Solving Practical Age Problems: A Case Study on Parent-Child Age Relations and Quadratic Equations
Age problems are a common type in mathematics, often providing a relatable and interesting challenge to students. One such problem involves the sum of a boy's age and his father's age and a condition related to their ages. Let's explore this problem in detail to find the solution.
Problem Statement and Initial Setup
Consider a scenario where the sum of a boy's age ((b)) and his father's age ((f)) is 24 years. Additionally, one-fourth of the product of their ages exceeds the boy's age by 9 years. We need to determine their ages using these conditions.
Formulating Equations
Based on the given information, we can set up the following equations:
The sum of their ages: (b f 24) One-fourth of the product of their ages exceeds the boy's age by 9 years: (frac{1}{4}bf b 9)From the first equation, we can express (f) in terms of (b): (f 24 - b).
Solving the Quadratic Equation
Next, we substitute (f 24 - b) into the second equation:
[frac{1}{4}b(24 - b) b 9]
Multiplying both sides by 4 to eliminate the fraction:
[b(24 - b) 4b 36]
Expanding and rearranging the equation:
[24b - b^2 4b 36]
[-b^2 24b - 4b - 36 0]
[-b^2 20b - 36 0]
Dividing the entire equation by -1 to simplify:
[b^2 - 20b 36 0]
Using the quadratic formula to solve for (b), where (a 1), (b -20), and (c 36):
[b frac{-B pm sqrt{B^2 - 4AC}}{2A}]
Substituting the values:
[b frac{20 pm sqrt{(-20)^2 - 4 cdot 1 cdot 36}}{2 cdot 1}]
[b frac{20 pm sqrt{400 - 144}}{2}]
[b frac{20 pm sqrt{256}}{2}]
[b frac{20 pm 16}{2}]
This gives us two solutions for (b):
(b frac{36}{2} 18) (b frac{4}{2} 2)Determining the Corresponding Ages of the Father
Now, we will find the corresponding ages of the father for each case:
If (b 18): (f 24 - 18 6) (This is not a realistic age for a father) If (b 2): (f 24 - 2 22) (This is a realistic age for a father)Thus, the possible pairs of ages are:
(text{Boy: 18 years, Father: 6 years (not realistic)}) (text{Boy: 2 years, Father: 22 years (realistic)})Therefore, the boy's age is (2) years and the father's age is (22) years.
Conclusion and Verification
From the calculations, we have determined that the realistic solution to the age problem is:
(text{Boy: 2 years}) (text{Father: 22 years})The problem also uses quadratic equations, which are fundamental in algebra and have wide applications in various fields, including physics and engineering. Understanding such problems enhances your problem-solving skills and analytical abilities.