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On my first day of University five years ago, whilst shopping for dorm supplies, I bought this large beer-and-hockey themed piggy bank. Since then, I have deposited any coins smaller than a quarter (ie pennies, nickels and dimes). Today, just 2 weeks shy of moving to the US to start grad school, was time to finally cash in.

Before counting, I wanted to see if I could reasonably estimate how much money there would be. I considered doing a “random sample” approach, counting the value of a small portion and scaling up to the full weight. However, unfortunately the only means at my disposal to weigh the samples was a bathroom scale inteded for weighing people in 0.1kg increments, so I didn’t think this would be accurate enough. Instead I weighed the entire piggy-bank (which came out to an impressive 4.2kg) and made some simple estimations of what the relative proportions of the coins would be as so:

In considering purchases which would result in me receiving small change, I assume that all values of “cents” are equally likely. I further assume that change is almost always given with the minimum number of coins heuristic. Making these two assumptions, and considering the possible values of change between $0.01 and $0.24 (since all other values will be isomorphic with the addition of some number of dollars and quarters), results in the following relative proportions: 25% dimes, 12.5% nickels and 62.5% pennies.

From this, and from considering the weights of each coin (dime = 1.75g, nickel = 3.95g, penny = 2.35g according to Wikipedia) I can estimate the total number of coins thus:

Using the estimated proportions, I therefore get the following estimations for the number of value of each coin type:

Dimes | Nickels | Pennies | |

Estimated Number: |
437.5 | 218.75 | 1093.75 |

Estimated Proportion: |
0.250 | 0.125 | 0.625 |

Estimated Value: |
$43.75 | $10.94 | $10.94 |

Total Estimated Value: |
$65.63 |

$65.63! That’s a lot of money! Let’s see how my estimations stack up to reality:

Final counts:

Dimes | Nickels | Pennies | |

Counted Number: |
462 | 280 | 812 |

Counted Proportion: |
0.297 | 0.180 | 0.523 |

Counted Value: |
$46.20 | $14.00 | $8.12 |

Total Counted Value: |
$68.32 |

So, the final value was quite close (<5% error), but I think we can put that down to fluke. Obviously, my estimated proportions were quite a bit off, although I was correct in predicting that pennies would be significantly more numerous than dimes, which were in turn significantly more numerous than nickels. Most saliently, the proportion of pennies was significantly less than predicted by my simple *a priori* method. This can likely be attributed to the tendency of pennies to be discarded or ignored which receiving change. I’m not sure why nickels would be tend to be relatively more numerous relative to dimes than I had predicted. Also interesting is that if I use the Wikipedia-listed weights of the coins multiplied by the real counts, the weight should only be around 3.8kg. Perhaps my scale was inaccurate at such a low weight (as it is designed for weighing humans).

Anyways, it was quite interesting to see how much small-change I had accumulated in 5 years. Tomorrow I’ll be off to the bank with 4kg of coins!

malc

said:my goddness. what a math geek. i wish i had the energy to imagine what was in my piggy bank. cool beans!

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Fletch

said:I did this same calculation before and found a vastly depressed number of quarters in my inventory, but the other proportions were right. But had done a lot of laundry and bought sodas out of the vending machine…