Question : P, Q, and R can complete a work alone in 12,15, and 20 days respectively. They started the work together. P left the work 8 days before the work was completed and Q left the work 5 days after P had left. R completed the remaining work alone. How many days will be required to complete the whole work?
Option 1: $\frac{32}{3}$ days
Option 2: $10$ days
Option 3: $\frac{28}{3}$ days
Option 4: $\frac{25}{3}$ days
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Correct Answer: $\frac{28}{3}$ days
Solution :
Given that P can complete the work in 12 days, Q can complete the work in 15 days, and R can complete the work in 20 days.
P left the work 8 days before it was completed, Q left the work 5 days after P had left, and R completed the remaining work alone.
Total work = LCM$(12, 15, 20) = 60$
Efficiency of P = $\frac{60}{12} = 5$
Efficiency of Q = $\frac{60}{15} = 4$
Efficiency of R = $\frac{60}{20} = 3$
Let the total work be done in $t$ days.
According to the question,
⇒ $5(t - 8) + 4(t - 3) + 3t = 60$
⇒ $5t - 40 + 4t - 12 + 3t = 60$
⇒ $12t = 112$
⇒ $t = \frac{112}{12} = \frac{28}{3}$ days
Hence, the correct answer is $\frac{28}{3}$ days.
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