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How many wins do I need to get from Gold 1 to Emerald?

20 wins

With a 55% win rate, you would need to play 200 games, you would win 110 of them and lose 90.

With a 60% win rate, you would need to play 100 games, you would win 60 of them and lose 40.

With a 65% win rate, you would need to play 67 games, you would win 44 of them and lose 23.

With a 70% win rate, you would need to play 50 games, you would win 35 of them and lose 15.

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.