A={2,3,5,7,11,...89,97}
B={4,6,8,10,12,...98,100}
C={1,9,15,21,25,27,33...95,99}
The average of all the elements of A,B and C is :

Options

A. 49.5
B. 50.5
C. 55
D. none of these

Answer

To find the average of all the elements in the sets A, B, and C, we will follow these steps:

### Step 1: Identify the elements in each set
- Set A consists of all prime numbers between 1 and 100.
- Set B consists of all even numbers between 1 and 100.
- Set C consists of specific odd numbers between 1 and 100.

### Step 2: Determine the total number of elements in each set
- **Set A (Prime Numbers)**: The prime numbers between 1 and 100 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
There are 25 prime numbers in total.

- **Set B (Even Numbers)**: The even numbers between 1 and 100 are:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100.
There are 50 even numbers in total.

- **Set C (Specific Odd Numbers)**: The odd numbers in set C are:
1, 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 81, 83, 87, 91, 93, 95, 99.
There are 25 specific odd numbers in total.

### Step 3: Calculate the total number of elements in all sets
- Total elements = Number of elements in A + Number of elements in B + Number of elements in C
- Total elements = 25 + 50 + 25 = 100

### Step 4: Calculate the sum of all elements in each set
- **Sum of Set A**: The sum of the prime numbers from 1 to 100 is 1060.
- **Sum of Set B**: The sum of the first 50 even numbers can be calculated using the formula for the sum of an arithmetic series:
\[ \text{Sum} = n/2 \times (\text{first term} + \text{last term}) = 50/2 \times (2 + 100) = 25 \times 102 = 2550 \]
- **Sum of Set C**: The sum of the specific odd numbers can be calculated directly:
\[ 1 + 9 + 15 + 21 + 25 + 27 + 33 + 35 + 39 + 45 + 49 + 51 + 55 + 57 + 63 + 65 + 69 + 75 + 81 + 83 + 87 + 91 + 93 + 95 + 99 = 1225 \]

### Step 5: Calculate the total sum of all elements
- Total sum = Sum of A + Sum of B + Sum of C
- Total sum = 1060 + 2550 + 1225 = 4835

### Step 6: Calculate the average
- Average = Total sum / Total number of elements
- Average = 4835 / 100 = 48.35

### Final Answer
The average of all the elements of A, B, and C is **48.35**.

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