Certainly! Let's solve the problem step-by-step by determining the value of [tex]\( k \)[/tex].
### Given Information
1. Sum of the product of frequencies and midpoints (Σfm): [tex]\( 72 + 8k \)[/tex]
2. Mean (X): [tex]\( 6 \)[/tex]
3. Sum of frequencies (Σf): [tex]\( 16 + k \)[/tex]
### Relationship Between Mean, Sum of Frequencies, and Sum of Product of Frequencies and Midpoints
The mean [tex]\( X \)[/tex] is given by the formula:
[tex]\[ X = \frac{\sum fm}{\sum f} \][/tex]
### Substituting the Given Values
We are given:
[tex]\[ X = 6 \][/tex]
[tex]\[ \sum fm = 72 + 8k \][/tex]
[tex]\[ \sum f = 16 + k \][/tex]
So, substituting into the formula for the mean, we have:
[tex]\[ 6 = \frac{72 + 8k}{16 + k} \][/tex]
### Solving for [tex]\( k \)[/tex]
We need to solve the equation:
[tex]\[ 6 = \frac{72 + 8k}{16 + k} \][/tex]
First, cross-multiply to clear the fraction:
[tex]\[ 6(16 + k) = 72 + 8k \][/tex]
Distribute the 6 on the left side:
[tex]\[ 96 + 6k = 72 + 8k \][/tex]
Now, isolate the terms involving [tex]\( k \)[/tex] on one side. Subtract [tex]\( 6k \)[/tex] from both sides:
[tex]\[ 96 = 72 + 2k \][/tex]
Next, subtract 72 from both sides:
[tex]\[ 24 = 2k \][/tex]
Finally, solve for [tex]\( k \)[/tex] by dividing both sides by 2:
[tex]\[ k = \frac{24}{2} \][/tex]
[tex]\[ k = 12 \][/tex]
So, the value of [tex]\( k \)[/tex] is [tex]\( 12 \)[/tex].
