The average age of a class was 15 years. When 5 more boys, whose average age was 12 years 6 months were admitted in the class, the average age of the whole class was reduced by 6 months. How many students were there in the class originally?

Points to Remember:

• This is a word problem requiring the application of averages.
• We need to solve for the original number of students in the class.
• The problem involves calculating weighted averages.

Introduction:

This question is a classic example of a weighted average problem. Weighted averages are used when different groups have different sizes and average values. Understanding weighted averages is crucial in various fields, from statistics and finance to demographics and resource allocation. The core concept is that the overall average is influenced by the size of each group contributing to it. This problem uses the concept to determine the original class size based on a change in the average age after adding new students.

Body:

1. Defining Variables and Setting up Equations:

Let’s denote:

• x as the original number of students in the class.
• The sum of the ages of the original students is 15x (since the average age was 15).

When 5 boys with an average age of 12 years 6 months (12.5 years) are added:

• The total age of the 5 new boys is 5 * 12.5 = 62.5 years.
• The total number of students becomes x + 5.
• The sum of the ages of all students (original + new) is 15x + 62.5.
• The new average age is 15 – 0.5 = 14.5 years.

Therefore, we can set up the equation:

(15x + 62.5) / (x + 5) = 14.5

2. Solving the Equation:

To solve for x, we can follow these steps:

1. Multiply both sides by (x + 5): 15x + 62.5 = 14.5(x + 5)
2. Expand the right side: 15x + 62.5 = 14.5x + 72.5
3. Subtract 14.5x from both sides: 0.5x + 62.5 = 72.5
4. Subtract 62.5 from both sides: 0.5x = 10
5. Divide both sides by 0.5: x = 20

Therefore, there were originally 20 students in the class.

3. Verification:

Let’s verify our answer:

• Original total age: 20 * 15 = 300 years
• Total age after adding 5 boys: 300 + 62.5 = 362.5 years
• New number of students: 20 + 5 = 25 students
• New average age: 362.5 / 25 = 14.5 years (which is 15 – 0.5 = 14.5, as expected)

Conclusion:

By setting up and solving a weighted average equation, we determined that there were originally 20 students in the class. The problem highlights the importance of understanding weighted averages in solving real-world problems involving different groups with varying characteristics. This approach can be applied to various scenarios involving population demographics, resource allocation, and financial analysis. Accurate calculations in such scenarios are crucial for effective planning and decision-making. The solution emphasizes the power of mathematical modeling in solving seemingly complex problems with a clear, systematic approach.

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