(i) The median of the observations 11, 12, 14, 18, (x + 4), 30, 32, 35, 41 arranged in ascending order is 24. Find the value of x.

(ii) If 10, 13, 15, 18, x + 1, x + 3, 31, 36, 38, 42 are the observations arranged in ascending order with median 28, find the value of x.

(i) Set of numbers are arranged in ascending order,

11, 12, 14, 18, (x + 4), 30, 32, 35, 41

Given,

Median = 24.

Here,

Number of observations (n) = 9, which is odd.

$\Rightarrow \text{Median} = \dfrac{9 + 1}{2} \text{th} \text{ observation} \\[1em] \Rightarrow 24 = \dfrac{10}{2} \text{th} \text{ observation} \\[1em] \Rightarrow 24 = 5 \text{ th} \text{ observation} \\[1em] \Rightarrow 24 = \text{x} + 4 \\[1em] \Rightarrow \text{x} = 24 - 4 \\[1em] \Rightarrow \text{x} = 20$

Hence, the value of x = 20.

(ii) Set of numbers are arranged in ascending order,

10, 13, 15, 18, x + 1, x + 3, 31, 36, 38, 42

Given,

Median = 28.

Here,

Number of observations (n) = 10, which is even.

$\Rightarrow \text{Median} = \dfrac{\Big(\dfrac{\text{n}}{2}\Big) \text{th} \text{ term} + \Big(\dfrac{\text{n}}{2} + 1\Big)\text{th} \text{ term}}{2} \\[1em] \Rightarrow 28 = \dfrac{\Big(\dfrac{10}{2}\Big) \text{th} \text{ term} + \Big(\dfrac{10}{2} + 1\Big)\text{th} \text{ term}}{2} \\[1em] \Rightarrow 28 = \dfrac{5 \text{th} \text{ term} + (5 + 1)\text{th} \text{ term}}{2} \\[1em] \Rightarrow 28 = \dfrac{5 \text{th} \text{ term} + 6\text{th} \text{ term}}{2} \\[1em] \Rightarrow 28 \times 2 = (\text{x} + 1) + (\text{x} + 3) \\[1em] \Rightarrow 56 = 4 + 2\text{x} \\[1em] \Rightarrow 2\text{x} = 56 - 4 \\[1em] \Rightarrow 2\text{x} = 52 \\[1em] \Rightarrow \text{x} = \dfrac{52}{2} \\[1em] \Rightarrow \text{x} = 26.$

Hence, the value of x = 26.