The Numbers of Cars Sold at a Certain Dealership on Six of GMAT Problem Solving

Question: 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) a 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

1. II only
2. III only
3. I and II only
4. II and III only
5. I, II, and III

Correct Answer: B

Solution and Explanation:
Approach Solution 1:
The problem statement states that:
Given:

• 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.
• The number of cars sold on the seventh business day was either 2, 4, or 5.

Asked:

• Find the value for which the average number of cars sold per business day for the 7 business days equals the median number of cars sold per day for 7 days.

It is a question on the topic of statistics.
Some important formulas for statistics are given below:
The arithmetic mean of n numbers = sum of n numbers / n
Median of n numbers = middle element when n is odd and sorted in ascending or descending order.
Therefore firstly we need to calculate the median of the cards sold for the 7 days.
As 7 is odd, so if all the numbers are arranged in the sorted order, the n the fourth element will be the median. Therefore it is clear that the median is an integer.
Now, the total number of cars sold in 6 days is 4 + 7 + 2 + 8 + 3 + 6 = 30
Now, let the no of cars sold on the seventh day be x
Therefore, the average number of cars sold in 7 days =\(\frac{(30+x)}{7}\).

It is said the average should be equal to the median, then the average should also be an integer.
\(\frac{(30+x)}{7}\)= integer for only one value of x, i.e, x= 5 (for x=2 and x = 4 it is not integer)
 

Therefore, for x = 5, the average number of cars sold per business day for the 7 business days equals the median number of cars sold per day for 7 days. 

Approach Solution 2:
The problem statement informs that:
Given:

• 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.
• The number of cars sold on the seventh business day was either 2, 4, or 5.

Find out:

• The value for which the average number of cars sold per business day for the 7 business days equals the median number of cars sold per day for 7 days.

The problem can be solved by this quicker and easier approach.

To solve the question, we need to arrange the numbers in ascending order that is:
2,3,4,x,6,7,8,

If we analyse the number closely, we will find that it is an evenly-spaced series with one missing number.

It is required to note that for an evenly-spaced series, the mean is equal to the median.

Therefore, to satisfy the condition of mean = median x has to be 5 only.

Hence, for x = 5, the average number of cars sold per business day for the 7 business days equals the median number of cars sold per day for 7 days. 

Approach Solution 3:
The problem statement informs that:
Given:

• 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.
• The number of cars sold on the seventh business day was either 2, 4, or 5.

Find out:

• The value for which the average number of cars sold per business day for the 7 business days equals the median number of cars sold per day for 7 days.

In order to solve the problem, we need to use the number given in each Roman numeral to find the average and median. It will also help us to determine whether they are equal.

I. 2

From least to greatest order, the 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 the list.

Therefore, the median is 4.

By calculating the average, we get:
As per the formula, average = sum/quantity
Average = (2 + 2 + 3 + 4 + 6 + 7 + 8)/7
Average = 32/7
We find that the average does not equal the median.
Therefore, the answer choice I is not correct.

So, we can eliminate answer choices C and E.

II. 4

From least to greatest order, the 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 the list. 
By calculating the average, we get:
As per the formula, average = sum/quantity
Average = (2 + 3 + 4 + 4 + 6 + 7 + 8)/7
Average = 34/7
We find that the average does not equal the median.
Therefore, the answer choice II is not correct.

So, we can eliminate answer choices A and D.

We are now confirmed that the correct answer choice is B. However, we should still check.

III. 5

From least to greatest order, the 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 the list. 
By calculating the average, we get:
As per the formula, average = sum/quantity
Average = (2 + 3 + 4 + 5 + 6 + 7 + 8)/7
Average = 35/7 = 5
We can find that the average does equal the median.
Therefore, answer choice III is correct.

“The numbers of cars sold at a certain dealership on six of”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from 501 GMAT Questions. GMAT Problem Solving questions enable the candidates to evaluate information and crack numerical problems. GMAT Quant practice papers help the candidates to analyse several sorts of questions that will enable them to improve their mathematical understanding.

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