My company compares office rental price in multiple currency and units. Users can choose to view the price in USD per square foot or EUR per square metre or GBP per square yard, etc.
For the sake of simplicity, let's say we are trying to convert the price from USD per square foot to EUR per square foot. In the specs, the business analyst wrote the conversion equation like this:
$$ \frac{\text{USD}}{\text{ft}^2} \times \frac{\text{EUR}}{\text{USD}} = \frac{\text{EUR}}{\text{ft}^2} $$
The equation looks fine to me. Now, let's further assume that
$$ 1 \text{ EUR} = 1.14 \text{ USD} \Rightarrow{} \frac{\text{EUR}}{\text{USD}} = 1.14 $$
But substituting this into the first equation gives incorrect result. For the same piece of real estate, the price in euro must be lower than the price in US dollars because a euro is worth more than the US dollar. The correct factor to use is:
$$ \frac{\text{EUR}}{\text{USD}} = \frac{1}{1.14} $$
What did I do wrong?
Let's say you have something that sells for $10 \text{ EUR}/m^2$. We'll call this $P$.
Let's assume the exchange rate is $1.14 \text{ USD/EUR}$. This is, for all intents and purposes, the number $1$, since it's one thing divided by the same thing.
To change the units, you multiply (or divide) $P$ by this conversion factor such that you're left with the units you want:
$$\require{cancel} \\ 10 \frac{\cancel{EUR}}{m^2} \cdot 1.14 \frac{\text{USD}}{\cancel{EUR}} = 11.40 \frac{\text{USD}}{m^2}.$$
This may be fast and breezy from a pure math standpoint, but it should get you to where you need.
Every conversion factor like this is essentially $1$. You don't want to change the absolute rate of what you're selling, just the units. So, to figure out the correct conversion, put equivalent quantities in numerator and denominator, and it will act like the number $1$.
So, if $1 \text{ EUR}$ and $1.14 \text{ USD}$ are the same amount of money (as in, will buy the same cup of coffee) then the conversion factor is either
$$\frac{1 \text{ EUR}}{1.14 \text{ USD}}$$
or
$$\frac{1.14 \text{ USD}}{1 \text{ EUR}}.$$
If you want to go from USD to EUR, use the first one. The USD units will cancel, and you'll be left with EUR (a smaller number, because each EUR is worth more than each USD).
If you want to go from EUR to USD, use the second one. The EUR units will cancel, and you'll be left with USD (a larger number, because each USD is worth less than each EUR).