Welcome to the world of aquaponics Rponda..... so many questions... that's cool
I'll try and answer what I can, someone else will jump in where I can't ... or will do the metric/imperial conversions before I do ....
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How long do I let the tank establish itself before I try something like lettuce?
You can plant out the growbeds immediately... but you may not get any significant growth until the system has cycled sufficiently to start providing the nitrates for the plant growth.... then it'll explode...
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You speak of 'cycling' the tank before adding fish. What does that mean?
"Cycling" the tank describres the process of establishing the nitrifying bacterial processes to convert ammonia => nitrites => nitrates and returning the water back to the fish tank cleaned of solids and nutrients.
There are various methods of providing the ammonia to begin cycling, one of which is "seeding" with existing water from another system ... as you have done
I suggest you read through the sections and FAQ posts on the portal page.
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How many inches of fish are a good balance for a 100 gallon tank?
At a rough calculation (need to check my conversions) I'd say aim for 10 - 15 fish at edible plate size...
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Is there any reason to add hydroponics fertilizer while the fish are increasing in number and size?
Absolutely none... and doing so will probably kill your fish... AP is essentially totally organic.
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Can one mix goldfish and tilapia in a tank?
I believe so
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Am I correct in thinking that I do not need to add a filter as the plants will serve that purpose?
Correct.... although if you are running hydroponic channels/nft or drippers then you will probably need to setup a filter
before the hydro channels to remove the majority of the solids.... otherwise they will cling to the root ball and/or drippers
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My pump is a 170 gph pump for the approximately 100 gallon tank. Is that sufficient? I also have a 1,000 gph pump but it makes the tank look like a washing machine mid cycle.
Will depend somewhat on how often you can turn over your tank volume.... that depends on the capacity of your growbeds and pump cycles....
More elaboration will follow from others I'm sure.
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I thought that I understood from the postings that the ratio of grow beds to tank is 4:1. The diagrams I have viewed however show tanks that tend to be twice the size of the grow beds. Do I have the ratio reversed?
No you have the ratio in the correct order, but it is in fact 2:1 not 4:1
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I read several white papers on polyculture farming that describe the use of ducks, pigs, etc. waste as part of the cycle that creates the fish food. I have a vermicultre box going. Is there any value to tossing a handfull of the worms, castings and semi composted gunk into the tank? I know the worms will feed the fish. Will the other elements help or hurt the tank biosystem
There are numerous discussions and threads on the forum about polyculture and coexistant systems and inputs....
Worms I'd put straight into your growbeds.... vegetative scraps, well Talapia will dispose of most of them...
Just be aware that fish are extremely sensitive to their environment.... in general avoid adding anything to your system that ultimately will find it's way to the fish tank.... unless you know it's safe...
Other than the addition of some safe trace elements and some organic pest controls you don't have any need to add anything to an AP system