Is it just me or is the convention for how exchange rates are written “backwards”?
E.g. here in the UK the rate between pounds and US dollars is written “£/$”, e.g.
£/$ = 1.56
In other words, there are 1.56 dollars per pound.
But the expression “£/" *looks as though* it should read "**pounds per dollar**", by analogy with "m/s" (metres per second), and so on. And the number of pounds per dollar is of course the reciprocal of 1.56, or about 0.64.
But then looking at it another way, I can see how "£/” would be correct:
I don’t think it’s a mathematical construct, but a left-to-right reading shortcut, i.e., £/$ is a shortcut for “pounds into dollars.” If old type had included the → symbol it might not have been a big deal.
Perhaps, but the / symbol is so commonly read as “per” that it seems pretty confusing to me: m/s means metres per second, but £/$ doesn’t mean pounds per dollar, it means the exact opposite!
(1 pound) / (1 dollar) = 1.56 - if you divide the value of a single pound by the value of a single dollar, the result is 1.56.
You can use it like a constant, like gravitational acceleration - X =1.56 £/$ . That way, when you’re doing conversion, the units will work out right.
You’ve got $100, how much in pounds?
$100 * X = 100 * 1.56 £/ = 100 * 1.56 * ( * £ / ) = 100 £ - there’s a $ in the numerator and denominator so they cancel.
You’ve got £100 how much in dollars?
£100 * 1/X = £100 * (1/(1.56 £/)) = £100 * (0.641 /£) = (100 * .641) * (£ * $ / £) = $64.1 - this time, there’s a £ in the numerator and denominator, so they cancel.
But your examples have it backwards, which is exactly what I mean! (I assume you made a typo in the first example and you meant the answer to be “156 £” as the sum indicates.
$100 = £64, not £156.
£100 = $156, not $64.
That’s what I meant when I said, shouldn’t it be written “$/£”? That way, the sums work:
Not a useful answer, but one could imagine that the symbols for dollars and pounds represent a constant that represents the intrinsic worth of the currency against some independent reference. For instance, stepping back 100 years, how much gold will one unit of each currency buy. Then it works.
1USD buys 0.02517 grams of gold. Thus 1$ = 0.02517
1GBP buys 0.03934 grams of gold. Thus 1£ = 0.03934
Then indeed £/$ = 1.5629
(OK, this is contrived, but it does at least make sense.)
I wonder where it is you are seeing this. I’m going to say it’s not representing a ratio but shorthand for a currency pair, though I’ve never seen it written with currency symbols instead of the three letter ISO codes for currencies.
In the financial markets, the exchange rate between Pounds Sterling and US Dollars is quoted as “GBP/USD”, where the first currency stated is the “base currency” and the second one is the “counter currency” or “quote currency”. The rate is quoted in terms of how many units in the counter currency you need to pay to get back one unit of the base currency. Also conventionally, GBP “outranks” USD in the market conventional ordering for quoting FX spot rates by base currency, so one would always quote GBP/USD and not USD/GBP.
As of right now, the current GBP/USD spot rate on Yahoo! Finance is 1.5609. This means that I’d have to pay USD 1.56 to get back one Pound Sterling. Whether I was long or short GBP as a currency trader (buying or selling USD versus Sterling), this is how I would ask for the going rate.
However, quoting FX spot rates is typically reserved for people in the capital markets. AFAIK, the little booths at the airport or around town that do currency exchanges always give exchange rates in both directions as ratios, to make it easier for people doing transactions either way who aren’t currency traders or portfolio managers hedging FX exposure…
Normally, when you see a ratio, the two sides are equal. If I say “2 kids per family”, then “2 kids” is equivalent to “1 family”. But with currency, the top and bottom are not equal in value, they’re equal in number. The rate is 1X/1Y.
In your example, it’s 1p/$1 = $1.56/$1 = 1.56.
Think of a bag of Doritos. They cost $1. If you have 1pound, that’s 1.56 bags of Doritos. If you have $1, that’s one bag.
So:
P/$
1.56 bags/1 bags = 1.56
Your error is coming from the idea that the ratio on the left hand side is the unit of the number on the right hand side. It’s not. The number on the right is unitless and the unit on the left is numberless. If you wanted to get a unit for the number, you’d have to multiply by the reciprocal.