Question : P, Q, and R can finish a work in 5 days, 10 days, and 15 days, respectively, working alone. P and Q work on the first day, P and R work on the second day, P and Q work on the third day and so on till the work is completed. In how many days the work will be completed?
Option 1: $\frac{13}{2}$
Option 2: $\frac{9}{2}$
Option 3: $\frac{7}{2}$
Option 4: $\frac{5}{2}$
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Correct Answer: $\frac{7}{2}$
Solution :
Time taken by P to complete the work = 5 days
Part of work done by P in a day = $\frac{1}{5}$
Time taken by Q to complete the work = 10 days
Part of work done by Q in a day = $\frac{1}{10}$
Time taken by R to complete the work = 15 days
Part of work done by R in a day = $\frac{1}{15}$
Part of work done in first two days = $\frac{1}{5}$ + $\frac{1}{10}$ + $\frac{1}{5}$ + $\frac{1}{15}$ = $\frac{6+3+6+2}{30}$ = $\frac{17}{30}$
Remaining work = $\frac{13}{30}$
Part of work done in third day = $\frac{1}{5}$ + $\frac{1}{10}$ = $\frac{2+1}{10}$ = $\frac{3}{10}=\frac{9}{30}$
Work left to be done = $\frac{13-9}{30}$ = $\frac{4}{30}$
Part of work done in fourth day = $\frac{1}{5}$ + $\frac{1}{15}$ = $\frac{3+1}{15}$ = $\frac{8}{30}$
Time taken to finish the work left in fourth day= $\frac{4}{8}$ =$\frac{1}{2}$day
Total time taken to complete the work = $3\frac{1}{2}=\frac{7}{2}$ days
Hence, the correct answer is $\frac{7}{2}$.
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