The mean marks obtained by a class of 40 students is 65. The mean marks of…

Concept: For any group, the total of all values equals the mean multiplied by the number of members, so total = mean × count. When a group is split into parts, the totals of the parts add up to the total of the whole group. Rearranging that, the mean of one part = (total of the whole − total of the other part) ÷ number of members in that part.

Application to this question

1. The class has 40 students with a mean of 65, so the total marks of the class = 65 × 40 = 2600.

2. Half of the class is 40 ÷ 2 = 20 students, and their mean is 45, so their total marks = 45 × 20 = 900.

3. The remaining students number 40 − 20 = 20, and their total marks = 2600 − 900 = 1700.

4. Therefore the mean marks of the remaining students = 1700 ÷ 20 = 85.

Cross-check: Combining the two halves back gives (900 + 1700) ÷ 40 = 2600 ÷ 40 = 65, which matches the class mean stated in the question. A faster deviation check gives the same result: the first half sits 65 − 45 = 20 marks below the class mean, and because the two halves are of equal size their deviations must cancel, so the other half sits 20 marks above the class mean, at 65 + 20 = 85.

Result: the mean marks of the remaining 20 students is 85.