To solve for [tex]\( n \)[/tex], we begin with the given conditions:
1. [tex]\(\Sigma(x - 200) = 446\)[/tex]
2. [tex]\(\Sigma x = 6846\)[/tex]
The notation [tex]\(\Sigma\)[/tex] represents the summation of the terms.
Firstly, we recognize that the summation [tex]\(\Sigma(x - 200)\)[/tex] can be expanded as:
[tex]\[
\Sigma(x - 200) = \Sigma x - \Sigma 200
\][/tex]
Given that [tex]\(\Sigma 200\)[/tex] is simply [tex]\(200n\)[/tex] (since it adds 200 for each of the [tex]\(n\)[/tex] terms), we can rewrite the equation as:
[tex]\[
\Sigma(x - 200) = \Sigma x - 200n
\][/tex]
Substituting the given summations into the equation:
[tex]\[
446 = 6846 - 200n
\][/tex]
We need to solve this equation for [tex]\( n \)[/tex]. To do this, let's isolate [tex]\( n \)[/tex]:
[tex]\[
446 = 6846 - 200n
\][/tex]
Subtract 6846 from both sides to simplify:
[tex]\[
446 - 6846 = -200n
\][/tex]
[tex]\[
-6400 = -200n
\][/tex]
Next, divide both sides of the equation by -200 to solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{6400}{200}
\][/tex]
[tex]\[
n = 32
\][/tex]
Thus, the value of [tex]\( n \)[/tex] is [tex]\( 32 \)[/tex].
