Question

Suppose that the mean and median of the non-negative numbers 21, 8, 17, \(a\), 51, 103, \(b\), 13, 67, \((a>b)\), are 40 and 21, respectively. If the mean deviation about the median is 26, then \(2a\) is equal to:

• A. 109
• B. 117
• C. 161
• D. 131

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We are given 9 numbers (including $a$ and $b$). The mean is the sum divided by 9, and the median is the middle value when the data is ordered. Mean deviation about the median is the average of the absolute differences between each data point and the median.

Step 2: Key Formula or Approach:
1. Total observations $n = 9$. 2. Mean = $\frac{\sum x_i}{9} = 40 \implies \sum x_i = 360$. 3. Sum of known values: $21+8+17+51+103+13+67 = 280$. 4. So, $a + b = 360 - 280 = 80$.

Step 3: Detailed Explanation:
1. Since the median is 21 and $n=9$, the 5th term in ascending order must be 21. 2. Given $a>b$ and median is 21, the arrangement must be $\{8, 13, 17, b, 21, a, 51, 67, 103\}$. (Since 21 is already in the list, $b$ must be $\leq 21$ and $a$ must be $\geq 21$). 3. Mean Deviation (M.D.) about Median (21): \[ \frac{|8-21| + |13-21| + |17-21| + |b-21| + |21-21| + |51-21| + |67-21| + |103-21| + |a-21|}{9} = 26 \] \[ 13 + 8 + 4 + (21-b) + 0 + 30 + 46 + 82 + (a-21) = 234 \] \[ 183 + a - b = 234 \implies a - b = 51 \] 4. We have $a + b = 80$ and $a - b = 51$. 5. Adding the equations: $2a = 131$.

Step 4: Final Answer:
The value of $2a$ is 131.