Hi Jim,
This would be straightforward to answer if there was a universal meaning of percentage markup. The confusion comes from the sloppy way we all use the word percentage. When we use percentage we should say percentage of what. So let's look at your question.
First interpretation:
I am the merchant and my cost for the part was $C. I want to mark it up 13.5%, that is I will mark it up 13.5% of my cost. Thus the markup is
13.5/100 $C = 0.135 $C
and hence your price $P is
$P = $C + 0.135 $C = (1.135) $C
You said your price was $810.00 so
$810.00 = (1.135) $C
and hence
$C = $810.00/1.135 = $713.66
Thus with this meaning of markup, if the merchant's cost is $C, the markup is m% and the merchant's price to you is $P then
$C = $P/(1 + m/100)
Second interpretation:
I am the customer and the merchant is charging me $P for the part. I know his markup is 13.5% and my interpretation is that 13.5% of what it costs me is the markup. That is 13.5% of $P is the markup so
the merchant's cost $C is (100%  13.5%) = 86.5% of $P so $C = 86.5% $P = 0.865 $P
Since you told me that $P = $810.0
$C = 0.865 $810.00 = $700.65.
Thus with this meaning of markup, if the merchant's cost is $C, the markup is m% and the merchant's price to you is $P then
$C = (1  m/100) $P
Hence determining the merchant's cost involves knowing how the markup was calculated.
Penny
