The median of the following observations arranged in ascending order is 64. Find the value of x :

27, 31, 46, 52, x, x + 4, 71, 79, 85, 90

1. 60

2. 61

3. 62

4. 66

No. of observations = 10, which is even.

Median = $\dfrac{\left(\dfrac{n}{2}\right)^\text{th}\text{term} + \left(\dfrac{n}{2} + 1\right)^\text{th}\text{term}}{2}$

$\left(\dfrac{n}{2}\right)^\text{th}\text{term} = \dfrac{10}{2}$ = 5^th term = x

$\left(\dfrac{n}{2} + 1\right)^\text{th}\text{term} = \dfrac{10}{2} + 1$ = 5 + 1 = 6^th term = x + 4

Given,

Median = 64

$\therefore \dfrac{x + x + 4}{2} = 64 \\[1em] \Rightarrow \dfrac{2x + 4}{2} = 64 \\[1em] \Rightarrow \dfrac{2(x + 2)}{2} = 64 \\[1em] \Rightarrow x + 2 = 64 \\[1em] \Rightarrow x = 64 - 2 \\[1em] \Rightarrow x = 62$

Hence, Option 3 is the correct option.