# Italian plumbing/2 – Numerical analysis

They read this blog. Yesterday we received the water bill and I discovered another strange, new world of mathematics.

The company running the water system charges the water on a range basis: they put thresholds of cubic meters of water per year, the consumer pays the lowest price for the first 80 cubic meters, then more (a little less than the double) for the next 70 (a total of 150/year) and the highest price if more than 150 cubic meters are used in a year.

This seems to me a fair system: the more you consume, the more you pay and if you use a large amount of water you pay more. (I do not agree with the prices, but I don’t know the maintenance expenses)

How they manage to transmute a logical system in an Italian system? They apply the usage cap to the single bill, the first one, then stop reading the actual quantity and make hypotheses, billing on the supposed quantity.

In example: the first bill report an usage of, say, 43 cubic meter in the first 3 months, based on the actual usage. Then they split 43 for the days, say 92, covered by the bill: 43/92 = 0.46739 cubic meter/day. (I don’t know if they make a rounding here, I think not)

Multiply this result by 365 days and obtain the infamous year usage: 0.46739 * 365 = 170,59. Round it: 171 cubic meters/year.

So, 171 is split in that way: 80 in the first price, 70 in the second price, 21 in the third. And now the “funny” part: they bill the first 43 cubic meters using these values. So 43 is split: 43*80/171 = 20.11, rounded to 20; 43*70/171 = 17.6, rounded to 18; 43*21/171 = 5.28, rounded to 5.

So, the consumer pays 5 cubic meter at the highest price, even if she stays in the house only these 3 months, using an actual total of 43 cubic meter/year. Also, the rounding is made on a bill basis, then when added together the errors are summed up.

20.11 * 4 = 80.44 rounded to 80 versus 20*4 = 80
17.6 * 4 = 70.4 rounded to 70 versus 18*4 = 72
5.28 * 4 = 21.12 rounded to 21 versus 5*4 = 20
Total: 171 versus 172.

Think about it next time you see a rounding: the more the additions, the bigger the error.

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