The senior classes at George Washington High School and Benjamin Franklin High School planned separate trips to an indoor climbing gym. The senior class at George Washington High School rented and filled 5 vans and 18 buses with 1034 students. Benjamin Franklin High School rented and filled 14 vans and 9 buses with 701 students. Each van and each bus carried the same number of students. Find the number of students in each van and in each bus.
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To find the number of students in each van and each bus, we need to set up a system of equations using the given information.
Let's assume that each van has "V" students and each bus has "B" students.
From the information given for George Washington High School:
Number of vans: 5
Number of buses: 18
Total number of students: 1034
We can set up the equation:
5V + 18B = 1034 (equation 1)
From the information given for Benjamin Franklin High School:
Number of vans: 14
Number of buses: 9
Total number of students: 701
We can set up the equation:
14V + 9B = 701 (equation 2)
Now we have a system of two equations with two variables. We can solve this system to find the values of V (number of students in each van) and B (number of students in each bus).
We can solve this system using the substitution method or the elimination method. Let's use the elimination method:
Multiply equation 1 by 14 and equation 2 by 5 to eliminate variable V:
(5V + 18B) * 14 = 1034 * 14 (equation 1)
(14V + 9B) * 5 = 701 * 5 (equation 2)
Simplifying the equations:
70V + 252B = 14476 (equation 3)
70V + 45B = 3505 (equation 4)
Now, we subtract equation 4 from equation 3 to eliminate variable V:
(70V - 70V) + (252B - 45B) = 14476 - 3505
207B = 10971
Divide both sides of the equation by 207:
B = 10971 / 207
B ≈ 53
Now we have the value of B, which represents the number of students in each bus. We can substitute this value back into equation 4 to solve for V:
70V + 45(53) = 3505
70V + 2385 = 3505
70V = 3505 - 2385
70V = 1120
Divide both sides of the equation by 70:
V = 1120 / 70
V = 16
Therefore, there are approximately 16 students in each van and 53 students in each bus.