Another way to look at this problem:
We know that James can paint 4 rooms in 10 hours; thus, in 30 hours he can paint 12 rooms.
We know that together they can paint 4 rooms in 6 hours. Thus, in 30 hours, they can point 20 rooms.
From this, we can see that June can paint 8 rooms in 30 hours. Thus, in order to paint just 4 rooms, it would take 15 hours.
James paints 4/10=0.4 rooms per hour
suppose June paints x rooms per hour.
Together they paint x+0.4 rooms per hour.
They paint 4 rooms in 6 hours so their combined rate is 4/6=2/3 rooms per hour
So x+0.4=0.66.., so x=2/3-2/5=4/15 rooms per hour.
Time for June to paint 4 rooms = 4/x =4*15/4=15 hours.
RonL