Simple currency problem

[Northbloke]

I enjoy those problems that look obvious but aren't necessarily and came across a real one with a pal yesterday. Simplified numbers to make the point.

He agreed to buy an item for 400 drogna when the exchange rate was 2.5 so £160.

However he wasn't completing the transaction for a couple of months so they verbally agreed that if the exchange rate changed they would split the difference and settle up on that.

That was yesterday and the rate was 2.0.

At what £ price should the item change hands?

I'm probably having a senior moment but I'll see what others say first.

[DocJock]

£180

[Mgd_uk]

£177.77

[langtounlad]

Splitting the difference of the exchange rate = 2.25 and so the £ conversion = £177.78.
That is strictly interpreting what you wrote but might not be what one or both parties intended.

[IJWS15]

"He agreed to buy an item for 400 drogna when the exchange rate was 2.5 so £160."

So the payment is 400 drogna. Which you can sell for £177.

[Northbloke]

Yes, it was only a friendly verbal agreement (and the words here may not be exact) but it turns out it was ambiguous.

Split the value difference = £180
Split the rate difference = £177.77

Not a big deal in this case so no problem but could have been if the rate change had been bigger.

But both my pal and I expected it to be the same by either method and were quite surprised that it isn't. (I've even stuck different rate changes into Excel to graph it).

I can't quite get my head around why it is different? Can anyone explain mathematically? I know you can't average averages and if something drops 20% you need to raise it 25% to get back to par. Is there something like that going on here?

[Toltec]

It is because the average of the reciprocal is not the same as the reciprocal of the average.


(1/X1 + 1/X2)/2 does not equal 2/(X1 + X2)

i.e.

(1/2 + 1/2.5)/2 = (0.5 + 0.4)/2 = 0.9/2 = 0.45

2/(2 + 2.5) = 2/4.5 = 1/2.25 = 0.44 recurring

Edited by Toltec on Tuesday 13th March 16:44