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How many wins do I need to get from Iron 3 to Plat?

60 wins

With a 55% win rate, you need to play 600 games, win 330 of them and lose 270.

With a 60% win rate, you need to play 300 games, win 180 of them and lose 120.

With a 65% win rate, you need to play 200 games, win 130 of them and lose 70.

With a 70% win rate, you need to play 150 games, win 105 of them and lose 45.

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.