If somebody gave you the option of choosing either the value of a kilo of coins or 40$, which will you pick?

Using CPLEX, we determine that the maximum possible value for 1000 grams of pennies, nickels, dimes, and quarters is 44.05$, so if you took the 40$, there is a small chance you will incur a loss. If the bag was packed with 400 pennies (weighs exactly a kilo), you'll make a 36$ profit. If the coins types were uniformly distributed, its value would be about 26$, and you would make a 14$ profit.

Say you believe you have 100$ worth of coins in a jar that you want to cash out at a Coinstar machine. How much would that weigh? A lower bound for that answer is 2.268 Kilos. If the coin types were uniformly distributed in the jar, you would carry about 3.7 Kilos.

What kind of coin ratios does a vending machine carry? This online post author made enough purchases to empty out a vending machine and found this:

Quarters:Dimes:Nickels in the ratio 1:2:1.175

Assuming that the vending machine has to satisfy exact change for any value between 5c and 95c in 5c increments, a minimum weight solution is:

Nickels:1, Dimes:9 (10 coins, weight 25.412g) with zero quarters. However, by adding a

__secondary__objective of also minimizing the number of coins (limited coin capacity?), we obtain the following alternative optimal solution that threw away 5 dimes and picked up 2 quarters:

Nickel:1, Dime:4, Quarter:2 (7 coins, 25.412g)

Quarters and Dimes are now present in the same ratio as that in the vending machine. However, our model doesn't like nickels much (although a melted nickel was once worth more than its value).

I have a digital counting jar that displays the current value of the coins currently in the jar. It currently reads $20.05, and its weight (excluding the empty jar) is about 700 grams. For this weight, the maximum value is 30.85$. If my coins were uniformly distributed in the jar, the model predicts a "maximum" value of 18.7$, which means that I have relatively more dimes and/or quarters in my mix. A dime and a nickel have the exact same bang-to-buck ratio, i.e. value/weight ratio, which is a source of the alternative optimal solutions discussed in part 2.

US Mint Production

Averaged over 2010-2011, the coins are produced in the approximate ratio:

p:n:d:q

12:1.5:3.5:1

Quite different from our preferred ratios calculated in part-1 and part-2. At this rate, my 700 gram jar would contain only about 15$ and a kilo of coins in this ratio would be worth about 18$.

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