1. A cake was divided into three parts. If two-thirds are cut from the first part and added to the second, then the third part will be twice as large as the first, but half the size of the second. What fraction of the cake do the cut pieces represent?
Answer: $3 / 7, 2 / 7, 2 / 7$.

## Solution:

Let the cake be divided into parts $x, y$, and $z$.

According to the problem: $x+y+z=1, \frac{1}{3} x \cdot 2=z, \frac{2}{3} x+y=2 z$.

Sequentially express everything in terms of $x: z=\frac{2}{3} x, \frac{2}{3} x+y=2 \cdot \frac{2}{3} x \Rightarrow y=\frac{2}{3} x$.

Substitute everything into the equation $x+y+z=1$, we get: $x+\frac{2}{3} x+\frac{2}{3} x=1$, from which $\frac{7}{3} x=1$ and $x=\frac{3}{7}$.

It remains to express the other shares: $y=z=\frac{2}{3} x=\frac{2}{3} \cdot \frac{3}{7}=\frac{2}{7}$.

## Criteria:

4 points for the answer with verification, 1 point for the answer without verification.