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 conduct weeks and good conduct weeks students need to earn more than 12 gold stars, we first need to establish the relationship between weeks of perfect conduct and good conduct and the total number of gold stars earned.

Let:

• \( p \) be the number of weeks with perfect conduct.
• \( g \) be the number of weeks with good conduct.

The total number of gold stars earned can be represented by the equation:

\[ \text{Total Gold Stars} = 3p + 2g \]

We want this total to be greater than 12 gold stars:

\[ 3p + 2g > 12 \]

Now, we can analyze this inequality. To make it easier to understand, we can rearrange it. For example, if we want to express \( g \) in terms of \( p \):

\[ 2g > 12 - 3p \] \[ g > \frac{12 - 3p}{2} \]

This means for a given number of perfect conduct weeks \( p \), the number of good conduct weeks \( g \) must be greater than \( \frac{12 - 3p}{2} \).

Now we can plot this line for various values of \( p \) to find the area where the inequality holds true:

1. If \( p = 0 \): \[ g > \frac{12 - 0}{2} = 6 \]

2. If \( p = 1 \): \[ g > \frac{12 - 3(1)}{2} = \frac{9}{2} = 4.5 \quad (\text{so } g \geq 5) \]

3. If \( p = 2 \): \[ g > \frac{12 - 3(2)}{2} = \frac{6}{2} = 3 \quad (\text{so } g \geq 4) \]

4. If \( p = 3 \): \[ g > \frac{12 - 3(3)}{2} = \frac{3}{2} = 1.5 \quad (\text{so } g \geq 2) \]

5. If \( p = 4 \): \[ g > \frac{12 - 3(4)}{2} = 0 \quad (\text{so } g \geq 0) \]

6. If \( p = 5 \): \[ g > \frac{12 - 3(5)}{2} = -\frac{3}{2} \quad (\text{no restriction on } g) \]

Now, using this information to find the correct graph involves looking for a region where the area above the lines flows to the right, indicating higher values of \( g \) for increasing values of \( p \).

From these points, we infer that as \( p \) increases, the number of required good conduct weeks decreases. Look for a graph that also illustrates the area above the line corresponding to the equation \( 3p + 2g = 12 \).

If you have graphs W, X, Y, and Z available, identify the one that reflects this understanding of the relationship between \( p \) and \( g \) according to the calculations above. The correct graph should show a downward sloping line, with an area above it indicating the combinations of \( p \) and \( g \) that satisfy the condition \( 3p + 2g > 12 \).