To solve the problem, we need to find the absolute value of \( x \) given that the mean deviation about the median of the numbers \( x, 2x, 3x, 4x, 5x, 6x, 7x, 8x, 9x, 10x \) is 30.
### Step-by-step Solution:
1. **Identify the Numbers**: The numbers given are \( x, 2x, 3x, 4x, 5x, 6x, 7x, 8x, 9x, 10x \).
2. **Find the Median**:
- Since there are 10 numbers (even count), the median is the average of the 5th and 6th terms.
- The 5th term is \( 5x \) and the 6th term is \( 6x \).
- Therefore, the median \( M \) is:
\[
M = \frac{5x + 6x}{2} = \frac{11x}{2}
\]
3. **Calculate the Mean Deviation**:
- The mean deviation \( D \) about the median is given by:
\[
D = \frac{1}{n} \sum_{i=1}^{n} |x_i - M|
\]
- Here, \( n = 10 \) and \( M = \frac{11x}{2} \).
- The absolute deviations from the median are:
- \( |x - \frac{11x}{2}| = |-\frac{9x}{2}| = \frac{9|x|}{2} \)
- \( |2x - \frac{11x}{2}| = |-\frac{7x}{2}| = \frac{7|x|}{2} \)
- \( |3x - \frac{11x}{2}| = |-\frac{5x}{2}| = \frac{5|x|}{2} \)
- \( |4x - \frac{11x}{2}| = |-\frac{3x}{2}| = \frac{3|x|}{2} \)
- \( |5x - \frac{11x}{2}| = |-\frac{x}{2}| = \frac{|x|}{2} \)
- \( |6x - \frac{11x}{2}| = |\frac{x}{2}| = \frac{|x|}{2} \)
- \( |7x - \frac{11x}{2}| = |\frac{3x}{2}| = \frac{3|x|}{2} \)
- \( |8x - \frac{11x}{2}| = |\frac{5x}{2}| = \frac{5|x|}{2} \)
- \( |9x - \frac{11x}{2}| = |\frac{7x}{2}| = \frac{7|x|}{2} \)
- \( |10x - \frac{11x}{2}| = |\frac{9x}{2}| = \frac{9|x|}{2} \)
4. **Sum of Absolute Deviations**:
- The total sum of absolute deviations is:
\[
\text{Total} = \frac{9|x|}{2} + \frac{7|x|}{2} + \frac{5|x|}{2} + \frac{3|x|}{2} + \frac{|x|}{2} + \frac{|x|}{2} + \frac{3|x|}{2} + \frac{5|x|}{2} + \frac{7|x|}{2} + \frac{9|x|}{2}
\]
- This simplifies to:
\[
\text{Total} = \frac{9 + 7 + 5 + 3 + 1 + 1 + 3 + 5 + 7 + 9}{2}|x| = \frac{50}{2}|x| = 25|x|
\]
5. **Calculate the Mean Deviation**:
- The mean deviation is:
\[
D = \frac{25|x|}{10} = 2.5|x|
\]
- We know from the problem statement that \( D = 30 \). Therefore:
\[
2.5|x| = 30
\]
6. **Solve for \( |x| \)**:
- Dividing both sides by 2.5:
\[
|x| = \frac{30}{2.5} = 12
\]
### Final Answer:
Thus, the value of \( |x| \) is \( 12 \).
