To determine which teacher-to-student ratios maintain the proportional relationship described in the table, let's first understand the original relationship given in the table.
From the provided table:
- For grade 6: 1 teacher corresponds to 25 students.
- For grade 7: 2 teachers correspond to 50 students.
- For grade 8: 3 teachers correspond to 75 students.
We can see that the number of students per teacher [tex]\( \text{ratio} \)[/tex] is consistent across all grades:
[tex]\[
\text{Ratio} = \frac{\text{Students}}{\text{Teachers}} = 25 \text{ students per teacher}
\][/tex]
Now, let's evaluate each given set of teachers and students to see if it maintains this same ratio.
1. For 5 teachers and 125 students:
[tex]\[
\text{Ratio} = \frac{125}{5} = 25 \text{ students per teacher}
\][/tex]
This matches the original ratio.
2. For 4 teachers and 120 students:
[tex]\[
\text{Ratio} = \frac{120}{4} = 30 \text{ students per teacher}
\][/tex]
This does not match the original ratio.
3. For 4 teachers and 100 students:
[tex]\[
\text{Ratio} = \frac{100}{4} = 25 \text{ students per teacher}
\][/tex]
This matches the original ratio.
4. For 6 teachers and 130 students:
[tex]\[
\text{Ratio} = \frac{130}{6} \approx 21.67 \text{ students per teacher}
\][/tex]
This does not match the original ratio.
5. For 7 teachers and 125 students:
[tex]\[
\text{Ratio} = \frac{125}{7} \approx 17.86 \text{ students per teacher}
\][/tex]
This does not match the original ratio.
Based on these calculations, the sets that maintain the proportional relationship of 25 students per teacher are:
- 5 teachers and 125 students
- 4 teachers and 100 students
These are the correct answers.
