Whose birthday is it?

Question

A group of people have gathered for a birthday celebration. Their ages are related as follows:

The product of the 1st person's and the 2nd person's ages is $311frac{2}{3}$ plus the 3rd person's age.
The difference between the 1st person's and the 2nd person's ages is $2frac{31}{33}$ times the 3rd person's age.
The quotient of the product of all their ages and the sum of all their ages is $826frac{4}{29}$.
The sum of the 1st person's age and the quotient of the 3rd person's and the 2nd person's ages is $41frac{17}{24}$.
The square of the 3rd person's age is triple the 1st person's age.

Whose birthday is it? And what is each person’s age?
Hint:

hexomino · Accepted Answer

Let us denote the ages of Person 1, Person 2, Person 3 by $x,y,z$ respectively. We'll assume that $x,y,z$ are positive throughout.
The product of the 1st person's and the 2nd person's ages is $311 frac{2}{3}$ plus the 3rd person's age.

The sum of the 1st person's age and the quotient of the 3rd person's and the 2nd person's ages is $41 frac{17}{24}$

Subtracting the first equation from the second gives

The difference between the 1st person's and the 2nd person's ages is $2 frac{31}{33}$ times the 3rd person's age.

Now let us check the consistency with the other equations
The square of the 3rd person's age is triple the 1st person's age

The quotient of the product of all their ages and the sum of all their ages is $826 frac{4}{29}$

Whose birthday is it?

Whose birthday is it?

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