## 55. 100 years among the three

Patrick was walking with his father and grandfather. He wanted to imagine the time when all of them together would be 100 years old.

His father said on this matter: “I will be 28 years older than you, and you will be six-fifths of your current age.” And his grandfather added: “I will be twice as old as your father would have been at the time of your birth, if you had been born one and a half years later.”

Patrick thought deeply, trying to figure out how much time would pass until they could celebrate their combined century.

Please help him.
55. Let $e$ be Patrick's age, $p$ be the father's age, and $g$ be the grandfather's age; let $x$ be the time interval after which all three will be 100 years old. Then the following equations hold:
1) $(g+x)+(p+x)+(e+x)=100$
2) $e+x=(6 / 5) e$,
3) $(p+x)-(e+x)=28$,
4) $g+x=2(p-e+1.5)$.

From the last two equations, we get

$$
g+x=59 \text {. }
$$

Then from equation (1), we find

$$
59+(p+x)+(e+x)=100
$$

Since

$$
p+x=28+(e+x) \text { and } e+x=(6 / 5) e \text {, }
$$

it follows that

$$
59+28+(12 / 5) e=100
$$

Therefore, $e=5$ years 5 months, and $x=e / 5=1$ year 1 month.

Thus, all three will be 100 years old together in 1 year and 1 month (or 13 months).