Question

Sunil takes $5$ days more than Anil to complete a certain work. $4$ day after starting the work, Anil left the work. The remaining work was done by Sunil in $5$ days. Find how many days each will take to complete the work.

Answer 1

Let Anil takes $x$ number of days to complete the work

$\Rightarrow 1$ day $=\frac{1}{x}$ of the work.

Sunil takes $5$ more days than Anil to complete the work.

$\Rightarrow x+5$ days

$\Rightarrow 1$ day $=\frac{1}{(x+5)}$ of the work

Together:

$1$ day $=\frac{1}{x}+\frac{1}{x+5}$

$1$ day $=\frac{(x+5+x)}{x(x+5)}$

$1$ day $=\frac{(2x+5)}{(x^2+5x)}$ of work done

To find $4$ days of work done:

$1$ day $=\frac{(2x+5)}{(x^2+5x)}$

$4$ days$=\frac{4(2x+5)}{(x^2+5x)}$

$4$ days $=\frac{(8x+20)}{(x^2+5x)}$ of work done

To find the amount of work left to be done:

Work left $=1-\frac{(8x+20)}{(x^2+5x)}$

Work left $=\frac{(x^2+5x-8x-20)}{(x^2+5x)}$

Work left $=\frac{(x^2-3x-20)}{(x^2+5x)}$

Solving for $x$:

Give that Anil needs $5$ days to complete the rest of the work

$\frac{(x^2-3x-20)}{(x^2+5x)}\times (x+5)$

$=\frac{(x^2-3x-20)}{x}$

$\Rightarrow 5x^2-3x-20=5x$

$x²-8x-20=0$

$(x-10)(x+2)=0$

$\therefore x=10$ or $x=-2$ (Rejected, number of days cannot be negative)

To find the number of days each take to complete the work

Anil $=x=10$ days

Sunil $=x+5=10+5=15$ days

Thus, Anil takes $10$ days and Sunil takes $15$ days to complete the work.