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An Arena With Many Vacant Seats

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by pkw209 » Fri Apr 16, 2010 12:45 pm
At a certain arena, with five levels, level I has 30 vacant seats, level II has 45 vacant seats, level III has 40 vacant seats, level IV has 25 vacant seats and level V has 10 vacant seats. If vacant seats are filled randomly, what is the minimum number of seats that must be filled to ensure that 3 levels are completely filled?

a) 115

b) 139

c) 140

d) 148

e) 149
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by eaakbari » Fri Apr 16, 2010 1:14 pm
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by liferocks » Fri Apr 16, 2010 5:22 pm
pkw209 wrote:At a certain arena, with five levels, level I has 30 vacant seats, level II has 45 vacant seats, level III has 40 vacant seats, level IV has 25 vacant seats and level V has 10 vacant seats. If vacant seats are filled randomly, what is the minimum number of seats that must be filled to ensure that 3 levels are completely filled?

a) 115

b) 139

c) 140

d) 148

e) 149
is the ans [spoiler]d)148 ?[/spoiler]

I have approached like this..
the maximum number of people can occupy the levels without completely filling any of it is 29+44+39+24+9=145

now if one more people come one of the levels will be completely filled definitely.

so the minimum number of number of seats that must be filled to ensure that 3 levels are completely filled 145+3=148
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by Stuart@KaplanGMAT » Fri Apr 16, 2010 5:43 pm
pkw209 wrote:At a certain arena, with five levels, level I has 30 vacant seats, level II has 45 vacant seats, level III has 40 vacant seats, level IV has 25 vacant seats and level V has 10 vacant seats. If vacant seats are filled randomly, what is the minimum number of seats that must be filled to ensure that 3 levels are completely filled?

a) 115

b) 139

c) 140

d) 148

e) 149
We want the minimum number to guarantee 3 full levels.

Well, we could put 29 in level 1, 44 in level 2, 39 in level 3, 24 in level 4 and 9 in level 5 without filling any levels at all; that's:

29 + 44 + 39 + 24 + 9 = 145 seats so far.

Any seat will fill will complete a level at this point; since we want to fill 3 levels, we need to fill 3 more seats:

145 + 3 = 148... choose (D).
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by hachette » Fri Apr 16, 2010 7:32 pm
pkw209 wrote:At a certain arena, with five levels, level I has 30 vacant seats, level II has 45 vacant seats, level III has 40 vacant seats, level IV has 25 vacant seats and level V has 10 vacant seats. If vacant seats are filled randomly, what is the minimum number of seats that must be filled to ensure that 3 levels are completely filled?
a) 115 b) 139 c) 140 d) 148 e) 149
In both the approach 145 is the maximum number of people occupying all levels without filling any level. If 148 is the minimum then what is the maximum number of seats that must be filled to ensure that 3 levels are completely filled ?
since 150 is the total no of seats. Thanks.
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by eaakbari » Fri Apr 16, 2010 7:32 pm
Stuart Kovinsky wrote:
pkw209 wrote:At a certain arena, with five levels, level I has 30 vacant seats, level II has 45 vacant seats, level III has 40 vacant seats, level IV has 25 vacant seats and level V has 10 vacant seats. If vacant seats are filled randomly, what is the minimum number of seats that must be filled to ensure that 3 levels are completely filled?

a) 115

b) 139

c) 140

d) 148

e) 149
We want the minimum number to guarantee 3 full levels.

Well, we could put 29 in level 1, 44 in level 2, 39 in level 3, 24 in level 4 and 9 in level 5 without filling any levels at all; that's:

29 + 44 + 39 + 24 + 9 = 145 seats so far.

Any seat will fill will complete a level at this point; since we want to fill 3 levels, we need to fill 3 more seats:

145 + 3 = 148... choose (D).
Hey Stuart, can you explain the logic to me?
To me it seemed like the minimum number of people he needed to fill 3 levels , so the 3 levels with minimum number of seats
Hence 30 + 25+10 = 65

What am I inferring wrong?
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by liferocks » Fri Apr 16, 2010 8:40 pm
eaakbari wrote: To me it seemed like the minimum number of people he needed to fill 3 levels , so the 3 levels with minimum number of seats
Hence 30 + 25+10 = 65

What am I inferring wrong?
In the question it is mentioned that the seats are filled up randomly.So if 65 seats are filled up it may or may not complete 3 levels. 65 people can seat in any of the total 150 seats and this is the case till 145 seats are filled up i.e total 145 seats can be filled and still none of the levels are completely filled.

after 145 if any more seat at least one of the levels has to be filled as there will be no other option left .hence we can certainly say that 3 of the levels are filled when minimum 145+3=148 seats are filled
hope this explains why 65 is not the minimum.
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by hachette » Fri Apr 16, 2010 11:19 pm
@ liferocks Random selection means without bias or preference.

So if there are 50 people in the arena , 10 seats in each level should be occupied.
If each of the 50 people randomly select a level without preference , in a big sample size each level should have 10 people. Probability of choosing any level is 1/5.

This will fill up level V.

If there are 110 people in the arena , then level V (capacity 10) , level IV capacity (25) will be filled. I , II and III will have 25.

The trouble starts with the next 5. If you want minimun they have to be in level I. But there is a bias.
since 115 is an option IMO A.
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by Testluv » Fri Apr 16, 2010 11:47 pm
This:
To me it seemed like the minimum number of people he needed to fill 3 levels , so the 3 levels with minimum number of seats
Hence 30 + 25+10 = 65

What am I inferring wrong?
would have been the correct approach had the question been:

What is the minimum number of seats that must be filled so that (any) 3 levels are completely filled?


But the question is asking for the MINIMUM number of seats that must be filled that will ENSURE that three rows are filled up. So, you're looking for how many people there have to be in order to guarantee that 3 levels are completely filled up.

Thus, you should think about the MAXIMUM number of seats you can fill WITHOUT filling up any of the rows at all...which is what Stuart did.

This question is also a good reminder of two things:

1) determine precisely what the question asks (ie, step 1 of the Kaplan method);

2) In minimum problems you usually have to maximize something (while in maximum problems, which are rarer, you usually have to minimize something).
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by eaakbari » Sat Apr 17, 2010 12:06 am
@ liferocks , Testluv, Stuart

Thanks for the help.
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by pkw209 » Tue Apr 20, 2010 12:11 pm
Initially, I thought it was A but that can't be the answer. Try and allocate 115 people across all 5 levels and you can still not have 3 levels filled.
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