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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

Ask by Powell Rojas. in Nigeria
Mar 13,2025

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Answer

The two friends are 17 and 19 years old.

Solution

1. We are given that the sum of their ages is \[ x + y = 36. \] 2. In five years, their ages will be \( x + 5 \) and \( y + 5 \). The product of their ages in five years is given by \[ (x+5)(y+5) = 528. \] 3. Expanding the product, we have \[ (x+5)(y+5) = xy + 5x + 5y + 25. \] 4. From step 1, since \( x + y = 36 \), we substitute to obtain \[ xy + 5(36) + 25 = xy + 180 + 25 = xy + 205. \] Setting the expression equal to 528, we write \[ xy + 205 = 528. \] 5. Solving for \( xy \), we subtract 205 from both sides: \[ xy = 528 - 205 = 323. \] 6. We now have a system with the sum and product of \( x \) and \( y \): \[ x + y = 36 \quad \text{and} \quad xy = 323. \] The corresponding quadratic equation whose roots are \( x \) and \( y \) is \[ t^2 - (x+y)t + xy = 0, \] or \[ t^2 - 36t + 323 = 0. \] 7. We solve the quadratic equation using the quadratic formula \[ t = \frac{36 \pm \sqrt{36^2 - 4 \cdot 323}}{2}. \] 8. First, compute the discriminant: \[ 36^2 - 4 \cdot 323 = 1296 - 1292 = 4. \] 9. Since the discriminant is positive, the roots are \[ t = \frac{36 \pm \sqrt{4}}{2} = \frac{36 \pm 2}{2}. \] 10. Calculating both solutions: - When using the positive square root: \[ t = \frac{36 + 2}{2} = \frac{38}{2} = 19. \] - When using the negative square root: \[ t = \frac{36 - 2}{2} = \frac{34}{2} = 17. \] 11. Thus, the two friends are \( 17 \) and \( 19 \) years old. So, the solution for \( x \) is either \( 17 \) or \( 19 \), and accordingly, \( y \) is \( 19 \) or \( 17 \).

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The Deep Dive

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.

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