In Mathematics paper, the number of students' obtained marks are distributed in the given figure and total marks of this paper is 200. What is percentage (round off) of student who are Ist division if 60% mark are set for first division?
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Notice that 185 out of 350 is slightly more than half because half of 350 is 175. Thus, the answer must be slightly greater than 50%. Looking at the options, only 53% satisfies this condition without any complex calculations.
• Step 1: Understanding the Question: This question is based on a histogram depicting frequency distribution of marks. We must find the total number of students and then the number of students who scored enough to secure a first division (at least 60% of the maximum marks).
• Step 2: Key Formula or Approach:
1. First Division Marks threshold:
\[ M_{\text{threshold}} = \text{Total Marks} \times 60\% \]
2. Sum the frequencies of all intervals to get the total number of students (\(N_{\text{total}}\)).
3. Sum the frequencies of intervals that are greater than or equal to the threshold mark (\(N_{\text{1st}}\)).
4. Calculate the required percentage:
\[ \text{Percentage} = \left(\frac{N_{\text{1st}}}{N_{\text{total}}}\right) \times 100 \]
• Step 3: Detailed Explanation:
- Step 3.1: Finding the threshold mark:
Total marks = 200
Threshold mark = \(200 \times 0.60 = 120\) marks.
Therefore, students scoring 120 marks or above are placed in the first division.
- Step 3.2: Finding the total number of students (\(N_{\text{total}}\)):
Let us sum the heights of all the bars in the given histogram:
- \(20\text{-}40\) interval: 10 students
- \(40\text{-}60\) interval: 18 students
- \(60\text{-}80\) interval: 32 students
- \(80\text{-}100\) interval: 45 students
- \(100\text{-}120\) interval: 60 students
- \(120\text{-}140\) interval: 75 students
- \(140\text{-}160\) interval: 55 students
- \(160\text{-}180\) interval: 40 students
- \(180\text{-}200\) interval: 15 students
Summing these frequencies:
\[ N_{\text{total}} = 10 + 18 + 32 + 45 + 60 + 75 + 55 + 40 + 15 = 350 \]
- Step 3.3: Finding the number of students in the first division (\(N_{\text{1st}}\)):
These are the students in the intervals starting from 120 up to 200:
- \(120\text{-}140\) interval: 75 students
- \(140\text{-}160\) interval: 55 students
- \(160\text{-}180\) interval: 40 students
- \(180\text{-}200\) interval: 15 students
Summing these frequencies:
\[ N_{\text{1st}} = 75 + 55 + 40 + 15 = 185 \]
- Step 3.4: Calculating the percentage:
\[ \text{Percentage} = \left(\frac{185}{350}\right) \times 100 \]
\[ \text{Percentage} = \frac{1850}{35} \approx 52.857\% \]
Rounding off to the nearest integer gives 53%.
• Step 4: Final Answer: The percentage of students in the first division is approximately 53%, which matches option (B).
