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

35 wins

With a 55% win rate, you would need to play 350 games, you would win 193 of them and lose 157.

With a 60% win rate, you would need to play 175 games, you would win 105 of them and lose 70.

With a 65% win rate, you would need to play 117 games, you would win 76 of them and lose 41.

With a 70% win rate, you would need to play 88 games, you would win 62 of them and lose 26.

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.