Question : A can complete one-third of a work in 10 days and B can do $\frac{3}{5}$ th of the same work in 24 days. They worked together for 10 days. The remaining work was completed by C alone in 15 days. In how many days can C alone do $\frac{2}{3}$rd of the same work?
Option 1: 27
Option 2: 24
Option 3: 30
Option 4: 32
Correct Answer: 24
Solution :
Time taken by A to do $\frac{1}{3}$rd of a work = 10 days
Time taken by A to complete the work = 3 × 10 = 30 days
Part of work done by A in a day = $\frac{1}{30}$
Time taken by B to do $\frac{3}{5}$th of a work = 24 days
Time taken by B to complete the work = $\frac{5 ×24}{3}$ = 40 days
Part of work done by B in a day = $\frac{1}{40}$
Part of work done by A and B together in a day $=\frac{1}{30}+\frac{1}{40} = \frac{4+3}{120} = \frac{7}{120}$
Part of the work done by A and B together in 10 days $=\frac{70}{120} = \frac{7}{12}$
Remaining work $=1-\frac{7}{12}=\frac{5}{12}$
Time taken by C to do $\frac{5}{12}$ of a work = 15 days
Time taken by C to complete the work = $\frac{15 ×12}{5}$ = 36 days
Time taken by C to do $\frac{2}{3}$ of a work = $\frac{2×36}{3}$ = 24 days
Hence, the correct answer is 24.
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