A is twice as old as B was when A was as old as B is now. When B is as old as A is now, their combined age will be 130 years. How old are A and B now?
I. solution: Let $A$'s age be $x$, and $B$'s age be $y$. When $A$ was $y$ years old, which was $x-y$ years ago, $B$ was $y-(x-y)=2y-x$ years old. According to the first statement of the problem,

$$
x=2(2y-x)
$$

from which

$$
3x-4y=0
$$

According to the second statement, $x-y$ years from now, their combined age will be 130 years, so

$$
x+(x-y)+y+(x-y)=130
$$

or

$$
3x-y=130
$$

From the difference of (1) and (2),

$$
3y=130, \quad \text{that is} \quad y=43 \frac{1}{3}
$$

follows, and from this

$$
x=57 \frac{7}{9}
$$

Thus, $A$ is $57 \frac{7}{9}$ years old, and $B$ is $43 \frac{1}{3}$ years old.

Marton Katalin (Bp. VI., Varga Katalin g. I. o. t.)

II. solution: Let $A$'s age be $x$, and $B$'s age be $x-n$. When $A$ was $x-n$ years old, $B$ was $x-2n$ years old. When $A$ will be $x+n$ years old, $B$ will be $x$ years old. Thus, according to the problem statement,

$$
x=2(x-2n)
$$

and

$$
x+x+n=130
$$

From the two equations,

$$
n=14 \frac{4}{9} \quad \text{and} \quad x=57 \frac{7}{9}.
$$

From this, we get the same solution for the two ages as before.

Pósch Margit (Bp, XIV., Teleki B. Ig. I o. t.)

III. solution: The problem is primarily made difficult to understand by its wording, but it becomes simpler if we visualize the relationships. Let $OP$ represent $A$'s age, and $OQ$ represent $B$'s age.

![](https://cdn.mathpix.com/cropped/2024_05_02_310e205811c34ded2ce9g-1.jpg?height=97&width=596&top_left_y=1884&top_left_x=751)

When $A$ was as old as $B$ is now, $A$ was $PQ$ years younger than now, so $B$'s age at that time is indicated by the point $R$ such that $RQ=QP$. The first condition of the problem can now be restated as $R$ being the midpoint of segment $OP$. Thus, $QP$ (or the age difference between $A$ and $B$) is one-fourth of $OP$ (or $A$'s age).

For the rest, the visualization is no longer necessary. When $B$ becomes as old as $A$ is now, $A$ will be one-fourth of $A$'s current age older, so the combined age of the two will be $\frac{9}{4}$ times $A$'s current age. This means that $A$'s current age is

$$
\frac{4}{9} \cdot 130 = \frac{520}{9} = 57 \frac{7}{9} \text{ years}
$$

$B$'s age is three-fourths of this, or

$$
\frac{3}{4} \cdot \frac{520}{9} = \frac{130}{3} = 43 \frac{1}{3} \text{ years.}
$$