Question 1199919
<pre>
A piggy bank contains only nickels, dimes, and quarters. There is a total of 105 coins and the total value of those coins is $10.75.

Find a general solution of the number of each coin.
Use the general solutions from part 1 to list 3 specific solutions (for n, d, & q).

n=nickel
d=dime
q=a quarter

Since number of nickels, dimes, and quarters are n, d, and q, we get the following COIN-COUNT equation:
  n + d + q = 105 --- eq (i), and total value-equation, .05n + .1d + .25q = 10.75, or 20(.05n + .1d + .25q) = 20(10.75), or
n + 2d + 5q = 215 --- eq (ii)
     d + 4q = 110 ----- Subtracting eq (i) from eq (ii) 
         4q = 110 - d 
         {{{matrix(1,3, q, "=", (110 - d)/4)}}} ----- Solving for q ----- eq (iii)

The SMALLEST number of dimes (d) that can be substituted         The next "d" integer-value that'll create the next      
into the above equation, and which also yields an                smaller integer in numerator that's a MULTIPLE of 4 is 6,
INTEGER-number of quarters is 2, since 2 leads to a              since 6 leads to a numerator of 104, a multiple of 4, and                       
numerator of 108, a MULTIPLE of 4 (the denominator).             which also yields an INTEGER-VALUE for q.
      We then get: {{{matrix(4,3, q, "=", (110 - d)/4, q, "=", (110 - 2)/4, q, "=", 108/4, q, "=", 27)}}}                                           We then get: {{{matrix(4,3, q, "=", (110 - d)/4, q, "=", (110 - 6)/4, q, "=", 104/4, q, "=", 26)}}}
Therefore, for the above, we get: Number of dimes: 2               Therefore, for the above, we get: Number of dimes: 6                              
Number of quarters: 27                                             Number of quarters: 26
Number of nickels, 105 - (2 + 27) = 105 - 29 = 76                  Number of nickels, 105 - (6 + 26) = 105 - 32 = 73
This gives us, (n, d, q) = (76, 2, 27)                             This gives us, (n, d, q) = (73, 6, 26)

                                          So far, we have: (n, d, q) = (76, 2, 27)
                                                           (n, d, q) = (73, 6, 26)

Looking at the above SEQUENCE, it's quite obvious that the nickels decrease by 3, the
                                                           dimes INCREASE by 4, and the
                                                           quarters DECREASE by 1. 

                               Thus, we get the following: (n, d, q) = (76,  2, 27)
                                                           (n, d, q) = (73,  6, 26)
                                                           (n, d, q) = (<font size  = 3><font color = blue><b>70, 10, 25</font></font></b>)
                                                           (n, d, q) = (<font size  = 3><font color = blue><b>67, 14, 24</font></font></b>)
                                                           (n, d, q) = (<font size  = 3><font color = blue><b>64, 18, 23</font></font></b>)

You don't need that many because you were asked for just 3, but you should get the "picture." CAPISCHE?</pre>