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Karen was the winner of a hot dog eating competition. There was a tota 12 Jul 2021, 03:51
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75% (hard)

Question Stats:

57% (02:47) correct 43% (03:06) wrong based on 124 sessions

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Karen was the winner of a hot dog eating competition. There was a total of 13 contenders at the competition. Comparing the number of eaten hot dogs by contender the median was 7 hot dogs and the average was 13 hot dogs. Karen ate more hot dogs than any other contender and the combination of eaten hot dogs is so that she ate the least number of hot dogs possible. How many hot dogs did Karen eat? (Consider hot dogs are only computed when eaten completely).

A) 20
B) 21
C) 22
D) 26
E) 75
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B
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Karen was the winner of a hot dog eating competition. There was a tota 12 Jul 2021, 09:56
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filipembribeiro
Karen was the winner of a hot dog eating competition. There was a total of 13 contenders at the competition. Comparing the number of eaten hot dogs by contender the median was 7 hot dogs and the average was 13 hot dogs. Karen ate more hot dogs than any other contender and the combination of eaten hot dogs is so that she ate the least number of hot dogs possible. How many hot dogs did Karen eat? (Consider hot dogs are only computed when eaten completely).

A) 20
B) 21
C) 22
D) 26
E) 75
Show more


The question can be rephrased to.

Median of 13 numbers is 7, and the average 13.
Find the least possible value of the greatest number, say x, if it is distinct from all others.

Total sum = 13*13 = 169.
As we are looking at the least max value, take all below median to be 7, so sum of these = 7*7=49.
All other except can be one less than x.

So 49+5(x-1)+x=169
5x-5+x=120
6x=125……20<x<21.

It cannot be 20, as then some other number will also be 20, so x=21 and the next 5 greatest will be 20,20,20,20,19.
49+21+4*20+19=49+21+80+19=169.

B
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Karen was the winner of a hot dog eating competition. There was a tota 16 Jul 2021, 06:01
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Could someone explain this one please? I didn't understand the explanation given here. Thanks in advance!
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Karen was the winner of a hot dog eating competition. There was a tota 16 Jul 2021, 06:47
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There are 13 people and the median is 7.

So we have a group of the first 6 people, Median, and a group of the next 6 people.

Average is 13 and total is 13 so the sum is 13 * 13 = 169

The first 6 people may eat any number of hotdogs but not more than 7. So, let us say all of them have eaten 7 each.

=> 7 * 7 = 49.

Remaining hot dogs are: 169 - 49 = 120

These 120 hot dogs will be eaten by the remaining group of 6 people so precisely each of them can eat 20 as 20 * 6 = 120.

But there is only 1 winner who ate the most number of hot dogs and that should be the least she should eat.

If Karen eats say 22, we need 98 more to be distributed among 5 people which can be done as = 19 + 19 +19 + 20 + 21.

But if Karen eats 21, we can still distribute 99 among 5 people as 20 + 20 + 20 + 20 + 19.

Answer B
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Karen was the winner of a hot dog eating competition. There was a tota 16 Jul 2021, 07:48
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MathRevolution
There are 13 people and the median is 7.

So we have a group of the first 6 people, Median, and a group of the next 6 people.

Average is 13 and total is 13 so the sum is 13 * 13 = 169

The first 6 people may eat any number of hotdogs but not more than 7. So, let us say all of them have eaten 7 each.

=> 7 * 7 = 49.

Remaining hot dogs are: 169 - 49 = 120

These 120 hot dogs will be eaten by the remaining group of 6 people so precisely each of them can eat 20 as 20 * 6 = 120.

But there is only 1 winner who ate the most number of hot dogs and that should be the least she should eat.

If Karen eats say 22, we need 98 more to be distributed among 5 people which can be done as = 19 + 19 +19 + 20 + 21.

But if Karen eats 21, we can still distribute 99 among 5 people as 20 + 20 + 20 + 20 + 19.

Answer B
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Thank you for clearing this up. While I easily arrived at the part that the top 6 ate 120 hot dogs, I was wondering if we have used some straight-up calculation/algebra going forward. I am still struggling with understanding where to stop using maths and start employing common sense to solve some questions.
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Karen was the winner of a hot dog eating competition. There was a tota 23 Mar 2024, 16:33
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