Join IDNLearn.com and start getting the answers you've been searching for. Our platform is designed to provide reliable and thorough answers to all your questions, no matter the topic.
Sagot :
Sure, let's solve this step-by-step.
### 1. Given Data
We are provided with the mean of the marks obtained, \(\bar{X} = 32.5\). The marks obtained and the number of students in each range are as follows:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline \text{Marks obtained} & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 \\ \hline \text{Number of students} & 5 & 10 & k & 35 & 15 \\ \hline \end{array} \][/tex]
### 2. Interval midpoints
First, we calculate the midpoints of each interval:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline \text{Interval} & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 \\ \hline \text{Midpoint} & 5 & 15 & 25 & 35 & 45 \\ \hline \end{array} \][/tex]
### 3. Total number of students
The total number of students can be expressed as:
[tex]\[ \text{Total number of students} = 5 + 10 + k + 35 + 15 = k + 65 \][/tex]
### 4. Total marks
The total marks can be calculated by multiplying the number of students in each interval by the interval midpoint and summing:
[tex]\[ \text{Total marks} = 5 \cdot 5 + 10 \cdot 15 + k \cdot 25 + 35 \cdot 35 + 15 \cdot 45 \][/tex]
[tex]\[ = 25 + 150 + 25k + 1225 + 675 = 25k + 2075 \][/tex]
### 5. Mean formula
The mean of the marks obtained is given as:
[tex]\[ \bar{X} = \frac{\text{Total marks}}{\text{Total number of students}} \][/tex]
Substituting the known values:
[tex]\[ 32.5 = \frac{25k + 2075}{k + 65} \][/tex]
### 6. Solve for \(k\)
Rewrite the equation:
[tex]\[ 32.5 (k + 65) = 25k + 2075 \][/tex]
Distribute on the left side:
[tex]\[ 32.5k + 2112.5 = 25k + 2075 \][/tex]
Isolate the \(k\) terms to one side:
[tex]\[ 32.5k - 25k = 2075 - 2112.5 \][/tex]
[tex]\[ 7.5k = -37.5 \][/tex]
Solve for \(k\):
[tex]\[ k = \frac{-37.5}{7.5} = -5 \][/tex]
So, the value of \(k\) is:
[tex]\[ k = -5 \][/tex]
### 1. Given Data
We are provided with the mean of the marks obtained, \(\bar{X} = 32.5\). The marks obtained and the number of students in each range are as follows:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline \text{Marks obtained} & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 \\ \hline \text{Number of students} & 5 & 10 & k & 35 & 15 \\ \hline \end{array} \][/tex]
### 2. Interval midpoints
First, we calculate the midpoints of each interval:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline \text{Interval} & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 \\ \hline \text{Midpoint} & 5 & 15 & 25 & 35 & 45 \\ \hline \end{array} \][/tex]
### 3. Total number of students
The total number of students can be expressed as:
[tex]\[ \text{Total number of students} = 5 + 10 + k + 35 + 15 = k + 65 \][/tex]
### 4. Total marks
The total marks can be calculated by multiplying the number of students in each interval by the interval midpoint and summing:
[tex]\[ \text{Total marks} = 5 \cdot 5 + 10 \cdot 15 + k \cdot 25 + 35 \cdot 35 + 15 \cdot 45 \][/tex]
[tex]\[ = 25 + 150 + 25k + 1225 + 675 = 25k + 2075 \][/tex]
### 5. Mean formula
The mean of the marks obtained is given as:
[tex]\[ \bar{X} = \frac{\text{Total marks}}{\text{Total number of students}} \][/tex]
Substituting the known values:
[tex]\[ 32.5 = \frac{25k + 2075}{k + 65} \][/tex]
### 6. Solve for \(k\)
Rewrite the equation:
[tex]\[ 32.5 (k + 65) = 25k + 2075 \][/tex]
Distribute on the left side:
[tex]\[ 32.5k + 2112.5 = 25k + 2075 \][/tex]
Isolate the \(k\) terms to one side:
[tex]\[ 32.5k - 25k = 2075 - 2112.5 \][/tex]
[tex]\[ 7.5k = -37.5 \][/tex]
Solve for \(k\):
[tex]\[ k = \frac{-37.5}{7.5} = -5 \][/tex]
So, the value of \(k\) is:
[tex]\[ k = -5 \][/tex]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.