There are two Classes A and B having 25 and 30 students respectively. In Class-A the highest score is 21 and lowest score is 17. In Class-B the highest score is 30 and lowest score is 22. Four students are shifted from Class-A to Class-B. Consider the following statements : 1. The average score of Class-B will definitely decrease. 2. The average score of Class-A will definitely increase. Which of the above statements is/are correct ?

(a) 1 only ✓ UPSC's answer
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

Why the answer is (a)

• Class A has 25 students with scores between 17 and 21, so any student shifted from Class A has a score $x$ where $17 \le x \le 21$.

• Class B has 30 students with scores between 22 and 30, so the minimum score in Class B is 22.

• Since the maximum possible score of a student from Class A (21) is strictly less than the minimum score in Class B (22), any student added to Class B will have a score lower than the current minimum of Class B.

• Adding a value lower than the current minimum to a set will always decrease the average of that set; thus, the average of Class B will definitely decrease.

• For Class A, removing a student with a score $x$ will increase the average if $x$ is below the current average of Class A, but decrease it if $x$ is above the current average. Since the current average of Class A is unknown (only the range 17-21 is given), the effect on Class A's average is not definite.

• Therefore, only Statement 1 is correct, making option (a) the right answer.

Why the other options are wrong

(b) 2 only: Statement 2 is incorrect because removing a student from Class A does not guarantee an increase in its average, as the removed student's score could be higher than the class average.
(c) Both 1 and 2: Statement 2 is incorrect because the change in Class A's average depends on the specific score of the shifted student relative to the class average, which is not fixed.
(d) Neither 1 nor 2: Statement 1 is correct because any student from Class A (max 21) is lower than the minimum score in Class B (22), which necessarily lowers Class B's average.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2021, held on 10 October 2021. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.