I think Cummings drives a Range Rover. The fuel capacity of those vehicles is 57 litres or 12.5 gallons.

Now the comsumption figure given by Range Rover for Extra Urban driving is 44.1 miles per gallon. (I've never managed to achieve manufacturers figures in any vehicle)

So 44.1 x 12.5 will give a maximum range of 551.25 miles

London to Durham          269 miles
Durham to London          269 miles
Durham to Barnard Castle 26 miles
Barnard Castle to Durham 26 miles

Total 590 miles ......………….Ooops!

Just one further point if he is unsure whether he filled up is a quick check on his bank statement. Would only take a couple of minutes.