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 8 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.

1. the senior classes at high school a and high school b planned separate trips to new york city. the senior class at high school a rented and filled 1 van and 6 buses with 372 students. high school b rented and filled 4 vans and 12 buses with 780 students. each van and each bus carried the same number of students. how many students can a van carry? how many students can a bus carry?

1. brendas school is selling tickets to a spring musical. on the first day of ticket sales the school sold 3 senior citizen tickets and 9 child tickets for a total of $75. the school took in $67 on the second day by selling 8 senior citizen tickets and 5 child tickets. what is the price each of one senior citizen ticket and one child ticket?

1. matt and ming are selling fruit for a school fundraiser. customers can buy small boxes of oranges and large boxes of oranges. matt sold 3 small boxes of oranges and 14 large boxes of oranges for a total of $203. ming sold 11 small boxes of oranges and 11 large boxes of oranges for a total of $220. find the cost each of one small box of oranges and one large box of oranges.

1. a boat traveled 336 miles downstream and back. the trip downstream took 12 hours. the trip back took 14 hours. what is the speed of the boat in still water? what is the speed of the current?

Explanation:

Problem 1

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 by 8: $v + b = 30$ → $b = 30 - v$

Step4: Substitute into second equation

$4v + (30 - v) = 54$
$3v + 30 = 54$
$3v = 24$ → $v = 8$

Step5: Solve for b

$b = 30 - 8 = 22$

Step1: Define variables

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

Step2: Set up equations

High School A: $v + 6b = 372$
High School B: $4v + 12b = 780$

Step3: Simplify second equation

Divide by 4: $v + 3b = 195$

Step4: Subtract equations

$(v + 6b) - (v + 3b) = 372 - 195$
$3b = 177$ → $b = 59$

Step5: Solve for v

$v = 372 - 6(59) = 372 - 354 = 18$

Step1: Define variables

Let $s$ = senior ticket price, $c$ = child ticket price.

Step2: Set up equations

Day 1: $3s + 9c = 75$
Day 2: $8s + 5c = 67$

Step3: Simplify first equation

Divide by 3: $s + 3c = 25$ → $s = 25 - 3c$

Step4: Substitute into second equation

$8(25 - 3c) + 5c = 67$
$200 - 24c + 5c = 67$
$-19c = -133$ → $c = 7$

Step5: Solve for s

$s = 25 - 3(7) = 25 - 21 = 4$

Answer:

8 students per van, 22 students per bus

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Problem 2