Question
These equations represent different possibilities for a waiter earning $11.25 for waiting on one table for one hour. The constant is the hourly rate and the coefficient represents the fraction of the bill, b, the customers left as a tip. Which equation correctly represents the hourly rate and tip for a bill of $30.00?
5.00 + One-sixthb = 11.25
5.25 +One-thirdb = 11.25
5.25 + One-fifthb = 11.25
6.00 + One-eighthb = 11.25
Answers
GPT-4o mini
To determine which equation correctly represents the hourly rate and tip for a bill of $30.00, we need to first identify how much of the total $11.25 is made up of the tip from the bill.
Let’s calculate the tip that corresponds to a bill of $30.00. Each equation can be thought of as a scenario where the waiter receives a certain base amount (the constant) plus a fraction of the bill as a tip. The coefficient represents the fraction of the bill, \( b \).
For each equation of the form:
\[ \text{Hourly Rate} + \left(\text{Fraction}\times \text{Bill}\right) = 11.25 \]
Where the bill \( b = 30.00 \), we would have:
\[ \text{Hourly Rate} + \left(\text{Fraction} \times 30\right) = 11.25 \]
Now, let's evaluate the equations:
1. **For the first equation:**
\[
5.00 + \frac{1}{6}b = 11.25
\]
\[
5.00 + \frac{1}{6} \times 30 = 11.25
\]
\[
5.00 + 5.00 = 11.25 \text{ (true)}
\]
2. **For the second equation:**
\[
5.25 + \frac{1}{3}b = 11.25
\]
\[
5.25 + \frac{1}{3} \times 30 = 11.25
\]
\[
5.25 + 10.00 = 15.25 \text{ (false)}
\]
3. **For the third equation:**
\[
5.25 + \frac{1}{5}b = 11.25
\]
\[
5.25 + \frac{1}{5} \times 30 = 11.25
\]
\[
5.25 + 6.00 = 11.25 \text{ (true)}
\]
4. **For the fourth equation:**
\[
6.00 + \frac{1}{8}b = 11.25
\]
\[
6.00 + \frac{1}{8} \times 30 = 11.25
\]
\[
6.00 + 3.75 = 9.75 \text{ (false)}
\]
Now, from the valid calculations:
- The first equation returns \( 11.25 \).
- The third equation also returns \( 11.25 \).
Thus, both the first and the third equations are correct. However, if we have to choose one based on the description of the hourly rate and fraction of the bill, the equation **5.00 + One-sixthb = 11.25** is a bit more common for a smaller tip fraction, while **5.25 + One-fifthb = 11.25** also works.
Ultimately, both equations, but specifically, **5.00 + One-sixthb = 11.25**, clearly aligns with the typical understanding of a smaller constant hourly fee.
Let’s calculate the tip that corresponds to a bill of $30.00. Each equation can be thought of as a scenario where the waiter receives a certain base amount (the constant) plus a fraction of the bill as a tip. The coefficient represents the fraction of the bill, \( b \).
For each equation of the form:
\[ \text{Hourly Rate} + \left(\text{Fraction}\times \text{Bill}\right) = 11.25 \]
Where the bill \( b = 30.00 \), we would have:
\[ \text{Hourly Rate} + \left(\text{Fraction} \times 30\right) = 11.25 \]
Now, let's evaluate the equations:
1. **For the first equation:**
\[
5.00 + \frac{1}{6}b = 11.25
\]
\[
5.00 + \frac{1}{6} \times 30 = 11.25
\]
\[
5.00 + 5.00 = 11.25 \text{ (true)}
\]
2. **For the second equation:**
\[
5.25 + \frac{1}{3}b = 11.25
\]
\[
5.25 + \frac{1}{3} \times 30 = 11.25
\]
\[
5.25 + 10.00 = 15.25 \text{ (false)}
\]
3. **For the third equation:**
\[
5.25 + \frac{1}{5}b = 11.25
\]
\[
5.25 + \frac{1}{5} \times 30 = 11.25
\]
\[
5.25 + 6.00 = 11.25 \text{ (true)}
\]
4. **For the fourth equation:**
\[
6.00 + \frac{1}{8}b = 11.25
\]
\[
6.00 + \frac{1}{8} \times 30 = 11.25
\]
\[
6.00 + 3.75 = 9.75 \text{ (false)}
\]
Now, from the valid calculations:
- The first equation returns \( 11.25 \).
- The third equation also returns \( 11.25 \).
Thus, both the first and the third equations are correct. However, if we have to choose one based on the description of the hourly rate and fraction of the bill, the equation **5.00 + One-sixthb = 11.25** is a bit more common for a smaller tip fraction, while **5.25 + One-fifthb = 11.25** also works.
Ultimately, both equations, but specifically, **5.00 + One-sixthb = 11.25**, clearly aligns with the typical understanding of a smaller constant hourly fee.