Murat and Mustafa can do a job together in fifteen days. After they have worked together for five days, Mustafa leaves the job. Murat completes the job in sixteen days. How long would it take Mustafa to do the job alone?
The answer is 40 days but I want the steps to solve this.
Let $\;x,y\;$ be the amount of days it takes to Murat, Mustafa respectively to complete the job alone. We're given that
$$15\left(\frac1x+\frac1y\right)=1$$
so that after 5 days they worked together they did one third of the job, and it then took Murat sixteen days to complete the two thirds that were still left:
$$\frac13+\frac{16}x=1\iff\frac x{16}=\frac32\implies x=24$$ and then
$$15\left(\frac1{24}+\frac1y\right)=1\implies\frac{15}y=\frac38\implies y=40$$
The classical way of solving this would be to find the speeds by which they both work (for convenience, in units of full jobs per day). Let's say Murat works at speed $v_1$ and Mustafa works at speed $v_2$. Then the fact that they finish in $15$ days if they cooperate means that $v_1 + v_2 = \frac1{15}$.
If they work together for five days, then a third of the work is done. That means Murat does two thirds of a job in $16$ days, or symbolically, that $$ v_1 = \frac{2/3}{16} = \frac1{24} $$ Now you should be able to do the rest.
Let $M$ be the fraction of the job that Murat can finish in one day. Let $m$ be the fraction of the job that Mustafa can finish in one day. Then the statement that they can finish the job together in 15 days is: $$M+m = \frac{1}{15}$$
After $5$ days then have finished $$5(M+m) = \frac{5}{15} = \frac13$$ a third of the job. Then we are told that for the remaining $\frac23$,
$$ 16M = \frac23 \implies M = \frac{2}{48} = \frac{1}{24} $$
So $$ \frac{1}{24} + m = \frac{1}{15}\\ m = \frac{1}{15}-\frac{1}{24} = \frac{24-15}{24\cdot 15}=\frac{9}{360} = \frac{1}{40} $$
So Mustapha finshes $\frac{1}{40}$ each day.
Let $x$ be the rate of work/day of Murat and $y$ the rate of work/day of Mustafa, and $w$ the unknown amount of work, then you can model the given information as: $$ 15 (x + y) = w \\ 5(x+y) + 16 x = w $$ The final equation is $$ d \, y = w $$ where $d$ is the sought for number of days.
These are three equations with three unknowns $d$, $x$, $y$ and an unkown workload $w$, which itself does not matter (except it should not be zero).
Substracting the second from the first equation gives $$ 10 (x+y) - 16 x = 0 \iff \\ 10 y = 6 x \iff \\ y = (3/5) x $$ and using this in the first one: $$ w = 15 x + 15 (3/5) x = 24 x \iff \\ x = w / 24 $$ Which leads to $$ y = (3/5) (w/24) = (1/40) w $$ Then using the last equation $$ d (1/40) w = w \iff \\ d = 40 $$
