MAT Exam 2020 Paper

Concept:Let the fraction of work done by A in one day be $A$ and by B in one day be $B$.Their combined daily work gives the total time.Use the changed speed condition to form a second equation.Explanation:Since A and B together finish the work in 5 days, their one-day work is:$$A + B = \frac{1}{5}$$If A works at twice the speed and B works at half the speed, they finish in 4 days.So, their new one-day work is:$$2A + \frac{B}{2} = \frac{1}{4}$$Multiply the first equation by $\frac{1}{2}$:$$\frac{A}{2} + \frac{B}{2} = \frac{1}{10}$$Subtract this equation from the second equation:$$\left(2A + \frac{B}{2}\right) - \left(\frac{A}{2} + \frac{B}{2}\right) = \frac{1}{4} - \frac{1}{10}$$This simplifies to:$$\left(2 - \frac{1}{2}\right)A = \frac{3}{20}$$$$\frac{3}{2}A = \frac{3}{20}$$$$A = \frac{3}{20} \times \frac{2}{3} = \frac{1}{10}$$Thus, A alone completes $\frac{1}{10}$ of the work in one day.Therefore, A alone would take 10 days to finish the whole work.Answer:A alone takes $\boxed{10}$ days.Correct option: A. 10 days.