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 have 9 numbers. Their mean is 40, their median is 21, and their mean deviation about the median is 26. We need to set up equations using these statistical formulas to find the values of \(a\) and \(b\).
Step 2: Key Formula or Approach:
1. Mean: \(\frac{\sum x_i}{n} = \mu \implies \sum x_i = 9 \times 40 = 360\)
2. Median: For 9 numbers, the median is the 5th number when arranged in ascending order.
3. Mean Deviation about Median: \(\frac{\sum |x_i - \text{Median}|}{n} = 26 \implies \sum |x_i - 21| = 234\)
Step 3: Detailed Explanation:
Let's find the sum of the known numbers:
\[ 21 + 8 + 17 + 51 + 103 + 13 + 67 = 280 \] Using the mean condition:
\[ 280 + a + b = 360 \implies a + b = 80 \] Next, analyze the median condition.
The known numbers sorted are: 8, 13, 17, 21, 51, 67, 103.
For 21 to be the 5th number in the overall sorted list of 9 elements, there must be exactly four numbers smaller than or equal to 21. We already have 8, 13, 17, and 21.
Therefore, of the unknown numbers \(a\) and \(b\), exactly one must be \(\le 21\) and the other must be \(\ge 21\).
Since \(a>b\), it must be true that \(b \le 21\) and \(a \ge 21\).
Now use the mean deviation about the median:
\[ \sum |x_i - 21| = 234 \] Substitute the known values:
\[ |8-21| + |13-21| + |17-21| + |21-21| + |51-21| + |67-21| + |103-21| + |a-21| + |b-21| = 234 \] \[ 13 + 8 + 4 + 0 + 30 + 46 + 82 + |a-21| + |b-21| = 234 \] \[ 183 + |a-21| + |b-21| = 234 \implies |a-21| + |b-21| = 51 \] Since \(a \ge 21\) and \(b \le 21\), we can resolve the absolute values: \(|a-21| = a-21\) and \(|b-21| = 21-b\).
\[ (a - 21) + (21 - b) = 51 \implies a - b = 51 \] Step 4: Final Answer:
We now have a system of two linear equations:
1. \(a + b = 80\)
2. \(a - b = 51\)
Adding the two equations together:
\[ 2a = 131 \]