We can calculate the success rate of every school, then find their average.
Success rate $ = \dfrac{\text{Number of students who passed the exams}}{\text{Number of appeared students}}$
- Success rate of school $P = \dfrac{280}{500} \times 100\% = 56\%.$
- Success rate of school $Q = \dfrac{330}{600} \times 100\% = 55\%.$
- Success rate of school $R = \dfrac{455}{700} \times 100\% = 65 \%.$
- Success rate of school $S = \dfrac{240}{400} \times 100\% = 60\%.$
Now, the average success rates of all the schools $ = \dfrac{56\% + 55\% + 65\% + 60\%}{4} = \dfrac{236\%}{4} = 59.0\%.$
So, the correct answer is $(C).$