bryan88 wrote:Michael arranged all his books in a bookcase with 10 books on each shelf and no books left over. After Michael acquired 10 additional books, he arranged all his books in a new bookcase with 12 books on each shelf and no books left over. How many books did Michael have before he acquired the 10 additional books?
(1) Before Michael acquired the 10 additional books, he had fewer than 96 books.
(2) Before Michael acquired the 10 additional books, he had more than 24 books.
We can use some Integer Properties rules to solve this one.
Let X be the original number of books.
Since we can place 10 books on each shelf with no books left over, we know that X is a multiple of 10.
Let Y be the new number of books, once we add 10 books (i.e., Y = X+10).
Since we can now place 12 books on each shelf with no books left over, we know that Y is a multiple of 12.
Important: If X is a multiple of 10, we also know that X+10 is a multiple of 10. In other words, we know that Y is a multiple of 10.
So, we know that Y is a multiple of 10 and 12.
Since 60 is the least common multiple of 10 and 12, we can conclude that Y is a multiple of 60.
So, some possible values of Y are 60, 120, 180, 240, etc.
This means that some possible values of X are
50, 110, 170, 230, etc.
The target question asks us to find the value of X
Statement 1: X < 96
So, the original number of books must be
50.
SUFFICIENT
Statement 2: X > 24
So, the original number of books could be
50, 110, 170, etc
NOT SUFFICIENT
So, the answer is
A
Cheers,
Brent
