Question 659025
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You have 25 coins, which have a total value of $1. what are the coins, and how many of each do you have ?
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        In his post,  tutor @Theo found out one solution  " 10 pennies,  12 nickels,  3 dimes and  0 quarters".

        He used the trial and error guessing.

        But the problem has two other solutions that eluded  Theo.

        Below I give a complete analysis to catch all possible solutions.



<pre>
Let x be the number of pennies, 
    y be the number of nickels,
    z be the number of dimes,
    w be the number of quarters.


We have two equations

    x +  y +   z +   w =  25,   (1)    for the total number of coins

    x + 5y + 10z + 25w = 100    (2)    for the total value.


From equation (2), subtract equation (1).  You will get

        4y + 9z  + 24w = 75,

        4y +       24w = 75 - 9z.    (3)


In the last equation, left side value is always a multiple of 4.
Hence, right side must be a multiple of 4.


z = 0  gives for the right side of (3)  the value  75 - 9*0 = 75, which is not a multiple of 4.

z = 1  gives for the right side of (3)  the value  75 - 9*1 = 66, which is not a multiple of 4.

z = 2  gives for the right side of (3)  the value  75 - 9*2 = 57, which is not a multiple of 4.

z = 3  gives for the right side of (3)  the value  75 - 9*3 = 48, which is a multiple of 4.



At z = 3,  from (3), we have
       
        4y +       24w = 48.    (3)



One possible solution for (3) is  (y,w) = (0,2).  Then   x + y + z + w = x + 0 + 3 + 2 = 25  implies  x = 20.

    So, one possible solution for the problem is  20 pennies, 0 nickels, 3 dimes and 2 quarters.



Second possible solution for (3) is  (y,w) = (6,1).  Then   x + y + z + w = x + 6 + 3 + 1 = 25  implies  x = 15.

    So, second possible solution for the problem is  15 pennies, 6 nickels, 3 dimes and 1 quarters.



Third possible solution for (3) is  (y,w) = (12,0).  Then   x + y + z + w = x + 12 + 3 + 0 = 25  implies  x = 10.

    So, third possible solution for the problem is  10 pennies, 12 nickels, 3 dimes and 0 quarters.


Thus the problem has three possible solutions:

    (1)  20 pennies,  0 nickels,  3 dimes and 2 quarters, 

    (2)  15 pennies,  6 nickels,  3 dimes and 1 quarters, 

    (3)  10 pennies, 12 nickels,  3 dimes and 0 quarters.


It is clear that there no other solutions.
</pre>

Solved completely.


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<H3>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Interesting fact:</H3>

Today, November 10, 2025, after completing my solution, I submitted  this problem to two artificial intelligence sites.

I was guided by my curiosity: it was interesting to me, how they will treat this problem ?


One AI was Google AI Overview. The second AI was math-gpt.org


Both produced incomplete answers.


Google AI Overview referred to the incomplete @Theo solution.


Math-gpt.org did not share his links with me.


But I made my conclusion: they both can not think (in the usual meaning of this word)
- they both re-write (compile) from existing solutions in their databases.



Surely, as usual, I informed the Google AI Overview about their mistake through their
feedback system with the reference to the current link.