League of Legends Guide
5 gold per 10 second items
written by MossmyrTable of Contents
Introduction  1 
Summary  2 
Notes  3 
Calculations  4 

Introduction
When are gold per sec items profitable? How long do I have to wait to sell it to make a profit? Should I even buy them?
These are questions I've asked myself for quite some time, and since I havn't found a good guide on the matter I've decided to make one myself.
Hope it helps you 
Summary
Avarice Blade
Cost: 750g (400g + 350g)
Should be upgraded after: 5:00
Can be sold for profit after: 12:30
[item=Heart of Gold]
Cost: 975g (475g + 300g + 200g)
Should be upgraded after: 0:06
Can be sold for profit after: 16:14
[item=Kage's Lucky Pick]
Cost: 765g (435g + 330g)
Should be upgraded after: 7:23
Can be sold for profit after: 12:14
Philosopher's Stone
Cost: 800g (180g + 435g + 185g)
Should be upgraded after: 15:06
Can be sold for profit after: 13:20 
Notes
I hope this made sense to you. I've tried to make it as correct as possible but there are a few flaws with the calculation:
First, I'm not taking into account the value of having all items put together to one (inventory slots are freed, which should be worth at least something). However, since you buy these items so early in the game, it isn't so important.
Second, what gold a certain aspect is worth differs somewhat. Ex. the Amplifying Tome gives you 20 ability power for 435g, but it can be upgraded to a lot of stuff (unlike doran's shield, blade and ring). Riot puts a higher price tag on stuff that can be upgraded to make better items. 
Calculations
All "Can be sold for profit after" calculations
([(Total item cost)  (Total item cost * 0.5)]/0.5)/60
(5 gold per 10 second items are sold for 50% of their original cost)
Avarice Blade
* Avarice Blade (750g) gives +12% critical strike chance and 5 gold per 10 seconds.
* Brawler's Gloves (400g) gives +8% critical strike chance.
 8% critical strike chance = 400g
> 12% critical strike chance = 600g
 Avarice Blade = (12% critical strike chance) + (5 gold per 10 seconds) = 750g
> 5 gold per 10 seconds = 150g
 [150g/(0.5g/s)]/60 seconds = 5:00
Heart of Gold
* Heart of Gold (975g) gives 200 health and 27 armor
* Ruby Crystal (475g) gives 180 health
* Cloth Armor (300g) gives 18 armor
 180 health = 475g
> 200 health = 528g
 18 armor = 300g
> 27 armor = 450g
 Heart of Gold = (200 health) + (27 armor) + (5 gold per 10 seconds) = 975g
> 5 gold per 10 seconds = 3g
 [3g/(0.5g/s)]/60 seconds = 0:06
Kage's Lucky Pick
* Kage's Lucky Pick (765g) gives 25 ability power and 5 gold per 10 seconds
* Amplifying Tome (435g) gives 20 ability power
 20 ability power = 435g
> 25 ability power = 544g
 Kage's Lucky Pick = (25 ability power) + (5 gold per 10 seconds) = 765g
> 5 gold per 10 seconds = 221g
 [221g/(0.5g/s)]/60 seconds = 7:23
Philosopher's Stone (800g) gives 22hp/5, 10mp/5 and 5gold/10
* Faerie Charm (180g) gives 3 mana per 5 second
* Regrowth Pendant (435g) gives 15 hit points per 5 second
 3mp/5 = 180g
> 10mp/5 = 600g
 15hp/5 = 435g
> 22.5hp/5 = 652.5g
 Philosopher's Stone = (10mp/5) + (22.5hp/5) + (5 gold per 10 seconds) = 800g
600g + 653g + (5 gold per 10 seconds) = 1253g
> 5 gold per 10 seconds = 453g
 [453g/(0.5g/s)]/60 seconds = 15:06