Question 5681
The first step would have to be set up a proportion. You know that 100% refers to the full 10.5 x 8.2. The problem is finding an expression for 80%.


So far we have the proportion {{{ 1/((10.5)(8.2)) = 0.80/E }}}. We need to work on finding the expression for E - the expression for the new dimensions when equal width strips are taken off all four sides.


Let w = the width of the strip to be taken off all four sides so that the resulting (reduced) size is 80% of the original background.

The full length is 10.5, right? And we're chopping off a strip from BOTH sides of it, so the reduced length is 10.5 - 2w. The full width is 8.2, so chopping off from both sides reduced the width to 8.2 - 2w. The area of the 80% would have to be (10.5 - 2w)(8.2 - 2w). It's just that we have to find the right w value so that the resulting area of the remaining background be 80% of the old background. Here's the full proportion:

{{{ 1/((10.5)(8.2)) = 0.80/((10.5 - 2w)(8.2 - 2w)) }}} 


{{{ 0.80*10.5*8.2 = (10.5 - 2w)(8.2 - 2w) }}} <---- Cross multiplied


{{{ 68.88 = 86.10 - 37.4x + 4x^2 }}} <---- simplified and distributed.


{{{ 4x^2 - 37.4x + 17.22 = 0 }}} <---- rearranged so that the right hand side is 0 so that we may solve for x.


Since the coefficients are not nice-looking numbers, we'll have to pull out our handy quadratic formula:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 


Let's evaluate first using the "positive" by plugging in the 4 for the a, the -37.4 for the b, and the 17.22 for the c:


{{{x = (-(-37.4) + sqrt( (-37.4)^2-4*4*17.22 ))/(2*4) }}} <---- once evaluated, this gives us x = 8.86 approximately.


Let's then compute the "negative":


{{{x = (-(-37.4) - sqrt( (-37.4)^2-4*4*17.22 ))/(2*4) }}} <----- once evaluated, this gives us x = 0.49 approximately.


Now, this proportion gave us two answers. Which one is right? I would say that the 0.49 cm would make more sense. It's impossible to "chop off" 8.87 centimeters from all sides of a background of only 10.5 x 8.2 cm^2.