I don't understand how you calculate it that way and why, but it seems wrong to me.
if I use your type in denarii for example with 20% bonus, i should have:
10*180/ (100-20) =22.5 minutes but i know for a fact that i gain 10 minutes per hour, so 30+ minutes per activation.
I think you got it wrong.
Here is how i calculate time in order to precise:
Basic idea is that time to sign agreement is the result of a fixed value / speed=> time=value/speed
Bonuses increase speed, thus time is decreased.
So, in your example, I would calculate it like this:
60 minutes + 10% * 60 minutes=66 minutes/ hour. This is your speed. So after 3 hours you gain 18 minutes not 20. And if you activate it 3 times, as it was a permanent boost you would get 18*3=54 minutes and your agreement would need 8 hours and 11 minutes. (time=9 hours * 60/66)
It is less than 54 because agreement ends before 9 hours.
This way, all bonuses increase the speed in fixed value :10%=6 minutes 20%=12 minutes per hour etc. and the actual impact is fixed per activation, no matter how much time you need for ending agreement.
The rest of the post explains how Amphitheater bonuses affect estimated time in different occasions.
Here things are fairly straightforward
example : denarii 1 hour with 20% boost=>speed=60*1.2=72
time=60/ 72= 50 minutes
example 2: skill 1 day with 50% boost=>speed=60*1.5=90
time = 254*60/90 so the time in activation will be 16 hours
in one activation we will sign 4.5 hours instead of 3. The distance we covered is 24-4.5=19.5
Here is the part that most guys get confused about.
1 sec before bonus ends time will be calculated with speed of 90 and will say 19.5*60/90=13 hours but when bonus ends time will be 19.5*60/60=>19.5 hours. We expected to gain 1 hour and 30 minutes and we did :)
This is why most guys think that game is cheating them.
Although the system works the same, we find here a big difference, the time we see when we try to sign an agreement is already being calculated with higher speed than 60 minutes / hour
ephorate adds up to 50% and academy up to 24%.
example: time we see is 16 hours, bonus from ephorate 40%, academy 10% and amphitheater 10%
In this example we know time and bonus but we don't know the initial value. So 10% amphitheater bonus wont change time to 16*60/66=14 hours and 33 minutes.
If we want to calculate that time we need to find the value=time*speed=16* 90/60=24 hours.
Our speed will be: 40% ephorate, 10% academy, 10% amphitheater=>60% boost for activation time => speed=60*1.6=96 minutes/ hour
so the time we see when we activate bonus is: 24*60/96=15 hours.
We saw that bonus is a fixed value and we expect to gain 18 minutes per activation.
Time 1 sec before bonus ends 15-3= 12 hours => value= 19 hours and 12 minutes
Without amphitheater boost we would have 16-3= 13 hours=> value=19 hours and 30 minutes
We gained exactly 18 minutes for initial time.
After boost ends, time = value/speed (without amphitheater bonus) = 19 hours and 12 minutes*60/ 90=12 hours and 48 minutes.
As we see we gained 18 minutes of initial time but because of already activated boosts the actual gain is 12 minutes.
Same way we estimate bonuses for troops building times and buildings construction times
Hope I helped.