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How many wins do I need to get from Silver 4 to Emerald?

48 wins

With a 55% win rate, you need to play 480 games, win 264 of them and lose 216.

With a 60% win rate, you need to play 240 games, win 144 of them and lose 96.

With a 65% win rate, you need to play 160 games, win 104 of them and lose 56.

With a 70% win rate, you need to play 120 games, win 84 of them and lose 36.

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.