The Mathematical Journey of Ages: A Father and Son’s Tale

Introduction

Today, we explore a classic problem in mathematics involving the ages of a father and his son. This engaging problem not only demonstrates the beauty of algebraic equations but also provides a clear understanding of the fundamental concepts that underpin age-related problems. Let's delve into the solution of this problem and uncover the intriguing relationships between the ages of Mr. Andrew and his son, Danny.

The Problem at Hand

Suppose we are given that Mr. Andrew is 4 times as old as his son Danny, who is 9 years old. The question is to determine in how many years Mr. Andrew will be twice as old as Danny. Intuitively, we can predict that Mr. Andrew must be 36 years old because 4 times 9 equals 36.

The Core of the Problem

Now, let's break down the problem mathematically using the ages and the relationship given. We can denote Mr. Andrew's current age as ( F ) and Danny's current age as ( S ). From the problem, we know: - ( F 4S ) - ( S 9 ) Substituting ( S ) into the first equation, we get: [ F 4 times 9 36 ] Thus, Mr. Andrew is currently 36 years old, and Danny is 9 years old. The problem now is to find out in how many years, denoted by ( t ), Mr. Andrew's age ( F t ) will be twice Danny's age ( S t ). This can be represented by the equation: [ F t 2(S t) ] Substituting ( F 36 ) and ( S 9 ) into the equation, we get: [ 36 t 2(9 t) ] Expanding the right side of the equation, we have: [ 36 t 18 2t ] Rearranging the terms to isolate ( t ), we get: [ 36 - 18 2t - t ] Simplifying, we find: [ 18 t ] Therefore, in 18 years, Mr. Andrew will be 54 years old, and Danny will be 27 years old, satisfying the condition that Mr. Andrew is twice as old as Danny.

Verification and Conclusion

To verify the solution, we can substitute ( t 18 ) back into the original conditions. Mr. Andrew's age in 18 years will be: [ 36 18 54 ] And Danny's age in 18 years will be: [ 9 18 27 ] Clearly, 54 is twice 27, confirming that the solution is correct. This problem not only teaches us about the practical application of algebra but also highlights the importance of step-by-step logical reasoning in solving real-world problems.

Conclusion

In conclusion, the solution to the problem "Mr. Andrew is 4 times as old as his son Danny who is 9 years old, in how many years will Mr. Andrew be twice as old as Danny?" is 18 years. This problem serves as an excellent example of how algebraic equations can be used to solve real-life problems and emphasizes the significance of consistent and logical thinking. It is a fundamental concept that underpins many mathematical and logical problems in our daily lives.

Sources:

- Example Math Problem Website - Example Algebra Resource