Understanding Age Ratios: A Case Study with a Father and Son

Introduction

When dealing with age-related problems, understanding how to break down and solve equations is a fundamental skill. In this article, we will walk through a detailed exploration of a problem where the ages of a man and his son are given in a specific ratio, and we need to find the son's age. We will solve this problem using multiple approaches and provide step-by-step explanations to ensure clarity and understanding.

Solving the Problem: The Average Age of a Man and His Son

Problem Statement:

The average age of a man and his son is 40 years. The ratio of their ages is 4:1 respectively. What is the son’s age?

Solution 1:

Let the age of the man be 4x and the age of the son be x.

The average age can be expressed as:

[ frac{4x x}{2} 40 ]

This simplifies to:

[ frac{5x}{2} 40 ]

To solve for x, multiply both sides by 2:

[ 5x 80 ]

Now divide by 5:

[ x 16 ]

Thus, the son's age is 16 years.

Alternative Solutions and Verifications

Solution 2:

Given that the age of the man is 36 years:

The ratio of the man's age to the son's age is 4:1. Therefore, the son's age is:

[ 36/4 9 ]

Thus, the son is 9 years old.

Solution 3:

Denote the ages of the father and son as F and S, respectively.

The average age of the father and son is 40 years:

[ frac{F S}{2} 40 ]

The sum of their ages is:

[ F S 80 ]

The ratio of their ages is 11:5:

[ frac{F}{S} frac{11}{5} ]

From the above ratio, we get:

[ F 11S/5 ]

Substitute into the sum equation:

[ 11S/5 S 80 ]

Multiplying by 5 on both sides:

[ 11S 5S 400 ]

Then:

[ 16S 400 ]

Finally:

[ S 25 ]

Hence, the son is 25 years old.

Conclusion

Through this comprehensive exploration, we have solved the problem of finding the son's age given the age of the father and the ratio between their ages. Each solution demonstrates a different approach to solving age-related problems, illustrating the power of algebra in solving real-world challenges. These techniques can be applied to a wide range of problems, making them valuable tools for students and professionals alike.

By mastering the concept of age ratios and their applications, one can more easily navigate various scenarios involving age-related questions. This knowledge is not only educational but can also enhance practical skills in everyday life and professional contexts, such as project timelines and employee management.