Question : A and B can complete a piece of work in 13 and 17 days respectively. A begins to do the work, and they work alternatively one at a time for one day each. The whole work will be completed in:
Option 1: $17 \frac{11}{17}$ days
Option 2: $17 \frac{17}{19}$ days
Option 3: $14 \frac{11}{17}$ days
Option 4: $11 \frac{11}{17}$ days
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Correct Answer: $14 \frac{11}{17}$ days
Solution :
Time taken by A to complete the work = 13 days
Part of work done by A alone in a day = $\frac{1}{13}$
Time taken by B to complete the work = 17 days
Part of work done by B alone in a day = $\frac{1}{17}$
Work done by A and B in the first 2 days if they work alternatively
= $\frac{1}{13}$ + $\frac{1}{17}$
= $\frac{13+17}{13×17}$
= $\frac{30}{221}$
Work done by A and B in 14 days = $\frac{7×30}{221}$ = $\frac{210}{221}$
Remaining work = $1-\frac{210}{221}=\frac{221-210}{221}$ = $\frac{11}{221}$
So, the time taken by A to complete the rest = $\frac{\frac{11}{221}}{\frac{1}{13}}$ = $\frac{11}{17}$
Thus, the total time taken by A and B to complete the work when they work alternatively
= $14\frac{11}{17}$ days
Hence, the correct answer is $14\frac{11}{17}$ days.
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