# Exchange rates on board

“Three teas, please,” said the passenger ahead of me in the queue. The *Armorique* was due in Plymouth any minute, and tea was of the essence.

“That’s £4.65, or €5.60 ((Prices correct as of June 2018)).”

Hang on a moment, I thought, remembering to order my own tea as well. 560 isn’t a multiple of 3. What’s going on?

“That’s £3.10, please, or €3.75.” That’s not even, either!

I didn’t ask - tea, as I mentioned, was of the essence.

It was obvious (to me) that the prices were being calculated in sterling, then converted into euros - but a moment’s calculation suggested that there was also some rounding going on, presumably to the nearest five cents.

### How come?

If £3.10 is €3.75 to the nearest cent, then - abusing notation slightly - €1.20806… ≤ £1 < €1.21129…

If £4.65 is €5.60, then €1.20323… ≤ £1 < €1.20537…

There is no overlap between the possible exchange rates!

### However…

If we’re working to the nearest *five* cents, then €1.20161… ≤ £1 < €1.21774… from the first piece of information, and €1.19892… ≤ £1 < €1.20968…

*Those* rates overlap - and we can deduce that the on-board exchange rate is roughly between €1.202… and €1.210… to the pound.