Question

To complete a job, Babu needs $6$ more days than Abu. If both of them do the job together it takes $4$ days to complete it. How many days Babu needs, if they do the job separately?

Answer 1

Let the number of days required for Abu to finish the job be $x$.
Then the number of days required for Babu to finish the job will be $x+6$.

That is, Abu finishes $\frac{1}{x}$ part of the job and Babu finishes $\frac{1}{x+6}$ part of the job in one day.

Thus, they can together finish $\frac{1}{x}+\frac{1}{x+6}$ part of the job in one day.

Given that they can together finish the job in $4days$, we have

$\frac{1}{x}+\frac{1}{x+6}=\frac{1}{4}$

$\Rightarrow \frac{(x+6)+x}{x(x+6)}=\frac{1}{4}$

$\Rightarrow 4(2x+6)=x^2+6x$

$\Rightarrow 8x+24=x^2+6x$

$\Rightarrow x^2-2x-24=0$

$\Rightarrow x^2-6x+4x-24=0$

$\Rightarrow (x-6)(x+4)=0$

$\Rightarrow x=6$ or $x=-4$

Here, $x$ is number of days required for Abu to finish the job, so it is not a negative number.

Thus, $x=-4$ is inadmissible.

Hence, Abu requires 6 days and Babu requires $x+6=6+6=12$ days to finish the job alone.