Claire M. answered • 06/06/19

Experienced Science and Math Tutor

Hi Carter! Looks like we're dealing with a system of linear equations here. We can work this out a few different ways.

When you say "same number of students in both vans and busses," we need to assume the problem means that all vans in both the first and second situation have the same number of students, and all busses in both the first and second situation have the same number of students. This problem doesn't make sense if we assume that vans and busses have the same number of students as each other! That would mean that, in situation one, 25 vehicles carry 844 students for 33.76 students per vehicle, and in situation two, 12 vehicles carry 555 students for 46.25 students per vehicle. Those numbers just don't match.

So let's look at these as equations! Let v=the number of students in one van, and b=the number of students in one bus.

14v + 11b = 844

3v + 9b = 555

We have two variables, and two separate equations, which means we can solve this system. There are a few different ways to do it, but in this case I'll demonstrate substitution.

First, we solve for one variable in one equation:

3v + 9b = 555

3v = 555 - 9b

v = 183 - 3b

We can use this new value for 'v' in our other equation:

14v + 11b - 844

14(183 - 3b) + 11b = 844

2562 - 42b + 11b = 844

2562 - 31b = 844

2562 - 844 = 31b

1718 = 31b

55.42 = b

Now that we have a value for 'b', we can put it back in our equation for v:

v = 183 - 3b

v = 183 - 3(55.42)

v = 183 - 166.26

v = 16.74

That means that there are 55.42 students per bus and 16.74 students per van. Obviously, we can't have a fraction of a student in a vehicle, so the reality of the word problem is a little different from the math, but we can say that *on average* 55.42 students ride in each bus, and *on average* 16.74 students ride in each van. You can use those numbers, and the equations we set up above, to calculate how many total students ride in buses and how many total students ride in vans on each trip.

Hope that helped!