Problem Solving

We can approach this problem by defining variables and building equations.

Let the capacity of Harold's old library be H, the capacity of Millicent's old library be M, and the capacity of their new library be N.

Harold has half as many books as Millicent: $$H=\frac{1}{2}M$$
The new library fits 1/3 of Harold's books and 1/2 of Millicent's books: $$N=\frac{1}{3}H+\frac{1}{2}M$$
Plugging in (1/2)M for H gives: $$N=\frac{1}{3}\left(\frac{1}{2}M\right)+\frac{1}{2}M$$ $$N=\frac{1}{6}M+\frac{3}{6}M$$ $$N=\frac{4}{6}M$$ $$N=\frac{2}{3}M$$
So the capacity of the new library is 2/3 the capacity of Millicent's old library, which is answer choice B.

swerve wrote: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 capacity?

A. 1/2
B. 2/3
C. 3/4
D. 4/5
E. 5/6

We can let the number of books Harold has in his old library be 30 and thus Millicent has 60 books in her old library. We see that Harold will bring 10 of his books and Millicent will bring 30 of her books to the new library, making their new library a capacity of 40. Since 40/60 = 2/3, their new library capacity is 2/3 of Millicent's old library capacity.

Answer: B