How much are you worth in quarters?

Imagine you are on a game show, and you’ve just won the grand prize. The prize is your own weight in coins!

The host gives you ten seconds to select what kind of coin you want your prize to be measured in. (Insert game show music here …)

Do you want to go home with your own weight in pennies? Do you want to go home with your own weight in nickels? How about dimes, quarters or dollar coins? What about half-dollars?

Quick, what do you select? (We’re assuming that your goal is to select the option that maximizes the prize money you take home). Obviously quarters are more valuable than dimes, but does their extra weight work against you since you since you’ll get less of them? Let's work it out …

US coins in circulation

To answer the prize question, we need to work out how many of each coin we get per pound (or kg, or Stone …), and then we can multiple this by the value of each coin.

There are six types of coin currently in US circulation. These are: the penny, nickel, dime, quarter, half-dollar, and the dollar coin. Here they are:

(Whilst there is more than one version of the dollar coin, they all have the same mass).

Value Density

Below is a table showing each of these coins and their respective weights.

Next we can calculate how many of each of these coins there are in one Kilogram (To my non-metric friends, one kg is approximately 2.2 lbs, though it does not really matter what weight we select when comparing the coins as we are really only looking at the ratio – as long as we use the same for each coin it does not matter).

Penny* Nickel Dime Quarter Half-Dollar Dollar Weight (g) 2.500 5.000 2.268 5.670 11.340 8.100 Coins/kg 400.000 200.000 440.917 176.367 88.183 123.457

* The Penny changed in 1982 from 3.11g when the Copper content was reduced.

Next we multiply the number of coins per kg by the value of each coin to give the result of Value per kg.

Penny Nickel Dime Quarter Half-Dollar Dollar Weight (g) 2.500 5.000 2.268 5.670 11.340 8.100 Coins/kg 400.000 200.000 440.917 176.367 88.183 123.457 Value $0.01 $0.05 $0.10 $0.25 $0.50 $1.00 Value/kg $4.000 $10.000 $44.092 $44.092 $44.092 $123.457

Results

There are some very interesting results. Firstly, the correct answer is dollar coins. You'll earn more than $123 for every kg you weigh if you select this coin. This is over thirty times your value in pennies. The next interesting thing is the equivalence of dimes, quarters and half-dollars. The ratio of their masses is exactly the same as the ratio of their values. They have the same value per weight! This means that it would not matter which of these coins you selected, if you elected to be paid in these coins; your prize would be the same.

In fact we can go further, since these coins have exactly the same value density, if we used just these three coins it does not matter the mix of the coins. Any random combination of these coins that weigh the same has the same value. That’s right – if you were handed a bucket of coins containing a random mixture of dimes, quarters and half-dollars you could determine the total value of money simply by weighing them, and would not need to count or sort the coins! (After subtracting the weight of the bucket, obviously!)

Why does this happen? Well, It goes back to the days when a coin’s value was simply the value of the metal that the coin was made out of. When coins were made out of Silver, the value of the Silver in the coin was the value of the coin. It makes sense that a coin containing twice as much Silver would be worth twice as much!

So, how much is the prize?

In North America, the average body weight is 80.7 kg, which would be a prize of just less than $10k if you selected dollar coins (cf. just $322.80 if you elected for pennies). If you are interested in another random way a body can be valued check out this article about splitting the body into elemental components.

What about other countries?

We can perform similar calculations for the coinage of other countries and regions.

Below is a table showing similar calculations for the UK.

One Pence Two Pence Five Pence Ten Pence Twenty Pence Fifty Pence One Pound Two Pound Five Pound Weight (g) 3.560 7.130 3.250 6.500 5.000 8.000 9.500 12.000 28.280 Coins/kg 280.899 140.252 307.692 153.846 200.000 125.000 105.263 83.333 35.361 Value £0.01 £0.02 £0.05 £0.10 £0.20 £0.50 £1.00 £2.00 £5.00 Value/kg £2.809 £2.805 £15.385 £15.385 £40.000 £62.500 £105.263 £166.667 £176.803

Here is a table for the coins used in connection with the Euro.

One Centime Two Centime Five Centime Ten Centime Twenty Centime Fifty Centime One Euro Two Euro Weight (g) 2.300 3.000 3.900 4.100 5.700 7.800 7.500 8.500 Coins/kg 434.783 333.333 256.410 243.902 175.439 128.205 133.333 117.647 Value €0.01 €0.02 €0.05 €0.10 €0.20 €0.50 €1.00 €2.00 Value/kg €4.348 €6.667 €12.821 €24.390 €35.088 €64.103 €133.333 €235.294

My final example is coins used in Switzerland.

Five Centime Ten Centime Twenty Centime Fifty Centime One Franc Two Franc Five Franc Weight (g) 1.800 3.000 4.000 2.200 4.400 8.800 13.200 Coins/kg 555.556 333.333 250.000 454.545 227.273 113.636 75.758 Value SFr 0.05 SFr 0.10 SFr 0.20 SFr 0.50 SFr 1.00 SFr 2.00 SFr 5.00 Value/kg SFr 27.778 SFr 33.333 SFr 50.000 SFr 227.273 SFr 227.273 SFr 227.273 SFr 378.788

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