If a person's teeth bleed while fasting and the blood doesn't enter the throat, their fast remains valid. However, if the the blood does reach the throat, the quantity of saliva compared to the quantity of blood will be taken into account. If the quantity of blood is equal to or more than the quantity of saliva, the fast is invalidated. If the quantity of blood is less than the saliva, the fast will remain valid.
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دلائل:
الفتاوی الھندیۃ: (203/1، ط: دار الفکر)
الدم اذا خرج من الا سنان ودخل حلقہ ان کانت الغلبۃ للبزاق لا یضرہ وان کانت الغلبۃ للدم یفسد صومہ وان کانا سواء افسد ایضا استحساناً
Allah the Almighty knows best.
Darul ifta Al-Ikhlas, Karachi.