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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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