Question

Mean of 17,4,8,6,15 is m and median of 8,14,10,5,7,20,19,n is (m-1).
Find the values of m and n.
​

Answer 1

hey mate !!!!

Here is your answer ________________________________

hope it helps.....

plz mark it as brainiest.....

Answer 2

The mean of observation is the sum of the observations divided by the no. of the observations.

Consider the observations 17, 4, 8, 6, 15.

Given that the mean of these observations is m. The no. of observations is 5.

Thus,

$\displaystyle \begin{aligned}&\frac{17+4+8+6+15}{5}=m \\ \\ \\ \Longrightarrow\ \ &\frac{50}{5}=m \\ \\ \\ \Longrightarrow\ \ &\bold{10=m}\end{aligned}$

Thus the value of m is 10.

Now consider the observations 8, 14, 10, 5, 7, 20, 19, n.

Given that the median of these observations is  m - 1,  which implies 10 - 1 = 9.

Thus the median is 9.

The no. of these observations is 8.  So the median is the average of the 4th and 5th terms when arranged in ascending order.

Now we have to find in how many ways the median, i.e., 9, can be obtained as the average of two observations.

$\displaystyle \begin{aligned}&\frac{T_4+T_5}{2}=9 \\ \\ \\ \Longrightarrow\ \ &T_4+T_5=9 \times 2 \\ \\ \\ \Longrightarrow\ \ &T_4+T_5=18\end{aligned}$

From this we get that the sum of the 4th and 5th terms should be 18 to get the median 9.

According to this, we're going to find how many pairs of observations among them give the sum 18.

Some inferences are given below:

⇒  There are 8 and 10 among the observations so that the 4th and 5th term can be 8 and 10 respectively.

$\_ \begin{quote}\displaystyle \frac{8+10}{2}=9\ \ \ ; \ \ \ 8+10=18$

⇒  There's no 4 among the observations to take 14 to get the median 9. Even if there was, it wouldn't be 4th term because there are other observations greater than 4.

⇒  There's no 13 among the observations to take 5 to get the median 9. Even if there was, it wouldn't be 5th term because there are other observations which values between 5 and 13.

⇒  There's no 11 among the observations to take 7 to get the median 9. Even if there was, it wouldn't be 5th term because there are other observations between 7 and 11.

⇒  19 and 20 can't be taken because they're greater than 18.

But there's no need to find out these inferences!

Because we've already got two observations among the 8 ones which give sum 18, which are 8 and 10, "and these are the nearest integers to the median." Thus there won't be other observations for the median 9, if all the observations are positive integers.

Hence we can say that the 4th and 5th observations among them are 8 and 10 respectively when arranged in ascending order.

Okay, arranging the observations except n in ascending order,

5, 7, 8, 10, 14, 19, 20

Here the positions of 8 and 10 are 3rd and 4th respectively. So, to get it there at 4th and 5th, n should be before 8.

To get the observations 8 and 10 as 4th and 5th terms respectively, the position of n should be one among the following three cases:

Case 1:    n, 5, 7, 8, 10, 14, 19, 20

Case 2:   5, n, 7, 8, 10, 14, 19, 20

Case 3:   5, 7, n, 8, 10, 14, 19, 20

In case 1, the possible values of n are '1, 2, 3, 4 and 5'.

In case 2, the possible values of n are '5, 6 and 7'.

In case 3, the possible values of n are '7 and 8'.

Thus the total possible values of n are 1, 2, 3, 4, 5, 6, 7 and 8.

n ∈ {1, 2, 3, 4, 5, 6, 7, 8}