Given that rectangle mnop ~ rectangle stuv, what is the length of tu

Knowing that similar polygons have sides in proportion we can set up a scale factor between the small rectangle and large rectangle. This is basically a ratio of one side of the small rectangle and the corresponding side of the larger rectangle.

scale factor: 4/6 = 2/3
Therefore all sides on the smaller rectangle and all corresponding sides on the larger rectangle are in the proportion 2/3. 

We can apply this formula to find TU. 

2/3 = 10/TU
Cross multiply as always
2TU=30
TU = 15

The method to find the female dormitory is 2-d=800