1 penny = 1 cent

1 nickle = 5 cents

1 dime = 10 cents

1 quarter = 25 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

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

A piggy bank contains 52 coins worth $5.10 = 510 cents

p + n + d + q = 52

p + 5n + 10d + 25q = 510

There are an equal number of pennies and dimes:

p = d

and twice as many nickels as dimes:

n = 2d

pug p = d and n = 2d into

p + n + d + q = 52 and

p + 5n + 10d + 25q = 510

we gt

d + 2d + d + q = 52 or 4d + q = 52

d + 5*2d + 10d + 25q = 510 or 21d + 25q = 510

by solving the system of equations:

4d + q = 52

21d + 25q = 510

we find

d = 10 dimes

q = 12 quarters

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there are 12 quarters in the bank.