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The numbers of cars sold at a certain dealership on six of the last seven business days were 4, 7, 2, 8, 3, and 6, respectively. If the number of cars sold on the seventh business day was either 2, 4, or 5, for which of the three values does the average (arithmetic mean) number of cars sold per business day for the seven business days equal the median number of cars sold per day for the seven days?

I. 2 II. 4 III. 5

(A) II only (B) III only (C) I and II only (D) II and III only (E) I, II, and III

Practice Questions Question: 12 Page: 153 Difficulty: 600

The numbers of cars sold at a certain dealership on six of the last seven business days were 4, 7, 2, 8, 3, and 6, respectively. If the number of cars sold on the seventh business day was either 2, 4, or 5, for which of the three values does the average (arithmetic mean) number of cars sold per business day for the seven business days equal the median number of cars sold per day for the seven days?

I. 2 II. 4 III. 5

(A) II only (B) III only (C) I and II only (D) II and III only (E) I, II, and III

The median of a set with odd number of elements is the middle element when arranged in ascending/descending order.

Hence, the median number of cars sold for the 7 (odd) days, is the fourth greatest number of cars sold in these 7 days, therefore the median must be an integer.

Next, the total number of cars sold in 6 days is 4+7+2+8+3+6=30.

The average number of cars sold in 7 days is \(\frac{30+x}{7}\). Since we need the average to be equal to the median, then the average must also be an integer.

\(\frac{30+x}{7}=integer\) only if \(x=5\) (for \(x=2\) or \(x=4\) it's not an integer).

Since there are a odd number of days the median will be an integer, if the average has to be equal to median then it must be an integer also. Only 5 does this.

The numbers of cars sold at a certain dealership on six of the last seven business days were 4, 7, 2, 8, 3, and 6, respectively. If the number of cars sold on the seventh business day was either 2, 4, or 5, for which of the three values does the average (arithmetic mean) number of cars sold per business day for the seven business days equal the median number of cars sold per day for the seven days?

I. 2 II. 4 III. 5

(A) II only (B) III only (C) I and II only (D) II and III only (E) I, II, and III

The median of a set with odd number of elements is the middle element when arranged in ascending/descending order.

Hence, the median number of cars sold for the 7 (odd) days, is the fourth greatest number of cars sold in these 7 days, therefore the median must be an integer.

Next, the total number of cars sold in 6 days is 4+7+2+8+3+6=30.

The average number of cars sold in 7 days is \(\frac{30+x}{7}\). Since we need the average to be equal to the median, then the average must also be an integer.

\(\frac{30+x}{7}=integer\) only if \(x=5\) (for \(x=2\) or \(x=4\) it's not an integer).

Re: The numbers of cars sold at a certain dealership on six of [#permalink]

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19 Aug 2012, 14:39

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Best Shortcut for this question:- If we arrange the numbers in ascending order 2,3,4,x,6,7,8,& observe closely it is a evenly spaced series with one missing number. For a evenly spaced series mean is equal to median. Thus to fulfill the condition of mean = median , x has to be 5 only. It took app 1 min 10 sec for me
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Re: The numbers of cars sold at a certain dealership on six of [#permalink]

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11 Jan 2014, 23:47

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Re: The numbers of cars sold at a certain dealership on six of [#permalink]

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09 Sep 2014, 22:44

Bunuel wrote:

The numbers of cars sold at a certain dealership on six of the last seven business days were 4, 7, 2, 8, 3, and 6, respectively. If the number of cars sold on the seventh business day was either 2, 4, or 5, for which of the three values does the average (arithmetic mean) number of cars sold per business day for the seven business days equal the median number of cars sold per day for the seven days?

I. 2 II. 4 III. 5

(A) II only (B) III only (C) I and II only (D) II and III only (E) I, II, and III

Practice Questions Question: 12 Page: 153 Difficulty: 600

if the reading part is done carefully, then we could get the solution in 30 secs.

2+3+4+6+7+8 = 30 so if u add 2 then mean= 32/7 not equal to 4(median). if 4 then 34/7 not equal to median (4). if % then 35/7 equal to median 5. set will be like [2,3,4,5,6,7,8,]..

Re: The numbers of cars sold at a certain dealership on six of [#permalink]

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12 Sep 2015, 19:05

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: The numbers of cars sold at a certain dealership on six of [#permalink]

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28 Apr 2016, 20:32

This is fairly simple, you don't have to do much work.

First, add up the numbers in your head (30), look at the 1,2,3 choices, and realize only 5 will be divisible by 7. you don't even need to arrange the numbers around to figure the median. time saver for the win

The numbers of cars sold at a certain dealership on six of the last seven business days were 4, 7, 2, 8, 3, and 6, respectively. If the number of cars sold on the seventh business day was either 2, 4, or 5, for which of the three values does the average (arithmetic mean) number of cars sold per business day for the seven business days equal the median number of cars sold per day for the seven days?

I. 2 II. 4 III. 5

(A) II only (B) III only (C) I and II only (D) II and III only (E) I, II, and III

Practice Questions

Question: 12 Page: 153 Difficulty: 600

To solve, we will use the number given in each Roman numeral to determine the average and median and determine whether they are equal.

I. 2

In order, from least to greatest, our values for the number of cars sold for the 7 days are:

2, 2, 3, 4, 6, 7, 8

We know that the median is the middle number of our list, so our median is 4. Next we calculate the average.

Average = sum/quantity

Average = (2 + 2 + 3 + 4 + 6 + 7 + 8)/7

Average = 32/7

We see that the average does not equal the median.

Answer choice I is not correct. We can eliminate answer choices C and E.

II. 4

In order, from least to greatest, our values for the number of cars sold for the 7 days are:

2, 3, 4, 4, 6, 7, 8

We know that the median is the middle number of our list, so our median is 4. Next we calculate the average.

Average = sum/quantity

Average = (2 + 3 + 4 + 4 + 6 + 7 + 8)/7

Average = 34/7

We see that the average does not equal the median.

Answer choice II is not correct. We can eliminate answer choices A and D. We know now that the correct answer choice is B, but we should still check.

III. 5

In order, from least to greatest, our values for the number of cars sold for the 7 days are:

2, 3, 4, 5, 6, 7, 8

We know that the median is the middle number of our list, so our median is 5. Next we calculate the average.

Average = sum/quantity

Average = (2 + 3 + 4 + 5 + 6 + 7 + 8)/7

Average = 35/7 = 5

We can see that the average does equal the median.

Answer choice III is correct.

The answer is B.
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Re: The numbers of cars sold at a certain dealership on six of [#permalink]

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26 Dec 2016, 01:26

Nice Question. Here is what i did in this one ->

Mean=30+x/7 = Median Since #=7 => Median will be in the series. Hence median must be an integer. So 30+x must be divisible by 7 Only 5 out of 2,4,5 will make that possible .

Re: The numbers of cars sold at a certain dealership on six of [#permalink]

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01 Apr 2017, 09:09

When the numbers are consecutive integers, then the mean of the set is equal to the median. (This is one case where median= mean) Here median will be the 4th integer in the set (after arranging the values in ascending/descending order). Therefore the answer is 5.
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