The expression 'precipitation ratio', by itself and out of context, is not meaningful: a ratio involves two quantities, so you need to sharpen your question first.
Rainfall intensity is a measure of the rate at which precipitation is occurring. For instance, if the total amount of water accumulation during a storm is measured at 6 liters per square meter (or mm), for instance, and if the storm actually lasts 3 hours, then the average rainfall rate (or intensity) for the whole event will be 2 liters per square meter and per hour. However, you may notice that the rain is initially very hard, and that the rate at which water drops then tapers off in time: in this case, one would say that the rainfall intensity is larger (perhaps 4 liters per square meter and per hour) initially, and only 0.5 liters per square meter and per hour in the latter part of the storm.
In summary, rainfall intensity is calculated by dividing the amount of rain accumulated by the period of time during which this happens. Rainfall intensity is to rainfall amount what speed is to the distance traveled in a mechanical analogue.
It is also worth being careful with terminology. "Rainfall intensity" is an expression that implies an instantaneous rate, that could change moment by moment. If you collect the rainfall in a gauge for an hour or a day, and then calculate the rate of accumulation during that hour or day, then that is correctly referred to as a "rainfall rate". So it is possible to speak of a (brief or momentary) "rainfall intensity of 5 mm per hour" and it is equally appropriate to refer to a "rainfall rate of 5 mm per hour" when this is the calculated average rate during the period of time when the rain was recorded.
Could you specify in which context, or perhaps in which document, you have encountered the expression 'precipitation ratio'? As I said in my first response, that expression is meaningless in isolation. We can't help you unless you provide some additional information on the nature of your question.
Could you please set a definition of the "precipitation ration", is it a ratio of the total rainfall dept to temporary time depth along the total rainfall duration, or what ?
Rainfall intensity is a ratio of the rainfall depth and the corresponding rainfall duration, either observed / measured or chosen such as designed.
Both Michel and David have provided good descriptions of rainfall intensity. I would add that the value of rainfall intensity is very much dependent on the time duration used. In general, the longer the time duration, the lower the intensity. In the field, this becomes important when attempting to quantify for instance, canopy interception, surface runoff and proportion of rainfall that enters the soil water store. For example, 30 mm of rainfall received in 30 minutes, will have very different physical outcomes compared with 30 mm of rainfall in 120 minutes.
As Michel and Jovan have mentioned, we need to know in what context you are using "precipitation ratio'? For example, to calculate the depth of rainfall from depth of snowfall, a precipitation ratio can be used (e.g. average snow 10:1, wet snow 12.5:1, or powder snow 7:1). Therefore, when a snowfall depth of 100 mm is measured for normal snow, this equates to 10 mm depth of rainwater per unit area.
Dear colleagues, I think that "precipitation ratio" means the total quantity of water harvested per day and that the “rainfall intensity” is the strength of a precipitation that means an averse, or a bruin or a huge (diluvium) rain, but it is also related to a certain amount of time, since we have different regimes of a rain even in one day.