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

12 wins

With a 55% win rate, you would need to play 120 games, you would win 66 of them and lose 54.

With a 60% win rate, you would need to play 60 games, you would win 36 of them and lose 24.

With a 65% win rate, you would need to play 40 games, you would win 26 of them and lose 14.

With a 70% win rate, you would need to play 30 games, you would win 21 of them and lose 9.

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.