To determine the equation that correctly models the relationship between a salesperson's weekly sales ([tex]\(s\)[/tex]) and their weekly earnings ([tex]\(e\)[/tex]), we start by analyzing the components of their earnings:
1. The salesperson earns a fixed amount of [tex]\( \$200 \)[/tex] per week regardless of their sales.
2. Additionally, they earn a [tex]\( 4\% \)[/tex] commission on their sales [tex]\( s \)[/tex].
We need to express this relationship mathematically.
Step-by-Step Solution:
1. Fixed Earnings:
The salesperson earns [tex]\( \$200 \)[/tex] per week as a base salary. This is a fixed amount added to their total earnings.
2. Commission Earnings:
The salesperson earns a commission of [tex]\( 4\% \)[/tex] of their sales [tex]\( s \)[/tex]. To find [tex]\( 4\% \)[/tex] of [tex]\( s \)[/tex]:
[tex]\[
\text{Commission} = 0.04 \times s
\][/tex]
This represents the additional amount earned based on sales.
3. Total Earnings:
The total weekly earnings [tex]\( e \)[/tex] is the sum of the fixed earnings and the commission earnings. Therefore, we add the fixed salary to the commission:
[tex]\[
e = 200 + 0.04 \times s
\][/tex]
This equation captures both the fixed earnings and the variable earnings from the commission.
4. Matching the Equations to Options:
We need to check which of the given options matches the derived equation [tex]\( e = 200 + 0.04s \)[/tex]:
- Option 1: [tex]\( e = \frac{200s + 4}{100} \)[/tex]
This option does not correctly represent the relationship.
- Option 2: [tex]\( e = 200s + 4 \)[/tex]
This option incorrectly scales the fixed earnings with sales.
- Option 3: [tex]\( e = 4s + 200 \)[/tex]
This option uses a [tex]\(4\)[/tex] instead of [tex]\(0.04\)[/tex], which would be incorrect.
- Option 4: [tex]\( e = \frac{4}{100} s + 200 \)[/tex]
Simplifying [tex]\(\frac{4}{100}\)[/tex]:
[tex]\[
\frac{4}{100} = 0.04
\][/tex]
Therefore:
[tex]\[
e = 0.04s + 200
\][/tex]
This matches our derived equation.
Thus, the correct equation that models the relationship between the salesperson's weekly sales [tex]\( s \)[/tex] and their weekly earnings [tex]\( e \)[/tex] is:
[tex]\[
e = \frac{4}{100}s + 200
\][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{\frac{4}{100} s + 200} \][/tex]
