In a factory, a certain product was produced $m$ times over $n$ days on two occasions. The first time with $a$ faster working workers, and the second time with $b$ less skilled workers. Now they need to produce $p$ units of this product again within $q$ days. For this, they can assign $c$ faster working workers, but this is not enough. How many less skilled workers do they need to assign in addition, to ensure the production is completed on time?
I. solution. According to the two previous production results, a well-trained worker produces $m / n a$ units per day, and a less trained worker produces $m / n b$ units of the product in question. According to the new demand, they need to produce $p / q$ units per day. Thus, the $c$ number of well-trained workers will produce $m c / n a$ units, and the remaining $\frac{P}{q}-\frac{m c}{n a}$ units need to be produced by less trained workers. Therefore, the number of slower workers to be assigned is

$$
\frac{\frac{p}{q}-\frac{m c}{n a}}{\frac{m}{n b}}=\frac{(p n a-q m c) b}{q m a}
$$

Erdődy Gabriella (Budapest, Villányi úti ált. isk. VI. o. t.)

II. solution. First, we calculate how many slower workers would be needed to meet the new task, and then how many of these are offset by the $c$ more experienced workers. If the factory wanted to complete the new order with only less trained workers, the previous $b$ workforce would need to be increased by the factor of how many more units need to be produced compared to the previous amount, i.e., $b$ would need to be multiplied by $p / m$; and also by the factor of how much longer the old working time is compared to the new one (or how much shorter and more rushed the new one is compared to the old one), i.e., it would also need to be multiplied by $n / q$. Thus, the number of people required would be given by the expression $\frac{p}{m} \cdot \frac{n}{q} \cdot b$.

From the two previous production periods, we see that the number of faster workers produces as much in the same amount of time as $b$ number of slower workers - where it is clearly $am$ and $q<n$. However, the calculation is valid even under different magnitude relationships.)