Ali is arranging the books on his bookshelves. He puts half his books on the bottom shelf and two-thirds of what remains on the second shelf. Finally he splits the rest of his books over the other two shelves so that the third shelf contains four more books than the top shelf. There are three books on the top shelf. How many books are on the bottom shelf？ 
Overseas Competition->Knowledge Point->Word Problem Modules->Solving Problems Involving Fractions and Percentages->Finding the Base
Since Ali places half his books on the bottom shelf and $$\frac{2}{3}$$ of the remainder on the second shelf, he places $$\frac{2}{3}\times\frac{1}{2}=\frac{1}{3}$$ of his books on the second shelf, leaving $$\left(1-\frac{1}{2}-\frac{1}{3}\right) =\frac{1}{6}$$~ of his books for the top two shelves. There are three books on the top shelf and four more, so seven books, on the third shelf. Therefore these $$10$$ books represent $$\frac{1}{6}$$ of the total number of books on the bookshelves. Hence there are $$60$$ books on the bookshelves and half of these, or $$30$$ books, on the bottom shelf. 