http://www.rubinghscience.org/surv/euroweights1.html
Sep 2004

Using Euro coins as standard weights

(c) 2004 Menno Rubingh
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Here is a cheap way to obtain a set of standard weights, for measuring weights up to about 100 gram, to an accuracy of about 0.5 gram : Use the coins of the currency of your country. Thus one can make use of the fact that these coins are fabricated in mass production, and because of that have not only a constant shape and size, but also a constant and accurately fixed weight.

Weights composed of these coins can be used to measure the weight of small objects (e.g. letters). The accuracy of the weight of the (Euro) coins is good enough so that in measuring the weight of objects an accuracy 0.5 gram is easily obtained in a cheap and quick way.(*)

(*) One cheap and quick way to construct things for weighing to up to 0.5 gram accuracy with the Euro coins is as follows:
1. Use a very simple home-made scales, i.e. a balance, simply consisting of a straight rod suspended exactly in the middle, with hanging from each end a 'plate' (or 'pan') on which to place the object to be weighed and the standard weights. A straight length of e.g. aluminium, two small identical pieces of board, and a small amount of supple thread suffice to construct a very serviceable scales.
2. Create the weights by holding the combinations of Euro coins together with narrow paper bands glued around them. The weight of the paper bands and the glue is much smaller than 0.5 g and is therefore negligible. (A sheet of A4 copying paper weighs about 5 gram.)

In the rest of this text, I'll use the Euro coins, to illustrate the idea. Also, I'll use the gram (abbreviated: 'g') as the unit of weight; that is, I'll assume that the weight of the objects to be measured should be determined in gram units. The weight of the Euro coins is as follows:

 Monetaryvalue of coin[Euro] Weight[g] 0.01 2.30 0.02 3.06 0.05 3.92 0.10 4.10 0.20 5.74 0.50 7.80 1.00 7.50 2.00 8.50
Table 1: Weights of the Euro coins
(Data obtained from: http://www.euroswapper.com/euro_coins.html)

Remarkably, the 1 Euro coin has a nice round weight of exactly 7.5 g; but in general, the makers of the coins have obviously not used it as a design criterion to give the coins a weight of a nice round number of grams...   These awkward numerical values of the weights of the individual coins seems the biggest hurdle to easy use of coins as standard weights (for weighing things cheaply and with a resolution of better than 7.5 g).

However, it is easy (with the Euro coins at least) to combine small numbers of coins together into combinations weighing a round number of grams. For example, the combination consisting of one 0.02 Euro coin (3.06 g), two 0.05 Euro coins (3.92 g each), and one 0.10 Euro coin (4,10 g), adds up to the round weight of exactly 15.00 g.

It is possible (with the Euro coins) to compose in this way a very satisfactory set of "standard weights", each consisting of a combination of a small number of coins. The following are the most useful combinations:

 Weight ofcombination[g] Number of coins of type Cost ofcombination[Euro] 0.01 0.02 0.05 0.10 0.20 0.50 9.96 3 1 0.05 10.04 2 1 0.09 15.00 1 2 1 0.22 15.50 2 1 2 0.14 16.00 2 1 0.70 16.50 2 1 1 0.62 17.00 1 2 1 0.42 17.50 2 1 1 1 0.34 17.50 3 1 0.35 18.00 4 1 1 0.26 20.00 3 3 1 0.14 25.00 3 1 1 1 0.71
Table 2: The most useful combinations of Euro coins
to obtain weights of a round number of grams.
An empty cell means that the coin is not used in the combination.

The combinations of 9.96 g and 10.04 g included in Table 2 above are the closest it is possible to get to a weight of 10.00 g.

In Table 2, the 15 g and 20 g weights are the mainstay of the weight set. With them, by placing them both at the left and at the right side in the scales, one can measure to a resolution of 5 g. The, relatively expensive, 25 g weight is not really strictly necessary, but I've added it for convenience.

By adding the 17.5 g weight to one's set of weights, the resolution is improved to 2.5 g. (The two possible coin combinations for the 17.5 g weight, both shown in the table, differ in cost only by 0.01 Euro.)

The remaining weights in Table 2 are included for those who want to improve their weight measuring resolution to 1.0 or 0.5 g.

In general, the larger the (desired) weight of the combination, the larger is the number of ways in which the coins can be put together to yield that given total weight. Small weights (made up of only a few coins) can only be composed in one or a very few ways; for large weights (made up of larger numbers of coins) of any given desired weight, in general a larger number of coin combinations exists that sum to that weight.

For desired weights of   n * 0.5 g (where n = whole number)
-- that is, weights of a whole number of grams or a whole number of grams plus 0.5 g --
it is, with the Euro coins, not possible to find combinations under a total weight of 10.50 g. The smallest possible combinations summing to   n * 0.5 g are:

10.50
11.00
11.50
15.00
15.50
16.00
16.50
17.00
17.50
18.00
20.00
From 20.00 g upwards, all combinations with a weight of   n * 0.5 g are possible.

To obtain weights of 30 g or larger with a weight of   n * 5 gram (n = whole number), that is, 30 g, 35 g, 40 g, 45 g, 50 g and so on, just put them together out of the combinations for 15 gram and for 20 gram (from Table 2). For example,

``` 50 g  =  (2 * 15 g) +      20 g        [ = 0.58 Euro ]
75 g  =       15 g  + (3 * 20 g)       [ = 0.64 Euro ]
100 g  =                5 * 20 g        [ = 0.70 Euro ]
```
... and so on.

Note (from Table 1), that the ratio of weight to monetary value of the Euro coins drops uniformly with increasing monetary value of the coin. That is, the smaller the coin, the more weight it gives you for the smallest cost. In order to keep things as cheap as possible, I have in the above therefore preferably composed my weights from from the smaller coins, as far as possible. For the same reason, I've used none of the coins of 1 Euro or more in my combinations.

When composing the larger   n * 5 gram weights from the 15 g and 20 g combinations, note that the 20 g combination, composed in relatively larger part from lower-value coins, is cheaper than the 15 g combination. (The 20 g costing 0.14 Euro, and the 15 g costing 0.22 Euro.)