To determine the missing values that could fit the proportional relationship between the number of stars Roberto collects and his total points, we first need to establish the ratio of points to stars from the given data:
- For 8 stars and 12 points:
[tex]\[
\frac{12}{8} = 1.5
\][/tex]
- For 10 stars and 15 points:
[tex]\[
\frac{15}{10} = 1.5
\][/tex]
- For 14 stars and 21 points:
[tex]\[
\frac{21}{14} = 1.5
\][/tex]
We observe that the ratio (points to stars) is consistent across the given data; it is always 1.5. This means that for any number of stars [tex]\( s \)[/tex] in the game, the corresponding points [tex]\( p \)[/tex] will be:
[tex]\[
p = 1.5 \times s
\][/tex]
Next, we need to check each of the options provided to see if they adhere to this ratio. We ensure to use this consistent ratio of 1.5 to verify whether each option is valid.
1. 2 stars and 3 points:
[tex]\[
\frac{3}{2} = 1.5
\][/tex]
This fits the ratio. So, this option is valid.
2. 6 stars and 4 points:
[tex]\[
\frac{4}{6} = 0.6667
\][/tex]
This does not fit the ratio. So, this option is not valid.
3. 16 stars and 24 points:
[tex]\[
\frac{24}{16} = 1.5
\][/tex]
This fits the ratio. So, this option is valid.
4. 12 stars and 18 points:
[tex]\[
\frac{18}{12} = 1.5
\][/tex]
This fits the ratio. So, this option is valid.
5. 4 stars and 5 points:
[tex]\[
\frac{5}{4} = 1.25
\][/tex]
This does not fit the ratio. So, this option is not valid.
Therefore, the three answers that could be used as the missing values in the table are:
- A 2 stars and 3 points
- C 16 stars and 24 points
- D 12 stars and 18 points
