Sure, let's tackle the question step-by-step.
### Step 1: Calculate the mean mark for Paper 1
The marks for Mathematical Literacy Paper 1 are:
[tex]\[ 120, 118, 115, 114, 87, 94, 113, 104, 113, 96, 105, 119, 120, 109 \][/tex]
To find the mean mark:
1. Sum all the marks.
2. Divide by the number of marks.
Sum of marks:
[tex]\[
120 + 118 + 115 + 114 + 87 + 94 + 113 + 104 + 113 + 96 + 105 + 119 + 120 + 109 = 1527
\][/tex]
Number of marks:
[tex]\[
14
\][/tex]
Mean mark for Paper 1:
[tex]\[
\text{Mean} = \frac{1527}{14} = 109.07142857142857
\][/tex]
### Step 2: Calculate the percentage mean for Paper 1
Since the paper was out of 150, the percentage mean is calculated as follows:
[tex]\[
\text{Percentage Mean} = \left( \frac{\text{Mean}}{150} \right) \times 100
\][/tex]
Substitute the calculated mean:
[tex]\[
\text{Percentage Mean} = \left( \frac{109.07142857142857}{150} \right) \times 100 = 72.71428571428571\%
\][/tex]
### Step 3: Find the value of [tex]\( M \)[/tex]
The marks for Mathematical Literacy Paper 2 are:
[tex]\[
107, 115, 104, 83, 61, 100, 104, M, 113, 120, 108, 126, 124, 105
\][/tex]
The range of a data set is the difference between the maximum and minimum values. Given the data, the known maximum and minimum values in Paper 2 are:
[tex]\[ \max_{\text{without } M} = 126 \][/tex]
[tex]\[ \min_{\text{without } M} = 61 \][/tex]
[tex]\[ \text{Observed range without } M = 126 - 61 = 65 \][/tex]
We are given that the range should be 77.
Thus, let's calculate the additional range needed:
[tex]\[ \text{Additional range} = 77 - 65 = 12 \][/tex]
Therefore, [tex]\( M \)[/tex] should be:
[tex]\[ M = \max_{\text{without } M} + \text{Additional range} \][/tex]
[tex]\[ M = 126 + 12 = 138 \][/tex]
So, the value of [tex]\( M \)[/tex] is 138.
### Summary
1. The mean mark for Paper 1 is [tex]\( 109.07142857142857 \)[/tex].
2. The percentage mean for Paper 1, if the paper was out of 150, is [tex]\( 72.71428571428571\% \)[/tex].
3. The value of [tex]\( M \)[/tex] in Paper 2, given that the range is 77, is [tex]\( 138 \)[/tex].
I hope this helps you understand how we arrived at the answers! If you have any further questions, feel free to ask.
