If 12 cars go, then 2 vans are needed. The pair \( \mathrm{c}=14 \) and \( \mathrm{v}=4 \) is a solution to the equation. If 6 cars go and 11 vans go, there will be extra space. 10 cars and 8 vans isn't enough to transport all the students. If 20 cars go, no vans are needed. 8 vans and 8 cars are numbers that meet the constraints in this situation.

It seems like you're describing a problem involving the transportation of students using cars and vans, with certain constraints and relationships between the number of cars and vans needed. Let's break down the information you've provided: 1. **Car and Van Relationship**: If 12 cars are used, then 2 vans are needed. This suggests a ratio of cars to vans. 2. **Given Pair**: The pair \( c = 14 \) (cars) and \( v = 4 \) (vans) is a solution to the equation, which likely means that this combination meets the transportation needs. 3. **Extra Space**: If 6 cars and 11 vans are used, there will be extra space, indicating that this combination exceeds the required capacity. 4. **Insufficient Capacity**: Using 10 cars and 8 vans is not enough to transport all the students, meaning this combination does not meet the required capacity. 5. **No Vans Needed**: If 20 cars are used, no vans are needed, suggesting that 20 cars can accommodate all students on their own. 6. **Another Valid Pair**: The combination of 8 vans and 8 cars also meets the constraints. ### Analyzing the Information From the first point, we can derive a ratio of cars to vans. If 12 cars require 2 vans, then: \[ \text{Ratio of cars to vans} = \frac{12}{2} = 6 \text{ cars per van} \] This means for every 6 cars, 1 van is needed. ### Setting Up Equations Let \( c \) be the number of cars and \( v \) be the number of vans. Based on the ratio, we can express the number of vans needed in terms of cars: \[ v = \frac{c}{6} \] ### Capacity Constraints 1. **From the pair \( (c, v) = (14, 4) \)**: - This satisfies the equation since \( v = \frac{14}{6} \approx 2.33 \), but since we need whole numbers, we can assume that this is a valid combination based on the overall capacity. 2. **From the pair \( (c, v) = (20, 0) \)**: - This indicates that 20 cars can transport all students without needing any vans. 3. **From the pair \( (c, v) = (10, 8) \)**: - This combination does not meet the requirement, indicating that the total capacity is insufficient. 4. **From the pair \( (c, v) = (6, 11) \)**: - This combination provides extra space, meaning it exceeds the required capacity. ### Conclusion To summarize, the relationships and constraints suggest that the transportation needs can be met with various combinations of cars and vans, but certain combinations are insufficient or excessive. The key takeaway is the ratio of cars to vans and the specific combinations that meet the transportation requirements. If you have a specific question or need further analysis on this problem, please let me know!