Suppose I want to determine how much fuel my car uses, on average, per km.
What I would do is to fill the tank and reset the odometer. Next time I need to go to the gas station, I would take note of how much fuel it takes to fill the tank again (the filled tank being a point of reference for fuel quantity) and of the number of km on the odometer.
Distance [km] / refill [l] = km made with one liter.
Maybe I would repeat this measure several times, to take into account that the car uses more or less fuel depending on speed, city or extraurban cycles or even the number of red lights I must stop at, but in the end I would have an average.
Anyway, I don't really want to do this, because traveling with a full tank is wasteful (I'm carrying around extra weight).
Dad says that it doesn't matter if I don't have that fixed reference, because I can just note down how much fuel per refill and the distance made between each refill and after repeating this process a dozen times I could get an estimate.
I have no idea which math he means to do, and he's not willing to share because he says my reasonings make him sick and he doesn't want to talk about it anymore. He wants me to show that he's right by taking the data he told me to gather and somehow computing it into a value that I have no reason to believe will be correct.
Since I don't think there's any way to get a realistic value without a point of reference, I'm afraid his value will be wildly off and useless, therefore I'm asking you if there's any method that could really give me a good estimate of the real value.
Additional facts that might be useful to know:
- The tank starts with a low fill, between the low fuel nock and the first white nock on the meter.
- I will try to never reach the low fuel nock. I have no idea how much km I can make after it beeps before the car stops and I will not risk discovering it.
- I will buy in 5€ increments if possible, so I will not be able to always put in the same amount of fuel every time as the fuel price changes from week to week.
- Sometimes, dad will use the car. He will possibly buy fuel, and he will note down the same data he told me to keep track of (odometer value at the fuel station and liters of fuel bought, and for some reason refill date). So, keeping track of additional data is not possible.
Let us take the theoretical "mileage" is $M$ km/litre which we want to estimate.
Let us the tank Capacity is $C$ litre where there is no indication of the fuel left inside, except that there is low fuel indicator which means there is less than $c$ litre left.
When this Indicator is ON, assume $x0$ litre is left.
Write the Distance on the ODO Meter = $D0$.
Now you can fill the tank & write the fuel filling litre $L1$ (known).
Move around until low fuel indicator is ON.
Let the Current values be $x1$ (unknown) & $D1$ (known).
Keeping writing it over weeks or months.
You will have these numbers : $x0,x1,x2,x3,x4,....,xn$ (unknown) , $L0,L1,L2,L3,L4,....,Ln$ (known) , $D0,D1,D2,D3,D4,....,Dn$ (known).
You can calculate these values each time you fill :
$M = [x0+(L1-x1)]/[(D1-D0)]$
$M \approx [(L1)]/[(D1-D0)]$ (because $x0-x1$ is small)
$M = [x0+(L1-x1)+x1+(L2-x2)]/[(D1-D0)+(D2-D1)]$
$M \approx (L1+L2)/[(D2-D0)]$ (because $-x1+x1=0$ & $x0-x2$ is small)
$M = [x0+(L1-x1)+x1+(L2-x2)+x2+(L3-x3)]/[(D3-D2)+(D2-D1)+(D1-D0)]$
$M \approx (L1+L2+L3)/(D3-D0)$ (because $-x1+x1=0$ & $-x2+x2=0$ & $x0-x3$ is small)
Over time, you will see that $M$ converges to the Exact value (or oscillates around this) because the total fuel consumed is large & total Distance travelled is large while the unknown Part is very small.
The Current term is given by this formula :
$M = [x0+(L1-x1)+x1+(L2-x2)+x2+(L3-x3).....+(Ln-xn)]/[(Dn-)+....+(D3-D2)+(D2-D1)+(D1-D0)]$
$M \approx (L1+L2+L3+....Ln)/(Dn-D0)$ (because $xn-x3$ is small)
In other words, this means : take Difference of Initial Distance & final Distance to calculate total Distance. Take total fuel filling accumulative , ignoring the low fuel left over. Take the ratio to get the APPROXIMATE Mileage which will eventually get to the EXACT theoretical Mileage.