A cabinet has two sliding doors of different sizes. When the cabinet is fully closed, they overlap by two-fifths of the width of the smaller door. When both doors are slid to one side, the part of the longer door that is not behind the shorter one is half as wide as the original overlap, and the open portion of the cabinet is 36 inches wide. What is the width of the whole cabinet?

From the given, we can say that the shorter door covers that amount with 2/5 of it overlapping the large door.  So 3/5 of it is 36 inches long.  Setting up a proportion: 3/5 = 36/x

3x = 180

x = 60 So from this, the shorter door is 60 inches long.  The longer door is 60 inches plus 1/2 of the original overlap. Original overlap is calculated by = 60 - 36 = 24 So the longer door is calculated as: (60 + 12) 72 inches.  So the width of the cabinet is the longer door + the amount it doesn't cover. So computing for the width will give us: 72 + 36 = 108 inches