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How many wins do I need to get from Gold 3 to Emerald?

28 wins

With a 55% win rate, you would need to play 280 games, you would win 154 of them and lose 126.

With a 60% win rate, you would need to play 140 games, you would win 84 of them and lose 56.

With a 65% win rate, you would need to play 93 games, you would win 60 of them and lose 33.

With a 70% win rate, you would need to play 70 games, you would win 49 of them and lose 21.

Assuming an average LP gain of +/-25 LP per game below Emerald and +/-20 LP for Emerald+ as per Riot Phroxzon's tweets here and here.