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The eldest of these, and Bilbo’s favourite, was young Frodo Baggins. When Bilbo was ninety-nine he adopted Frodo as his heir, and brought him to live at Bag End; and the hopes of the Sackville- Bagginses were finally dashed. Bilbo and Frodo happened to have the same birthday, September 22nd. ‘You had better come and live here, Frodo my lad,’ said Bilbo one day; ‘and then we can celebrate our birthday-parties comfortably together.’ At that time Frodo was still in his tweens, as the hobbits called the irresponsible twenties between childhood and coming of age at thirty-three.

Twelve more years passed. Each year the Bagginses had given very lively combined birthday-parties at Bag End; but now it was understood that something quite exceptional was being planned for that autumn. Bilbo was going to be eleventy-one, 111, a rather curious number, and a very respectable age for a hobbit (the Old Took himself had only reached 130); and Frodo was going to be thirty- three, 33, an important number: the date of his ‘coming of age’.

Tongues began to wag in Hobbiton and Bywater; and rumour of the coming event travelled all over the Shire. The history and character of Mr. Bilbo Baggins became once again the chief topic of conversation; and the older folk suddenly found their reminiscences in welcome demand.

No one had a more attentive audience than old Ham Gamgee, commonly known as the Gaffer. He held forth at The Ivy Bush, a small inn on the Bywater road; and he spoke with some authority, for he had tended the garden at Bag End for forty years, and had helped old Holman in the same job before that. Now that he was himself growing old and stiff in the joints, the job was mainly carried on by his youngest son, Sam Gamgee. Both father and son were on very friendly terms with Bilbo and Frodo. They lived on the Hill itself, in Number 3 Bagshot Row just below Bag End.

‘A very nice well-spoken gentlehobbit is Mr. Bilbo, as I’ve always said,’ the Gaffer declared. With perfect truth: for Bilbo was very polite to him, calling him ‘Master Hamfast’, and consulting him constantly upon the growing of vegetables — in the matter of ‘roots’, especially potatoes, the Gaffer was recognized as the leading authority by all in the neighbourhood (including himself).

‘But what about this Frodo that lives with him?’ asked Old Noakes of Bywater. ‘Baggins is his name, but he’s more than half a Brandybuck, they say. It beats me why any Baggins of Hobbiton should go looking for a wife away there in Buckland, where folks are so queer.’

From
The Lord of the Rings
by J.R.R. Tolkien

(I thought I would post some fiction that gives equal support to your contention as what you posted. Thought it would help.)