Question : A can do $\frac{7}{8}$ of work in 28 days, and B can do $\frac{5}{6}$ of the same work in 20 days. The number of days they will take to complete if they do it together is:
Option 1: $15\frac{3}{7}$ days
Option 2: $17\frac{3}{5}$ days
Option 3: $14\frac{5}{7}$ days
Option 4: $13\frac{5}{7}$ days
Correct Answer: $13\frac{5}{7}$ days
Solution :
Time taken by A alone to complete $\frac{7}{8}$ of a work = 28 days
Time taken by A alone to complete the work = $28 × \frac{8}{7}$ days = 32 days
⇒ Part of work done by A alone in a day = $\frac{1}{32}$
Time taken by B alone to complete $\frac{5}{6}$ of the work = 20 days
Time taken by B alone to complete the work = $20 ×\frac{6}{5}$ = 24 days
⇒ Part of work done by B alone in a day = $\frac{1}{24}$
Let the time taken by A and B together to complete the work = $x$
⇒ Part of work done by A and B in a day = $\frac{1}{x}$
⇒ $\frac{1}{32}$ + $\frac{1}{24}$ = $\frac{1}{x}$
⇒ $\frac{3+4}{96}$ = $\frac{1}{x}$
⇒ $x$ = $\frac{96}{7}$ = $13 \frac{5}{7}$ days
Hence, the correct answer is $13\frac{5}{7}$ days.
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