Solving the Age Mystery: A Mathematical Journey with Liz and Ronald

Solving the Age Mystery: A Mathematical Journey with Liz and Ronald

In the world of puzzling riddles, age-related questions often present a unique challenge. This particular problem involves two individuals, Liz and Ronald, and their age difference and the product of their ages. Let's embark on a journey to solve this mystery and uncover the ages of Liz and Ronald.

The Puzzle in Detail

Liz is 2 years older than Ronald. At the same time, the product of their ages is 440. This simple statement of fact forms the basis of our mathematical challenge. We denote Liz's age as L and Ronald's age as R.

Setting Up the Equations

From the given information, we can establish the following equations: L R 2 LR 440

Substituting and Solving the Quadratic Equation

We can substitute the first equation into the second to create a quadratic equation in terms of R alone.

Substituting L R 2 into LR 440, we get:

begin{equation*}R(R 2) 440end{equation*}

Expanding and rearranging the equation, we obtain:

begin{equation*}R^2 2R 440end{equation*}

Further simplification leads to a standard quadratic equation:

begin{equation*}R^2 2R - 440 0end{equation*}

In this quadratic equation, the coefficients are as follows:

a 1 (coefficient of (R^2)) b 2 (coefficient of R) c -440 (constant term)

To solve the quadratic equation, we will use the quadratic formula:

begin{equation*}R frac{-b pm sqrt{b^2 - 4ac}}{2a}end{equation*}

Applying the Quadratic Formula

Substituting the values of a, b, and c into the formula, we get:

begin{equation*}R frac{-2 pm sqrt{2^2 - 4 times 1 times (-440)}}{2 times 1}end{equation*}

Simplifying inside the square root:

begin{equation*}R frac{-2 pm sqrt{4 1760}}{2}end{equation*}begin{equation*}R frac{-2 pm sqrt{1764}}{2}end{equation*}

Finding the square root of 1764:

begin{equation*}R frac{-2 pm 42}{2}end{equation*}

This gives us two potential solutions for R:

begin{equation*}R frac{-2 42}{2} frac{40}{2} 20end{equation*}begin{equation*}R frac{-2 - 42}{2} frac{-44}{2} -22end{equation*}

Since age cannot be negative, we discard the negative solution and take R 20.

Calculating Liz’s Age

With R 20, we can find L using the relationship L R 2:

begin{equation*}L 20 2 22end{equation*}

Verification

To ensure our solution is correct, let's verify by calculating the product of their ages:

begin{equation*}L times R 22 times 20 440end{equation*}

This confirms that our solution is indeed correct.

Conclusion

Through the use of quadratic equations and logical reasoning, we have determined that Liz is 22 years old and Ronald is 20 years old. This problem not only showcases the power of algebra but also highlights the importance of logical deduction in solving real-world problems.