Final answer:
To transport all 55 members with 10 chaperones in one trip, it is not possible with the given number of vehicles.
Step-by-step explanation:
To determine the number of each type of vehicle needed for transporting all 55 members in one trip, we need to first calculate the total number of people that can be seated in both the vans and sedans.
The chaperones will be driving 10 vehicles in total. Let's assume the number of vans needed is 'v' and the number of sedans needed is 's'. Since each van can seat 8 people and each sedan can seat 5 people, the total number of people that can be seated in the vans is 8v and the total number of people that can be seated in the sedans is 5s.
We can set up a system of equations to represent the given information:
8v + 5s = 55 (for the total number of people)
v + s = 10 (for the total number of chaperones)
By solving this system of equations, we can find the values of v and s that satisfy both equations. Once we have the values, we'll know how many of each type of vehicle are needed.
Using a method of substitution or elimination, we can solve for v and s. For simplicity, let's solve using elimination:
From the second equation, we can rewrite it as v = 10 - s.
Substituting this value of v into the first equation, we get 8(10 - s) + 5s = 55.
Simplifying and solving for s, we get 80 - 8s + 5s = 55. Combining like terms, we have -3s + 80 = 55. Solving for s, we find that s = 25.
Substituting this value of s back into v = 10 - s, we find v = 10 - 25 = -15. Since we can't have a negative number of vehicles, this solution is not valid.
Therefore, there must be a mistake in the given information or the problem itself, as it is not possible to transport all 55 members in one trip with the given number of vehicles.