2. Alberto, Barbara, and Ciro meet one day to prepare ravioli for a charity dinner in support of the mathematics Olympics. First, they decide to evenly distribute the working hours between the morning and the afternoon, and of course, they work simultaneously and for the same amount of time. Alberto is very reliable: he prepares 90 ravioli per hour for the entire workday. Barbara makes 110 ravioli per hour in the morning, but in the afternoon she is more distracted and prepares 70 ravioli per hour. Ciro makes $2 / 3$ of his ravioli at a rate of 140 ravioli per hour and the last third at only 50 ravioli per hour. Who made the most ravioli by the end of the day?
(A) Alberto
(B) Barbara
(C) Ciro
(D) Alberto and Barbara, in equal numbers.
(E) Alberto and Ciro, in equal numbers.
2. The answer is (D). Let $h$ be the number of hours worked during the morning (and thus also during the afternoon). We know that Alberto prepares $2 h \cdot 90=180 h$ ravioli, while Barbara prepares $h \cdot 110 + h \cdot 70 = 180 h$. Let $t_{1}$ be the time Ciro spends preparing the first $2 / 3$ of his ravioli, and $t_{2}$ the time he spends preparing the last third. If we call $r$ the total number of ravioli prepared by Ciro, we then have

$$
\left\{\begin{array}{l}
t_{1}+t_{2}=2 h \\
t_{1} \cdot 140=\frac{2 r}{3} \\
t_{2} \cdot 50=\frac{r}{3}
\end{array}\right.
$$

We can thus write $t_{1}=\frac{2 r}{3 \cdot 140}, t_{2}=\frac{r}{3 \cdot 50}$, and substituting in the first equation we find $r\left(\frac{1}{210}+\frac{1}{150}\right)=2 h$, from which $r=175 h$.