Question

Shwetha takes $6$ days less than the number of days taken by Ankitha to complete a piece of work. If both Shwetha and Ankitha together can complete the same work in $4$ days, in how many days will Ankitha alone complete the work?

Answer 1

Let the number of days Ankitha takes to complete the work be $x$.

It is given that Shwetha takes $6$ days less than the number of days taken by Ankitha to complete a piece of work, therefore, the number of days Shwetha takes to complete the work is $(x-6)$.

Now, amount of work done by Ankitha in $1$ day is $\frac{1}{x}$ and amount of work done by Shwetha in $1$ day is $\frac{1}{x-6}$.

It is also given that both Shwetha and Ankitha together can complete the same work in $4$ days, therefore, amount of work done by both in $1$ day is $\frac{1}{4}$.

Thus, we have

$\frac{1}{x}+\frac{1}{x-6}=\frac{1}{4}\Rightarrow \frac{x-6+x}{x(x-6)}=\frac{1}{4}\Rightarrow \frac{2x-6}{x^2-6x}=\frac{1}{4}\Rightarrow x^2-6x=4(2x-6)\Rightarrow x^2-6x=8x-24\Rightarrow x^2-6x-8x+24=0\Rightarrow x^2-14x+24=0\Rightarrow x^2-12x-2x+24=0\Rightarrow x(x-12)-2(x-12)=0\Rightarrow (x-2)=0,(x-12)=0\Rightarrow x=2,x=12$

$x=2,$ can not be taken because $x-6$ will become $-4$

thus, $x=12$.

Hence, the number of days Ankitha alone takes to complete the work is $12$