This seems like a really silly thing to be confused by!
Recently we have purchased a house (yay!).
Stamp duty was calculated as 1% of (price-VAT). Where price is 450,000 and VAT is 13.5%.
My step by step way to calculate this was
VAT cost = (450,000/100)*13.5 = 60,750
Cost - VAT cost= 450,000 - 60,750 = 389,250
Stamp at 1% of that is 3,892.50
However, what was actually requested was calculated as follows. I don't understand this method:
450,000 / 113.5 = 3,964.75
This is matched by, for example independent websites like https://switcher.ie/mortgages/guides/stamp-duty/
How can there be two seemingly valid answers to the same question? What am I missing about the calculation?
(Stamp duty is already paid at this point so there are no legal ramifications in this!)
Which one is right, depends on whether the $450,000$ already include VAT.
Does not include VAT
If it does not then the first equation is right. Because then VAT$=13.5\%\cdot 450,000$ as you calculated.
Does include VAT
But if VAT is already included, then you have to do something else to calculate it. Because then
$$ \text{VAT} = 13.5\% \cdot x $$ and $$ 450,000 = \text{total} = x+\text{VAT} = (100 + 13.5)\% \cdot x = 1.135x $$ So now the price without VAT is $x = 450,000/1.135$ and 1% of that means dividing by 100 again, so you end up with $$ 450,000/(1.135*100)=450,000/(113.5) $$ as is the formula you found.
Putting together our formula for $x$ and our formula in this case impies $$ \text{VAT} = 13.5\% \cdot (450,000/1.135) = (13.5/113.5)\cdot 450,000 $$ and since $13.5/113.5 \neq 13.5/100 = 13.5\%$, this is in general not equal to the first case.