8th grade teachers and students from M. I.T went on a field trip to an zoo. Tickets were $5.00 each for teachers and$3.50 each for students, and the group paid
$51.50 in total. A few weeks later, the same group visited a natural history
museum where the tickets cost $20.00 each for teachers and $8.50 each for
students, and the group paid $156.50 in total.
Find the number of teachers and students on the field trips.
A 4 Teachers and 9 Students
4 Teachers and 10 Students
3 Teachers and 9 Students
D None of the above

As there are multiple answers going with trial and error seems good in this case.

A

5.00 + 5.00 + 5.00 + 5.00 = 20.00

3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 = 31.5

20.00 + 31.5 = 51.5

B

5.00 + 5.00 + 5.00 + 5.00 = 20.00

3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 = 35.00

20.00 + 35.00 = 55.00

C

5.00 + 5.00 + 5.00 = 15.00

3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 + 3.50 = 31.5

15.00 + 31.5 =46.5

Knowing that for the first one all A,B, and C were wrong we can determine that D is the answer. But wait if I made one mistake and didn't understand if $156.5 is equivalent to $156.50 that would mean A is correct.

Always recheck your answers and good luck,

Jon

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4