## Task B-2.6.

Students Marko and Luka worked on a part-time basis, Marko in a tourist office, and Luka in a hotel. Marko earned a total of 200 kn for his work. Luka worked 5 hours less than Marko and earned 125 kn. If Marko had worked as many hours as Luka, and Luka as many hours as Marko, then Luka would have earned 150 kn more than Marko. How many hours did Marko work, and how many did Luka, and what was the hourly wage for each of them?
## Solution.

$n=$ the number of hours Marko worked, $n>5$.

$x=$ Marko's hourly wage, $y=$ Luka's hourly wage

Then we have the following system of equations

$$
\begin{gathered}
n x=200 \\
(n-5) y=125 \\
n y-(n-5) x=150
\end{gathered}
$$

From the first and second equations, express $x, y$ :

$$
x=\frac{200}{n}, \quad y=\frac{125}{n-5}
$$

Substitute into the third equation:

$$
\begin{gathered}
n \cdot \frac{125}{n-5}-(n-5) \cdot \frac{200}{n}=150 \\
125 n^{2}-200(n-5)^{2}=150 n(n-5) \\
5 n^{2}-8(n-5)^{2}=6 n(n-5) \\
9 n^{2}-110 n+200=0
\end{gathered}
$$

from which we get $n=10$ or $n=\frac{20}{9}$.

The solution $n=\frac{20}{9}$ does not satisfy the condition $n>5$, so $n=10$.

Marko worked 10 hours, and Luka $n-5=5$ hours.

Marko earns $\frac{200}{10}=20 \mathrm{kn}$ per hour, and Luka $\frac{125}{5}=25 \mathrm{kn}$.