1 dime = 10 cents

1 quarter = 25 cents

let

d = the number of dimes, d>=0

q = the number of quarters, q>=0

a coin bank has 125 coins, quarters, and dimes

d + q = 125

they are worth $8.75 = 875 cents

10d + 25q = 875

on the other hand 10d + 25q > 10*d + 10q = 10(d + q) = 10*125 = 1250 > 875 that is we have

10d + 25q > 875 and 10d + 25q = 875 follows there are no-negative integer solutions d and q that satisfy the system of equations

d + q = 125

10d + 25q = 875

this problem has not solutions.