Question 1189395
let x = the number of books the dealer had.
let h = the number of hard cover books.


1/5 of the books were hard cover.
therefore:
h = 1/5 * x


the dealer sold 2/3 of the books that included 3/4 of the hard cover books.


this means the dealer sold 3/4 * h.


this means the dealer did not sell 1/4 * h.


the fraction of the books that were not sold = 1/3 * x.


the ratio of the number of hard cover books that were not sold to the number of books that were not sold is (1/4 * h) / (1/3 * x).


since h = 1/5 * x, the ratio becomes (1/4 * 1/5 * x) / (1/3 * x).


that becomes (1/20 * x) / (1/3 * x).


that becomes (1/20) / (1/3) which is equal to (1/20) * 3 which is equal to 3/20.


the fraction of the unsold books that were hard cover was 3/20.


to test this out, i created an example.
in the example, i assumed the total number of books was 120.
since the number of hard cover books was 1/5 * the total number of books, then the number of hard cover books was 24.


2/3 of the books were sold, including 3/4 of the hard cover books.


what was sold was:
2/3 * 120 = 80 total books.
this included 3/4 * 24 = 18 hard cover books.


what was not sold was:
1/3 * 120 = 40 total books.
this included 1/4 * 24 = 6 hard cover books.


the ratio of unsold hard cover books to unsold books was 6/40 = 3/20.


this confirms the fraction is correct.


your solution is that the fraction of unsold books that were hard cover books is 3/20.