Average Age Problem: Find Coach's Age

This problem involves calculating the age of a coach by understanding how their inclusion affects the average age of a group of players.

Player Average Age Basics

We are given that the average age of four players is 18.5 years. The average age is calculated by summing up all the ages and dividing by the number of people.

Calculating Total Player Age

To find the total age of the four players, we multiply the number of players by their average age.

Step 1: Calculate the total age of the four players.

Total Age of Players = Number of Players $\times$ Average Age

Using LaTeX for the calculation:

\( \text{Total Age} = 4 \times 18.5 \text{ years} = 74 \text{ years} \)

So, the combined age of the four players is 74 years.

Age Increase Calculation

When the coach's age is included, the group now consists of 5 people (4 players + 1 coach). The problem states that the average age increases by 20%.

Step 2: Calculate the amount by which the average age increases.

Increase in Average Age = 20% of the original average age

Using LaTeX for the calculation:

\( \text{Increase} = 20\% \times 18.5 \text{ years} \)

\( \text{Increase} = \frac{20}{100} \times 18.5 \text{ years} \)

\( \text{Increase} = 0.20 \times 18.5 \text{ years} = 3.7 \text{ years} \)

Step 3: Calculate the new average age after the increase.

New Average Age = Original Average Age + Increase in Average Age

Using LaTeX for the calculation:

\( \text{New Average Age} = 18.5 \text{ years} + 3.7 \text{ years} = 22.2 \text{ years} \)

The new average age for the group of five people is 22.2 years.

New Total Age with Coach

Now that we know the new average age and the new number of people, we can calculate the total age of the group including the coach.

Step 4: Calculate the new total age of the group (players + coach).

New Total Age = New Number of People $\times$ New Average Age

Using LaTeX for the calculation:

\( \text{New Total Age} = 5 \times 22.2 \text{ years} \)

\( \text{New Total Age} = 111 \text{ years} \)

The combined age of the four players and the coach is 111 years.

Coach's Age Determination

Finally, to find the coach's age, we subtract the total age of the players from the new total age which includes the coach.

Step 5: Calculate the coach's age.

Coach's Age = New Total Age - Total Age of Players

Using LaTeX for the calculation:

\( \text{Coach's Age} = 111 \text{ years} - 74 \text{ years} \)

\( \text{Coach's Age} = 37 \text{ years} \)

Therefore, the age of the coach is 37 years.