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A salesperson earns [tex]$\$[/tex]200[tex]$ a week plus a $[/tex]4\%[tex]$ commission on her sales. Which equation models the relationship between her sales in a given week, $[/tex]s[tex]$, and her weekly earnings, $[/tex]e[tex]$?

A. $[/tex]e = \frac{200s + 4}{100}[tex]$

B. $[/tex]e = 200s + 4[tex]$

C. $[/tex]e = 4s + 200[tex]$

D. $[/tex]e = \frac{4}{100}s + 200$

Sagot :

GinnyAnswer
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]
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