1 penny = 1 cent

1 nickle = 5 cents

1 dime = 10 cents

let

p = the number of pennies, p>0, p is integer

n = the number of nickels, n>0, n is integer

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

there are twice as many nickels as pennies:

n = 2p

and 5 times as many dimes as pennies:

d = 5p

in all she has $1.22 = 122 cents:

p + 5n + 10d = 122

by solving the system of equations:

n = 2p

d = 5p

p + 5n + 10d = 122

we find:

p = 2 pennies

n = 4 nickles

d = 10 dimes

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Joan has 2 pennies, 4 nickles and 10 dimes.