Question : A alone can do $\frac{1}{10}$th part of a work in 8 hours. B alone can do $\frac{1}{5}$th part of the same work in 4 hours. If they work together, then how many hours will they take to complete $\frac{9}{16}$th part of the same work?
Option 1: 9 hours
Option 2: 5 hours
Option 3: 8 hours
Option 4: 4 hours
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Correct Answer: 9 hours
Solution : Let the rate at which A and B can complete the work as $R_a$ and $R_b$, respectively. Given that A can do $\frac{1}{10}$th part of the work in 8 hours. ⇒ $R_a=\frac{1}{10×8}=\frac{1}{80}$ work/hour Given that B can do $\frac{1}{5}$th part of the work in 4 hours. ⇒ $R_b=\frac{1}{5×4}=\frac{1}{20}$ work/hour When A and B work together, their combined rate $R_{ab}$ is the sum of their individual rates: ⇒ $R_{ab}=R_a + R_b=\frac{1}{80}+\frac{1}{20}=\frac{1}{16}$ work/hour. $\therefore$ The time it takes for them to complete $\frac{9}{16}$th part of the work = $\frac{\frac{9}{16}}{\frac{1}{16}}$ = 9 hours Hence, the correct answer is 9 hours.
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