A student appeared in 6 papers. The maximum marks are the same for each paper. His marks in these papers are in the proportion of 5 : 6 : 7 : 8 : 9 : 10. Overall he scored 60%. In how many number of papers did he score less than 60% of the maximum marks ?

(a) 2
(b) 3 ✓ UPSC's answer
(c) 4
(d) 5

Why the answer is (b)

• Let the maximum marks for each paper be $M$. The total maximum marks for 6 papers is $6M$.

• The student's marks are in the ratio 5:6:7:8:9:10. Let the common multiplier be $x$. The total marks obtained are $5x + 6x + 7x + 8x + 9x + 10x = 45x$.

• The student scored 60% overall, so $45x = 0.60 \times 6M = 3.6M$.

• Solving for $x$, we get $x = \frac{3.6M}{45} = 0.08M$.

• The marks in each paper are $5(0.08M) = 0.40M$, $6(0.08M) = 0.48M$, $7(0.08M) = 0.56M$, $8(0.08M) = 0.64M$, $9(0.08M) = 0.72M$, and $10(0.08M) = 0.80M$.

• Comparing these to 60% of maximum marks ($0.60M$), the papers with scores less than 60% are the first three (40%, 48%, and 56%). Thus, he scored less than 60% in 3 papers.

Why the other options are wrong

(a) 2: Option (a) is incorrect because the student scored less than 60% in three papers (40%, 48%, and 56%), not two.
(c) 4: Option (c) is incorrect because the fourth paper's score is 64%, which is greater than 60%, so only three papers are below the threshold.
(d) 5: Option (d) is incorrect because the last three papers (64%, 72%, and 80%) all exceed 60%, meaning only three papers are below the threshold.

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.