let

h = the money Hannah has

f = the money Francine has

p = the money Peter has

Hannah and Francine have $120:

h + f = 120

Hannah and Peter have $230:

h + p = 230

Peter has 6 times as much money as Francine:

p = 6f

by solving the system of equations:

h + f = 120

h + p = 230

p = 6f

we find:

h = $98

click here to see the step by step solution of the system of equations:

Hannah has $98.