An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\frac23$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?
[asy] // Diagram by TheMathGuyd  pair A,B,C; A=(0,1); B=(sqrt(3)/2,-1/2); C=-conj(B); fill(2B--3B--3C--2C--cycle,grey); dot(3A); dot(3B); dot(3C); dot(2A); dot(2B); dot(2C); draw(2A--2B--2C--cycle); draw(3A--3B--3C--cycle); draw(2A--3A); draw(2B--3B); draw(2C--3C); [/asy]
$\textbf{(A) } 1 : 3 \qquad \textbf{(B) } 3 : 8 \qquad \textbf{(C) } 5 : 12 \qquad \textbf{(D) } 7 : 16 \qquad \textbf{(E) } 4 : 9$
All equilateral triangles are similar. For the outer equilateral triangle to the inner equilateral triangle, since their side-length ratio is $\frac32,$ their area ratio is $\left(\frac32\right)^2=\frac94.$ It follows that the area ratio of three trapezoids to the inner equilateral triangle is $\frac94-1=\frac54,$ so the area ratio of one trapezoid to the inner equilateral triangle is \[\frac54\cdot\frac13=\frac{5}{12}=\boxed{\textbf{(C) } 5 : 12}.\] 
~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat, MRENTHUSIASM