Two variable equation

Question

I'm stuck with the following example (42.). Some help is much appreciated. Thank you.

Answer 1

This is a "speed" problem. The first person's speed is $v_1=1/x$ (jobs/day) and the second person's speed is $v_2=1/(x-3)$ (jobs/day).

If both work together, their collective speed is simply $v=v_1+v_2=\frac{1}{x}+\frac{1}{x-3}$. Now, at this collective speed they are able to finish $1$ job in $\frac25 (x+4)$ days. Thus,

$$\left(\frac{1}{x}+\frac{1}{x-3}\right)\left(\frac25 (x+4)\right)=1$$

the solutions of which are $x=1$ and $x=24$ (solutions to a quadratic equation). But since $x>3$, then $x=24$ days.

Thus, $v_1=\frac{1}{24}$ and $v_2=\frac{1}{21}$. If person $1$ works for $8$ days, then only $\frac13$ job is complete. That means that person $2$ must work for $14$ days to complete this formerly incomplete job.