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

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

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06 Aug 2012, 01:43
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Question Stats:

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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
[Reveal] Spoiler: OA

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

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06 Aug 2012, 01:43
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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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06 Aug 2012, 12:31
[Reveal] Spoiler:
B

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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19 Aug 2012, 13:39
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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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09 Sep 2014, 21:44
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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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28 Apr 2016, 19:32
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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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29 Apr 2016, 08:28
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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.

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

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22 Jul 2016, 04:19
I think the easiest way would be the smallest number (2) + the biggest number (8) = 10/2 = 5
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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, 00:26
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 .

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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01 Apr 2017, 08:09
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).
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Re: The numbers of cars sold at a certain dealership on six of [#permalink]

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22 Jun 2017, 04:05
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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25 Aug 2017, 08:19
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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12 Sep 2017, 19:14
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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Re: The numbers of cars sold at a certain dealership on six of [#permalink]

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04 Feb 2018, 11:34
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

[Reveal] Spoiler:
B

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

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04 Feb 2018, 18:35
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

Posted from my mobile device
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Re: The numbers of cars sold at a certain dealership on six of [#permalink]

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05 Feb 2018, 00:08
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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Re: The numbers of cars sold at a certain dealership on six of   [#permalink] 05 Feb 2018, 00:08
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