2016 AMC 10A Problem 7

The sum of the seven data values is $540+x$. If the mean equals $x$, then $$\frac{540+x}{7}=x,$$ so $540=6x$ and $x=90$. With $x=90$, the ordered data set is $40,50,60,90,90,100,200$. Its median is $90$, and its mode is also $90$, so the value works. Thus, the correct answer is D.