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Preservation Chance is a modifier which allows resources to be preserved when creating items, allowing for the creation of items at no cost to the player. When resource preservation triggers, the player will receive a notification stating "You managed to preserve your resources".
The generic "chance to preserve resources" only applies when creating items within a skill, it does not affect:
The preservation chance for all skills is capped at 80%, unless a modifier which increases the cap applies.
Like preservation, doubling helps create more output for the same input. But doubling does this by increasing the output for each action taken, while preservation increases the number of actions possible for each unit of input. For sources of doubling, see Skill Boosts#Doubling Boosts for global modifiers or the "Skill Boosts" section of a specific skill. See Double Loot for more about doubling in combat.
Chance Cap
The preservation chance for all skills is capped at 80%. For a few skills, there exists bonuses which can increase this cap:
Calculating Gains
Let's say you have 120 bars, 80% resource preservation, and 0% doubling chance. If a dagger takes one bar to create, how many daggers would you create on average out of 120 bars?
The answer is [math]\displaystyle{ \frac{120}{1 - 80/100} = \frac{120}{0.2} = 600. }[/math]
The main idea behind resource preservation is that you can reuse the preserved resources to make more items, but those resources when used will themselves be preserved also.
Using the same example with 120 bars, you take 120 actions producing 120 daggers, but with 80% preservation you are left with [math]\displaystyle{ 120 \times 0.8 = 96 }[/math] bars.
Now you can use those 96 bars to do the same thing, and you will be left with [math]\displaystyle{ 96 \times 0.8 = 76.8 }[/math] bars on average.
Repeat that to infinity, and you will get the average number of daggers that you will produce from 120 bars.
But how do we calculate this number?
So the number of daggers produced is [math]\displaystyle{ 120 + (120 \times 0.8) + (120 \times 0.8 \times 0.8) +\cdots + (120 \times 0.8^n) }[/math] where [math]\displaystyle{ n }[/math] goes to infinity.
If we generalize and represent the number of starting resources ([math]\displaystyle{ 120 }[/math]) with [math]\displaystyle{ a }[/math] and the probability ([math]\displaystyle{ 0.8 }[/math]) with [math]\displaystyle{ r }[/math], the number of actions performed with [math]\displaystyle{ a }[/math] action's worth of starting resources and a resource preservation of [math]\displaystyle{ r }[/math] (where [math]\displaystyle{ 0 \le r \lt 1 }[/math]) can be expressed as follows:
[math]\displaystyle{ a + (a \times r) + (a \times r \times r) + \cdots + (a \times r^n) = \sum\limits^{\infty}_{n=0} ar^n }[/math]
This expression is an infinite geometric series, which can simply be expressed as [math]\displaystyle{ \displaystyle \lim_{n \to \infty}\tfrac{a(1 - r^n)}{1 - r} = \frac{a}{1 - r} }[/math].
So, coming back to our example where [math]\displaystyle{ a = 120 }[/math] and [math]\displaystyle{ r = 0.8 }[/math], we get [math]\displaystyle{ \frac{120}{1 - 0.8} = 600 }[/math].
Sources
For a list of boosts that apply to all skills, see the Skill Boosts page.
Summoning tablet preservation
Magic
Ranged
Archaeology
Bear & 
Beaver synergy can increase the cap for 
Fletching by 5%
Fox synergy can increase the cap for 
Herblore by 5%
Herblore Preservation Scroll from 
Township can increase the cap for 
Combat:
