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

32 wins

With a 55% win rate, you would need to play 320 games, you would win 176 of them and lose 144.

With a 60% win rate, you would need to play 160 games, you would win 96 of them and lose 64.

With a 65% win rate, you would need to play 107 games, you would win 70 of them and lose 37.

With a 70% win rate, you would need to play 80 games, you would win 56 of them and lose 24.

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.