Alright, let's solve this problem step by step.
We have four equations representing different scenarios of a waiter earning [tex]$\$[/tex]11.25[tex]$ for waiting on one table for one hour. In each equation, the constant term represents the hourly rate, and the coefficient of $[/tex]b[tex]$ represents the fraction of the bill that the waiter receives as a tip. We are given that the bill, $[/tex]b[tex]$, is $[/tex]30.00[tex]$, and the total amount earned should equal $[/tex]11.25$.
Let's check each equation:
1. Equation 1:
[tex]\[
5.00 + \frac{1}{6} \times 30 = 11.25
\][/tex]
First, calculate the tip:
[tex]\[
\frac{1}{6} \times 30 = 5.00
\][/tex]
Add this to the hourly rate:
[tex]\[
5.00 + 5.00 = 10.00
\][/tex]
This does not equal [tex]$11.25$[/tex].
2. Equation 2:
[tex]\[
5.25 + \frac{1}{3} \times 30 = 11.25
\][/tex]
First, calculate the tip:
[tex]\[
\frac{1}{3} \times 30 = 10.00
\][/tex]
Add this to the hourly rate:
[tex]\[
5.25 + 10.00 = 15.25
\][/tex]
This does not equal [tex]$11.25$[/tex] either.
3. Equation 3:
[tex]\[
5.25 + \frac{1}{5} \times 30 = 11.25
\][/tex]
First, calculate the tip:
[tex]\[
\frac{1}{5} \times 30 = 6.00
\][/tex]
Add this to the hourly rate:
[tex]\[
5.25 + 6.00 = 11.25
\][/tex]
This equals [tex]$11.25$[/tex], so this equation fits our requirement.
4. Equation 4:
[tex]\[
600 + \frac{1}{8} \times 30 = 1125
\][/tex]
First, calculate the tip:
[tex]\[
\frac{1}{8} \times 30 = 3.75
\][/tex]
Add this to the hourly rate:
[tex]\[
600 + 3.75 = 603.75
\][/tex]
This does not equal [tex]$11.25$[/tex].
Based on these evaluations, the correct equation that properly represents the hourly rate and tip for a bill of [tex]$30.00$[/tex] is:
[tex]\[
5.25 + \frac{1}{5} \times 30 = 11.25
\][/tex]
Thus, it is the third equation.
