# 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 €5.601."

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.

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

1. Prices correct as of June 2018 []

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