SOLUTION: $1.65 is made up of pennies, nickels, and dimes. Half of the coins are nickels. List a set of coins that will solve the problem including why only the set of coins you chose worked

I'll just work the problem for you. You'll have to
write it up using complete sentences and proper grammar,
and filling in your own explanations.

Let p = the number of pennies
Let n = the number of nickels
Let d = the number of dimes

We have the equation:



or multiplying through by 100,



Since half the coins are nickels, we have





 

We have the system:



Substituting  for  in the first equation







Divide through by 3:







Since the left side is even,
the right side must be even also,
and positive. Therefore  
must be even and positive, too, 
Therefore d must be odd and no more 
than 11.

Therefore there are 6 possibilities:

d = 1, 3, 5, 7, 9, or 11

If , then

 becomes





Substituting  and 
into








So one possibility is 

, , 

-------

If , then

 becomes





Substituting  and 
into








So another possibility is 

, , 

-----------------

If , then

 becomes





Substituting  and 
into








So another possibility is 

, , 

-------------

If , then

 becomes





Substituting  and 
into








So another possibility is 

, , 

-------------

If , then

 becomes





Substituting  and 
into








So another possibility is 

, , 

-------------

If , then

 becomes





Substituting  and 
into







So another possibility is 

, , 

-------------

So all the 6 possibilities are 

0 pennies, 11 nickels, 11 dimes, 22 coins, half of which are 11 nickels.
5 pennies, 14 nickels, 9 dimes, 28 coins, half of which are 14 nickels.
10 pennies, 17 nickels, 7 dimes, 34 coins, half of which are 17 nickels.
15 pennies, 20 nickels, 5 dimes, 40 coins, half of which are 20 nickels.
20 pennies, 23 nickels, 3 dimes, 46 coins, half of which are 23 nickels.
25 pennies, 26 nickels, 1 dimes, 52 coins, half of which are 26 nickels.

Edwin