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rate of slower --- x units/min
rate of faster ---- y units/min
combined rate = x+y units/min
let the volume of the whole thing have 1 unit
1/(x+y) = 40
x+y = 1/40
1/x - 1/y = 60
(y - x)/xy = 60
60xy = y - x
60xy + x = y
x(60y+1) = y
x = y/(60y+1)
then in x+y = 1/40
y/(60y+1) + y = 1/40
multiply by 40(60y+1)
40y + 40y(60y+1) = 60y+1
40y + 240y^2 + 40y = 60y+1
2400y^2 + 20y - 1 = 0
60y - 1)(40y + 1) = -
y = 1/60 or y is negative, which is not possible
so if y = 1/60, then x = (1/60)/(2) = 1/120
so time of slower = 1/(1/120) = 120 minutes
time for faster = 1/(1/60) = 60 minutes
check:
combined rate = 1/60 + 1/20
= 1/40
time at combined rate = 1/(1/40) = 40 minutes, as given