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Math Expert V
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Question Stats: 68% (01:39) correct 32% (01:49) wrong based on 250 sessions

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On a certain Saturday, flights arrived to XYZ airport every hour for 24 hours. Is the total number of flights that arrived on Saturday greater than 180?

(1) On that Saturday, the median number of flight arrivals every hour is 17.

(2) On that Saturday, the highest number of flight arrivals in an hour was 30.

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Official Solution:

(1) On that Saturday, the median number of flight arrivals every hour was 17. The median of a set with even number (24) of terms is the average of two middle terms when arranged in ascending/descending. So, the median of 17 means that 17 is the average of 12th and 13th greatest numbers of arrivals. Hence, at least 12 hours of arrivals (from 13th greatest to 24th greatest) had at least 17 flights, which means that the total number of arrivals on Saturday was at least $$12*17=204 \gt 180$$. Sufficient.

(2) On that Saturday, the highest number of flight arrivals in an hour was 30. Clearly insufficient: if each hour had 30 arrivals then the total number of arrivals on Saturday would be $$24*30 \gt 180$$, but if 23 hours had 1 arrival and one hour had 30 arrivals then the total number of arrivals on Saturday would be $$23*1+30 \lt 180$$. Not sufficient.

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I think this question is good and helpful.
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It is one of the gmatclub test questions which I always get wrong. Thanks Bunuel for the beautiful explanation.
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Bunuel wrote:
Official Solution:

(1) On that Saturday, the median number of flight arrivals every hour was 17. The median of a set with even number (24) of terms is the average of two middle terms when arranged in ascending/descending. So, the median of 17 means that 17 is the average of 12th and 13th greatest numbers of arrivals. Hence, at least 12 hours of arrivals (from 13th greatest to 24th greatest) had at least 17 flights, which means that the total number of arrivals on Saturday was at least $$12*17=204 \gt 180$$. Sufficient.

(2) On that Saturday, the highest number of flight arrivals in an hour was 30. Clearly insufficient: if each hour had 30 arrivals then the total number of arrivals on Saturday would be $$24*30 \gt 180$$, but if 23 hours had 1 arrival and one hour had 30 arrivals then the total number of arrivals on Saturday would be $$23*1+30 \lt 180$$. Not sufficient.

Hi Bunuel,

For statement 1, shouldn't the minimum number of flights be 221?
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Bunuel wrote:
Official Solution:

(1) On that Saturday, the median number of flight arrivals every hour was 17. The median of a set with even number (24) of terms is the average of two middle terms when arranged in ascending/descending. So, the median of 17 means that 17 is the average of 12th and 13th greatest numbers of arrivals. Hence, at least 12 hours of arrivals (from 13th greatest to 24th greatest) had at least 17 flights, which means that the total number of arrivals on Saturday was at least $$12*17=204 \gt 180$$. Sufficient.

(2) On that Saturday, the highest number of flight arrivals in an hour was 30. Clearly insufficient: if each hour had 30 arrivals then the total number of arrivals on Saturday would be $$24*30 \gt 180$$, but if 23 hours had 1 arrival and one hour had 30 arrivals then the total number of arrivals on Saturday would be $$23*1+30 \lt 180$$. Not sufficient.

Hi Bunuel,

For statement 1, shouldn't the minimum number of flights be 221?

I think at least 232 flights as it was said "flights arrived to XYZ airport every hour for 24 hours" which mean there is at least one flight arrived every hour.

So, the least number is 11 * 1 flights before the median PLUS 13*17 flights from the median and after. That makes 232 flights.
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Bunuel wrote:
Official Solution:

(1) On that Saturday, the median number of flight arrivals every hour was 17. The median of a set with even number (24) of terms is the average of two middle terms when arranged in ascending/descending. So, the median of 17 means that 17 is the average of 12th and 13th greatest numbers of arrivals. Hence, at least 12 hours of arrivals (from 13th greatest to 24th greatest) had at least 17 flights, which means that the total number of arrivals on Saturday was at least $$12*17=204 \gt 180$$. Sufficient.

(2) On that Saturday, the highest number of flight arrivals in an hour was 30. Clearly insufficient: if each hour had 30 arrivals then the total number of arrivals on Saturday would be $$24*30 \gt 180$$, but if 23 hours had 1 arrival and one hour had 30 arrivals then the total number of arrivals on Saturday would be $$23*1+30 \lt 180$$. Not sufficient.

I read it as if ever hour, the median number of arrival flights was 17. And thought that about every 30min 17 flights arrive :/
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Bunuel wrote:
Official Solution:

(1) On that Saturday, the median number of flight arrivals every hour was 17. The median of a set with even number (24) of terms is the average of two middle terms when arranged in ascending/descending. So, the median of 17 means that 17 is the average of 12th and 13th greatest numbers of arrivals. Hence, at least 12 hours of arrivals (from 13th greatest to 24th greatest) had at least 17 flights, which means that the total number of arrivals on Saturday was at least $$12*17=204 \gt 180$$. Sufficient.

(2) On that Saturday, the highest number of flight arrivals in an hour was 30. Clearly insufficient: if each hour had 30 arrivals then the total number of arrivals on Saturday would be $$24*30 \gt 180$$, but if 23 hours had 1 arrival and one hour had 30 arrivals then the total number of arrivals on Saturday would be $$23*1+30 \lt 180$$. Not sufficient.

Hi Bunuel,

For statement 1, shouldn't the minimum number of flights be 221?

I agree. In order for the median value to be 17, the number of flights for the 12th hour must also be 17; so, the number of hours that had 17 flights would be (24-12+1) = 13, which would result in 221 flights (17*13). Bunuel please clarify.
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Not really "the number of flights for the 12th hour must also be 17". The 12th can be 16, then the 13th can be 18 (so that the median can be 17). The thing we can be sure is that the number of flights for the 13th hour must be at least 17 to guarantee that the average of 12th (which has lower or equal number of flights as 13th does, assuming in ascending order) and 13th can be 17.

In the case of descending order, you can also prove that at least 12 hours of arrivals had at least 17 flights.
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Hi all, can anyone explain, the below line in detail;
"On that Saturday, the median number of flight arrivals every hour is 17."

I actually assumed the median of arrivals per hr is 17 every hr; which would mean totally 33 arrivals per hr; easily >180 for 24 hrs.
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When we are given the median of a set of numbers, do we ASSUME that the ordering of the numbers is either ascending or descending?
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The sets are always in ascending order if not stated.
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bansi99 wrote:
The sets are always in ascending order if not stated.

A set, by definition, is a collection of elements without any order. While, a sequence, by definition, is an ordered list of terms.
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Bunuel chetan2u PKN GMATPrepNow niks18 KarishmaB pushpitkc

Can you please explain below quote?

Quote:
the median of 17 means that 17 is the average of 12th and 13th greatest numbers of arrivals.

My understanding:
Median can be decimal or an integer for even number of elements in a sequence.
e.g. S1: {1,2,2,3}
Here, mean = (2+2)/ 2 = 2
S2: {1,2,3,4}
Mean = (2+3) / 2 = 2.5

In above quote, am I supposed to find mean of 12 and 13? How is it equal to 17?
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Bunuel chetan2u PKN GMATPrepNow niks18 KarishmaB pushpitkc

Can you please explain below quote?

Quote:
the median of 17 means that 17 is the average of 12th and 13th greatest numbers of arrivals.

My understanding:
Median can be decimal or an integer for even number of elements in a sequence.
e.g. S1: {1,2,2,3}
Here, mean= (2+2)/ 2 = 2
S2: {1,2,3,4}
Mean= (2+3) / 2 = 2.5

In above quote, am I supposed to find mean of 12 and 13? How is it equal to 17?

first of all the highlighted part is incorrect. I think you meant median there.

Median means a mid-value between numbers and 17 is greater than 12 or 13 so it cannot be the median. here you are give that total elements are 24 (24 hours) so the median will be between 12th & 13th element. You are mistaking the 12th & 13th position to number.
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Quote:
the median of 17 means that 17 is the average of 12th and 13th greatest numbers of arrivals.
My understanding:
Median can be decimal or an integer for even number of elements in a sequence.
e.g. S1: {1,2,2,3}
Here, mean = (2+2)/ 2 = 2
S2: {1,2,3,4}
Mean = (2+3) / 2 = 2.5

Absolutely correct.

Quote:
In above quote, am I supposed to find mean of 12 and 13? How is it equal to 17?

What is said in the question:-
flights arrive at XYZ airport every hour for 24 hours. It means, every hour corresponds to certain number of flights( minimum no of flights being 1).
We have 24 hours in a day. So, we have a minimum of 24*1=24 numbers of flights arriving at XYZ.
Maximum number of flights could be any number depending upon additional info provided in the question.

Question stem:- Is # of flights in 24 hrs>180 ?

St1:- On the same Saturday, the median number of flight arrivals every hour is 17.
We have total no of hours=24 ( If we consider it a set, then it has 24 data points and each data point has certain frequency(here no of flights))
Median number of flights is equal the mean number of flights at 12th and 13th hours or data point when the data points are arranged in ascending or descending fashion.
No of hours ---------------.No of flights/frequency
1st---------------------------- $$\geq1$$
2nd---------------------------- $$\geq1$$
.
.
.
11th---------------------------$$\geq1$$
12th--------------------------- $$\leq17$$ & $$\geq1$$
Median-------------------------17
13th---------------------------- $$\geq17$$ & $$\leq33$$
.
.
.
24th--------------------------- $$\geq17$$

We can take the above data in descending order too.

Minimum # of flights arrived in 24 hrs=11*1+17*1+17*1+11*17=11+17+17+187=232>180
Sufficient.

P.S:- You have to take the average of no of flights in 12th hour and no of flights in 13th hour. Not the average of 12 and 13. 12 and 13 represent the hour number not the no of flights in 12th and 13th hours respectively.

Hope it helps.
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I think this is a high-quality question and I agree with explanation.
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Bunuel wrote:
Official Solution:

(1) On that Saturday, the median number of flight arrivals every hour was 17. The median of a set with even number (24) of terms is the average of two middle terms when arranged in ascending/descending. So, the median of 17 means that 17 is the average of 12th and 13th greatest numbers of arrivals. Hence, at least 12 hours of arrivals (from 13th greatest to 24th greatest) had at least 17 flights, which means that the total number of arrivals on Saturday was at least $$12*17=204 \gt 180$$. Sufficient.

(2) On that Saturday, the highest number of flight arrivals in an hour was 30. Clearly insufficient: if each hour had 30 arrivals then the total number of arrivals on Saturday would be $$24*30 \gt 180$$, but if 23 hours had 1 arrival and one hour had 30 arrivals then the total number of arrivals on Saturday would be $$23*1+30 \lt 180$$. Not sufficient.

I need some general clarification here. The statement is considered sufficient as long as we can answer the question that is asked. So as long as the statement resolves to a conclusive number then we say whether it is sufficient or not. It is not as if the result has to be more than 180 for the statement to be sufficient right? Re: M05-30   [#permalink] 17 Sep 2019, 08:47
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