Which Alice?

ALICE IN PUZZLE-LAND
Hare had one-quarter, and since three-quarters is three times as
much as one-quarter, the Hatter did indeed have three times as
much as the March Hare to start with."
"Good," said the Gryphon. "Now then, after he gave the March
Hare one tart, he had twice as many tarts as the March Hare. What
fraction of all the tarts did he then have?"
"Let's see," said Alice. "Reasoning the same way, he had two parts
to the March Hare's one part—in other words, he had two tarts to
every one of the March Hare's, or, out of every three tarts, the
Hatter had two and the March Hare had one. This means that the
Hatter had two-thirds of the tarts and the March Hare had onethird."
"Quite right," said the Gryphon.
"Well, where do we go from there?" asked Alice.
"Ah," said the Gryphon, "the whole point is that by giving the
March Hare just one tart, the Hatter reduced his share from threequarters
to two-thirds. Now, how much reduction is that? In other
words, what fraction of the tarts when taken away from threequarters
leaves two-thirds?"
"I'm not sure I understand you," replied Alice.
"What I'm really asking is how much is three-quarters minus twothirds?
That's the amount needed to take away from three-quarters
to get two-thirds!"
"Oh, I understand!" said Alice. "Let's see now, three-quarters
minus two-thirds? I think we better first reduce everything to
twelfths."
"You sure better!" replied the Gryphon.
"Well, three-quarters is nine-twelfths, and two-thirds is eighttwelfths,
so the difference is one-twelfth."
"Right," said the Gryphon. "And now can you solve the puzzle?"
"I still don't see how!" replied Alice.
"Then you've missed the whole point!" replied the Gryphon.
"The point is that by giving the March Hare one tart, he gave him
one-twelfth of the total number of tarts. So one tart is one-twelfth
the number of tarts. Therefore—"
"Therefore there were twelve tarts!" interrupted Alice excitedly
"That means the Hatter originally had nine—which is three-
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