A grandfather and his granddaughter have their ages in the ratio 9 : 2. Sum of their ages is a prefect square. The difference in their ages is a multiple of 11. What are their ages?

Age ratio of grandfather to granddaughter = 9 : 2.

Sum of their ages is a perfect square.

Difference in their ages is a multiple of 11.

Let's denote the ages of the grandfather and granddaughter as9 x and 2 x respectively, where x is a positive integer representing the common factor.
According to the given information:
The sum of their ages is a perfect square.
The difference in their ages is a multiple of 11.
Let's first find the possible values of x that satisfy the condition that the sum of their ages is a perfect square.
The sum of their ages is 9 x+2x = 11 x, which must be a perfect square. Since the sum of their ages is a perfect square, 11x must be a multiple of 11 squared, which is 12.
So, 11 x = 121 => x = $$\frac{121}{11}$$​ = 11.
Now, we can find their ages:
Grandfather's age: 9x = 9×11 = 99
Granddaughter's age: 2x = 2 × 11 = 22
Therefore, the grandfather is 99 years old, and the granddaughter is 22 years old.