Ranked

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How many ranked games in a series do you need to win to join higher division?

PROMOTIONS AND SERIES Once you're placed in a division, you'll earn League Points (LP) for each game you win, and lose them for games you lose. Once you collect 100 LP in a tier, you'll enter a promotional series. To advance a division, you'll need to win 2 out of 3 games. To advance a tier (from Silver I to Gold V), you need to win 3 of 5.

Promotion series up to Gold I don’t expire, but you’ll only have 28 days to complete a Platinum promo series and 5 days to complete Diamond and Master promotions. Dodging a promo series counts as a loss.

Source: https://support.riotgames.com/hc/en-us/articles/204010760-Ranked-Play-FAQ#h2q2

To move up one level in ranked (silver 5 to silver 4) you have to get 100 lp. depending on your elo you get 20-30 lp for a win and loose 10-20 for a loss. Once you have 100 lp you get put into promos where you have to win the majority of games in to move up... do that 5 times and you will move from bronze to silver... or silver to gold