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## OG confusing question

tagged by: harshitpuri

This topic has 3 expert replies and 1 member reply
harshitpuri Junior | Next Rank: 30 Posts
Joined
30 Aug 2017
Posted:
10 messages

#### OG confusing question

Thu Nov 30, 2017 10:55 am
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?
a. Before Michael acquired the 10 additional books, he had fewer than 96 books.
b. Before Michael acquired the 10 additional books, he had more than 24 books.

how can be this a ds question when u can calculate the no. of shelves 5 as 10n + 10=12.
the info seems irrelevant.

### GMAT/MBA Expert

GMATGuruNY GMAT Instructor
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Thu Nov 30, 2017 11:41 am
harshitpuri 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?
a. Before Michael acquired the 10 additional books, he had fewer than 96 books.
b. Before Michael acquired the 10 additional books, he had more than 24 books.
Since the original number of books can be arranged with 10 books per shelf, the original number of books is a multiple of 10.
When 10 books are added to this multiple of 10, the resulting number of books must also be a multiple of 10.
Since the resulting number of books can be arranged with 12 books per shelf, the resulting number of books must also be a multiple of 12.
Implication:
Since the resulting number of books must be a multiple of both 10 and 12 -- and the LCM of 10 and 12 is 60 -- the resulting number books must be a multiple of 60:
60, 120, 180...
Thus, the number of books BEFORE 10 books are added must be 10 less than the values in the list above:
50, 110, 170...

Statement 1:
Of the blue values above, only 50 is less than 96.
Thus, the original number of books = 50.
SUFFICIENT.

Statement 2:
All of the blue values above are greater than 24.
Thus, the original number of books can be any of the blue values above.
INSUFFICIENT.

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harshitpuri Junior | Next Rank: 30 Posts
Joined
30 Aug 2017
Posted:
10 messages
Thu Nov 30, 2017 11:15 pm
GMATGuruNY wrote:
harshitpuri 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?
a. Before Michael acquired the 10 additional books, he had fewer than 96 books.
b. Before Michael acquired the 10 additional books, he had more than 24 books.
Since the original number of books can be arranged with 10 books per shelf, the original number of books is a multiple of 10.
When 10 books are added to this multiple of 10, the resulting number of books must also be a multiple of 10.
Since the resulting number of books can be arranged with 12 books per shelf, the resulting number of books must also be a multiple of 12.
Implication:
Since the resulting number of books must be a multiple of both 10 and 12 -- and the LCM of 10 and 12 is 60 -- the resulting number books must be a multiple of 60:
60, 120, 180...
Thus, the number of books BEFORE 10 books are added must be 10 less than the values in the list above:
50, 110, 170...

Statement 1:
Of the blue values above, only 50 is less than 96.
Thus, the original number of books = 50.
SUFFICIENT.

Statement 2:
All of the blue values above are greater than 24.
Thus, the original number of books can be any of the blue values above.
INSUFFICIENT.

Got the solution,
but to move from 100 to 120 it has to add 20 books , the questions says no. of books added are 10. i.e the only one option it has is to move from 50 to 60, if the questions has added the no. of books added is in multiple of 10, the question would have been clear, but i find it confusing right now.

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GMATGuruNY GMAT Instructor
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Fri Dec 01, 2017 5:20 am
harshitpuri wrote:
Got the solution,
but to move from 100 to 120 it has to add 20 books , the questions says no. of books added are 10. i.e the only one option it has is to move from 50 to 60, if the questions has added the no. of books added is in multiple of 10, the question would have been clear, but i find it confusing right now.
The entire point of the problem is that an addition of EXACTLY 10 BOOKS brings the total from a multiple of 10 to a multiple of 12.
If the number of added books could be ANY MULTIPLE OF 10, then the original number of books would not be constrained.

We could start with 10 books (a multiple of 10), add 50 books (also a multiple of 10), and yield a total of 60 books (a multiple of 12).
We could start with 20 books (a multiple of 10), add 40 books (also a multiple of 10), and yield a total of 60 books (a multiple of 12).
We could start with 30 books (a multiple of 10), add 30 books (also a multiple of 10), and yield a total of 60 books (a multiple of 12).
We could start with 40 books (a multiple of 10), add 20 books (also a multiple of 10), and yield a total of 60 books (a multiple of 12).
We could start with 50 books (a multiple of 10), add 10 books (also a multiple of 10), and yield a total of 60 books (a multiple of 12).

If ANY multiple of 10 could be added to the original number of books, then all 5 cases above would satisfy Statement 1.
Because EXACTLY 10 BOOKS must be added, only the case in blue satisfies Statement 1.

An algebraic approach:

Since the original number of books must be a multiple of 10, let the original number of books = 10a, where a is a positive integer.
After 10 books are added, the new total = 10a + 10.
Since the new total must be a multiple of 12, we get:
10a + 10 = 12b, where b is a positive integer.

Simplifying the resulting equation, we get:
10(a + 1) = 12b
a + 1 = (12b)/10
a + 1 = (6/5)b
a = (6/5)b - 1.

For the value of a to be a positive integer, b must be a multiple of 5.
The following options are yielded:
Case 1: b=5 and a = (6/5)(5) - 1 = 5, with the result that the original number of books = 10a = 10*5 = 50.
Case 2: b=10 and a = (6/5)(10) - 1 = 11, with the result that the original number of books = 10a = 10*11 = 110.
Case 3: b=15 and a = (6/5)(15) - 1 = 17, with the result that the original number of books = 10a = 10*17 = 170.
And so on.

Statement 1 is satisfied only by Case 1, in which case the original number of books = 50.
SUFFICIENT.
Statement 2 is satisfied by all 3 cases, with the result that the original number of books can be different values.
INSUFFICIENT.

_________________
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Scott@TargetTestPrep GMAT Instructor
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Fri Dec 08, 2017 10:50 am
harshitpuri 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 are given that Michael arranged all his books in a bookcase with 10 books on each shelf and no books left over, and that after he acquired 10 additional books, he arranged all his books in a new bookcase with 12 books on each shelf and no books left over. We need to determine how many books Michael had before he acquired 10 additional books.

Using the given information we can determine that Michael originally had a total number of books that was a multiple of 10, and after he acquired 10 new books, he had a total number of books that was a multiple of 12.

Statement One Alone:

Before Michael acquired the 10 additional books, he had fewer than 96 books.

The information in statement one is sufficient to answer the question. Since we know the original number of books was a multiple of 10, the number of books could have been the following:

10, 20, 30, 40, 50, 60, 70, 80, or 90.

Using the above numbers, he could have had the following number of books after acquiring 10 more:

20, 30, 40, 50, 60, 70, 80, 90, or 100.

Remember, after acquiring the 10 new books, the total number of books was a multiple of 12. Of the numbers above, only 60 is a multiple of 12. Thus, Michael originally had 50 books. Statement one is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

Before Michael acquired the 10 additional books, he had more than 24 books.

We can analyze statement two in a similar way to how we analyzed statement one. Michael could have originally had any one of the following numbers of books:

30, 40, 50, 60, 70, 80, 90, 100, 110, 120, and so on.

Using the above numbers, he could have had any one of the following numbers of books after acquiring 10 more:

40, 50, 60, 70, 80, 90, 100, 110, 120, 130, and so on.

Once again, remember that after acquiring the 10 new books, the total number of books was a multiple of 12. Of the numbers above, 60 and 120 are both multiples of 12. Thus, Michael could have originally had 50 or 110 books. Statement two is not sufficient to answer the question.

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Scott Woodbury-Stewart Founder and CEO

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