Let's organize our solution step-by-step.
### Step 1: Calculate the points per star
Erin earns 20 points for collecting 6 stars in the game. To find out how many points she earns per star, we divide the total points by the number of stars:
[tex]\[
\text{Points per star} = \frac{20 \text{ points}}{6 \text{ stars}} \approx 3.\!33 \text{ points per star}
\][/tex]
### Step 2: Calculate the number of stars needed for 10 points
Using the points per star, we can calculate how many stars Erin needs to collect to earn 10 points:
[tex]\[
\text{Stars for 10 points} = \frac{10 \text{ points}}{3.\!33 \text{ points per star}} \approx 3 \text{ stars}
\][/tex]
### Step 3: Verify the number of stars needed for 20 points
We know that Erin normally earns 20 points for 6 stars. For accuracy, we can verify this based on our points per star calculation:
[tex]\[
\text{Stars for 20 points} = \frac{20 \text{ points}}{3.\!33 \text{ points per star}} \approx 6 \text{ stars}
\][/tex]
This confirms the given data point that 20 points correspond to 6 stars.
### Step 4: Calculate the number of stars needed for 40 points
Similarly, we calculate the number of stars Erin needs to earn 40 points:
[tex]\[
\text{Stars for 40 points} = \frac{40 \text{ points}}{3.\!33 \text{ points per star}} \approx 12 \text{ stars}
\][/tex]
### Step 5: Calculate the points earned for 9 stars
Lastly, we have a special condition where Erin collects 9 stars:
[tex]\[
\text{Points for 9 stars} = 9 \text{ stars} \times 3.\!33 \text{ points per star} \approx 30 \text{ points}
\][/tex]
With these calculations, the results obtained are:
1. Points per star: [tex]\(\approx 3.\!33\)[/tex]
2. Stars for 10 points: [tex]\(\approx 3\)[/tex]
3. Stars for 20 points: [tex]\(\approx 6\)[/tex]
4. Stars for 40 points: [tex]\(\approx 12\)[/tex]
5. Points for 9 stars: [tex]\(\approx 30\)[/tex]
These results match the numerical answers provided earlier.
