Question : When A alone does a piece of work, he takes 25 days more than the time taken by A and B to do the work together. On the other hand, B alone takes 16 days more than the time taken by A and B to do the work together. How many days will A and B, working together, take to do the work?
Option 1: 24
Option 2: 20
Option 3: 23
Option 4: 22
Correct Answer: 20
Solution : Let the total days to complete the work be $x$. One day work of A working alone = $\frac{1}{x+25}$ One day work of B working alone = $\frac{1}{x+16}$ According to the question, $\frac{1}{x+25}+\frac{1}{x+16}=\frac{1}{x}$ ⇒ $\frac{x+16+x+25}{(x+25)(x+16)}=\frac{1}{x}$ ⇒ $\frac{2x+41}{(x^2+16x+25x+400)}=\frac{1}{x}$ ⇒ $\frac{2x+41}{(x^2+41x+400)}=\frac{1}{x}$ ⇒ $2x^2+41x=x^2+41x+400$ ⇒ $x^2=400$ $\therefore x=20$ Hence, the correct answer is 20.
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