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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