Problem 2. Three professors wrote a collection of math problems and divided the honorarium in the ratio 8:6:5. If the honorarium had been divided in the ratio 7:5:4, one of the professors would have received 250 denars more than he received in the first division. How much did each of the authors receive?
Solution. Let the total honorarium received by the three professors be $x$ denars. Then the first professor received $\frac{8}{19} x$ denars, the second $\frac{6}{19} x$ denars, and the third $\frac{5}{19} x$ denars. If the honorarium is divided in the ratio $7: 5: 4$, then the first would receive $\frac{7}{16} x$, the second $\frac{5}{16} x$, and the third $\frac{4}{16} x$ denars. Since $\frac{8}{19} < \frac{7}{16}$ and $\frac{5}{19} > \frac{4}{16}$, it follows that with the second distribution, the first professor would receive 250 denars more than with the first distribution, i.e.,
$\frac{7}{16} x - \frac{8}{19} x = 250$). From this, $x = 15200$. Therefore, the first professor received 6400 denars, the second 4800 denars, and the third 4000 denars.