Question

the state fair 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 6 buses with 240 students. high school b rented and filled 4 vans and 1 bus with 54 students. every van had the same number of students in it as did the buses. find the number of students in each van and in each bus.

Explanation:

Step1: Define variables

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

Step2: Set up equations

High School A: $8v + 8b = 240$
High School B: $4v + b = 54$

Step3: Simplify first equation

Divide $8v + 8b = 240$ by 8:
$v + b = 30$ → $v = 30 - b$

Step4: Substitute into second equation

Substitute $v = 30 - b$ into $4v + b = 54$:
$4(30 - b) + b = 54$
$120 - 4b + b = 54$
$120 - 3b = 54$

Step5: Solve for $b$

$-3b = 54 - 120$
$-3b = -66$
$b = \frac{-66}{-3} = 22$

Step6: Solve for $v$

Substitute $b=22$ into $v=30-b$:
$v = 30 - 22 = 8$

Answer:

Number of students in each van: 8
Number of students in each bus: 22