Question 1170563
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n = number of nickels
d = number of dimes
q = number of quarters
n,d, and q are nonnegative whole numbers


1 nickel = 5 cents
n nickels = 5n cents


1 dime = 10 cents
d dimes = 10d cents


1 quarter = 25 cents
q quarters = 25q cents


(n nickels) + (d dimes) + (q quarters) = (5n cents) + (10d cents) + (25q cents)
(n nickels) + (d dimes) + (q quarters) = (5n+10d+25q) cents
5n+10d+25q = 410
Since $4.10 is equivalent to 410 cents


We can divide all terms of this equation by 5 to go from 
5n+10d+25q = 410
to
n+2d+5q = 82


We are told "There are twice as many quarters as dimes", so,
q = 2d
Whatever the number of dimes there are, we double it to get the number of quarters.


Lastly, we have 28 coins total that are either nickels, dimes or quarters
This means n+d+q = 28. Solve for n to get n = 28-d-q. 


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The system of equations we can have is
n+2d+5q = 82
q = 2d
n = 28-d-q


Let's substitute equation (3) into equation (1)
n+2d+5q = 82
28-d-q+2d+5q = 82 ... replace n with 28-d-q
28+d+4q = 82
d+4q = 82-28
d+4q = 54


Then plug in q = 2d
d+4q = 54
d+4(2d) = 54
d+8d = 54
9d = 54
d = 54/9
d = 6 .... there are 6 dimes


Use this value of d to find q
q = 2d
q = 2(6)
q = 12 .... there are 12 quarters


We can then find the number of nickels
n = 28-d-q
n = 28-6-12
n = 22-12
n = 10 .... there are 10 nickels


So we get
n = 10
d = 6
q = 12


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Let's check the answers
n = 10 nickels leads to 5n = 5*10 = 50 cents
d = 6 nickels leads to 10d = 10*6 = 60 cents
q = 12 quarters leads to 25q = 25*12 = 300 cents
We get a total of 50+60+300 = 410 cents = $4.10
That takes care of one condition


Another condition is that we must have 28 coins
n+d+q = 10+6+12 = 16+12 = 28
So that condition is satisfied as well.


Finally, the number of quarters must be double that of the dimes. This is true since q/d = 12/6 = 2. Or we can see that (d,q) = (6,12) satisfies the equation q = 2d by plugging in the values. 
q = 2d
12 = 2*6
12 = 12


All three conditions are met, so we've confirmed the answers. 
 
 
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Answers: 
10 nickels
6 dimes
12 quarters


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