Mrs. Brown has a reward program in her classroom for students who have perfect or good weekly conduct. Each student that earns perfect conduct for a week gets three gold stars, and each student that earns good conduct for a week gets two gold stars. At the end of each six-weeks, Mrs. Brown provides breakfast for all students that have earned more than 12 gold stars.

Question

Which of the following graphs represents the number of perfect conduct weeks and good conduct weeks students need?

W.
X.

Y.
Z.

Answer 1

To determine which graph represents the number of perfect and good conduct weeks that students need to earn more than 12 gold stars, we need to establish the relationship between perfect conduct weeks (let’s denote this as \( P \)) and good conduct weeks (denote this as \( G \)).

Each week of perfect conduct earns a student 3 gold stars, and each week of good conduct earns 2 gold stars. Therefore, we can express the total number of gold stars earned by a student as:

\[ \text{Total Gold Stars} = 3P + 2G \]

To find the combinations of \( P \) and \( G \) that result in more than 12 gold stars, we need to set up the inequality:

\[ 3P + 2G > 12 \]

Next, we can rewrite this inequality to find the boundary line:

\[ 2G > 12 - 3P \] \[ G > 6 - \frac{3}{2}P \]

The graph of this line will feature a slope of \( -\frac{3}{2} \) and intercept \( 6 \) on the G-axis.

To find valid combinations, we can consider a few values for \( P \) to see what \( G \) would need to be in order for the total to exceed 12:

• When \( P = 0 \): \[ G > 6 \implies G \geq 7 \]
• When \( P = 1 \): \[ 3(1) + 2G > 12 \implies 2G > 9 \implies G > 4.5 \implies G \geq 5 \]
• When \( P = 2 \): \[ 3(2) + 2G > 12 \implies 2G > 6 \implies G > 3 \implies G \geq 4 \]
• When \( P = 3 \): \[ 3(3) + 2G > 12 \implies 2G > 3 \implies G > 1.5 \implies G \geq 2 \]
• When \( P = 4 \): \[ 3(4) + 2G > 12 \implies 2G > 0 \implies G \geq 0 \]

Now that we have established the relationship and relevant sample points, we need to look for the graph that matches this inequality. The area above the line \( G = 6 - \frac{3}{2}P \) should represent all combinations of \( P \) and \( G \) that yield more than 12 gold stars.

Now, without access to the actual graphs W, X, Y, and Z, I recommend looking for a graph that shows a line with a negative slope (downward as you move right) and includes the area above this line, indicating combinations of \( P \) and \( G \) that are valid (more than 12 gold stars).

If any specific graphs are given, please refer to the one that visually matches this description.