Míša and Jana both have their birthdays today, together they are 84 years old. Míša is twice as old as Jana was when Míša was as old as Jana is today.

How old is Míša and how old is Jana?

(M. Volfová)
Information from the problem can be expressed using one unknown, for example, as follows:

|  | today | earlier |
| :---: | :---: | :---: |
| Míša | $2 x$ | $84-2 x$ |
| Jana | $84-2 x$ | $x$ |

The age difference is the same in both rows (the difference between "today" and "earlier"), which leads to the equation

$$
2 x-84+2 x=84-2 x-x,
$$

whose solution is

$$
\begin{aligned}
7 x & =168 \\
x & =24 .
\end{aligned}
$$

From this, we get $2 x=48$ and $84-2 x=36$, meaning Míša is 48 years old and Jana is 36 years old.

Another solution. The relationships from the problem can be expressed with multiple unknowns, for example, as follows:

|  | today | earlier |
| :---: | :---: | :---: |
| Míša | $2 x$ | $y$ |
| Jana | $y$ | $x$ |

Míša and Jana together are 84 years old, and the age difference is the same in both rows. This leads to a system of two equations

$$
\begin{aligned}
& 2 x+y=84, \\
& 2 x-y=y-x,
\end{aligned}
$$

which is equivalent to the system

$$
\begin{aligned}
4 x+2 y & =168, \\
3 x & =2 y .
\end{aligned}
$$

Substituting the second equation into the first, we get

$$
\begin{aligned}
7 x & =168 \\
x & =24 .
\end{aligned}
$$

From this, it follows that $2 x=48$ and $y=\frac{3}{2} \cdot 24=36$, meaning Míša is 48 years old and Jana is 36 years old.

Grading suggestion. 2 points for expressing the relationships contained in the initial table; 2 points for setting up and solving the equation, or system of equations; 1 point each for calculating Míša's and Jana's ages.

Note. The unknown can also be the difference between "today" and "earlier" or the difference between Míša's and Jana's ages; in such a case, the initial relationships together with other conditions would lead to a different system of equations with a similar solution, and thus also grading.

From the problem, it follows that Míša is older than Jana and that Míša's age is an even number. The sum of their ages is 84 years, so instead of solving the equations, it is possible to sequentially check which of the following options meets the other requirements from the problem:

| Míša | 44 | 46 | 48 | $\ldots$ |
| :--- | :--- | :--- | :--- | :--- |
| Jana | 40 | 38 | 36 | $\ldots$ |

This way, it can be easily seen that the third option is the correct one. Without justification that it is the only possible solution, grade such a solution no higher than 4 points. Award an additional 2 points for a well-reasoned explanation of the uniqueness of the solution, based on the quality of the commentary.