13.053. A crystal, while in the formation stage, uniformly increases its mass. Observing the formation of two crystals, it was noted that over a year, the first crystal increased its initial mass by $4 \%$, while the second increased by $5 \%$, at the same time the mass increase of the first crystal over 3 months was equal to the mass increase of the second crystal over 4 months. What were the initial masses of these crystals, if it is known that after each of them increased by 20 g, the ratio of the mass of the first crystal to the mass of the second crystal reached the number 1.5?
Solution.

Let $a$ be the initial mass of the first crystal; $b$ be the second; and let $x$ be the mass increase of the first crystal over 3 months. Over a year, the mass increase of the first crystal is $4x$; of the second crystal - $3x$. According to the problem, $a+4x=1.04a, b+3x=1.05b$. According to the problem, $\frac{a+20}{b+20}=1.5$. Solving the system $\left\{\begin{array}{l}a+4x=1.04a \\ b+3x=1.05b \\ a+20=1.5(b+20),\end{array}\right.$ we find that $a=100 \text{ g}; b=60 \text{ g}$.

Answer: 100 and 60 g.