Let n = number of new students.

[[model:questions/solutions/q10247.png]]

Total marks before = class size × original average.
• Original total = 15 × 78 = 1170 marks
• New total = (15 + n) × 82 = 1230 + 82n
• New students' marks = n × 90 = 90n
• Equation: 1170 + 90n = 1230 + 82n
• 90n − 82n = 1230 − 1170
• 8n = 60
• n = 60 ÷ 8 = 7.5

Final Answer: 7.5