Composting Ratios

The ratio is 30:1  I know this is really hard to achieve  but I had a general question
if I have a 5 gallon bucket of food scraps.   That I throw onto my compost pile.   How many 5 gallon buckets of shredded cardboard should I also be throwing onto the pile?  

The short answer is two parts green to one part brown, by the weight.
In theory, here is the calculation to get the number:
Material      C%   N%   C/N
Cardboard  40      0.1   400
Scraps         10-20 1-2     10

Assuming you have 10 lbs of Kitchen scraps containing 10% carbon and 1% nitrogen, to reach a C:N ratio of 30:1 in the compost, you will need n lbs of cardboard:

(10×0.1+0.4n)÷(10×0.01+0.001n)=30
n=5.4

In practice, it's less likely for people to weigh the materials out precisely. So just mix them up by the roughly 2:1 ratio and see what happens. since the cardboard is dry, I would make a slurry of kitchen scraps and soak the cardboard in to maximize the interacting surface area. The compost pile also needs to be protected somehow to avoid lost in moisture and heat. If it's not heating up then there is too much carbon; if it smells like ammonia, there is too much nitrogen.

I have found for myself a 3:1 is the most appropriate for my compost.

Moisture really is key, and I don't turn an awful lot but I can see my piles dropping volume quickly.

I like May's suggestion of 2 parts green manure and one part browns.

Just keep things not too wet and not too dry.

Turning whenever it seems right.

I ripped some cardboards in to pieces and mixed in my hot compost pile to test. Since a hot pile is bacteria dominated and lignocellulose is more readily digested by fungi, I didn't expect the cardboard to breakdown fast but they basically remained unchanged after over a month  (materials added constantly to maintain a hot pile).
I am wondering if certain additive in cardboard manufacturing is preventing it from microbial decomposition. I observed similar things while composting feathers and wools. Feathers from my chicken are halfway gone in less than a week while wool fabric scraps are still there after 3 years through several compost piles then buried in ground.

Adding a picture of stuffs I just fished out of the pile:
Feather added one week ago vs intact feather
Cardboard added in July still recognizable

Composting can get as complicated or as simple as the person wanting to make it wants to.

Some people get REALLY deep into the percentages and how much they put in.

I think a diversity of things both carbon rich and nitrogen rich gives me the best product instead of trying to keep its ingredients limited.

If I remember correctly, food scraps generally have a C:N ratio someware around 20:1... Shredded cardboard its closer to 300:1, you would need to add roughly 2 five-gallon buckets of shredded cardboard per every 1 five-gallon bucket of food scraps. This helps balance the nitrogen content of the food scraps with the carbon content of the cardboard, aiding in proper composting.