Question 1199919
<font color=black size=3>
n = number of nickels
d = number of dimes
q = number of quarters
These are nonnegative integers {0,1,2,3,etc}


Given info: <font color=blue>There is a total of 105 coins and the total value of those coins is $10.75</font>


Break that into 2 pieces
Given info: <font color=blue>There is a total of 105 coins</font> and <font color=blue>the total value of those coins is $10.75</font>


The first part lets us say
n+d+q = 105


and the second fact lets us say
0.05n+0.10d+0.25q = 10.75
which is equivalent to
5n+10d+25q = 1075
after moving the decimal point two spots to the right for each item. 
We multiplied both sides by 100.


We have this system of equations
n+d+q = 105
5n+10d+25q = 1075


We have 2 equations, but 3 unknowns.
Unfortunately there isn't enough information to determine the values of n,d,q.


The next section goes into greater detail as to why this is the case.


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n+d+q = 105 solves to n = 105-d-q


Substitute that into the other equation
5n+10d+25q = 1075
5(105-d-q)+10d+25q = 1075
525-5d-5q+10d+25q = 1075
525+5d+20q = 1075
5(105+d+4q) = 1075
105+d+4q = 1075/5
105+d+4q = 215
d = 215-105-4q
d = 110-4q


If q = 10, then,
d = 110-4q
d = 110-4*10
d = 70
and,
n = 105-d-q
n = 105-70-10
n = 25
Showing that (n, d, q) = (25, 70, 10) is one ordered triple solution.
Check:
n+d+q = 25+70+10 = 105 coins total
5n+10d+25q = 5*25+10*70+25*10 = 1075 cents total value
1075 cents = 1075/100 = $10.75



If q = 11, then,
d = 110-4q
d = 110-4*11
d = 66
and,
n = 105-d-q
n = 105-66-11
n = 28
Showing that (n, d, q) = (28, 66, 11) is another ordered triple solution.
Check:
n+d+q = 28+66+11 = 105 coins total
5n+10d+25q = 5*28+10*66+25*11 = 1075 cents total value


Try out other values of q to see what happens.



Here are all the ordered triple solutions (n,d,q) such that we focus on positive integers only.
<pre>
(n, d, q) = (1, 102, 2)
(n, d, q) = (4, 98, 3)
(n, d, q) = (7, 94, 4)
(n, d, q) = (10, 90, 5)
(n, d, q) = (13, 86, 6)
(n, d, q) = (16, 82, 7)
(n, d, q) = (19, 78, 8)
(n, d, q) = (22, 74, 9)
(n, d, q) = (25, 70, 10)
(n, d, q) = (28, 66, 11)
(n, d, q) = (31, 62, 12)
(n, d, q) = (34, 58, 13)
(n, d, q) = (37, 54, 14)
(n, d, q) = (40, 50, 15)
(n, d, q) = (43, 46, 16)
(n, d, q) = (46, 42, 17)
(n, d, q) = (49, 38, 18)
(n, d, q) = (52, 34, 19)
(n, d, q) = (55, 30, 20)
(n, d, q) = (58, 26, 21)
(n, d, q) = (61, 22, 22)
(n, d, q) = (64, 18, 23)
(n, d, q) = (67, 14, 24)
(n, d, q) = (70, 10, 25)
(n, d, q) = (73, 6, 26)
(n, d, q) = (76, 2, 27)
</pre>
I used computer software to quickly generate this list. 
This is to do so efficiently and to avoid error. 
It's also to ensure that the list is fully exhaustive.


Once again, there isn't enough information to nail down the single exact value of each n, d, and q value.
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