Q:

What ratio of rice is needed for lentils to be a "complete protein"?

From what I understand Lentil needs Rice for the maximum benefit of its high protein quantity, of course that will then increase the carbs. What is the minimum ratio of rice to lentils to maximize protein to carb ratio while making the protein complete?

Incomplete vs. Complete Protein
  Complete proteins contain all nine essential amino acids. The main sources of complete proteins are from animal based food, such as meat, seafood, eggs and dairy. Soy and quinoa are also complete proteins. Although incomplete proteins sounds like they are lacking and not as nutritious, they just need to be paired with another type of protein. For instance, adding peanut butter to bread creates a complete protein. According to Columbia University, the proteins do not need to be eaten together to receive the health benefits but any time within 24 hours will suffice.
Lentils and Protein
  Lentils are rich in amino acids and high in protein. However, regular lentils are lacking in two essential amino acids. Because lentils do not have enough of these nutrients, the healthy legumes are an incomplete protein. However, lentils can be sprouted which changes their nutritional components. Sprouted lentils have an increase in all nine amino acids, although the exact increase is variable. To sprout your lentils, soak the seeds in cool water for eight to 12 hours. Rinse the lentils and store in a jar in a cool place for a few days, rinsing every eight hours.

Source: http://www.livestrong.com/article/527529-myth-of-lentils-as-an-incomplete-protein/

Grains
  Grains, such as rice, oats, wheat, rye and corn, can act as complementary proteins for legumes such as lentils. They contain the cysteine and methionine that lentils lack, and lentils provide the lysine that grains do not contain enough of. Whole grains are a more nutritious option than refined grains. Legumes and grains are one of the most common pairings for complementary protein. Indian daal with rice is an example of a lentil and grain dish.

Source: http://healthyeating.sfgate.com/foods-mixed-lentils-provide-complete-source-amino-acids-1195.html

A:

Preamble
I will act in the interest of the OP and not close the question. I assume that my premise that the ratio doesn't matter (explained in the answer proper) is an established fact (at least I don't know of any source claiming the opposite). If we find a claim to the contrary, we will have to close the whole question, because it means that there is no undisputed answer even at that very general level, and the question falls completely in the field of "unverifiable" nutrition. 
Answer
There is no such ratio, as complete proteins are not about ratios. They are about minimum intake of essential amino acids. This means that, if you would only eat rice and lentils and nothing else (or at least no other sources of protein), there is a minimal amount of rice you have to eat per day to not get deficiency symptoms. If this amount of rice is X gram, then it doesn't matter if you eat X gram of rice and 2X gram of lentils, or X gram of rice and 10X gram of lentils. The ratio of rice to lentils is irrelevant. 
The more interesting question for you is probably what is the minimal amount of rice you have to eat per day. There are freely available nutrition value databases such as the one derobert linked. Once you have determined that you want to eat at least Y mg of a given amino acid per day (supposedly it will be an amino acid not contained in lentils), you can use a simple rule of three calculation to find out how much rice you need to eat in order to get Y mg of the amino acid in question. 
But how do you decide how much you want to eat? There are government approved RDAs, as well as thousands of sources at all levels of credibility which recommend their own intake levels, usually pointing out that RDAs are intentionally conservative. We cannot help you with that decision. It is completely off topic for us, and it is up to you to find such sources and decide which one to believe. Consulting a nutrition professional is a very good option if you are going on a highly restricted diet; they will probably be able to give you such a value. Personally, I wouldn't trust a number given to me by a stranger on the Internet, and would try to keep in mind that the minimal amount (under which you get deficiency symptoms) and the optimal amount are most likely not the same, but many sources conflate them. 

