1. Ana, her mother, and her grandmother are celebrating their birthday on June 23rd. Today, June 23rd, 2021, the sum of their ages is 127. When Ana was 4 years old, her grandmother was twice as old as her mother. Two years ago, her mother was three times as old as Ana. In which year was Ana's mother born? How old will Ana be when she and her mother together have the same number of years as her grandmother? In which year will that be?
## Solution.

This year, Ana is $A$ years old, her mother is $M$ years old, and her grandmother is $B$ years old. It is given that $A+M+B=127$.

Ana was 4 years old $x$ years ago. Therefore, $x=A-4$.

Furthermore, $x$ years ago, her mother was $M-x$ years old, and her grandmother was $B-x$ years old. According to the condition of the problem,

$$
\begin{aligned}
B-x=2 \cdot(M-x) \\
B-(A-4)=2 \cdot(M-(A-4)) \\
B-A+4=2 M-2 A+8 \\
A+B-2 M=4
\end{aligned}
$$

From the first equation, we have $A+B=127-M$, so

$$
\begin{gathered}
127-M-2 M=4 \\
3 M=123 \\
M=41
\end{gathered}
$$

Ana's mother is 41 years old, which means she was born in 1980.

Two years ago, her mother was 39 years old and three times as old as Ana. This means that Ana was $39 \div 3 = 13$ years old two years ago, and she is now 15 years old ($A=15$).

The grandmother is $127-41-15=71$ years old ($B=71$).

In $y$ years, Ana and her mother together will be as old as the grandmother. It holds that

$$
\begin{gathered}
15+y+41+y=71+y \\
56+y=71 \\
y=15
\end{gathered}
$$

Ana will be 30 years old when she and her mother together will be as old as the grandmother.

This will be in the year 2036.