Solving the Average Age Problem for Students

This question asks us to find the average age of a group of new students admitted to a class, given information about the class's average age before and after their admission.

The concept of average is fundamental here. The average of a set of values is the sum of the values divided by the number of values. We can use this relationship to find the total sum (or total age in this case) if we know the average and the number of items.

The formula for average is:

\(\text{Average} = \frac{\text{Sum of values}}{\text{Number of values}}\)

From this, we can derive the formula for the sum of values:

\(\text{Sum of values} = \text{Average} \times \text{Number of values}\)

Step-by-Step Calculation of Student Ages

Let's break down the problem into steps using the given information about the students and their ages.

Initial Situation (Before New Students)

• Number of students initially = 15
• Average age of initial students = 19 years
• Total age of initial 15 students = Initial Average Age \(\times\) Number of Initial Students
• Total age of initial 15 students = \(19 \times 15\) years
• Total age of initial 15 students = \(285\) years

Situation After New Students are Admitted

• Number of new students admitted = 5
• Total number of students after admission = Initial students + New students
• Total number of students = \(15 + 5 = 20\)
• Average age of the class after admission = 18.5 years
• Total age of all 20 students = New Average Age \(\times\) Total Number of Students
• Total age of all 20 students = \(18.5 \times 20\) years
• Total age of all 20 students = \(370\) years

Finding the Total Age of the New Students

The total age of the 20 students is the sum of the total age of the initial 15 students and the total age of the 5 new students.

Total age of 5 new students = (Total age of all 20 students) - (Total age of initial 15 students)

Total age of 5 new students = \(370 - 285\) years

Total age of 5 new students = \(85\) years

Calculating the Average Age of the New Students

Now we can find the average age of the 5 new students using their total age and their number.

Average age of 5 new students = \(\frac{\text{Total age of 5 new students}}{\text{Number of new students}}\)

Average age of 5 new students = \(\frac{85}{5}\) years

Average age of 5 new students = \(17\) years

Therefore, the average age of the 5 newly admitted students is 17 years.

Revision Table: Key Calculations for Average Age

Additional Information on Averages and Weighted Averages

This problem is a classic example where understanding the relationship between average, sum, and count is crucial. It can also be viewed in terms of weighted averages, though the direct calculation of total sums is often simpler.

A weighted average is used when different items have different "weights" or importance. In this case, the two groups of students (initial and new) have different numbers, which act as weights when calculating the overall average. The overall average age (18.5) is a weighted average of the initial group's average (19) and the new group's average (let's call it \(A_{new}\)).

The formula for weighted average would be:

\(\text{Overall Average} = \frac{(\text{Weight}_1 \times \text{Average}_1) + (\text{Weight}_2 \times \text{Average}_2)}{\text{Weight}_1 + \text{Weight}_2}\)

Here, the weights are the number of students:

\(18.5 = \frac{(15 \times 19) + (5 \times A_{new})}{15 + 5}\)

\(18.5 = \frac{285 + 5 \times A_{new}}{20}\)

Multiplying both sides by 20:

\(18.5 \times 20 = 285 + 5 \times A_{new}\)

\(370 = 285 + 5 \times A_{new}\)

Subtracting 285 from both sides:

\(370 - 285 = 5 \times A_{new}\)

\(85 = 5 \times A_{new}\)

Dividing by 5:

\(A_{new} = \frac{85}{5}\)

\(A_{new} = 17\)

This confirms the result obtained by calculating total ages. Both methods rely on the core relationship between average, sum, and the number of items.