Harold and Millicent are getting married and need to combine their already-full libraries. If Harold, who has 1/2 as many books as Millicent, brings 1/3 of his books to their new home, then Millicent will have enough room to bring 1/2 of her books to their new home. What fraction of Millicent's old library capacity is the new home's library capacit

Let x be the number of books that Millicent has. So the number of books that Harold has is x/2. And the number of book that Harold is bringing to the new home is (x/2)/3 = x/6.

The home has enough room for another half of Millicent's books, which is x/2. So the total books capacity of the new home is

[tex]\frac{x}{6} + \frac{x}{2} = \frac{x}{6} + \frac{3x}{6} = \frac{4x}{6} = \frac{2x}{3}[/tex]

Since x is also the number of books in Millicent's old library capacity, we can conclude that the fraction of Millicent's old library capacity to the new home's library capacity is 2/3, or 2 thirds