## Task 1 - 340811

Anja, Bernd, and Christina have their birthdays on the same day.

On this day, Anja says to Christina: " $\frac{3}{4}$ of your age is the same number of years as $\frac{2}{3}$ of my age." Bernd says to Christina: " $\frac{3}{4}$ of your age is the same number of years as half of my age."

Christina says: "The sum of our three ages expressed in years is 58."

Show that these statements uniquely determine how old Anja, Bernd, and Christina are! State these three age values!
From the given information, for the age $a$ respectively $b$ respectively $c$ of Anja respectively Bernd respectively Christina:

$$
\frac{3}{4} c=\frac{2}{3} a \quad \text { (1) } \quad \frac{3}{4} c=\frac{1}{2} b \quad \text { (2) } \quad a+b+c=58
$$

From (1) it follows that $9 c=8 a$, so $a=\frac{9}{8} c$ (4), from (2) it follows that $3 c=2 b$, so $b=\frac{3}{2} c$. (5)

Thus, from (3)

$$
\frac{9}{8} c+\frac{3}{2} c+c=58 \quad \Rightarrow \quad c=\frac{58 \cdot 8}{29}=16
$$

and then from (4), (5) further $a=18$ and $b=24$.

Therefore, it is uniquely determined by the given information: Anja is 18 years old, Bernd is 24 years old, Christina is 16 years old.