Let the ages of the two colleagues be x and y years.

Deriving Age Ratio from Reciprocal Condition

The first condition states that the sum of the reciprocals of their ages is nine times the difference of the reciprocals of their ages.

Mathematically, this can be written as:

$ \frac{1}{x} + \frac{1}{y} = 9 \left| \frac{1}{x} - \frac{1}{y} \right| $

Assuming $y > x$, which means $\frac{1}{x} > \frac{1}{y}$, the equation becomes:

$ \frac{y+x}{xy} = 9 \left( \frac{y-x}{xy} \right) $

Multiplying both sides by $xy$ (since ages $x, y$ are non-zero):

$ x + y = 9(y-x) $

$ x + y = 9y - 9x $

Rearranging the terms to group $x$ and $y$:

$ x + 9x = 9y - y $

$ 10x = 8y $

$ 5x = 4y $

This implies the ratio of the ages is:

$ \frac{x}{y} = \frac{4}{5} $

We can represent the ages as $x = 4k$ and $y = 5k$ for some positive constant $k$. This satisfies the condition that $y > x$.

Solving for Ages using Product-Sum Ratio

The second condition states that the ratio of the product of their ages to the sum of their ages is 100:9.

$ \frac{xy}{x+y} = \frac{100}{9} $

Now, substitute the expressions for $x$ and $y$ in terms of $k$ into this equation:

$ \frac{(4k)(5k)}{4k+5k} = \frac{100}{9} $

$ \frac{20k^2}{9k} = \frac{100}{9} $

Simplify the left side by canceling out one $k$ (since $k$ must be positive):

$ \frac{20k}{9} = \frac{100}{9} $

To solve for $k$, multiply both sides by 9:

$ 20k = 100 $

Divide by 20:

$ k = \frac{100}{20} = 5 $

Now, calculate the ages using the value of $k$:

Age $x = 4k = 4 \times 5 = 20$ years.

Age $y = 5k = 5 \times 5 = 25$ years.

Final Answer

The ages of the two colleagues are 20 years and 25 years.