A and B can do a piece of work in 72 days; B and C can do it in 120 days; A and C can do it in 90 days. In what time can A alone do it ?

120 days

Let the daily work rates of A, B, and C be $a, b,$ and $c$, respectively.

Given:

$a + b = \frac{1}{72}$

$b + c = \frac{1}{120}$

$a + c = \frac{1}{90}$

Adding all three equations:

$2(a + b + c) = \frac{1}{72} + \frac{1}{120} + \frac{1}{90}$

Taking LCM = 360:

$2(a + b + c) = \frac{5 + 3 + 4}{360} = \frac{12}{360} = \frac{1}{30}$

$a + b + c = \frac{1}{60}$

Now,

$a = (a + b + c) - (b + c) = \frac{1}{60} - \frac{1}{120} = \frac{1}{120}$

Hence, A alone can complete the work in: $\frac{1}{120}$