Hi and welcome to the forum
How did you set the tank up?
eg: did you use any cleaning products to clean it originally?
Have you tested the tank water for ammonia, nitrite, nitrate and pH?
If yes, what are the results in numbers?
Did you use tap water to fill the tank up?
Did you add a dechlorinator when you set the tank up?
Did you use dedicated fish buckets for cleaning the tank and filling it, or any bucket around the house?
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Fish can usually spend an hour or more in a bag and not have any problems during transit. However, if the fish have only come into the shop in the last day or two, they will be stressed and won't travel as well to your house. Likewise if the shop has done water changes on their tanks in the last 24-48 hours, the stress of the new water from the water change, and then going into your tank with possible different water, can stress and kill fish.
If the shop assistant chases the fish around for 30 or 40 seconds it can stress the fish and they die on the way home or a day later.
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Test your tank water and make sure it is safe and suitable for fish. Make sure it has no chlorine or chloramine in.
Call the local pet shop and ask them when they get new fish in and when they do water changes.
You want to buy fish the day before they do a water change (or just before they do them), and if possible a week or more after the fish have come in. Unfortunately most shops get new fish in every week so that second part is hard to do.
eg: The shop does water changes on Monday and Tuesday and new fish come in on Wednesday. You go to the shop and buy fish Sunday or Monday before the water changes are done. This gives the fish a few days to recover from being shipped to the shop, and you get them before they get a water change.
You also want to get them before they are fed if possible. Most shops feed the fish in the morning so either get there before they feed them or go in later that day. Make sure the fish have not eaten in the last 3 or 4 hours so there will be less chance of them regurgitating the food in the bag.
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If you are travelling long distances, ask the shop if they can put oxygen in the bag. Some shops have bottles of oxygen and can put that in the bag instead of just air from the room.
Ask the shop to put the fish into a couple of bags (eg: 4 tetras in each bag). If the shop asks why, just tell them you have been losing fish the day or day after you get them and you want to minimise the stress on them by having fewer fish in each bag.
Put the fish into a dark carry bag and wrap them in a blanket or towel and put that in an esky (cooler). Put some more blankets or towels in the esky to stop the bag/s moving around. Put the lid on the esky and head home.
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When you get home, open the esky and remove the blankets but leave the fish in the carry bag for 5-10 minutes. This will let them have some light without being taken from complete blackness to bright light instantly. While this is happening, you turn the aquarium lights off and leave them off for the rest of the day. You can turn the tank light back on the following day.
Take the bags of fish and float them in the tank for about 30 minutes. After 5 minutes, add 1/2 to 1 cup of tank water to the bag. Do this every 5 minutes for the 30 minute period. Then gently pour the fish into the tank.
If the main display tank already has other fish in, then put the new fish into a quarantine tank so they don't introduce diseases into the main tank.
Don't feed the new fish until the day after you get them. Just leave the tank light off and let them settle down and recover from the trip.
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The following day open the curtains or turn the room light on and wait at least 30 minutes before turning the tank light on. This will give the fish a chance to wake up slowly and without being blinded by bright light.
At night turn the room light on, then turn the tank light off. Wait at least 30 minutes before turning the room light off. This gives the fish a chance to settle down for the night.
Use this method for turning the tank light on and off every day on any tank you have.
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Make sure the aquarium has a picture on the back to make the fish feel more secure. You can buy plastic aquarium backings (pictures) from any pet shop or online. You tape these to the back of the aquarium, on the outside. You can use newspaper, coloured card or even a plastic bin liner. As long as there is something across the back to make the fish feel safe.