Solving the Age Puzzle: When was the Product of Their Ages 96?

Let's dive into a fascinating age-related puzzle. A man is 37 years old, and his child is 8 years old. The question is, how many years ago was the product of their ages 96?

The Equation

Let x be the number of years ago when the product of their ages was 96. At that time, the man's age would have been 37 - x and the child's age would have been 8 - x. The equation representing the product of their ages is:

(37 - x) * (8 - x) 96

Expanding and Solving the Equation

Expanding the left side:

296 - 37x - 8x^2 96

Combining like terms:

x^2 - 45x 200 0

This is a quadratic equation. We can solve it using the quadratic formula:

x frac{-b ± sqrt{b^2 - 4ac}}{2a}

For this equation, a 1, b -45, and c 200. First, we calculate the discriminant:

1225 (-45)^2 - 4 * 1 * 200 2025 - 800

Now, taking the square root of the discriminant:

sqrt{1225} 35

Substituting back into the quadratic formula:

x frac{45 ± 35}{2}

Calculating the two possible values for x:

x frac{80}{2} 40

x frac{10}{2} 5

Verifying the Solution

Since we are looking for a realistic number of years ago, we take x 5. Thus, 5 years ago the product of their ages was 96.

To verify:

The man's age 5 years ago: 37 - 5 32

The child's age 5 years ago: 8 - 5 3

Calculating the product:

32 * 3 96

This confirms our solution.

Alternative Method

Another way to look at the problem is by recognizing that the solution is a factor pair of 96 that can be derived from the given conditions. The quadratic equation can be factored as:

(x - 40)(x - 5) 0

Thus, x 5 and x 40. Since the ages would be negative if x 40, the valid solution is x 5.

5 years ago, the man was 32, and the child was 3, and the product of their ages is indeed 96.

Reflecting on the solution, it's quite amazing how quickly one can solve such a problem with some logical reasoning and mathematical prowess. Patting myself on the back and wishing others good luck with their intellectual pursuits!