Can base currency influence absolute value for quoted currency

I am approaching this question from a Computer Science perspective (Developer, Programming etc.), so apologies if the question is trivial. I have been searching the web for a while, but I am afraid I am too unfamiliar with the domain to yield useful answers, so a brief explanation and maybe some references / pointers would be welcome:

Let’s say I have currency ABC, DEF and XYZ. (Like USD, EUR, GBP etc.) Now if I want to convert ABC into any of these, I’ll end up having something like this, using ABC as the base:

1 ABC = 1.5 DEF
1 ABC = 0.75 XYZ

Now if math holds true, then the if I change the base from ABC to DEF and XYZ, I should see this:

1 DEF = 2/3 ABC
1 DEF = 0.5 XYZ

1 XYZ = 2 DEF
1 XYZ = 4/3 ABC

Now, the questions:

1. Is this correct? or is it possible that the rates are different? let’s say 1 DEF = 0.56 XYZ instead of 0.5, in the above example?
2. If it’s not the case, what is a reliable source to find out the exact rates for different base currencies? (I am looking for a bit more data than a simple google exchange calculator though)
3. Also, can the difference be so significant that after doing a loop I end up with more money? (e.g 100 ABC --> 150 DEF ==> 80 XYZ –> 106 ABC)

Edit

As @DStanley pointed out, reciprocal values are not the same, so let me clarify further.

In other words, where I am going at is something like this: is this ‘base currency’ a thing in exchange or not really?

• If you have a ‘base’ in CAD, the USD --> EUR ratio is 1.27
• If you have a ‘base’ in CHF, the USD --> EUR ratio is 1.28
• If you have a ‘base’ in USD, the USD --> EUR ratio is 1.30
• If you have a ‘base’ in GBP, the USD --> EUR ratio is 1.32

Here’s an ‘example’ maybe this helps to conway what I want to know.

Lets say you have a 130 USD in bank notes and you want to convert it to EUR. You get the EUR sell price, and you’ll be able to buy 1 EUR for 1.3 USD, so you’ll get 100 EUR. Now, if you want to buy USD with your 100 note, you’ll only get 120 USD for the 100 EUR. (so 10 bucks is the profit for the intermidiery. So far so good)

Now what if, you have the same 130 USD in bank notes, you still want EUR but you are from Canada. Is there a mandatory USD --> CAD, CAD --> EUR transition first, or can the canadian institute work with the 1.3 ratio as the US counter-part. Or, and this is the most important bit I guess, can they give an EUR for 1.27, instead of the 1.3 because of some ‘magic’?

Apologies I am struggling to formulate the question itself. It really highliths my lack of domain knwoledge and agan, thanks for the help!