Students Number Calculation Using Averages

This problem requires us to find the total number of students based on given average ages. We can solve this using the concept of weighted averages. Let's break down the calculation step-by-step.

Average Age Data

• Average age of students = 13 years
• Number of teachers = 14
• Average age of teachers = 34 years
• Combined average age (students + teachers) = 14 years

Calculation Steps Explained

We need to find the number of students. Let's denote the number of students as '$n_s$'.

1. Define Variables:

   • Number of students = '$n_s$'
   • Average age of students = '$A_s = 13$' years
   • Number of teachers = '$n_t = 14$'
   • Average age of teachers = '$A_t = 34$' years
   • Combined average age = '$A_{total} = 14$' years

2. Calculate Total Age Sums:

   • The sum of the ages of all students is the number of students multiplied by their average age: Total Age (Students) = '$n_s \times A_s = 13n_s$'
   • The sum of the ages of all teachers is the number of teachers multiplied by their average age: Total Age (Teachers) = '$n_t \times A_t = 14 \times 34$' Calculating this gives: $14 \times 34 = 476$ years.

3. Formulate the Combined Average Equation:

   The combined average age is the total sum of ages (students + teachers) divided by the total number of people (students + teachers).

   $$ A_{total} = \frac{\text{Total Age (Students)} + \text{Total Age (Teachers)}}{\text{Number of Students} + \text{Number of Teachers}} $$ Substituting the values we have: $$ 14 = \frac{13n_s + 476}{n_s + 14} $$

4. Solve for the Number of Students ('$n_s$'):

   To find '$n_s$', we rearrange the equation:

   • Multiply both sides by '$ (n_s + 14) $': $$ 14 \times (n_s + 14) = 13n_s + 476 $$
   • Distribute the 14 on the left side: $$ 14n_s + (14 \times 14) = 13n_s + 476 $$ $$ 14n_s + 196 = 13n_s + 476 $$
   • Group the '$n_s$' terms on one side and the constants on the other. Subtract '$13n_s$' from both sides: $$ (14n_s - 13n_s) + 196 = 476 $$ $$ n_s + 196 = 476 $$
   • Subtract 196 from both sides to isolate '$n_s$': $$ n_s = 476 - 196 $$ $$ n_s = 280 $$

So, the calculation shows that the number of students is 280.