Expert: Paul Klarreich - 7/8/2007

Question
I was wondering how I can calculate how opaque two sheets should be to form a single opaque sheet with the same opacity.

Example:
I have one sheet which is 50 opaque. How opaque should two combined sheets be in order to make it look like its only one 50 opaque sheet.

This is not for school or anything. Was just working on something and the question came up and I could not find the answer.
If you cant answer this question, could you tell me what category of math this is, because I wouldn't have a clue.

Answer
Questioner:   Dennis
Category:  Advanced Math
 
Subject:  Opacity
Question:  I was wondering how I can calculate how opaque two sheets should be to form a single opaque sheet with the same opacity.

>> I think you mean 'with a GIVEN opacity.'

Example:
I have one sheet which is 50 opaque. How opaque should two combined sheets be in order to make it look as if it's only one 50 opaque sheet.

This is not for school or anything. Was just working on something and the question came up and I could not find the answer.
If you can't answer this question, could you tell me what category of math this is, because I wouldn't have a clue.
..............................
Hi, Dennis,

You are not defining your terms very well, so I'll try.  I assume you mean that '50 opaque' means 'blocks 50% of the light'.  In general, I'll assume that 'x opaque' means 'blocks x% of the light.'

(Therefore transmits  (100-x)% of the light.)

...... example .....
For example, suppose you have  '20 opaque' paper.  Then it blocks 20% of the light and transmits 80%, or 4/5 of the light.

Now put two sheets together.  The first passes 4/5 of the light to the second sheet, which passes 4/5 OF THAT 4/5, or (4/5)^2 = 16/25 of the light.  

That is 64% passed, or 36% blocked, so it becomes the equivalent of '36 opaque.'  
..............................

So suppose you have paper such that two sheets of it block 50% of the light, or 1/2 of it.  

Let x be the fraction of light transmitted by each sheet.
Then x^2 is the fraction transmitted by two 'stacked' sheets.

But you want 50% or 1/2 to be transmitted, so:

x^2 = 1/2

x = 1/sqrt(2) ~~ 1/1.4142 = 0.7071, or about 70.7% transmittance.

Then the 'opacity' is 29.3%.  That's your answer -- 29.3 opaque.