Question

high school a and high school b planned separate trips to new york city. the senior class at high school a rented and filled 16 vans and 5 buses with 417 students. high school b rented and filled 10 vans and 8 buses with 480 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?

Explanation:

Step1: Define variables

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

Step2: Set up equations

For School A: $16v + 5b = 417$
For School B: $10v + 8b = 480$

Step3: Eliminate one variable

Multiply first equation by 8: $128v + 40b = 3336$
Multiply second equation by 5: $50v + 40b = 2400$
Subtract the two equations:
$$(128v + 40b) - (50v + 40b) = 3336 - 2400$$
$$78v = 936$$

Step4: Solve for $v$

$$v = \frac{936}{78} = 12$$

Step5: Solve for $b$

Substitute $v=12$ into $10v + 8b = 480$:
$$10(12) + 8b = 480$$
$$120 + 8b = 480$$
$$8b = 480 - 120 = 360$$
$$b = \frac{360}{8} = 45$$

Answer:

A van can carry 12 students, and a bus can carry 45 students.