Page "Ghost Festival" Paragraph 9

Maudgalyāyana then sought the Buddha's advice to help his mother gain a human birth.

The Buddha established a day after the traditional summer retreat ( the 14th day of the seventh month in the lunar calendar, usually mid-to-late August ) on which Maudgalyāyana was to offer food and robes to five hundred bhikkhus.

Through the merits created, Maudgalyāyana's mother finally gained a human birth.

which Buddhist sutra, did the Buddha set on the 14th day of 7th lunar month ???

Ullambana or Zhong Yuan Jie is in fact on the fullmoon day ( 15th day ) of the 7th month ); the reason the rituals practiced today is on the 14th day is because it's customarily practised on the midnight of the 15th, very much like the celebration of Jade Emperor's birthday which starts on the eve of the ninth day of the first lunar month.

Due to long-hour ceremony of ancestral worship and charitable rituals for the hungry ghost, many started to do it on the 14th day instead.