A trapezoid with area 417/16 has three of its sides on the x-axis, the line x=3, and the line x=6. The fourth side is contained in the line y=mx+5/2. The value of m can be written as a/b where a and b are coprime positive integers. Find a+b.

the height at x = 3 is 3m+5/2

the height at x = 6 is 6m+5/2

area = (1/2)(sum of two parallel sides)(distance between them)
(1/2)(3m+5/2 + 6m+5/2)(3) = 417/16
(3/2)(9m + 5) = 417/16
times 16
24(9m+5) = 417
216m + 120 = 417
216m = 297
m = 297/216 = 11/8

comparing with a/b, a=11 and b=8
then a+b = 19