The problem is based on a pie chart showing the distribution of marks scored by 200 pupils. We need to determine: (a) The number of pupils who scored between 41 and 50 marks and the number of pupils who scored above 80 marks. (b) The fraction of pupils who scored at most 50 marks. (c) The modal class. However, the actual pie chart and relevant data (percentages or degrees for each category) are missing from the image. To solve this problem, I need the information from the pie chart. Assuming the pie chart is provided separately, here is how you would approach each part:

1. Problem Description

The problem is based on a pie chart showing the distribution of marks scored by 200 pupils. We need to determine:

(a) The number of pupils who scored between 41 and 50 marks and the number of pupils who scored above 80 marks.

(b) The fraction of pupils who scored at most 50 marks.

(c) The modal class.

However, the actual pie chart and relevant data (percentages or degrees for each category) are missing from the image. To solve this problem, I need the information from the pie chart. Assuming the pie chart is provided separately, here is how you would approach each part:

2. Solution Steps

(a) Determining the number of pupils:

(i) Between 41 and 50 marks:

* Find the percentage (or angle in degrees) corresponding to the marks between 41 and 50 in the pie chart. Let's assume it is $x$%.

* Calculate the number of pupils: Number of pupils = $\frac{x}{100} \times 200$.

(ii) Above 80 marks:

* Find the percentage (or angle in degrees) corresponding to the marks above 80 in the pie chart. Let's assume it is $y$%.

* Calculate the number of pupils: Number of pupils = $\frac{y}{100} \times 200$.

(b) Determining the fraction of pupils scoring at most 50 marks:

* Identify all sections of the pie chart that represent scores "at most 50 marks". This means scores equal to or less than

5

0. * Find the percentage (or angle in degrees) of each section and sum these. Let's assume the sum of the percentages is $z$%.

* Calculate the number of students = $\frac{z}{100} \times 200$.

* The required fraction = $\frac{\text{Number of pupils scoring at most 50 marks}}{\text{Total number of pupils}} = \frac{(\frac{z}{100} \times 200)}{200} = \frac{z}{100}$. Simplify this fraction if possible.

(c) Determining the modal class:

* The modal class is the class interval (range of marks) with the highest frequency, i.e., the largest sector in the pie chart.

* Identify the section of the pie chart with the largest percentage (or largest angle in degrees).

* The class interval represented by that section is the modal class.

3. Final Answer

Since the pie chart is missing, I cannot provide numerical answers. You need to provide the percentage or angle information from the chart to calculate the actual numbers.

For example, if the pie chart said:

41-50 Marks = 25%

Above 80 Marks = 10%

At most 50 marks = 60%

Modal Class = 41-50 Marks

Then, the answers would be:

(a) (i) $\frac{25}{100} \times 200 = 50$ pupils

(ii) $\frac{10}{100} \times 200 = 20$ pupils

(b) $\frac{60}{100} = \frac{3}{5}$

(c) 41-50 Marks