To determine the equation that models the relationship between the sales in a given week, [tex]\( s \)[/tex], and the weekly earnings, [tex]\( e \)[/tex], we need to take into account the two components of the salesperson's earnings:
1. A base salary of $200 per week.
2. A commission of [tex]\( 4\% \)[/tex] on the sales, [tex]\( s \)[/tex].
First, let's convert the percentage commission into a decimal for ease of calculation:
[tex]\[ 4\% = \frac{4}{100} = 0.04 \][/tex]
Now we can write the total earnings, [tex]\( e \)[/tex], as the sum of the base salary and the sales commission.
The commission part of the earnings is:
[tex]\[ \text{Commission} = 0.04s \][/tex]
So the total weekly earnings, [tex]\( e \)[/tex], can be expressed as:
[tex]\[ e = 200 + 0.04s \][/tex]
Now, let's compare this equation with the given options:
1. [tex]\( e = \frac{200s + 4}{100} \)[/tex]
Simplifying this equation:
[tex]\[
\frac{200s + 4}{100} = 2s + 0.04
\][/tex]
This does not match our derived equation.
2. [tex]\( e = 200s + 4 \)[/tex]
This equation indicates that the earnings are proportional to the sales plus an arbitrary constant addition, which does not match our derived form.
3. [tex]\( e = 4s + 200 \)[/tex]
This equation indicates that the commission rate is much higher than 4%, which is not in accordance with the given problem.
4. [tex]\( e = \frac{4}{100}s + 200 \)[/tex]
Simplifying this equation:
[tex]\[
\frac{4}{100}s = 0.04s
\][/tex]
Therefore,
[tex]\[
e = 0.04s + 200
\][/tex]
This matches our derived equation exactly.
Hence, the correct equation that models the relationship between the salesperson's sales and weekly earnings is:
[tex]\[ e = \frac{4}{100}s + 200 \][/tex]
