If T=number of teachers and P=number of pupils, T/P=13/5. This ratio is unrealistic for teacher:pupil, so perhaps it's female:male teachers? 13:5 means there are 13 teachers for every 5 pupils. That would mean that there would be 1,300 teachers for 500 pupils!
If F=number of female teachers and M=number of M teachers, F=M+48.
F/M=13/5, so F=13M/5. Therefore 13M/5=M+48, 13M=5M+240, 8M=240, M=30 and F=78.
Total number of teachers=30+78=108.