Question

1. 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 7 vans and 10 buses with 332 students. high school b rented and filled 4 vans and 15 buses with 459 students. each van and each bus carried the same number of students. find the number of students in each van and in each bus.

Explanation:

Step1: Define variables for van/bus capacity

Let $v$ = students per van, $b$ = students per bus.

Step2: Set up equations from given data

For High School A: $7v + 10b = 332$
For High School B: $4v + 15b = 459$

Step3: Eliminate one variable (multiply equations)

Multiply first equation by 3: $21v + 30b = 996$
Multiply second equation by 2: $8v + 30b = 918$

Step4: Subtract equations to solve for $v$

Subtract second new equation from first:
$$(21v + 30b) - (8v + 30b) = 996 - 918$$
$$13v = 78$$
$$v = \frac{78}{13} = 6$$

Step5: Substitute $v$ to find $b$

Plug $v=6$ into $7v + 10b = 332$:
$$7(6) + 10b = 332$$
$$42 + 10b = 332$$
$$10b = 332 - 42 = 290$$
$$b = \frac{290}{10} = 29$$

Answer:

Each van carries 6 students, and each bus carries 29 students.