​Mean of the numbers 2, 4, 5, 8, 2 and 3 is m. The numbers 4, 3, 3, 5, m, 3 and p have mean m + 1, median q and mode r. What is the value of (p + q – r)?​

Correct option is D

Given:

First set of numbers: $2,4,5,8,2,3$

Mean of the first set = $m$

Second set of numbers: $4,3,3,5,m,3,p$

Mean of the second set = $m+1$

Median = $q$

Mode = $r$

Find  $p+q-r$.

Formula Used:

Mean=Sum of the numbersCount of the numbers\text{Mean} = \frac{\text{Sum of the numbers}}{\text{Count of the numbers}}$$\text{mean} = \frac{\text{Sum of the numbers}}{\text{Count of the numbers}}$$ 

Median = Middle value of the sorted data (or average of two middle values if count is even).

​Solution:

The first set is 2,4,5,8,2,3

2,4,5,8,2,32, 4, 5, 8, 2, 3Using the mean formula:​

Mean =$$\frac{2+4+5+8+2+3​ }{6}$$ = 4

Thus, $m\; =4$.

The second set is 

$4,3,3,5,m,3,p$. Substituting $m=4$, the set becomes:

$$4,3,3,5,4,3,p$$

The mean of the second set is 

$m+1=4+1=5$. Using the mean formula:

5 = $$\frac{\text{Sum of the numbers}}{\text{Count of the numbers}}$$

5 = $$\frac{4+3+3+5+4+3+p​}{7}$$

35 = 22+ p

Thus, 

$p=13$.

The sorted second set is:

$$3,3,3,4,4,5,13$$

The median is the middle value (4th number in the sorted list):

$$q=4$$

Thus, $q=4$.

The mode is the number that appears most frequently. In the set 

$3,3,3,4,4,5,13$, the number 333 appears three times, which is the highest frequency.

Thus, $r=3$.

Substitute the values of 

$p,q,r$:

$$p=13,q=4,r=3$$

p+q−r=13+4−3=14p + q - r = 13 + 4 - 3 = 14p + q − r = 13 + 4 − 3 = 14

Thus, option (d) is right.

​