Should metric or KPI be a ratio or whole number ? Several popular books like lean analytics say a metric/kpi should be a ratio. In reality, all the top companies have whole number a KPI or metric: Facebook - MAU, eBay - GMV, Airbnb - number of bookings.
IMO ratios often involve 2 components. Ex: if denominator goes up the overall ratio will go down.
Eg total searches vs searches per unique user. The latter is useful to understand if user behavior is shifting to searching more (perhaps you want to measure the impact of a new search feature). It also normalizes for changes in user growth during the period you’re measuring.
That said, I don’t know why you have to limit yourself to just one metric and one type of metric.