Answer: 62

Using trial and improvement

 50 correct:   200 points

50 incorrect: - 50 points

        total:     150 points, too small

  75 correct:   300 points

25 incorrect: - 25 points

        total:      275 points, too big

  60 correct:   240 points

40 incorrect: - 40 points

        total:      200 points, too small

  62 correct:   248 points

38 incorrect: - 38 points

        total:      210 points, perfect

Starting from 100 correct

100 correct: 400 points

Each question incorrect instead of correct: Lose 4 points for not correct

                                                              And 1 more for incorrect

                                                              Total lose 5

400 - 210 = 190

190$\div$5 = 38

38 questions were incorrect and not correct

The other 62 were correct

Using coding

If 1 is added to the score for each question, then, for candidates who answer all of the questions, then the scoring system becomes 5 points for each correct answer and none for each incorrect answer.

Adding 1 to the score for each question is the same as adding 100 to the total score, so under the scoring system of 5 points for each correct answer and none for each incorrect answer, Sarah would have got 210 + 100 = 310 points.

310$\div$5 = 62, so Sarah answered 62 questions correctly.

Using simultaneous equations

Suppose Sarah answered $s$ questions correctly and $t$ questions incorrectly. Then $s+t=100$, because there are $100$ questions altogether.

Sarah would score $4s$ points for the $s$ correct answers, and lost $t$ points for the $t$ incorrect answers, so she would score $4s-t$ points. So $4s-t=210$.

Adding these two equations gives $s+t+4s-t=100+210 \Rightarrow 5s=310 \Rightarrow s=62$.