We are given that:
h = 60% of all the games were at-home games
w = 40% of all games were wins
35% of all games were at-home wins
To clearly understand the problem and number of games, let us set as a basis of calculation that the total number of games played is 10 (just for the sake of calculation). Therefore:
10 total games
h = 6 home games
w = 4 total wins
3.5 of these games were won at-home
Since we can now clearly see the values, the proportion of at-home games that were wins is simply:
Proportion of at-home games won = 3.5 / 6
Proportion of at-home games won = 0.58