We are given that the total number of students is 150.
The number of students who speak English is |E| = 55.
The number of students who speak French is |F| = 85.
The number of students who speak neither English nor French is 30.
Let's use the principle of inclusion-exclusion to find the number of students who speak at least one of the languages.
The total number of students is the sum of those who speak English, those who speak French, those who speak both, and those who speak neither.
Total students = |E ∪ F| + |(E ∪ F)'|
150 = |E ∪ F| + 30
|E ∪ F| = 150 - 30 = 120
The principle of inclusion-exclusion states:
|E ∪ F| = |E| + |F| - |E ∩ F|
120 = 55 + 85 - |E ∩ F|
120 = 140 - |E ∩ F|
|E ∩ F| = 140 - 120 = 20
Therefore, the number of students who speak both English and French is 20.
The question asks for the number of students who speak both English and French. The answer is $\boxed{20}$ (option c).