Question

the table shows data from a concession stand, representing weekly sales (x) in dollars and profit (y) in dollars.
when using the median - fit method with summary points (1,485, 905), (1,950, 1,175), and (2,535, 1,535), what is the approximate slope of the best - fit model?
options: (\frac{18}{31}), (\frac{3}{5}), (\frac{5}{3}), (\frac{31}{18})

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). We can use two of the summary points to calculate the slope. Let's use \((1485, 905)\) and \((1950, 1175)\) first, or \((1950, 1175)\) and \((2535, 1535)\). Let's take \((1485, 905)\) and \((1950, 1175)\) to start.

Step2: Calculate the difference in y and x

For points \((x_1,y_1)=(1485, 905)\) and \((x_2,y_2)=(1950, 1175)\):
\( \Delta y = 1175 - 905 = 270 \)
\( \Delta x = 1950 - 1485 = 465 \)

Step3: Simplify the slope fraction

Simplify \( \frac{270}{465} \). Divide numerator and denominator by 15: \( \frac{270\div15}{465\div15}=\frac{18}{31} \). Wait, but let's check with another pair, say \((1950, 1175)\) and \((2535, 1535)\):
\( \Delta y = 1535 - 1175 = 360 \)
\( \Delta x = 2535 - 1950 = 585 \)
Simplify \( \frac{360}{585} \), divide by 15: \( \frac{24}{39}=\frac{8}{13} \)? Wait, no, maybe I made a mistake. Wait, the summary points are (1,485, 905), (1,950, 1,175), and (2,535, 1,535). Wait, maybe the first point is (1485, 905), second (1950, 1175), third (2535, 1535). Let's use (1485, 905) and (2535, 1535):
\( \Delta y = 1535 - 905 = 630 \)
\( \Delta x = 2535 - 1485 = 1050 \)
Simplify \( \frac{630}{1050}=\frac{3}{5} \)? Wait, no, 630÷210=3, 1050÷210=5. Wait, maybe I misread the points. Wait the problem says summary points (1,485, 905), (1,950, 1,175), and (2,535, 1,535). So x values: 1485, 1950, 2535; y values: 905, 1175, 1535. Let's calculate slope between (1485,905) and (2535,1535):
\( m=\frac{1535 - 905}{2535 - 1485}=\frac{630}{1050}=\frac{3}{5} \) (dividing numerator and denominator by 210: 630÷210=3, 1050÷210=5). Wait, but earlier with (1485,905) and (1950,1175): \( \frac{1175 - 905}{1950 - 1485}=\frac{270}{465}=\frac{18}{31}\approx0.58 \), and \( \frac{3}{5}=0.6 \), \( \frac{18}{31}\approx0.58 \), but maybe the intended points. Wait, let's check the options: \( \frac{18}{31} \), \( \frac{3}{5} \), \( \frac{5}{3} \), \( \frac{31}{18} \). Wait, maybe I used the wrong points. Wait the summary points are (1,485, 905), (1,950, 1,175), (2,535, 1,535). Wait, 1,485 is 1485, 1,950 is 1950, 2,535 is 2535. Let's take (1485,905) and (1950,1175):
\( y_2 - y_1 = 1175 - 905 = 270 \)
\( x_2 - x_1 = 1950 - 1485 = 465 \)
Simplify \( \frac{270}{465} \): divide numerator and denominator by 15: 270÷15=18, 465÷15=31. So \( \frac{18}{31} \). Wait, but let's check with (1950,1175) and (2535,1535):
\( y_2 - y_1 = 1535 - 1175 = 360 \)
\( x_2 - x_1 = 2535 - 1950 = 585 \)
Simplify \( \frac{360}{585} \): divide by 15: 24/39, divide by 3: 8/13≈0.615. No, that's not matching. Wait, maybe the first two points are (1485,905) and (1950,1175), giving 270/465=18/31≈0.58, which is one of the options. So the slope is \( \frac{18}{31} \)? Wait, no, wait the problem says "approximate slope". Wait, maybe I made a mistake in the points. Wait the summary points are (1,485, 905), (1,950, 1,175), (2,535, 1,535). Let's recalculate the slope between (1485,905) and (1950,1175):
\( m = \frac{1175 - 905}{1950 - 1485} = \frac{270}{465} = \frac{18}{31} \) (dividing numerator and denominator by 15: 270 ÷15=18, 465÷15=31). So that's one of the options.

Answer:

\(\frac{18}{31}\)