Unraveling the Secrets of Team Age Averages: A Mathematical Exploration

Understanding the average age dynamics within a team is pivotal for a wide range of applications, such as team composition, strategic planning, and demographic analysis. This article delves into the mathematical intricacies of determining the average age of a team given specific conditions, using a compelling example involving a team captain and a keeper. We will walk through the process step-by-step and provide a comprehensive solution to the problem.

Introduction to the Problem

Consider a team of 11 members where the captain is 25 years old and the keeper, who is 5 years older, is 30 years old. The key challenge is to determine the average age of the entire team, given that the average age of the remaining 9 players (excluding the captain and the keeper) is 1 year less than the average age of the whole team.

Solution Approach

To solve this problem, let's denote the average age of the whole team as ( A ). We will use algebraic expressions to represent the given conditions and derive the solution step by step.

Step 1: Define the Variables and Expressions

Captain's Age: ( 25 ) years

Keeper's Age: ( 25 5 30 ) years

Total Number of Team Members: ( 11 )

Total Age of the Team: ( 11A )

Total Age of Remaining Players (excluding captain and keeper): ( 11A - 55 )

Average Age of Remaining Players: ( frac{11A - 55}{9} )

Step 2: Apply the Given Condition

The problem states that the average age of the remaining players is 1 year less than the average age of the whole team:

[ frac{11A - 55}{9} A - 1 ]

Step 3: Solve the Equation

Multiplication: Multiply both sides by 9 to eliminate the fraction:

[ 11A - 55 9A - 9 ]

Expansion: Expand the right side:

[ 11A - 55 9A - 9 ]

Rearrangement: Rearrange the equation:

[ 11A - 9A 55 - 9 ]

[ 2A 46 ]

[ A 23 ]

Therefore, the average age of the team is 23 years.

Alternative Approach

Let's explore an alternative approach to confirm our solution:

Denote ( x ) as the sum of the ages of all players except the captain and the keeper. The average age of the whole team is:

[ frac{25 30 x}{11} frac{55 x}{11} ]

The average age of the remaining 9 players is:

[ frac{x}{9} ]

According to the problem, it is 1 year less than the average age of the whole team:

[ frac{x}{9} frac{55 x}{11} - 1 ]

Solving the above equation:

[ frac{x}{9} frac{55 x - 11}{11} ]

[ frac{x}{9} frac{44 x}{11} ]

Multiply both sides by 99 (LCM of 9 and 11) to eliminate the fractions:

[ 11x 9(44 x) ]

[ 11x 396 9x ]

[ 2x 396 ]

[ x 198 ]

Substitute ( x 198 ) back into the equation for the average age of the whole team:

[ frac{55 198}{11} frac{253}{11} 23 ]

Thus, confirming that the average age of the team is 23 years.

Conclusion

Through meticulous algebraic manipulation and alternative strategies, we have deduced that the average age of the team is 23 years. This problem not only highlights the importance of mathematical reasoning but also offers practical insights into team composition and age distribution.

For more detailed and complex problems, similar methods can be applied, providing a solid foundation for understanding and solving intricate age-related questions in team settings.