Two friend are \( x \) and \( y \) years old. The sum of their age is 36 . In five yeas time the Product of their age will be 528 . Fund \( x \) and

Let's say \( x \) is the current age of friend one, and \( y \) is the current age of friend two. We know that \( x + y = 36 \) and \( (x + 5)(y + 5) = 528 \). To solve, we can substitute \( y \) from the first equation into the second: 1. \( y = 36 - x \) 2. Substitute into the product equation: \( (x + 5)(36 - x + 5) = 528 \) 3. This simplifies to \( (x + 5)(41 - x) = 528 \) 4. Expanding gives \( 41x - x^2 + 205 - 5x = 528 \) 5. Rearranging leads to the quadratic equation: \( x^2 - 36x + 323 = 0 \) Using the quadratic formula, you can find the values of \( x \) and \( y \). Now, on to some historical context: Many famous mathematicians, such as Euclid and Diophantus, took on similar algebraic problems in ancient times. These puzzles were not just academic exercises but were often related to determining land boundaries, trade benefits, and familial relationships, reflecting the practical scenarios of their day. For a fun twist, consider this scenario: if your own age and your friend's age added up to 36, being able to predict future ages just like the problem presented offers a fun glimpse into the mystery of growing older! It often inspires us to think about how age differences play a role in friendships and life experiences.