algebraic way? OG Quant

[bryan88]

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.

[Brent@GMATPrepNow]

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

[Brent@GMATPrepNow]

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.

Alternate approach: Listing and looking for a pattern

Notice that, if we can place 10 books on each shelf with no books left over, then the original number of books must be a multiple of 10.
Once we add 10 books, we can place 12 books on each shelf with no books left over. So, the new number of books must be a multiple of 12.

Let's list possible numbers of original books, and then see what happens when we add 10 books.

10 originally --> 20 after the addition (20 isn't a multiple of 12)
20 originally --> 30 after the addition (30 isn't a multiple of 12)
30 originally --> 40 after the addition (40 isn't a multiple of 12)
40 originally --> 50 after the addition (50 isn't a multiple of 12)
50 originally --> 60 after the addition (60 is a multiple of 12) WORKS!
60 originally --> 70 after the addition (70 isn't a multiple of 12)
70 originally --> 80 after the addition (80 isn't a multiple of 12)
80 originally --> 90 after the addition (90 isn't a multiple of 12)
90 originally --> 100 after the addition (100 isn't a multiple of 12)
100 originally --> 110 after the addition (110 isn't a multiple of 12)
110 originally --> 120 after the addition (120 is a multiple of 12) WORKS!
.
.
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We can see a pattern here. The original number of books can be 50, 110, 170, 230, etc.

Statement 1: the original number of books is less than 96
So, the original number of books must be 50
SUFFICIENT

Statement 2: the original number of books is greater than 24
So, the original number of books could be 50, 110, 170, etc
NOT SUFFICIENT

So, the answer is A

Cheers,
Brent