P and Q can complete a job in 24 days working together. P can alone complete it in 32 days. Both of them worked together for 8 days and then P left. The number of days Q will take to complete the remaining job is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 64 days
In time-and-work questions, a person who finishes a job in \(n\) days does \(\frac{1}{n}\) of it in one day, and when people work together their one-day work is added. Q’s one-day work is therefore found by subtracting P’s from the joint figure.
P and Q together: 24 days, so one-day work \(=\frac{1}{24}\)
P alone: 32 days, so one-day work \(=\frac{1}{32}\)
Q’s one-day work \(\displaystyle =\frac{1}{24}-\frac{1}{32}=\frac{4-3}{96}=\frac{1}{96}\)
Work done by both in 8 days \(\displaystyle =8\times\frac{1}{24}=\frac{1}{3}\)
Work left \(\displaystyle =1-\frac{1}{3}=\frac{2}{3}\)
Days Q needs for it \(\displaystyle =\frac{2}{3}\div\frac{1}{96}=\frac{2}{3}\times96=64\)
Shortcut: Take the whole job as 96 units, the LCM of 24 and 32.
P and Q together: \(96\div24=4\) units a day
P alone: \(96\div32=3\) units a day
Q alone: \(4-3=1\) unit a day
Done by both in 8 days: \(8\times4=32\) units
Left: \(96-32=64\) units, which Q finishes in 64 days