The numbers 24, 45, a, 35, 59, 83, 46, b, 29, 74 are serially numbered as they appear· in the sequence. When each number is added to its serial number, then the average of the new numbers formed is 55. The average of the missing numbers (a and b) is:

Options

A. 50
B. 62
C. 38
D. 58

Answer

To solve the problem step by step, we will follow the instructions given in the question and perform the necessary calculations.

### Step 1: Identify the numbers and their serial positions
The numbers given are:
1. 24
2. 45
3. a
4. 35
5. 59
6. 83
7. 46
8. b
9. 29
10. 74

### Step 2: Add the serial number to each number
We will add the serial number (position) to each number:
- For the 1st number: \( 24 + 1 = 25 \)
- For the 2nd number: \( 45 + 2 = 47 \)
- For the 3rd number: \( a + 3 \)
- For the 4th number: \( 35 + 4 = 39 \)
- For the 5th number: \( 59 + 5 = 64 \)
- For the 6th number: \( 83 + 6 = 89 \)
- For the 7th number: \( 46 + 7 = 53 \)
- For the 8th number: \( b + 8 \)
- For the 9th number: \( 29 + 9 = 38 \)
- For the 10th number: \( 74 + 10 = 84 \)

### Step 3: Write the equation for the average
The average of these new numbers is given as 55. The formula for average is:
\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
In this case, the number of observations is 10. Therefore, we can write:
\[ \frac{(25 + 47 + (a + 3) + 39 + 64 + 89 + 53 + (b + 8) + 38 + 84)}{10} = 55 \]

### Step 4: Simplify the equation
First, we can simplify the sum in the numerator:
\[ 25 + 47 + 39 + 64 + 89 + 53 + 38 + 84 + 3 + 8 + a + b \]
Calculating the constant terms:
\[ 25 + 47 = 72 \]
\[ 72 + 39 = 111 \]
\[ 111 + 64 = 175 \]
\[ 175 + 89 = 264 \]
\[ 264 + 53 = 317 \]
\[ 317 + 38 = 355 \]
\[ 355 + 84 = 439 \]
Now including the terms with \( a \) and \( b \):
\[ 439 + 3 + 8 + a + b = 450 + a + b \]
Thus, we have:
\[ \frac{450 + a + b}{10} = 55 \]

### Step 5: Solve for \( a + b \)
Multiplying both sides by 10 gives:
\[ 450 + a + b = 550 \]
Subtracting 450 from both sides:
\[ a + b = 100 \]

### Step 6: Find the average of \( a \) and \( b \)
The average of \( a \) and \( b \) is given by:
\[ \text{Average of } a \text{ and } b = \frac{a + b}{2} = \frac{100}{2} = 50 \]

### Final Answer
The average of the missing numbers \( a \) and \( b \) is **50**.