Questions: (11) Yellowstone National Park is a popular field trip destination. This year the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 8 vans and 5 buses with 420 students. High School B rented and filled 10 vans and 10 buses with 750 students. Every van had the same number of students in it as did the buses. How many students can a van carry? How many students can a bus carry?

Solution Steps

To find the number of students each van and bus can carry, we set up a system of linear equations based on the information given:

1. High School A rented 8 vans and 5 buses for a total of 420 students: \[ 8v + 5b = 420 \]

2. High School B rented 10 vans and 10 buses for a total of 750 students: \[ 10v + 10b = 750 \]

We can simplify the second equation by dividing all terms by 10: \[ v + b = 75 \]

We now have the following system of equations:

1. \(8v + 5b = 420\)
2. \(v + b = 75\)

To solve for \(v\) and \(b\), we can express \(b\) from the second equation: \[ b = 75 - v \]

Substitute \(b = 75 - v\) into the first equation: \[ 8v + 5(75 - v) = 420 \]

Simplify and solve for \(v\): \[ 8v + 375 - 5v = 420 \] \[ 3v = 45 \] \[ v = 15 \]

Substitute \(v = 15\) back into the equation \(b = 75 - v\): \[ b = 75 - 15 = 60 \]

Final Answer