It is a good question and in fact it is answered only by reading between the lines.
The Halachic terms are Ikkar ("main component" in the question) and Taffel (a lesser component).
From the Shulchan Aruch (Siman 212), the Achronim have concluded that the Taffel actually does require a Beracha but that it is included in the Beracha of the Ikkar. The evidence for that claim is that the Shulchan Aruch says that if you eat the Taffel before the Ikkar you actually have to bless it's Beracha (and later of course you also have to bless on the Ikker it's Bracha).
From that observation Achronim have concluded that the blessing determines the Ikker. So, to answer your question, you would bless on the carrot and pees (Boire Pri HaAdama) and the rice will not require it's own Beracha.
An interesting complication (to which I don't know the answer) would be if 3 components are present, each with a different blessing and none is over 50%...
Another note is that if you want to comply with the rule of the Chayei Adam who thinks that you should bless for each component if it is distinguishable (Nickar) then you should squash the dish and then make the (one) proper blessing (you cannot bless for each of the components because we fear the Issur of "Bal Tosif").
Note also the exception that if the smaller component is of the "5 grain types" (חמשת מיני דגן), even so, it is considered Ikkar (and will cause the suiting Beracha, which is Mezonot) providing that it gives its taste to the dish (as opposed to cases where it is just "a glue" for the other components etc).