A female cockroach can produce 80 offspring at a time. In half of the offspring produced by a certain female are female and each female produces 80 offspring, how many cockroaches are there in the third generation.

128000 cockroaches

Explanation:

If the a cockroach produces 80 offspring and 40 are female in the first generation.

The second generation will therefore be:

2nd generation = 40 x 80

                          =  3200 cockroaches

considering that half of the population are female, half of 3200 is 1600

3rd generation = 1600 x 80

                         = 128000 cockroaches

Hence, the total number of cockroaches in the 3rd generation is 128,000