# A takes 6 days less than the time taken by B to finish a piece of work. If both A and B together can finish it in 4 days, how many days will B take to finish the work?

Dec 3, 2016

We call the time it takes for B to finish the piece of work $x$.

For problems like these, we must consider how much work can be done in one day.

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

$\frac{x - 6}{x \left(x - 6\right)} + \frac{x}{x \left(x - 6\right)} = \frac{1}{4}$

$4 \left(x - 6 + x\right) = {x}^{2} - 6 x$

$4 \left(2 x - 6\right) = {x}^{2} - 6 x$

$8 x - 24 = {x}^{2} - 6 x$

$0 = {x}^{2} - 14 x + 24$

$0 = \left(x - 12\right) \left(x - 2\right)$

$x = 12 \mathmr{and} 2$

If the time it takes $B$ is $2$ days, then the time it takes $A$ is $- 4$ days, which has no sense.

Therefore, it would take $B$ a period of $12$ days to finish the job.

Hopefully this helps!