I have tried many ways to rank up fast in factions so here what i found
1. Mote of light ( very expensive)
2. Coin exchange for heavy ammo when xur arrives ( best way but needs more grinding)
3. Materials exchange ( not effective cause it takes time to collect and needs legendary marks to buy it and legendary marks is so important for beginners )
4. Weapon parts and armor materials ( more important to use than wasting it on ranking )
5. Playint vanguard heroic strikes ( most effective way cause playing with a vangaurd class item and an active booster you will get at least 130+ reputation for the faction for each strike )
So if we combined step 2 and step 5 you will be rank 25 in 2 weeks
Good luck
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Don't forget that you can also Exchange ether seeds (1) and glimmer (250) with Variks for heavy ammo synthesis Which outside of Xur's going rate of 3 synthesis for 1 strange coin is perhaps the best option Here's great little site that many might not know about [url]http://destiny.neonblack.moe/reputation[/url] It will show you a complete break-down of each of your factions and what it will take across the various ways in leveling up a particular faction