user3149: I aways thought that the king and passed pawn vs. king and rook endgame was always drawn. But I had two passed pawns in the below game,with Black to move, I was unable to draw the game. Why is that?  
[FEN "8/1R6/8/5pp1/3K1k2/8/8/8 b - - 0 1"]


user17: What I was taught was that a king and two connected passed pawns was "drawable" if the pawns were both on the SIXTH rank, with the pawnholder to move. Assuming that the two kings and rook were all in a position to support/oppose the pawns.
If the pawn are both on the seventh rank, you might be able to win by "queening" one of them. Unless the opponent can sacrifice the rook for BOTH pawns.
In your case, your pawns were on the fourth rank, not far advanced enough to be "threatening," giving the rook too much time (and space to maneuver).
Pawns gain value the further advanced they are. A protected passed pawn on the sixth rank is worth TWO points (not one) and two connected passed pawns are worth more than four points because of the "synergy," that is, about a rook.
A protected passed pawn on the seventh rank is worth about three points, because the opponent will have to give up at least a minor piece to prevent it from queening.

user3831: There is no absolute result for this type of endgame-it all depends on the position of the pieces. In fact, any result is possible.
Objectively, this particular position with black to move is a loss-you didn't make a mistake.I suggest you consult endgame tablebase, which announce a loss in 24 moves with best play by Black. Here is an example line.
[FEN "8/1R6/8/5pp1/3K1k2/8/8/8 b - - 0 1"]

1... g4 2. Kd3 Kf3 3. Kd2 g3 4. Ke1 Ke3 5. Rb3+ Kf4 6. Kf1 Kg4 7. Kg2 f4 8. Rb4 Kf5 9. Kf3 Ke5 10. Rxf4 Kd5 11. Kxg3 Ke5 12. Kf3 Kd5 13. Re4 Kc5 14. Re5+ Kd4 15. Rf5 Kc3 16. Rf4 Kd2 17. Rd4+ Kc2 18. Ke2 Kc3 19. Re4 Kb2 20. Kd2 Kb3 21. Rd4 Kb2 22. Rd3 Ka1 23. Rb3 Ka2 24. Kc2 Ka1 25. Ra3#

In the game, the White king is able to get and front of the Black pawns after Black tries to push them for promotion. White is then able to scoop them up as Black attempts to defend them with their king. Once Black is down to their king, it is a simple king and rook endgame for White. Win.
Some nice online resources about this endgame type:
General ideas and examples
More general ideas, examples and statistics
Real games collection

3149: <p>I aways thought that the king and passed pawn vs. king and rook endgame was always drawn. But I had two passed pawns in the below game,with Black to move, I was unable to draw the game. Why is that?  </p>

<pre><code>[FEN "8/1R6/8/5pp1/3K1k2/8/8/8 b - - 0 1"]
</code></pre>
 17: <p>What I was taught was that a king and two connected passed pawns was "drawable" if the pawns were both on the SIXTH rank, with the pawnholder to move. Assuming that the two kings and rook were all in a position to support/oppose the pawns.</p>

<p>If the pawn are both on the seventh rank, you might be able to win by "queening" one of them. Unless the opponent can sacrifice the rook for BOTH pawns.</p>

<p>In your case, your pawns were on the fourth rank, not far advanced enough to be "threatening," giving the rook too much time (and space to maneuver).</p>

<p>Pawns gain value the further advanced they are. A protected passed pawn on the sixth rank is worth TWO points (not one) and two connected passed pawns are worth more than four points because of the "synergy," that is, about a rook.</p>

<p>A protected passed pawn on the seventh rank is worth about three points, because the opponent will have to give up at least a minor piece to prevent it from queening.</p>
 3831: <p>There is no absolute result for this type of endgame-it all depends on the position of the pieces. In fact, any result is possible.</p>

<p>Objectively, this particular position with black to move is a loss-you didn't make a mistake.I suggest you consult <a href="https://www.shredderchess.com/online-chess/online-databases/endgame-database.html" rel="nofollow noreferrer">endgame tablebase</a>, which announce a loss in 24 moves with best play by Black. Here is an example line.</p>

<pre><code>[FEN "8/1R6/8/5pp1/3K1k2/8/8/8 b - - 0 1"]

1... g4 2. Kd3 Kf3 3. Kd2 g3 4. Ke1 Ke3 5. Rb3+ Kf4 6. Kf1 Kg4 7. Kg2 f4 8. Rb4 Kf5 9. Kf3 Ke5 10. Rxf4 Kd5 11. Kxg3 Ke5 12. Kf3 Kd5 13. Re4 Kc5 14. Re5+ Kd4 15. Rf5 Kc3 16. Rf4 Kd2 17. Rd4+ Kc2 18. Ke2 Kc3 19. Re4 Kb2 20. Kd2 Kb3 21. Rd4 Kb2 22. Rd3 Ka1 23. Rb3 Ka2 24. Kc2 Ka1 25. Ra3#
</code></pre>

<p>In the game, the White king is able to get and front of the Black pawns after Black tries to push them for promotion. White is then able to scoop them up as Black attempts to defend them with their king. Once Black is down to their king, it is a simple king and rook endgame for White. Win.</p>

<p>Some nice online resources about this endgame type:</p>

<p><a href="http://www.chess-insights.com/rook-vs-two-connected-pawns-part-four.html" rel="nofollow noreferrer">General ideas and examples</a></p>

<p><a href="http://likesforests.blogspot.dk/2007/08/rook-vs-2-pawns.html" rel="nofollow noreferrer">More general ideas, examples and statistics</a></p>

<p><a href="http://www.chessgames.com/perl/chesscollection?cid=1003156" rel="nofollow noreferrer">Real games collection</a></p>
