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A box contains bags of marbles

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by outty » Wed May 28, 2014 4:04 pm
A box contains bags of marbles. All of the bags hold the same number of marbles except one bag, which holds one marble more than each of the other bags hold. If the box contains a total of 2001 marbles, how many bags are in the box?

(1) The number of bags is between 13 and 23 inclusive

(2) There is an even number of bags, and there is an even number of marbles in the bag containing the extra marble.

c
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Source: — Data Sufficiency |

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by [email protected] » Wed May 28, 2014 4:57 pm
Hi outty,

This DS question is ultimately about factoring and multiples.

We're told that there are a certain number of bags in a box, and they all hold the same number of marbles each, EXCEPT for one bag that holds an extra marble. We know that the total number of marbles = 2001. Ignoring the "1 extra" marble for a moment, we have 2000 marbles spread evenly across all of the bags.

This means that the number of bags is a factor of 2,000. I won't list all of the possibilities here, but you COULD figure that out if you had no choice.

For reference, I will prime factor 2000....2000 = 2x2x2x2x5x5x5

The question asks for the number of bags...

Fact 1: The number of bags is between 13 and 23, inclusive.

The number of bags could be 16 (2x2x2x2) or 20 (2x2x5)
Fact 1 is INSUFFICIENT

Fact 2: There is an EVEN number of bags and an even number of marbles in the bag that has the extra marble.

This means that all of the other bags have an ODD number of marbles. So we need (EVEN)(ODD) = 2,000

We could have 16 (2x2x2x2) x 125 (5x5x5) ---> 16 BAGS
We could also have 80 (2x2x2x2x5) x 25 (5x5) ---> 80 BAGS
Fact 2 is INSUFFICIENT

Combined, Fact 1 limits us to 2 options (16 and 20), but 20 does NOT fit the given information in Fact 2 (we'd have 20 x 100 ---> this number of marbles is NOT odd). There's only one answer: 16.
Combined, SUFFICIENT

Final Answer: C

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by Brent@GMATPrepNow » Wed May 28, 2014 6:50 pm
outty wrote:A box contains bags of marbles. All of the bags hold the same number of marbles except one bag, which holds one marble more than each of the other bags hold. If the box contains a total of 2001 marbles, how many bags are in the box?

(1) The number of bags is between 13 and 23 inclusive

(2) There is an even number of bags, and there is an even number of marbles in the bag containing the extra marble.
Hey, I made up that question :-)
https://www.beatthegmat.com/a-box-of-bag ... 29612.html

My solution is similar to Rich's but introduces some variables.

Target question: How many bags are in the box?

Let B = # of bags
Let M = # of marbles in MOST bags
So, M + 1 = # of marbles in the bag containing the extra marble.

This means that BM + 1 = 2001
Or we can say that BM = 2000
Let's find the prime factorization of 2000.
We get 2000 = (2)(2)(2)(2)(5)(5)(5)
In other words, BM = (2)(2)(2)(2)(5)(5)(5)

Statement 1: The number of bags is between 13 and 23 inclusive
If BM = (2)(2)(2)(2)(5)(5)(5), and B ranges from 13 to 23 inclusive, there are EXACTLY TWO POSSIBLE CASES:
Case a: B = 16 and M = 125
Case b: B = 20 and M = 100
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: There is an even number of bags, and there is an even number of marbles in the bag containing the extra marble.
In other words, B is EVEN and (M+1) is EVEN
Or we can say that B is EVEN and M is ODD
If BM = (2)(2)(2)(2)(5)(5)(5), there are several possible case. Here are two:
Case a: B = 16 and M = 125
Case b: B = 80 and M = 25
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that EITHER B = 16 and M = 125 OR B = 20 and M = 100
Statement 2 tells us that M is ODD.
So, it must be the case that B = 16 and M = 125
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent
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by GMATGuruNY » Thu May 29, 2014 2:37 am
outty wrote:A box contains bags of marbles. All of the bags hold the same number of marbles except one bag, which holds one marble more than each of the other bags hold. If the box contains a total of 2001 marbles, how many bags are in the box?

(1) The number of bags is between 13 and 23 inclusive

(2) There is an even number of bags, and there is an even number of marbles in the bag containing the extra marble.

c
If we remove the extra marble from the last bag, each bag will contain the same number of marbles.
Let b = the number of bags and m = the number of marbles per bag after the extra marble is removed.
Since removing the extra marble decreases the total number of marbles by 1 -- from 2001 to 2000 -- we get:
bm = 2000.

Factor pairs of 2000:
1*2000
2*1000
4*500
5*400
8*250
10*200
16*125
20*100
25*80
40*50

Statement 1: The number of bags is between 13 and 23 inclusive
Of the factor pairs listed above, ONLY TWO are viable:

Case 1: b=16, m=125
In this case, there are 16 bags, each with 125 marbles -- yielding a total of 2000 marbles -- except for 1 bag that contains 126 marbles, bringing the total to 2001.

Case 2: b=20, m=100
In this case, there are 20 bags, each with 100 marbles -- yielding a total of 2000 marbles -- except for 1 bag that contains 101 marbles, bringing the total to 2001.

Since it's possible that b=16 or b=20, INSUFFICIENT.

Statement 2: There is an even number of bags, and there is an even number of marbles in the bag containing the extra marble.
Case 1 also satisfies statement 2.
Of the factor pairs listed above, the following case also is viable:

Case 3: b=2000, m=1
In this case, there are 2000 bags, each with 1 marble -- yielding a total of 2000 marbles -- except for 1 bag that contains 2 marbles, bringing the total to 2001.

Since it's possible that b=16 or b=2000, INSUFFICIENT.

Statements combined:
Only Case 1 satisfies both statements.
Thus, b=16.
SUFFICIENT.

The correct answer is C.
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