## Going to Cambridge

Errors, Accuracy, Precision.

Dad drove to Cambridge last weekend. He filled the petrol tank on the way home from work on Friday before he set off and he filled up again when he reached Cambridge. He says it is 320 miles to Cambridge. He says he put 32 litres in on Friday and another 30 in on Saturday afternoon on the edge of Cambridge. What do you think the car’s consumption is in miles per gallon?

Yes, you think this is a trivial problem. Okay, so do it already. Then read what follows.

I don’t believe you did a calculation yet….

Why not accept the explicit challenge - to come up with an answer, and the implicit one to make it a good answer?

i) Lower school methods: 320/30 is miles per litre and this answer *4.54 is m.p.g. A good student observes that the Friday figure of 32 litres is irrelevant. A good student at this level rounds their answer back to an appropriate precision, probably to 2 s.f. - two significant figures.

ii)  Higher order thinking: 320 miles is 2 s.f. implied precision, so the distance is between 315 and 325 miles*. When he fills the tank, what does ‘full’ mean? Is it the same ‘full’ every time? Well, probably not, in any exact sense. Since he filled the tank with 32 litres on Friday, we might assume he gives the 30 litre figure to 2 s.f. too. We need to consider that the previous 32 litres’ ‘full’ is not the same as the 30 litres’ ‘full’, just similar. Experience suggests he might well have a variability of 10%, i.e. he used 27 to 33 litres; you might say he’ll be closer than that, 29-31 litres. So the calculation has, as the lower bound, 315/33*4.54 and as the upper bound 325/27*4.54. However  you do it, his figures give a range of answers and the true value is somewhere in between, but not exactly halfway between the limits.

iii)  So, when Dad starts sounding off about the car’s improved consumption on a long run, he needs to be challenged on his assumptions, doesn’t he? There is not enough evidence – unless and until he takes more care with recording the process of filling the tank: he needs to go from absolutely full and return to that state. He could go from empty to empty, but that produces hazardous driving!

Maybe Dad will let you fill the tank; the nozzle you stick into the car has a sensor that notices when there’s back pressure, so as to avoid over-filling the tank. If the pump is pushing hard enough, this sensor operates frequently when you push the end of the pipe all the way into the car. Frequent users hold the nozzle a little way back so that the sensor operates properly and stops when the fuel reaches up the feeder pipe, not just when it is struggling to go down that pipe. Dad quite possibly ‘fills up’ to within the last possible litre, or perhaps instead to teh first time the pump indicates the tank is almost full. Over-filling the tank (filling the whole of the feeder pipe) is likely to cause leakage around the tank cap and staining of the paintwork. Motorcyclists probably fill with less variation but a bigger % range, since their bike tanks are usually ridiculously small. You might explore the range (distance between tank-filling) of a car, bike, or lorry. I suggest typical figures would be 200, 400, 600 miles, though there is enough variety for many to disagree with me. I ended up being so annoyed with the size of the tank on my first bike it became a deciding factor in buying the next one, and then similarly with cars. On a bike, you don’t mind stopping so much as you need to stretch; my second bike had a range over 250 and the bigger cars I owned might do 600 miles.

Yes, I used gallons. Dad still thinks in those, not litres. When he says 32 litres, he doesn’t mean 35 - we assume he is giving us good information.

(ii*) 320 miles could be 3s.f. but then Dad probably would have said “320 exactly”; he didn’t, so it is assumed to be “about 320”. Since Dad said 30 litres and 32 litres, we will assume he is using the same precision, because taking ‘thirty’ as between 25 and 35 is most unhelpful and makes Dad inconsistent with his numbers. Since you’re reading this, we will assume that your Dad is reasonably good with figures, too.

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