Question

Person $A$ takes $6$ days less than the time taken by the person $B$ to finish a piece of work. If both $A$ and $B$ together can finish it in $4$ days, find the time that $B$ would take to finish this work by himself.

Answer 1

Let the time taken for $B$ to finish that work in $x$ days.
$\therefore$ time taken for $A$ to finish the same work in $(x-6)$ days
One day $B$ can finish the work is $\frac{1}{x}$
One day $A$ can finish the work is $\frac{1}{x-6}$
Given $A$ and $B$ are join together to finish that work in $4$ days
i.e. $\frac{1}{x}+\frac{1}{x-6}=\frac{1}{4}$
$\Rightarrow \frac{x-6+x}{x(x-6)}=\frac{1}{4}$
$\Rightarrow \frac{2x-6}{x^2-6x}=\frac{1}{4}$
$\Rightarrow x^2-6x=8x-24$
$\Rightarrow x^2-14x+24=0$
$\Rightarrow (x-12)(x-2)=0$
$\Rightarrow x=2,12$
Therefore, $x=12(x=2$ is not possible since $2-6<0)$
The number of days $B$ finish that work in $12$ days.