Simple Average — Mains Questions

A company has three departments with 20, 30, and 50 employees respectively. The average salaries in these departments are ₹40,000, ₹50,000, and ₹60,000 respectively. If the company decides to give a uniform bonus of ₹5,000 to all employees, analyze the impact on departmental and overall average salaries. Also, determine what percentage increase in the smallest department's average salary would be needed to match the overall company average after the bonus.

In a school's annual examination, the average marks of students in Mathematics across 5 sections are 72, 68, 75, 80, and 65 respectively, with each section having 40 students. Due to a paper evaluation error, it was discovered that 10 marks were incorrectly deducted from each student's score in one section. If the corrected overall average became 73, identify which section was affected and analyze the mathematical principles involved in this correction process.

A research study involves collecting data from three age groups: 18-25 years (100 participants), 26-35 years (150 participants), and 36-45 years (200 participants). The average scores on a cognitive test are 85, 78, and 72 respectively. Analyze how the overall average would change if: (a) 20 participants from the youngest group withdraw, (b) the middle group's average increases by 5 points due to retesting, and (c) both changes occur simultaneously. Discuss the mathematical concepts involved and their research implications.