All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
When the following conditions are met, Zakah of trading goods must be paid.
1) When the value of all the trading goods combined accumulated with the cash that a person has reaches the level of Nisaab (85 grams of pure gold).
2) A complete lunar year should pass after owning the above.
As for how to pay the Zakah, the merchant should estimate the value of the goods according to the market rate at the time of Zakah regardless of their original price, then he should add profits he has gained till the time of obligation and also the loans that he is certain will be repaid to him and then he should pay 2.5% of the entire amount.
If a merchant wants to pay the Zakah before completing a whole year he may pay it, but must have the intention of Zakah. Imaam At-Tirmithi said: ‘The majority of Muslim scholars are of the opinion that paying Zakah before the completion of one year is permissible.’
Imaam Ibn Taymiyyah said: ‘It is allowed to pay the Zakah before it becomes obligatory (completing a year) and after reaching the Nisaab.’
Therefore, one may pay the Zakah in installments and at the end of the year he should check how much he has paid in Zakah. If he has paid less than what he had to pay then he should pay the remaining amount as Zakah. If he had paid more than what is obligatory on him then the remaining amount is considered as charity.
As for giving Zakah to a poor relative for whom spending is not obligatory upon you, this is permissible; it is even likable since the Prophet said: "Charity given to a poor man is just charity, but when given to a relative it serves a double purpose, being both charity and maintaining kin relations”. [Ahmad, An-Nasaa’ee, Ibn Hibbaan]
Therefore, you may give your Zakah to your aunt who cannot bear her medical expenses.
Allaah knows best.