Random Variables · Superchallenge problem

A clinic must screen 100 people for an infection; each person is infected with probability $0.1$, independently. It splits them into pools of equal size $k$ (with $k$ a divisor of 100, $k \ge 2$) and tests each pool's combined sample once. A negative pool clears all its members; a positive pool (at least one infected member) is followed by a separate test for each of its $k$ members. With the best choice of $k$, what is the expected total number of tests? Round to the nearest hundredth.