What is the area, in square units, of a trapezoid bounded by the lines $y = x$, $y = 10$, $y = 5$ and the $y$-axis? Express your answer as a decimal to the nearest tenth.
The vertices of the trapezoid are $(5,5)$, $(10,10)$, $(0,10)$, and $(0,5)$.  Its bases are $5$ and $10$ units long, and its height is $5$ units.  Averaging the bases and multiplying by the height, we find an area of $\left(\frac{5+10}{2}\right)(5)=\boxed{37.5}$ square units.

[asy]
unitsize(2mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));
dotfactor=4;

fill((5,5)--(10,10)--(0,10)--(0,5)--cycle,gray);

draw((-12,-12)--(14,14),Arrows(4));
draw((-14,10)--(14,10),Arrows(4));
draw((-14,5)--(14,5),Arrows(4));

draw((-15,0)--(15,0),Arrows(4));
draw((0,-15)--(0,15),Arrows(4));

label("$y=x$",(14,14),NE);
label("$y=10$",(14,10),E);
label("$y=5$",(14,5),E);[/asy]