Is it permissible to give zakaah to a family fund to help the poor?

Whoever has money that reaches the nisaab (minimum amount at which zakaah becomes due) and a full hijri year has passed (since acquiring it) has to pay zakaah on it. 

This nisaab for money is the equivalent of 85 grams of gold or 595 grams of silver, whichever is lower in value.  

See al-Sharh al-Mumti’, 6/103-104); Majmoo’ Fataawa Ibn ‘Uthyameen, 18/248.

Based on this, the amount that you have – 5000 pounds – reaches the nisaab, so you have to pay zakaah. 

Secondly: The amount that you have to give is one-quarter of one-tenth, or 2.5%. With regard to the amount in question – 5000 pounds – the zakaah that is due is 125 pounds. 

Thirdly: If a hijri year has passed since you acquired the nisaab, then zakaah must be paid immediately and it is not permissible to delay it. See al-Sharh al-Mumti’, 6/186. 

Based on this, your delaying zakaah for half a year after the full year had passed is a shortcoming and you have to repent from that and hasten to give the zakaah. 

Fourthly: With regard to your question, “If anything is left of this 5000 after another year has passed, will I have to pay zakaah on it?”, the answer is: every time a year has passed and the nisaab (minimum amount) is still there, zakaah is due on it. If the amount of money falls below the niasaab then no zakaah is due. If the money reaches the nisaab again then zakaah is due again, and the reckoning of the year should start from the time when the money reaches the nisaab. 

Fifthly: With regard to giving the zakaah to the family fund, there are eight areas in which zakaah may be spent, which Allaah has described in the verse where He says (interpretation of the meaning): 

“As-Sadaqaat (here it means Zakaah) are only for the Fuqaraa’ (poor), and Al‑Masaakeen (the poor) and those employed to collect (the funds); and to attract the hearts of those who have been inclined (towards Islam); and to free the captives; and for those in debt; and for Allaah’s Cause (i.e. for Mujaahidoon — those fighting in a holy battle), and for the wayfarer (a traveller who is cut off from everything); a duty imposed by Allaah. And Allaah is All-Knower, All-Wise”

[al-Tawbah 9:60]

Giving zakaah to poor relatives is better than giving it to others, because the Prophet (peace and blessings of Allaah be upon him) said: “Charity given to a poor person is charity, and charity given to a relative is two things: charity and upholding the ties of kinship.” Narrated by al-Nasaa’i, 2582; classed as saheeh by al-Albaani in Saheeh al-Nasaa’i.

 There is nothing wrong with giving zakaah to the family fund, so long as you tell the person in charge of the fund that this is zakaah, so that he will spend it in one of the areas defined in the verse quoted above, and he will hasten to spend it immediately.  

And Allaah knows best.