The two tables below show the amount of tip, [tex]\( y \)[/tex], included on a bill charging [tex]\( x \)[/tex] dollars.

Restaurant A
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
10 & 1 \\
\hline
20 & 2 \\
\hline
30 & 3 \\
\hline
\end{array}
\][/tex]

Restaurant B
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
25 & 5 \\
\hline
50 & 10 \\
\hline
75 & 15 \\
\hline
\end{array}
\][/tex]

Which compares the slopes of the lines created by the tables?

A. The slope of the line for Restaurant B is [tex]\(\frac{1}{5}\)[/tex] times greater than the slope of the line for Restaurant A.
B. The slope of the line for Restaurant B is 2 times greater than the slope of the line for Restaurant A.
C. The slope of the line for Restaurant B is 5 times greater than the slope of the line for Restaurant A.
D. The slope of the line for Restaurant B is 10 times greater than the slope of the line for Restaurant A.



Answer :

Let's delve into this problem step-by-step to determine the slopes of the lines for Restaurant A and Restaurant B, and then compare them.

### Calculate the Slope for Restaurant A

For Restaurant A, we have the points [tex]\((10, 1)\)[/tex], [tex]\((20, 2)\)[/tex], and [tex]\((30, 3)\)[/tex].

To find the slope of the line (denoted as [tex]\(m_a\)[/tex]) passing through these points, we can use the formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Using the first two points [tex]\((10, 1)\)[/tex] and [tex]\((20, 2)\)[/tex]:
[tex]\[ m_a = \frac{2 - 1}{20 - 10} = \frac{1}{10} = 0.1 \][/tex]

### Calculate the Slope for Restaurant B

For Restaurant B, we have the points [tex]\((25, 5)\)[/tex], [tex]\((50, 10)\)[/tex], and [tex]\((75, 15)\)[/tex].

Using the first two points [tex]\((25, 5)\)[/tex] and [tex]\((50, 10)\)[/tex]:
[tex]\[ m_b = \frac{10 - 5}{50 - 25} = \frac{5}{25} = \frac{1}{5} = 0.2 \][/tex]

### Compare the Slopes

We now have the slopes:
- Slope for Restaurant A: [tex]\(m_a = 0.1\)[/tex]
- Slope for Restaurant B: [tex]\(m_b = 0.2\)[/tex]

To find out how many times greater the slope of Restaurant B ([tex]\(m_b\)[/tex]) is compared to the slope of Restaurant A ([tex]\(m_a\)[/tex]), we can divide [tex]\(m_b\)[/tex] by [tex]\(m_a\)[/tex]:
[tex]\[ \text{Comparison} = \frac{m_b}{m_a} = \frac{0.2}{0.1} = 2.0 \][/tex]

### Conclusion

The slope of the line for Restaurant B is 2 times greater than the slope of the line for Restaurant A.

Hence, the correct comparison is: The slope of the line for Restaurant B is 2 times greater than the slope of the line for Restaurant A.