Will watering tea down make it clearer? If I poured water into my tea, would I see more or less of the bottom of the tea-cup?
Intuitively, there would be as many particles blocking as many photons, and so I'd see the bottom just as clearly as before.
 A: There is some misconception in Tims question already: 

as many particles blocking as many photons, and so I'd see the bottom just as clearly as before

Blocking of photons will not impair clarity at all!  
One has to distinguish absorption and scattering! Tea is a solution of 
some dyestuffs in water (optically) this dyestuff absorbs light of 
certain wavelengths (making the tea looking yellow/brown) but the tea 
is not made turbid ("Unclear") by that absorption. (Think of some color filter 
for cameras!)
The absorption is governed by Lambert/Beer law. 
The law shows, that doubling the length of the liquid column is 
"neutralized" by reducing concentration to half of original value. 
Now the question of turbidity, In practice a tea may contain 
some particles from lime in the water, often some agglomerates from 
lime and some tanning agents in the tea. On top there is some 
fat/oil which drift as droplets or collect on the surface. 
Any particle floating in the tea, which is about 50 nm up to 
some µmeters and having a index of refraction different enough
from the index of water, will cause some scattering. 
This scattering will blur the picture. 
Most of the scattering in such suspensions like tea or milk 
are governed by Mie scattering. Scattering/concentrations/length 
is not as simple as Lambert-Beer, but at low concentrations 
one can assume an analog dependence. 
Some practical note: when having a good (low lime) water and 
and the leaves are in a bag with good filter action, and You 
wait for a moment until the oil accumulates at the surface, 
tea will be a nearly perfectly clear liquid. 
Georg 
A: You are right -- there is an approximated formula for this, called Lamber-Beer law:
$A=\epsilon l c$,
where $A$ is absorbance (log of quotient of light intensities before and after the solution), $\epsilon$ is substance dependent coefficient (constant), $l$ is the size of layer of solution and $c$ is the concentration. Let's now say that the glass cross-section is $S$, and initial height of fluid inside is $h$; adding water up to height $\alpha h$ will decrease concentration $c$ $\alpha$ times, but increase $l$ $\alpha$ times, so absorbance will stay the same.
A: Index of refraction is related to density, as your intuition predicted. The question is how would tea and water look like at a molecular level. Without a microscope I can only guess, of course, but the tea as being big solid chunks floating around in the water might explain.
