Coins

harpazo

Pure Mathematics
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A jar contains only nickels, dimes, and quarters. There is at least one of each type of coin in the jar. If the total value of the coins in the jar equals 60 cents, how many quarters are in the jar?

0.25x + 0.05x + 0.10x = 0.60

Correct set up?

 

MarkFL

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St. Augustine
No, you are assuming there is the same number of each type of coin. What you want is:

\(\displaystyle 25Q+10D+5N=60\)

or

\(\displaystyle 5Q+2D+N=12\)

This is a Diophantine equation, and so even though there are 3 variables and only 1 equation, you can find the possible solutions.
 
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MarkFL

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St. Augustine
By some experimentation, we find the possible solutions:

\(\displaystyle (Q,D,N)=(1,1,5),\,(1,2,3),\,(1,3,1)\)

All 3 possible solutions have 1 quarter in the jar. Now we could simply reason as follows:

We know there must be at least 1 quarter in the jar, and there cannot be 3 or more since that's more than 60 cents. So, there must be 1 or 2. If there are 2, then that leaves 10 cents for the dimes and nickels, but with at least one of each, that's a minimum of 15 cents that must be left over. Thus, there can only be 1 quarter in the jar.
 
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harpazo

Pure Mathematics
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By some experimentation, we find the possible solutions:

\(\displaystyle (Q,D,N)=(1,1,5),\,(1,2,3),\,(1,3,1)\)

All 3 possible solutions have 1 quarter in the jar. Now we could simply reason as follows:

We know there must be at least 1 quarter in the jar, and there cannot be 3 or more since that's more than 60 cents. So, there must be 1 or 2. If there are 2, then that leaves 10 cents for the dimes and nickels, but with at least one of each, that's a minimum of 15 cents that must be left over. Thus, there can only be 1 quarter in the jar.
Can you show the expirementation that was done here?