5. When drying, apricots lose $10 \%$ of their mass, and grapes lose $30 \%$ of their mass. Determine the proportion in which apricots and grapes must be taken so that after drying, the mixture contains twice as much dried apricot as raisins. In your answer, indicate the ratio of the initial mass of apricots to the mass of grapes as a decimal, rounding it to two decimal places if necessary.
Solution. If we take $x$ (kg) of apricots and $y$ (kg) of grapes, then after drying, the apricots will weigh $0.9 x$, and the grapes will weigh $0.7 y$. Since the first mass should be twice the second, then $0.9 x = 1.4 y$, which means $\frac{x}{y} = \frac{14}{9} \approx 1.56$.

Answer: 1.56.

## Set of Creative Problems.

I. Find the smallest positive root of the equation

$$
\sin ^{2 l} m x^{\circ}=\sin ^{2 l} n x^{\circ}
$$

( $x^{\circ}$ means $x$ degrees). Represent $x$ as an irreducible fraction, and write the sum of its numerator and denominator in the answer.

Solution. Let's solve the equation:

$$
\begin{gathered}
2 \sin ^{2} m x^{\circ}=2 \sin ^{2} n x^{\circ} \\
\cos 2 m x^{\circ}=\cos 2 n x^{\circ} \\
2 m x^{\circ}= \pm 2 n x^{\circ}+360^{\circ} k \\
x^{\circ}=\frac{180 k}{m \pm n}, \quad k \in \mathbb{Z}
\end{gathered}
$$

The smallest positive root is obtained by choosing the sign “+” and $k=1$:

$$
x=\frac{180}{m+n}
$$