Question

The number of cars sold at certain dealership on six of the last seven business days were $4,{\text{ }}7,{\text{ }}2,{\text{ }}8,{\text{ }}3$ and $6$ respectively. If the number of cars sold on the seventh business day we either $2,{\text{ }}4$ 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). i and iii only

Answer 1

Hint: Here it is given the information of sold cars at certain car dealers of the last seven days. We have to find which is the value of the average number of cars sold per business day for the seven business days equal to the median number of cars sold per day for the seven days. We solve the problem by checking if the given values satisfied the required condition. Then we will get the answer.

Formula used: Mean is the sum of the data values divided by the total number of data values. Median is the middle value of the data values after arranging the data values in the ascending or descending order.
If $n$ is odd, then the median is ${\left( {\dfrac{{n + 1}}{2}} \right)^{th}}$ value.
If $n$ is even, then the median is $\dfrac{{{{\left( {\dfrac{n}{2}} \right)}^{th}} + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}}}{2}$ value.

Complete step-by-step solution:
It is given that, number of cars sold on six days is $4,{\text{ }}7,{\text{ }}2,{\text{ }}8,{\text{ }}3,{\text{ }}6$
Let us consider the first term to check,
i. If the number of cars sold on the seventh day is $2$
Hence,
Total cars sold on seven days after arranging in ascending order are $2,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}6,{\text{ }}7,{\text{ }}8$
First we have to find the mean,
Mean is $\dfrac{{2 + 2 + 3 + 4 + 6 + 7 + 8}}{7}$
Adding the terms we get,
$ \Rightarrow \dfrac{{32}}{7}$
$ \Rightarrow 4.57$
Now we have to find the median,
Hence, total number of values $ = 7$ (odd) then
The median is ${\left( {\dfrac{{7 + 1}}{2}} \right)^{th}}$ value $ = {4^{th}}$ value $ = 4$
If the number of cars sold on the seventh day is $2$ then ${\text{Mean}} \ne {\text{Median}}$
ii. If the number of cars sold on the seventh day is $4$
Hence,
Total cars sold on seven days after arranging in ascending order are $2,{\text{ }}3,{\text{ }}4,{\text{ }}4,{\text{ }}6,{\text{ }}7,{\text{ }}8$
First we have to find the mean,
Mean is $\dfrac{{2 + 3 + 4 + 4 + 6 + 7 + 8}}{7}$
Adding the terms we get,
$ \Rightarrow \dfrac{{34}}{7}$
$ \Rightarrow 4.85$
Now we have to find the median,
Hence, total number of values $ = 7$ (odd) then
The median is ${\left( {\dfrac{{7 + 1}}{2}} \right)^{th}}$ value $ = {4^{th}}$ value $ = 4$
If the number of cars sold on the seventh day is $4$ then ${\text{Mean}} \ne {\text{Median}}$
iii. If the number of cars sold on the seventh day is $5$
Hence,
Total cars sold on seven days after arranging in ascending order are $2,{\text{ }}3,{\text{ }}4,{\text{ }}5,{\text{ }}6,{\text{ }}7,{\text{ }}8$
First we have to find the mean,
Mean is $\dfrac{{2 + 3 + 4 + 5 + 6 + 7 + 8}}{7}$
Adding the terms we get,
$ \Rightarrow \dfrac{{35}}{7}$
$ \Rightarrow 5$
Now we have to find the median,
Hence, total number of values $ = 7$ (odd) then
The median is ${\left( {\dfrac{{7 + 1}}{2}} \right)^{th}}$ value $ = {4^{th}}$ value $ = 5$
If the number of cars sold on the seventh day is $4$ then ${\text{Mean}} = {\text{Median}}$
$\therefore 5$ is the average (arithmetic mean) number of cars sold per business day for the seven business days equal to the median number of cars sold per day for the seven days.

Option B is the correct answer.

Note: We have to remember that, the arithmetic mean is the simplest and most widely used measure of a mean, or average. It simply involves taking the sum of a group of numbers, then dividing that sum by the count of the numbers used in the series.