13. A confectionery company produces two types of nougat, using the same white paste and hazelnuts, but in different proportions. In type A nougat, hazelnuts represent $30 \%$ of the weight and $40 \%$ of the volume; in type B nougat, hazelnuts represent $60 \%$ of the weight. What percentage of the volume do the hazelnuts represent in type B nougat?
13. The answer is 70. Let $P_{A}$ be the weight of torrone A, $V_{A}$ the volume of torrone A, $P_{B}$ the weight of torrone B, and $V_{B}$ the volume of torrone B. It is given that $30\% P_{A}$ occupies $40\% V_{A}$, so the specific weight of the hazelnuts is $\frac{3}{4} \frac{P_{A}}{V_{A}}$. Considering the weight and space occupied by the white paste in torrone A, we find that its specific weight is $\frac{7}{6} \frac{P_{A}}{V_{A}}$. Therefore, to calculate the volume occupied by the hazelnuts and the white paste in torrone B, we divide their weight, i.e., $60\% P_{B}$ for the hazelnuts and $40\% P_{B}$ for the white paste, by their specific weight, obtaining that the former occupy $\frac{4}{5} \frac{P_{B}}{P_{A}} V_{A}$ and the latter $\frac{12}{35} \frac{P_{B}}{P_{A}} V_{A}$. Adding these two volumes gives the total volume, i.e., $V_{B}=\frac{4}{5} \frac{P_{B}}{P_{A}} V_{A}+\frac{12}{35} \frac{P_{B}}{P_{A}} V_{A}=\frac{8}{7} \frac{P_{B}}{P_{A}} V_{A}$. To calculate the percentage of the volume occupied by the hazelnuts, we divide the volume they occupy by the total volume of the torrone:

$$
\frac{\frac{4}{5} \frac{P_{B}}{P_{A}} V_{A}}{\frac{8}{7} \frac{P_{B}}{P_{A}} V_{A}}=70\%
$$

Therefore, the hazelnuts occupy $70\%$ of the volume of torrone B.

## SECOND SOLUTION

In the first torrone, with $30\%$ of the weight, the hazelnuts made up $40\%$ of the volume, and thus the paste, with $70\%$ of the weight, made up $60\%$ of the volume. Therefore, if the "voluminosity" of the hazelnuts is $4 / 3$ that of the paste, it results in $6 / 7$. Therefore, the volume share of the hazelnuts in the second torrone is

$$
\frac{60 \cdot 4 / 3}{60 \cdot 4 / 3 + 40 \cdot 6 / 7} = \frac{80}{80 + 240 / 7} = \frac{560}{800} = 70\%
$$