1 dime = 10 cents

1 quarter = 25 cents

1 half-dollar = 50 cents

let

d = the number of dimes, d is integer, d>=0

q = the number of quarters, q is integer, q>=0

h = the number of half-dollars, h>=0

A coin purse contains 24 coins in dimes, quarters, and half-dollars

d + q + h = 24

there are five more dimes than quarters

d = q + 5

the total value of the coins is $5.45 = 545 cents

10d + 25q + 50h = 545

by solving the system of equations

d + q + h = 24

d = q + 5

10d + 25q + 50h = 545

we find

d = 12 dimes

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there are 12 dimes in the coin purse.