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?

144 days

The correct answer is Option (1) → 144 days

Step 1: Let the work rates

Let total work = 1 unit.

• P + Q together can do the work in 60 days → work rate:

$P + Q = \frac{1}{60}$​

• Q + R together can do the work in 120 days → work rate:

$Q + R = \frac{1}{120}$​

• P + R together can do the work in 90 days → work rate:

$P + R = \frac{1}{90}$​

Step 2: Add all three equations

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

$2P + 2Q + 2R = ?$

Compute RHS (LCM = 360):

$\frac{1}{60} = \frac{6}{360}, \quad \frac{1}{120} = \frac{3}{360}, \quad \frac{1}{90} = \frac{4}{360}$

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

$(P + Q + R) = \frac{13}{360} \quad \Rightarrow \quad P + Q + R = \frac{13}{720}$

Step 3: Find Q alone

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

$\frac{1}{90} = \frac{8}{720} \quad \Rightarrow \quad Q = \frac{13}{720} - \frac{8}{720} = \frac{5}{720}$

• Work rate of Q = 5/720 per day → time taken = $\frac{1}{5/720} = \frac{720}{5} = 144$