What are the frequency and duty cycle when a periodic signal waveform has a pulse width of 25s and a period of 150s?

Frequency: F=1/P =1/150=6.66milliHertz or .00666 Hz which is a very low frequency.

Duty Cycle: DC=PW/P=25/150=16.6%

Your example uses seconds. it is more common to see examples in microseconds or less. The math is the same.