We are given a set of 9 non-negative numbers: {21, 8, 17, $a$, 51, 103, $b$, 13, 67}. We are also given that $a > b$.

Calculate Mean and Sum

The sum of the given numbers is $21 + 8 + 17 + a + 51 + 103 + b + 13 + 67 = 280 + a + b$.

The number of observations is $N=9$.

The mean is given as 40. Using the formula Mean $= \frac{\text{Sum}}{\text{N}}$:

$40 = \frac{280 + a + b}{9}$

$360 = 280 + a + b$

$a + b = 360 - 280$

$a + b = 80 \quad (1)$

Determine Median and Variable Constraints

The median is given as 21. For 9 numbers, the median is the $\frac{9+1}{2} = 5$th value in the sorted list.

The sorted list must have 21 as the 5th element. This means there are 4 numbers less than or equal to 21, and 4 numbers greater than or equal to 21.

The known numbers sorted are: {8, 13, 17, 21, 51, 67, 103}.

Numbers less than 21: {8, 13, 17}.

Numbers greater than 21: {51, 67, 103}.

Since the 5th element is 21, the 4 elements before it must be $\le 21$. These must be {8, 13, 17} and one of $a$ or $b$. This implies the other variable must be $\ge 21$.

Given $a > b$, the only possibility is that $b \le 21$ and $a \ge 21$.

Calculate Mean Deviation

The mean deviation (MD) about the median (M=21) is given as 26.

The formula for MD is: $MD = \frac{1}{N} \sum_{i=1}^{N} |x_i - M|$

$26 = \frac{1}{9} \sum |x_i - 21|$

$\sum |x_i - 21| = 26 \times 9 = 234$

Let's calculate the sum of deviations for the known numbers:

• $|8 - 21| = 13$
• $|13 - 21| = 8$
• $|17 - 21| = 4$
• $|21 - 21| = 0$
• $|51 - 21| = 30$
• $|67 - 21| = 46$
• $|103 - 21| = 82$

Sum of known deviations = $13 + 8 + 4 + 0 + 30 + 46 + 82 = 183$.

The total sum of deviations includes the deviations of $a$ and $b$: $183 + |a - 21| + |b - 21| = 234$ $|a - 21| + |b - 21| = 234 - 183$ $|a - 21| + |b - 21| = 51 \quad (2)$

Solve for $a$ and $b$

Using the constraints $b \le 21$ and $a \ge 21$ derived from the median condition:

• $|a - 21| = a - 21$ (since $a \ge 21$)
• $|b - 21| = -(b - 21) = 21 - b$ (since $b \le 21$)

Substitute these into equation (2):

$(a - 21) + (21 - b) = 51$

$a - b = 51 \quad (3)$

Now we have a system of two linear equations:

1. $a + b = 80$
2. $a - b = 51$

Add equation (1) and (3):

$(a + b) + (a - b) = 80 + 51$

$2a = 131$

$a = \frac{131}{2} = 65.5$

Substitute $a = 65.5$ into equation (1):

$65.5 + b = 80$

$b = 80 - 65.5 = 14.5$

Check conditions: $a=65.5, b=14.5$. $a > b$ (65.5 > 14.5 True). $b \le 21$ (14.5 <= 21 True). $a \ge 21$ (65.5 >= 21 True). The values satisfy all conditions.

Final Calculation

The question asks for the value of $2a$.

$2a = 2 \times 65.5 = 131$