Given:
Anil and Karan together can do a work in 20 days.
Karan and Himanshu together can do the same work in 25 days.
Himanshu and Anil together can do the same work in 50 days.
Formula Used:
Work rate formula: If A and B can do a work in D days, their combined work rate is 1/D.
Individual work rate can be found using the system of equations derived from combined work rates.
Calculation:
Let Anil's work rate be A, Karan's work rate be K, and Himanshu's work rate be H.
From the given information:
A + K = 1/20
K + H = 1/25
H + A = 1/50
Adding all three equations:
(A + K) + (K + H) + (H + A) = 1/20 + 1/25 + 1/50
2A + 2K + 2H = 1/20 + 1/25 + 1/50
2(A + K + H) = 1/20 + 1/25 + 1/50
A + K + H = (1/20 + 1/25 + 1/50) / 2
⇒ A + K + H = (5/100 + 4/100 + 2/100) / 2
⇒ A + K + H = 11/100 / 2
⇒ A + K + H = 11/200
Now, to find Himanshu's work rate (H):
From A + K = 1/20:
H = (A + K + H) - (A + K)
⇒ H = 11/200 - 1/20
⇒ H = 11/200 - 10/200
⇒ H = 1/200
Therefore, Himanshu alone can do the work in:
Time = 1 / H = 1 / (1/200) = 200 days
Himanshu alone can do the same work in 200 days.
Alternate Method LCM (20,25,50) = 100 = Total work.
A and K do = 100/20 = 5 units of work in one day.
K and H do = 100/25 = 4 units of work in one day.
H and A do = 100/50 = 2 units of work in one day.
All together do = (5 + 4 + 2)/2 = 11/2, divided by 2 because everyone is coming two times.
H alone does = All - A and K work = 11/2 - 5 = 1/2 units of work.
H alone complete total work in 100 / (1/2) = 200 days.