Math Problem Statement
100 students of a class appeared for a test
that had 30 questions. Each question carried
1 mark for the correct answer and there were
no negative markings for wrong answers. The
number of students in the class who
attempted at least n questions wrong is given
by an = 60 – 2n (n > 0). The average marks
of the class can be:
(1) 20.3 (2) 21.3
(3) 22.3 (4) 25.3
Solution
To solve this problem, let's break it down step by step.
Given Information:
- 100 students appeared for a test.
- The test had 30 questions, and each question carries 1 mark.
- The number of students who attempted at least questions wrong is given by the formula:
We are tasked with finding the average marks of the class.
Step 1: Understanding the Formula for Students Attempting at Least Wrong
The formula tells us the number of students who attempted at least questions wrong. This number decreases as increases. For instance:
- If , the number of students who attempted at least 1 question wrong is .
- If , the number of students who attempted at least 2 questions wrong is , and so on.
This suggests that students are making more errors as increases.
Step 2: Total Marks per Student
If a student attempts questions wrong, then the number of correct answers is . The marks a student gets are equal to the number of correct answers, which is .
Step 3: Calculate the Average Marks
The number of students with at least wrong answers is . We need to find out how to use this to compute the average marks of the class.
- Let’s calculate the total number of students attempting wrong answers for each value of and the marks they score.
- Then, sum the total marks obtained by all students.
- Finally, divide the total marks by the total number of students (which is 100).
Step 4: Identify the Correct Option
Based on the provided choices and after calculating, the correct average turns out to be:
This is option (2).
Would you like me to go into more detail about any specific part of this problem?
Here are 5 questions for further exploration:
- How would the problem change if there were negative markings for wrong answers?
- What if the number of questions were increased to 50 instead of 30?
- How can we calculate the exact distribution of marks across the students?
- What would happen if the formula for changed to something nonlinear?
- How does the average marks change if a bonus is given for correct answers?
Tip: In any word problem, clearly identifying and labeling key information will help you develop a strategy for solving it efficiently.
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Math Problem Analysis
Mathematical Concepts
Algebra
Arithmetic Mean
Sequences and Series
Formulas
an = 60 - 2n (for n > 0): Represents the number of students with at least n wrong answers
Marks per student = 30 - n: Total marks if n questions are wrong
Average Marks = (Total Marks of All Students) / (Number of Students)
Theorems
-
Suitable Grade Level
Grades 9-12