The natural two-digit number $N$ envied its friend, the two-digit decimal number $X$, for its decimal point. So $X$ gave it to $N$. The number $N$ inserted the decimal point between its two digits and didn't mind being 567 tenths smaller than it was before. $X$ was also happy because now it was twice as close to its friend $N$ on the number line as it was before. Determine which two numbers $N$ and $X$ this story is about.

( $S$. Bednářová)
Let $N=10a+b$ and $X=c+0.1d$, where $a, b, c, d$ are digits. From the properties of the number $N$ it follows that

$$
10a+b-(a+0.1b)=56.7
$$

The only solution is $a=6$ and $b=3$. The sought number $N$ is therefore 63. From the properties of the number $X$ it follows that

$$
63-(c+0.1d)=2 \cdot(10c+d-6.3) .
$$

The only solution is $c=3$ and $d=6$. The sought number $X$ is therefore 3.6.