Question
A store sells shirts to the public at one pricing scale and wholesale at another pricing scale. The tables below describe the cost, y, of x shirts.
Public
A 2-column table with 3 rows. Column 1 is labeled x with entries 2, 5, 9. Column 2 is labeled y with entries 24, 60, 108.
Wholesale
A 2-column table with 3 rows. Column 1 is labeled x with entries 18, 35, 50. Column 2 is labeled y with entries 162, 315, 360.
How do the slopes of the lines created by each table compare?
The slope of the Public table is Three-fourths of the slope of the Wholesale table.
The slope of the Wholesale table is Three-fourths of the slope of the Public table.
The slope of the Public table is 12 times greater than the slope of the Wholesale table.
The slope of the Wholesale table is 12 times greater than the slope of the Public table.
Public
A 2-column table with 3 rows. Column 1 is labeled x with entries 2, 5, 9. Column 2 is labeled y with entries 24, 60, 108.
Wholesale
A 2-column table with 3 rows. Column 1 is labeled x with entries 18, 35, 50. Column 2 is labeled y with entries 162, 315, 360.
How do the slopes of the lines created by each table compare?
The slope of the Public table is Three-fourths of the slope of the Wholesale table.
The slope of the Wholesale table is Three-fourths of the slope of the Public table.
The slope of the Public table is 12 times greater than the slope of the Wholesale table.
The slope of the Wholesale table is 12 times greater than the slope of the Public table.
Answers
GPT-4o mini
To find the slopes of the lines created by each table, we will calculate the slope for each table using the formula for the slope between two points, which is:
\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
### Public Table
The Public table has the following entries:
- For the points (2, 24) and (5, 60):
\[
\text{slope} = \frac{60 - 24}{5 - 2} = \frac{36}{3} = 12
\]
- For the points (5, 60) and (9, 108):
\[
\text{slope} = \frac{108 - 60}{9 - 5} = \frac{48}{4} = 12
\]
Thus, the slope for the Public table is **12**.
### Wholesale Table
The Wholesale table has the following entries:
- For the points (18, 162) and (35, 315):
\[
\text{slope} = \frac{315 - 162}{35 - 18} = \frac{153}{17} = 9
\]
- For the points (35, 315) and (50, 360):
\[
\text{slope} = \frac{360 - 315}{50 - 35} = \frac{45}{15} = 3
\]
Thus, the slope for the Wholesale table is **9**.
### Comparing the Slopes
Now we will compare the slopes of the Public and Wholesale tables:
- Slope of Public: 12
- Slope of Wholesale: 9
Next, we can express the relationship between the two slopes:
\[
\text{slope of Public} = 12 \quad \text{and} \quad \text{slope of Wholesale} = 9
\]
Now we check if one is a multiple of the other:
\[
\frac{\text{slope of Public}}{\text{slope of Wholesale}} = \frac{12}{9} = \frac{4}{3}
\]
This indicates that the slope of the Public table is \(\frac{4}{3}\) times the slope of the Wholesale table or, equivalently, that the slope of the Public table is **1.333 times greater than the slope of the Wholesale table**.
Based on the answers you provided:
- The slope of the Public table is **12 times greater** than the slope of the Wholesale table (this is not correct).
- The slope of the Wholesale table is **12 times greater** than the slope of the Public table (this is not correct).
- The slope of the Public table is **three-fourths** of the slope of the Wholesale table (this is not correct).
- The slope of the Wholesale table is **three-fourths** of the slope of the Public table.
Thus, the correct relationship is that the slope of the Public table is **\( \frac{4}{3} \) times** greater than the slope of the Wholesale table, which isn't explicitly listed in your answer choices, but it is accurate to say that the slope of the Wholesale table is three-fourths of the slope of the Public table.
Therefore, the correct choice is:
**The slope of the Wholesale table is Three-fourths of the slope of the Public table.**
\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
### Public Table
The Public table has the following entries:
- For the points (2, 24) and (5, 60):
\[
\text{slope} = \frac{60 - 24}{5 - 2} = \frac{36}{3} = 12
\]
- For the points (5, 60) and (9, 108):
\[
\text{slope} = \frac{108 - 60}{9 - 5} = \frac{48}{4} = 12
\]
Thus, the slope for the Public table is **12**.
### Wholesale Table
The Wholesale table has the following entries:
- For the points (18, 162) and (35, 315):
\[
\text{slope} = \frac{315 - 162}{35 - 18} = \frac{153}{17} = 9
\]
- For the points (35, 315) and (50, 360):
\[
\text{slope} = \frac{360 - 315}{50 - 35} = \frac{45}{15} = 3
\]
Thus, the slope for the Wholesale table is **9**.
### Comparing the Slopes
Now we will compare the slopes of the Public and Wholesale tables:
- Slope of Public: 12
- Slope of Wholesale: 9
Next, we can express the relationship between the two slopes:
\[
\text{slope of Public} = 12 \quad \text{and} \quad \text{slope of Wholesale} = 9
\]
Now we check if one is a multiple of the other:
\[
\frac{\text{slope of Public}}{\text{slope of Wholesale}} = \frac{12}{9} = \frac{4}{3}
\]
This indicates that the slope of the Public table is \(\frac{4}{3}\) times the slope of the Wholesale table or, equivalently, that the slope of the Public table is **1.333 times greater than the slope of the Wholesale table**.
Based on the answers you provided:
- The slope of the Public table is **12 times greater** than the slope of the Wholesale table (this is not correct).
- The slope of the Wholesale table is **12 times greater** than the slope of the Public table (this is not correct).
- The slope of the Public table is **three-fourths** of the slope of the Wholesale table (this is not correct).
- The slope of the Wholesale table is **three-fourths** of the slope of the Public table.
Thus, the correct relationship is that the slope of the Public table is **\( \frac{4}{3} \) times** greater than the slope of the Wholesale table, which isn't explicitly listed in your answer choices, but it is accurate to say that the slope of the Wholesale table is three-fourths of the slope of the Public table.
Therefore, the correct choice is:
**The slope of the Wholesale table is Three-fourths of the slope of the Public table.**