How are the Gem values worked out?

There seems to be a "complicated" way to do this. WARNING: The following might differ for some games and versions. This is an estimate by this wikia.

Similarly, the gem value of a set's emoticons and backgrounds appears to be equal to the number of cards in the set * the gem value of normal cards / 2, rounded up to the nearest multiple of 20. The value of a foil is ten times (10x) the value of the normal card. This applies to all sets. The value of an emoticon is equal to the value of a background crafted from the same set.

Emoticons and backgrounds associated with Valve (Valve games and summer/winter sale trading cards) are always worth 100 gems. Free-to-play games (which drop cards based on how much one pays) generally have high-value items as well. In general, the more cards in a set, the lower the value of the items and cards themselves. The inverse is also true. Only trading cards, emoticons, and backgrounds can be turned into gems. The process is irreversible.

So to sum this up:

• Only trading cards, emoticons, and backgrounds can be converted into gems

• Emoticons and backgrounds that deals with Value will always be worth 100 gems.

• The gem value of a emoticon/background is based on the number of the cards in its respective set times the gem value of normal cards then divided by two and finally rounded to the nearest multiple of 20.

• The gem value of a foil card is 10x of its normal version

Formulas could be found in the wikia link above as well as more information. For cards:

Assuming that n = # cards in set, c = value of single card, g = value of emoticon/background, and b = gems to craft booster, any one of the four can be calculated if any two other values are known. This doesn't hold true for every game, but is accurate for the vast majority:

n = 6000/b
n = 2g/c
c = bg/3000
c = 2g/n
g = 3000c/b
g = n * c/2
b = 6000/n
b = g/3000c