Question

Prof. Suman takes a number of quizzes for a course. All the quizzes are out of 100. A student can get an A grade in the course if the average of her scores is more than or equal to 90. Grade B is awarded to a student if the average of her scores is between 87 and 89 Both include(D). If the average is below 87, the student gets a C grade. Ramesh is preparing for the last quiz and he realizes that he will score a minimum of 97 to get an A grade. After the quiz, he realizes that he will score 70, and he will just manage a B. How many quizzes did Prof. Suman take?

• A. 6
• B. 7
• C. 8
• D. 9
• E. None of these

Hint:

When quiz averages determine grades, form inequalities for both conditions Before and after the last quiz) and solve them simultaneously. This approach pinpoints the total number of quizzes.

Answer 1

The Correct Option is D

Step 1: Define variables.
Let the total number of quizzes be \(n\). Let the sum of Ramesh’s scores in the first \(n-1\) quizzes be \(S\). His last quiz score will be either \(97\) (to just get an (A) or \(70\) Actual scor(E).
Step 2: A-grade condition.
For an A grade, the average must be at least \(90\): \[ \frac{S + 97}{n} \geq 90. \] \[ S + 97 \geq 90n. \hfill (1) \] Step 3: B-grade condition with 70.
With a score of \(70\), the average should fall in \([87,89]\): \[ 87 \leq \frac{S + 70}{n} \leq 89. \] So, \[ 87n \leq S + 70 \leq 89n. \hfill (2) \] Step 4: Use inequalities.
From (1): \(S \geq 90n - 97.\)
From (2): \(87n - 70 \leq S \leq 89n - 70.\) For consistency: \[ 90n - 97 \leq 89n - 70. \] \[ n \leq 27. \] Also, \[ 90n - 97 \geq 87n - 70 ⇒ 3n \geq 27 ⇒ n \geq 9. \] Step 5: Check \(n=9\).
If \(n=9\):
From (1): \(S \geq 90(9) - 97 = 810 - 97 = 713.\)
From (2): \(87(9)-70 = 713 \leq S \leq 89(9)-70 = 801 - 70 = 731.\)
So \(713 \leq S \leq 731\), which is consistent. Hence \(n=9\). \[ \boxed{9} \]