let

n = the number of nurses employed

d = the number of doctors employed

a hospital employs a total of 77 nurses and doctors

n + d = 77

the ratio of nurses to doctors is 9:2

n : d = 9 : 2 cross multiply

2n = 9d

by solving the system of equations

n + d = 77

2n = 9d

we find

d = 14 doctors

n = 63 nurses

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there 63 nurses and 14 doctors employed at the hospital.