Finding Ages of Two Brothers: A Mathematical Puzzle

Today, we explore a classic mathematical problem involving the ages of two brothers. This problem, while seemingly simple, presents a unique challenge in solving for the ages of the brothers through algebraic equations. By carefully breaking down the problem and following a systematic approach, we can uncover the ages of both brothers.

Problem Statement: The difference in ages of two brothers is 6 years. The sum of their ages is 20 years. How old was the older brother 5 years ago?

Step-by-Step Solution

Solution 1: Using Basic Equations

Let's start with the most straightforward approach. We introduce two variables, x for the older brother's age and y for the younger brother's age. We can write the following equations based on the problem statement:

x - y 6 (Equation 1) x y 20 (Equation 2)

By adding these two equations, we get:

2x 26

x 13

Substituting x 13 into the first equation, we solve for y:

13 - y 6

y 7

So, the older brother is 13 years old, and the younger brother is 7 years old. To find out how old the older brother was 5 years ago, we subtract 5 from his current age:

13 - 5 8

The older brother was 8 years old 5 years ago.

Solution 2: Detailed Algebraic Steps

For a more detailed explanation, let's use algebra to solve the problem step-by-step:

Let the age of the younger brother be x. Then the age of the older brother is x 6. According to the problem, the sum of their ages is 20 years:

x (x 6) 20

Simplifying this equation:

2x 6 20

Subtracting 6 from both sides:

2x 14

Dividing by 2:

x 7

So, the younger brother is 7 years old, and the older brother is:

7 6 13

To find the age of the older brother 5 years ago:

13 - 5 8

The older brother was 8 years old 5 years ago.

Through these detailed steps, we can clearly see how to solve the problem using basic algebraic techniques.

Solution 3: Utilizing the Product of Ages

Another intriguing approach involves the product of their ages. We are given that the product of their ages is 21 and their sum is 22. Let's denote the ages of the younger brother as x and the older brother as x 6. The equations become:

x(x 6) 21 x (x 6) 22

Solving the quadratic equation:

x^2 6x - 21 0

Factoring the quadratic equation:

(x - 3)(x - 7) 0

The solutions are x 3 and x 7.

Since the sum is 22, the ages are 7 and 13. Thus, the older brother was 8 years old 5 years ago:

13 - 5 8

Conclusion

By using different methods, we have consistently arrived at the same conclusion: the older brother was 8 years old 5 years ago. This problem showcases the power of algebraic techniques in solving real-world problems and highlights the importance of systematic problem-solving approaches.