‘A’ and ‘B’ can complete a work in 2 hours and 55 minutes. ‘A’ alone can do the same work two hours faster than ‘B’ alone can. How long will ‘B’ take to do the work alone ?
2 hours and 10 minutes
4 hours and 10 minutes
5 hours
7 hours
Answer is Wrong!
Answer is Right!
This question was previously asked in
UPSC CISF-AC-EXE – 2017
Since A can do the same work two hours faster than B, the time taken by A alone is `x – 2` hours.
The work rate of B is 1/x work per hour.
The work rate of A is 1/(x-2) work per hour.
A and B together complete the work in 2 hours and 55 minutes.
2 hours 55 minutes = 2 + 55/60 hours = 2 + 11/12 hours = (24+11)/12 hours = 35/12 hours.
The combined work rate of A and B is 1 / (35/12) = 12/35 work per hour.
The combined work rate is also the sum of their individual rates: 1/(x-2) + 1/x = 12/35.
[x + (x-2)] / [x(x-2)] = 12/35
(2x – 2) / (x² – 2x) = 12/35
35(2x – 2) = 12(x² – 2x)
70x – 70 = 12x² – 24x
12x² – 94x + 70 = 0
Divide by 2: 6x² – 47x + 35 = 0
Using the quadratic formula x = [-b ± sqrt(b² – 4ac)] / 2a:
x = [47 ± sqrt((-47)² – 4 * 6 * 35)] / (2 * 6)
x = [47 ± sqrt(2209 – 840)] / 12
x = [47 ± sqrt(1369)] / 12
x = [47 ± 37] / 12
Two possible solutions for x:
x1 = (47 + 37) / 12 = 84 / 12 = 7
x2 = (47 – 37) / 12 = 10 / 12 = 5/6
If x = 5/6 hours, then A takes x – 2 = 5/6 – 2 = -7/6 hours, which is not possible as time cannot be negative.
So, x = 7 hours is the valid solution.
B takes 7 hours to do the work alone.