The median of the observations 11,12,14,18,x+2,x+4,30,32,35,41 arranged in ascending order is 24.  Find the value of x.A) 19B) 21C) 24D) 26

1. ### Break Down the Problem

We need to find the value of $x$ such that the median of the given observations is 24. The observations are $11, 12, 14, 18, x+2, x+4, 30, 32, 35, 41$.

To find the median:

• Since there are 10 observations, the median will be the average of the 5th and 6th terms when arranged in ascending order.
• We will denote the observations in increasing order and find suitable values of $x$ that position the terms correctly, ensuring the 5th and 6th terms average to 24.

2. ### Relevant Concepts

Median for a set of numbers:

• For an even number of terms, $\text{Median} = \frac{a_{n/2} + a_{(n/2) + 1}}{2}$

3. ### Analysis and Detail

1. We start by arranging the observations in ascending order:

   • Given numbers: $11, 12, 14, 18, x+2, x+4, 30, 32, 35, 41$

2. To find $x$, set the median condition:

   • According to the problem, $\frac{a_5 + a_6}{2} = 24$, which gives us $a_5 + a_6 = 48$.

3. We analyze different scenarios for $x$ considering that $x+2 < x+4$. Thus, we need to find placements for $x+2$ and $x+4$ among the other numbers:

   • For $x+2$:
     • If $x+2 < 30$, it means $x < 28$.
   • For $x+4$:
     • If $x+4 > 18$, it means $x > 14$.

4. Given the constraints above, we check a few integer values of $x$ within the range $14 < x < 28$.

4. ### Verify and Summarize

By substituting $x$ with possible values from options A) 19, B) 21, C) 24, D) 26:

• If $x = 19$:

  • $x+2 = 21$
  • $x+4 = 23$
  • New list: $11, 12, 14, 18, 21, 23, 30, 32, 35, 41$
  • Median: $\frac{21 + 23}{2} = 22$ (not 24).

• If $x = 21$:

  • $x+2 = 23$
  • $x+4 = 25$
  • New list: $11, 12, 14, 18, 23, 25, 30, 32, 35, 41$
  • Median: $\frac{23 + 25}{2} = 24$ (matches 24).

• If $x = 24$:

  • $x+2 = 26$
  • $x+4 = 28$
  • New list: $11, 12, 14, 18, 26, 28, 30, 32, 35, 41$
  • Median: $\frac{26 + 28}{2} = 27$ (not 24).

• If $x = 26$:

  • $x+2 = 28$
  • $x+4 = 30$
  • New list: $11, 12, 14, 18, 28, 30, 30, 32, 35, 41$
  • The median is $\frac{28 + 30}{2} = 29$ (not 24).

The only value of $x$ that ensures the median is 24 is $x = 21$.

Final Answer

The value of $x$ is 21 (Option B).