## 15. Puzzle

Martin once told Martina: “I am three times as old as you were when I was as old as you are now.”

And in response, Martina noted: “When I am as old as you are now, together we will be 77 years old.”

How old are Martin and Martina?
15. Let $x$ be Martin's age and $y$ be Martina's age. What Martin said to Martina can be expressed by the equation

$$
x=3[y-(x-y)]
$$

from which

$$
4 x=6 y \text{, or } 2 x=3 y \text{. }
$$

Martina's response can be written as the equation

$$
x+[x+(x-y)]=77
$$

i.e.

$$
3 x=y+77
$$

Substitute $y$ with $3 x-77$ in the first equation:

$$
2 x=3(3 x-77),
$$

from which

$$
7 x=3 \cdot 77, \text{ or } x=33
$$

Therefore,

$$
y=3 x-77=22
$$

Thus, Martin is 33 years old, and Martina is 22 years old.