The Father's Age: A Mathematical Puzzle

Age problems often require a bit of logical reasoning and mathematical skills. This article explores a classic age-related puzzle, where we need to determine the current age of a father based on future age conditions. Initially, we will analyze a complex problem involving father and son age relationships, and then solve it step-by-step with detailed explanations. Finally, we will explore a slightly modified version of the problem and provide a comprehensive solution using logical reasoning and algebraic equations.

Initial Problem: The Age Relationship Puzzle

The initial problem states: "Initial Mark is 5 years old. After 6 years, he will be three times younger than his father. How old is his father now?" This problem seems straightforward but requires a bit of algebraic manipulation to solve accurately.

Solving the Initial Problem

We need to determine the current age of the father based on the future conditions provided. Let's denote the son's current age as s and the father's current age as f.

Step-by-Step Solution

The problem states that the son is 5 years old now. After 6 years, the son will be 5 6 11 years old. At that time, the father will be three times his age (11 years).

We can set up the following equation based on the given information:

If the father's age after 6 years is three times his son's age after 6 years, we have: After 6 years, the father's age (f 6) 3 * (son's age 6) 3 * 11 33. So, f 6 33. Solving for f, we get: f 33 - 6 27.

Therefore, the father's current age is 27 years old.

This solution provides a clear step-by-step approach to solving the age problem. Let's now examine a similar but more complex problem.

The Generalized Problem: A More Detailed Analysis

The generalized problem states: "The age of the father is seven times that of his son. In the next 6 years, his age will be three times the age of his son. How old was the father 5 years ago?" We will solve this step-by-step using algebraic equations.

Formulating the Equations

Let's denote the current age of the son as s and the current age of the father as f.

From the given conditions, we know:

The father's age is seven times the son's age, so we can write: f 7s. In 6 years, the father's age will be three times the son's age, so we can write: f 6 3(s 6).

Substituting f 7s into the second equation, we get:

7s 6 3(s 6) → 7s 6 3s 18. Solving for s, we get: 4s 12 → s 3.

Now that we have the son's current age, we can find the father's age:

If s 3, then f 7 * 3 21.

The father's age 5 years ago is:

f - 5 21 - 5 16.

Thus, the father was 16 years old 5 years ago.

Conclusion and Final Answer

From the detailed analysis, we can conclude that the father's current age is 27 years old, and five years ago, he was 16 years old. This solution demonstrates the application of algebraic equations to solve complex age-related problems, further reinforcing the importance of logical reasoning and mathematical skills in solving real-world problems.