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Joined: 02 Sep 2009
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The numbers of cars sold at a certain dealership on six of  [#permalink]

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Question Stats: 89% (01:59) correct 11% (02:03) wrong based on 1737 sessions

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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

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Posts: 58415
Re: The numbers of cars sold at a certain dealership on six of  [#permalink]

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SOLUTION

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).

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

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

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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,]..

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

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

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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

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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I think the easiest way would be the smallest number (2) + the biggest number (8) = 10/2 = 5
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GRE 1: Q169 V154 Re: The numbers of cars sold at a certain dealership on six of  [#permalink]

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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 .

Hence only 5

Hence B

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

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

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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

Fastest way to solve a Roman Numeral Question -

1. DO NOT START WITH THE 1st ROMAN NUMERAL. Check the most occurring roman numeral in the 5 options and check for it first.

Here it is II, occurring in 4 options.

Checked. Did not work.
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Re: The numbers of cars sold at a certain dealership on six of  [#permalink]

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Bunuel wrote:
SOLUTION

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).

Perfect . This makes sense ! My approach was to find the mean and average by substituting the 3 numbers given. But this one saves time
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The numbers of cars sold at a certain dealership on six of  [#permalink]

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Bunuel niks18 amanvermagmat
Quote:
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.

I hope you simply mean to convey that median needs to be an integers since it represents no of cars and hence it can not be a fraction.

Eg. no of children / umbrella can not be fraction.

If i look closely the nos are evenly spaced, can I not conclude directly 5 as mean since all consecutive no
shall pull mean in same sequence.
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: The numbers of cars sold at a certain dealership on six of  [#permalink]

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Hi All,

This question is essentially a big 'concept' question with a little bit of math thrown in. We're asked which of the 3 Roman Numerals would give us a situation in which the average of the group would equal the median of the group.

Since we're dealing with a group of 7 numbers (the 6 that we're given and the 7th that is either 2, 4 or 5), the median will be the 4th number (once the values are put in ascending order).

The total number of cars sold in 6 days is 4+7+2+8+3+6=30.
In order, those numbers are 2, 3, 4, 6, 7, 8 ...and X, although we don't know what X is yet.

Since the median will be an integer, we need to add a number to the group that, when the sum of the group is divided by 7, gives us an integer also AND that integer has to match the median.

Looking at the 3 Roman Numerals, there's only one that fits everything we're looking for: 5

(30+5)/7 = 5 --> 5 is the average
2, 3, 4, 5, 6, 7, 8 --> 5 is the median

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

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Bunuel niks18 amanvermagmat
Quote:
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.

I hope you simply mean to convey that median needs to be an integers since it represents no of cars and hence it can not be a fraction.

Eg. no of children / umbrella can not be fraction.

If i look closely the nos are evenly spaced, can I not conclude directly 5 as mean since all consecutive no
shall pull mean in same sequence.

Yes we have a consecutive series if 5 is included. So mean=median

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

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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

In an AP with odd number of terms, the Median is always equal to the Middle Number.
I we arrange the numbers in ascending order, we have -
$$2,3,4,a,6,7,8.$$
Here, a can be only 5.
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The numbers of cars sold at a certain dealership on six of  [#permalink]

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In questions involving answer choices, which contain various combinations of three roman numerals, following technique is generally very helpful:

1 - See which of the roman numeral options is occuring the most in the five answer choices

2 - Guess that roman numeral option as the right answer and check it against the question by using the relavent technique such as back solving, and algebra etc.

3 - If the anwer is incorrect then remove all the answer choices, which contain that roman numeral

4 - repeat 1, 2, and 3 until you are down to only one answer choice

In this question, the most occuring roman numeral is II, which is found in four of the five choices.

Remember: if this question were a 650+ question then there would have been no more than two answer choices with the same roman numeral; but since this is a relatively easier question, so we see that by just guessing and checking on one roman numeral option i.e. II, we are left with single answer choice (B), which is the answer!

Solving for roman numeral option (II):
Median: 4
Average: 34/7 is greater than 4
So 4 cannot be the 7th value

Originally posted by FANewJersey on 07 Apr 2018, 07:53.
Last edited by FANewJersey on 07 Apr 2018, 07:58, edited 1 time in total.
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The numbers of cars sold at a certain dealership on six of  [#permalink]

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rahul16singh28 wrote:
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

In an AP with odd number of terms, the Median is always equal to the Middle Number.
I we arrange the numbers in ascending order, we have -
$$2,3,4,a,6,7,8.$$
Here, a can be only 5.

rahul16singh28 Not necessarily, "a" could also be 4. Kindly recheck your logic.
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