In a class, the number of students who scored 61 or more marks is 30 and there are at most 39 students who have scored below 61 marks.
If the average marks of the class is 48, then determine that what is the largest possible average marks of students who have scored below 61 marks?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 38
The total marks of the class are fixed by the class average. The less the 30 high scorers take out of that total, the more is left for the rest, so their average is largest when every high scorer has the least possible mark, 61, and when the class is as large as allowed.
Let \(n\) students score below 61 (\(n\le39\)).
The marks left for the \(n\) students (the class total less the least the high scorers can take):
Total marks of the class \(=48(30+n)=1440+48n\)
Least total of the 30 high scorers \(=30\times61=1830\)
Largest total left for the \(n\) students \(=1440+48n-1830=48n-390\)
Their largest average (this grows as \(n\) grows, so it is largest for \(n=39\)):
Largest average of the \(n\) students \(\displaystyle =\frac{48n-390}{n}=48-\frac{390}{n}\)
For \(n=39\): \(48-\frac{390}{39}=48-10=38\)