Question: In Dirac's Principles of QM following is stated:

$$
\langle x | A + B | x \rangle = \langle x | A | x \rangle + \langle x | B |x \rangle
$$

but

$$
\langle x | AB | x \rangle \ne \langle x | A | x \rangle \langle x | B | x \rangle
$$

and so $\langle x|A|x \rangle$ is not exact but average value of observable A otherwise in second relation both sides would have to be equal.
I don't understand the second relation. Shouldn't both side be equal like this,
$\langle x|AB|x \rangle = \langle x|A(B|x \rangle) = b \langle x|A|x \rangle = ba \langle x|x \rangle = ba = \langle x|A|x \rangle \langle x|B|x \rangle$ , where $a$ and $b$ are corresponding eigenvalues. What is wrong here?

Edit: It is embarrassing. I indeed was thinking $|x\rangle$ was eigenvector of both $A$ and $B$ out of sleep deprivation I suppose which I only realised this morning. So anyway I am going to ask moderator to delete this question.

Answer: 
When submerged, the coin displaces as much water as it has volume (logical).

When floating on the box, the coin displaces as much water as corresponds to its weight.

As metal has a higher density than water, it means that the coin in the box displaces more water than when the coin is submerged.
I'm not an expert so please check if this makes sense.

It seems like the "reason" for this is that when it's on the box, it's displacing water "using" the box. Since the box is less dense, the same weight occupies more space, so the weight of the coin gets amplified by the ratio of its specific weight to the box's specific weight. For example, if the box's specific weight is 10 times lighter than the coin's, the coin will now be pushing on 10 times the volume as its own (- the coin's) volume into the water. (Ignoring the difference in water level. I'm just making the point that what is pushed is "box", not "coin".)