Question 947343: A bag contains 15 coins. Each coin is a penny,
nickel, dime or quarter. The total value of these
coins is $2.16. There is at least one coin of
each denomination, and the number of pennies,
nickels, dimes and quarters are all distinct. If
there are 5 dimes, how many nickels are in the bag?
Answer by josgarithmetic(39618)   (Show Source):
You can put this solution on YOUR website! SEE BELOW, IMPROVEMENT/ALTERNATIVE
p, n, d, q;
p+n+d+q=15.
0.01p+0.05n+0.1d+0.25q=2.16.
d=5.
Simplify the money count.
and substitute for d;
You have a system,
which is
You can assume any one of p, or n, or q is constant, and then solve for the other two variables in terms of the chosen constant. This is because you are assuming a two-variable, linear system.
As a guess, the number of quarters is probably not too large compared to the 15 count of coins. Assign k=q.
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E1-E2,
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Try to find p based on this formula for n.
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What will make sense for k? In summary, we have a formula for p and a formula for n:
 .
We must have natural numbers for each of n, p, k. None of them be equal. None of them over 13.
Choose and find.
k__________n________p
1__________33_______-22
2__________27_______-15
3__________21_______-8
4__________15_______-1
5__________9________6, still not good because need some nickels, 9+6=15.
6__________3________13
7__________-3
FAILED TO RESOLVE THE PROBLEM...........................
IMPROVEMENT/ALTERNATIVE:
Re-examining the description, the two key equations are
which are based on the given assignment that d=5.
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Subtracting the coin count equation from the money count equation (both being in their just stated simplified form), we have
and p has been eliminated.
Using this simple equation relating n and q, try choosing different natural number values for q and find resulting n, and look for results which makes sense.
pick q____________find n
1_________________33
2_________________27
3_________________21
4_________________15
5_________________9
6_________________3--------THERE!!! Their sum is only 9, allowing the....five dimes.
ANSWER RESULT
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Combination appears to be:
p=?
n=3
d=5
q=6
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p=15-3-5-6=1, because total coins must be 15
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Notice all counts are unequal.
One penny, three nickels, five dimes, six quarters.
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