(Last minute edit, use 31.65857 ;-) )
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I get different values for the multiplication factor depending on whether I use your bill's figures, or mine ! It has to be down to rounding factors I guess but ...
Looking at my bill it worked out at about 31.7, but looking at the figures you give it looks more like 31.5: but I guess they are in the same ball park.
Your example where if the difference between your last meter reading and the present one, i.e. what you used, is 37, then :
37 * 31.5 = 1,165.5 kWhrs which ≈ 1166 so 31.5 seems close enough.
But using 31.7 it would have been
37 * 31.7 = 1209.9 kWhrs which ≈ 1210 which isn't as close but not that far out.
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I know what you mean about the complex equation they say they use. But it doesn't seem to me as if they use what they say they do. Let me see what my gas bill actually says regarding working it out :
meter reading (in m³)
* calories per m³ (39.2)
* fiddle factor called 'volume correction' (1.0226400)
÷ conversion to kWhs (3.6)
So... 39.2 * 1.02264 / 3.6 = 11.13541333
I don't see how they can possibly be claiming that they use about 11.
Unless .... just had a thought ... they are using a metric meter as an example and assume they read m³ directly. Whereas there are plenty of imperial meters out there reading in, well I'm not sure, cubic feet I'd suspect. So they'd need a further multiplication factor of some value.