P and Q together can do a piece of work in 60 days. Q and R together can do it in 120 days. P and R together can do it in 90 days. In what time can Q alone do the same work?

Let P, Q, and R represent the number of days each person takes to complete the work alone.

P and Q together finish in 60 days, so they complete $\frac{1}{60}$ of the work per day.

Q and R together finish in 120 days, so they complete $\frac{1}{120}$ of the work per day.

P and R together finish in 90 days, so they complete $\frac{1}{90}$ of the work per day.

This gives:

$\frac{1}{P} + \frac{1}{Q} = \frac{1}{60}$ ... (1)

$\frac{1}{Q} + \frac{1}{R} = \frac{1}{120}$ ... (2)

$\frac{1}{P} + \frac{1}{R} = \frac{1}{90}$ ... (3)

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Adding all three equations:

$(\frac{1}{P} + \frac{1}{Q}) + (\frac{1}{Q} + \frac{1}{R}) + (\frac{1}{P} + \frac{1}{R}) = \frac{1}{60} + \frac{1}{120} + \frac{1}{90}$

$2(\frac{1}{P}) + 2(\frac{1}{Q}) + 2(\frac{1}{R}) = \frac{1}{60} + \frac{1}{120} + \frac{1}{90}$

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The LCM of 60, 120, and 90 is 360.

$\frac{1}{60} = \frac{6}{360}$

$\frac{1}{120} = \frac{3}{360}$

$\frac{1}{90} = \frac{4}{360}$

$\frac{6}{360} + \frac{3}{360} + \frac{4}{360} = \frac{13}{360}$

Therefore:

$2(\frac{1}{P} + \frac{1}{Q} + \frac{1}{R}) = \frac{13}{360}$

$\frac{1}{P} + \frac{1}{Q} + \frac{1}{R} = \frac{13}{720}$

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From equation (3): $\frac{1}{P} + \frac{1}{R} = \frac{1}{90}$

Subtracting this from the sum:

$(\frac{1}{P} + \frac{1}{Q} + \frac{1}{R}) - (\frac{1}{P} + \frac{1}{R}) = \frac{13}{720} - \frac{1}{90}$

$\frac{1}{Q} = \frac{13}{720} - \frac{1}{90}$

Converting $\frac{1}{90}$ to denominator 720:

$\frac{1}{90} = \frac{8}{720}$

$\frac{1}{Q} = \frac{13}{720} - \frac{8}{720}$

$\frac{1}{Q} = \frac{5}{720}$

$\frac{1}{Q} = \frac{1}{144}$

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Therefore, Q alone can complete the work in 144 days.