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Dealing with US Dollars...

Assuming we know the weights of half-dollars, quarters, dimes, nickels, and pennies.

Also assuming the weights remain constant (don't change from one quarter to another quarter).

If we have a bag of these coins (could include any coin), and we weigh them to get a total weight, can we know how much money we have in the bag?

If we cannot know exactly, how close of an estimate can we get?

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It depends on the specific weights, and whether you have bounds for the minimum/maximum number of coins of each type. In general, what you will have a single diophantine equation of the form $a\mathbf{h} + b\mathbf{q} + c\mathbf{d}+d\mathbf{n}+e\mathbf{p}=k$, where $k$ is the weight, $\mathbf{h}$, $\mathbf{q}$, $\mathbf{d}$, $\mathbf{n}$, $\mathbf{p}$ the weight of half-dollars, quarters, dimes, nickels, and pennies, resp., and $a$, $b$, $c$, $d$, and $e$ being the unknowns (which must be nonnegative integers). By itself, it will usually have many solutions. – Arturo Magidin Jan 3 '11 at 0:49
Makes sense...thanks! – Sev Jan 3 '11 at 11:18
up vote 4 down vote accepted

According to the U.S. Mint, nickels weigh 5.000 grams and pennies 2.500 grams. So you generally cannot know exactly how much money is in the bag: for example, 10 grams of coins could be two nickels (10 cents) or four pennies (4 cents).

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On the other hand, if you restrict to dimes, quarters and half-dollars, you can determine the worth from the total weight since those coins all have the same value to weight ratio: 4.40917 cents per gram. – Matthew Conroy Jan 3 '11 at 1:19

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