Attach a unit to each number
Use hypothetical GBP/USD = 1.25 USD/GBP and AUD/USD = 0.625 USD/AUD. Dividing them cancels the dollar unit: 1.25 USD/GBP ÷ 0.625 USD/AUD = 2.00 AUD/GBP. Under these assumptions one pound is equivalent to two Australian dollars. These are invented prices, not a live quote.
Why not multiply?
Multiplying 1.25 by 0.625 produces 0.78125, but multiplying the stated units does not leave AUD/GBP. If the second input were quoted as USD/AUD instead, the appropriate operation would change. The direction of the quote determines the arithmetic.
Compare the synthetic and direct observations
Write the provider, observation time and price type for the direct GBP/AUD quote and separately for each input. If a hypothetical direct quote is 1.998, the relative difference is (2.000 / 1.998 − 1) × 100 ≈ 0.1001%. The difference alone does not establish an executable arbitrage. Bid/ask spreads, unsynchronised observations and order terms are missing.
Using a euro-based reference table
In a table such as the ECB's, rates are quoted per euro. Divide AUD per EUR by GBP per EUR to obtain AUD per GBP. Unit cancellation is the same method even though the input arrangement differs. ECB reference rates are information, not execution prices.
A compact audit note
Record input one unit, input two unit, resulting unit, provider and observation time. If the desired unit does not remain after cancellation, revisit the expression before interpreting its result.