This ratio measures the rate at which good I can be exchanged for good j and is what matters for economic choices. For example, the real wage is the wage paid for labour relative to the price of consumption and determines the trade-off between consumption and labour for a consumer. Two sets of prices represent the same set of relative prices if it is possible to change one set to the other by multiplying each price by a constant. This observation motivates the use of a numeraire commodity to allow all prices to be expressed relative to the price of the numeraire.
Reference: Oxford Press Dictonary of Economics, 5th edt.