A Mathematical Puzzle: Unraveling the Father-Son Age Riddle

One classic math problem involves creating an equation based on the given conditions to find the unknown ages of a father and his son. In this article, we will explore three different methods to solve a specific puzzle involving a father who is four times older than his son, and their combined age is 40 years. We'll provide step-by-step solutions to help you understand the logic and reasoning behind each method.

Method 1: Direct Substitution and Equation Solving

Lets assume the age of the son is x. According to the given information, the father is four times older than the son, so the father’s age is 4x. The sum of their ages is 40. We can then write the equation:

x 4x 40

Combining like terms, we get:

5x 40

Dividing both sides by 5:

x 8

Therefore, the son is 8 years old.

Method 2: Setting Up Simultaneous Equations

Let’s set up the problem using two equations. Denote the age of the son as s and the age of the father as k.

Equation 1: k 4s

Equation 2: k s 40

Substitute the first equation into the second:

4s s 40

Combine like terms:

5s 40

Divide both sides by 5:

s 8

Plugging s back into the first equation:

4(8) 32

Therefore, the son is 8 years old.

Method 3: Simplified Approach with Substitution

Let the age of the son be x. According to the problem, the father’s age is then 4 times that of the son, i.e., 4x. The combined age of both is 40 years.

We set up the equation:

x 4x 40

Simplify and solve for x:

5x 40

Divide both sides by 5:

x 8

Therefore, the son is 8 years old.

Conclusion and Verification

Using all three methods, we consistently derive that the son is 8 years old. To verify, the father’s age would be 32 (4 times 8), and the sum of their ages is 40 (8 32 40). This is indeed a classic example of how setting up and solving equations can yield clear and definitive results even in seemingly complex problems.

By exploring different approaches to solving this problem, we enhance our understanding of fundamental algebraic techniques and the importance of logical reasoning in mathematics. Whether you’re a student, teacher, or anyone looking to improve your mathematical skills, these methods provide a valuable learning experience.