If the median of the data 24, 25, 26, (x + 2), (x + 3), 30, 31, 34 is 27.5, then x is equal to :

Question

Accepted Answer

To determine the value of $ x $ such that the median of the data set 24, 25, 26, $ (x + 2) $, $ (x + 3) $, 30, 31, 34 is 27.5, we need to follow these steps:

1. **Understanding the Median**: The median is the middle value of a data set when it is ordered. If the number of data points is even, the median is the average of the two middle numbers.

2. **Data Set Analysis**: The given data set has 8 numbers: 24, 25, 26, $ (x + 2) $, $ (x + 3) $, 30, 31, 34. Since there are 8 numbers, the median will be the average of the 4th and 5th numbers when the data is arranged in ascending order.

3. **Arranging the Data**: The data is already partially ordered. We need to determine the position of $ (x + 2) $ and $ (x + 3) $ relative to the other numbers to find the correct order.

4. **Finding the Median**: The median is given as 27.5, which means:
   $$
   \frac{(x + 2) + (x + 3)}{2} = 27.5
   $$
   Solving this equation:
   $$
   \frac{2x + 5}{2} = 27.5
   $$
   $$
   2x + 5 = 55
   $$
   $$
   2x = 50
   $$
   $$
   x = 25
   $$

5. **Verification**: If $ x = 25 $, then $ (x + 2) = 27 $ and $ (x + 3) = 28 $. The ordered data set becomes 24, 25, 26, 27, 28, 30, 31, 34. The 4th and 5th numbers are 27 and 28, respectively. The average of these two numbers is:
   $$
   \frac{27 + 28}{2} = 27.5
   $$
   This confirms that the median is indeed 27.5.

6. **Explanation of Incorrect Options**:
   - **Option 1 (x = 27)**: If $ x = 27 $, then $ (x + 2) = 29 $ and $ (x + 3) = 30 $. The ordered data set would be 24, 25, 26, 29, 30, 30, 31, 34. The median would be the average of 29 and 30, which is 29.5, not 27.5.
   - **Option 3 (x = 28)**: If $ x = 28 $, then $ (x + 2) = 30 $ and $ (x + 3) = 31 $. The ordered data set would be 24, 25, 26, 30, 30, 31, 31, 34. The median would be the average of 30 and 30, which is 30, not 27.5.
   - **Option 4 (x = 29)**: If $ x = 29 $, then $ (x + 2) = 31 $ and $ (x + 3) = 32 $. The ordered data set would be 24, 25, 26, 30, 31, 31, 32, 34. The median would be the average of 30 and 31, which is 30.5, not 27.5.

Therefore, the correct answer is Option 2, where $ x = 25 $.