Solving the Football Team Average Age Problem

This problem asks us to find the average age of a football team based on information about the captain, goalkeeper, and the remaining players.

Understanding the Given Information

• Total number of players in the team: 11
• Captain's age: 28 years
• Goalkeeper's age: 4 years older than the captain

Calculating the Goalkeeper's Age

The goalkeeper is 4 years older than the captain (28 years).

Goalkeeper's age = Captain's age + 4 years

Goalkeeper's age = $28 + 4 = 32$ years

Information After Removing Captain and Goalkeeper

• Number of players remaining: Total players - (Captain + Goalkeeper) = $11 - 2 = 9$ players
• Ages removed: Captain's age (28) and Goalkeeper's age (32)
• Sum of ages removed: $28 + 32 = 60$ years

The average age of these remaining 9 players is 2 years less than the average age of the whole team (11 players).

Setting Up the Equation

Let's denote the average age of the whole team (11 players) as $A$ years.

The sum of the ages of the whole team is $11 \times A = 11A$ years.

The sum of the ages of the remaining 9 players is the total sum minus the sum of the captain's and goalkeeper's ages.

Sum of ages of remaining 9 players = Total sum of ages - Sum of ages of captain and goalkeeper

Sum of ages of remaining 9 players = $11A - 60$ years

The average age of the remaining 9 players is the sum of their ages divided by the number of players.

Average age of remaining 9 players = $\frac{11A - 60}{9}$

We are told that the average age of the remaining 9 players is 2 years less than the average age of the whole team ($A$).

So, we can write the equation:

$\frac{11A - 60}{9} = A - 2$

Solving for the Average Age of the Team

Now, we solve the equation for $A$:

Multiply both sides of the equation by 9 to remove the denominator:

$9 \times \left(\frac{11A - 60}{9}\right) = 9 \times (A - 2)$

$11A - 60 = 9A - 18$

Subtract $9A$ from both sides of the equation:

$11A - 9A - 60 = -18$

$2A - 60 = -18$

Add 60 to both sides of the equation:

$2A = -18 + 60$

$2A = 42$

Divide both sides by 2:

$A = \frac{42}{2}$

$A = 21$

So, the average age of the whole team is 21 years.

Verification

Let's check if this answer fits the condition.

• Average age of the team (11 players) = 21 years.
• Sum of ages of the team = $11 \times 21 = 231$ years.
• Captain's age = 28 years.
• Goalkeeper's age = 32 years.
• Sum of ages of captain and goalkeeper = $28 + 32 = 60$ years.
• Sum of ages of remaining 9 players = Total sum - Sum of captain and goalkeeper = $231 - 60 = 171$ years.
• Average age of remaining 9 players = $\frac{171}{9} = 19$ years.

The problem states that the average age of the remaining players (19 years) is 2 years less than the average age of the whole team (21 years). $19 = 21 - 2$, which is true. The answer is verified.

The average age of the team is 21 years.

Revision Table: Football Team Age Calculation

Additional Information: Average and Word Problems

Average is a fundamental concept in mathematics used to represent a typical value in a set of data. Word problems involving averages often require setting up equations based on the relationship between the total sum, the number of items, and the average.

• When items are removed from a group, the sum of the remaining items is the original sum minus the sum of the removed items.
• When items are added to a group, the sum of the new group is the original sum plus the sum of the added items.
• These types of problems often involve translating the given information into algebraic equations to solve for an unknown value, like the average or a specific value within the set.
• Carefully reading and identifying the quantities and relationships described is crucial for setting up the correct equation.