Two whole Positive Number-Math riddle

Explanation :  

Let the two whole, positive numbers be represented by $a$ and $b$. The given conditions are:

1. When multiplied, the result is a one-digit number: $a\times b=\text{Single Digit}$

2. When added, the result is a two-digit number: $a+b=\text{Two Digit Number}$

Now, let's consider the specific case of $a=1$ and $b=9$.

1. Multiplication: $1\times 9=9$

   The result is a single-digit number, which satisfies the first condition.

2. Addition: $1+9=10$

   The result is a two-digit number, fulfilling the second condition.

Thus, the numbers $a=1$ and  $b=9$ meet both criteria. This demonstrates the mathematical solution to the given puzzle, where the multiplication of 1 and 9 yields a single-digit result, and the addition of 1 and 9 produces a two-digit sum.