).
Time taken by Amar alone = 48 units/1.5 units/month
= 32 months
Correct Answer: 32