A={2,3,5,7,11,...89,97}
B={4,6,8,10,12,...98,100}
C={3,9,15,21,27,33...,99}
If a smallest and a greatest element of the Set B is transferred to Set A, then the average of A,B,C respectively :

Options

A. decreases,decreases,increases
B. decreases,constant and increased
C. increases,constant,constant
D. can't be determind

Answer

To solve the problem, we need to find the averages of the sets A, B, and C after transferring the smallest and greatest elements of set B to set A. Let's break this down step by step.

### Step 1: Identify the sets A, B, and C
- Set A consists of prime numbers from 2 to 97:
\( A = \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97\} \)

- Set B consists of even numbers from 4 to 100:
\( B = \{4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100\} \)

- Set C consists of numbers that are multiples of 6 up to 99:
\( C = \{3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99\} \)

### Step 2: Transfer elements from Set B to Set A
- The smallest element of Set B is 4, and the greatest element is 100.
- After transferring these elements to Set A, the new Set A becomes:
\( A' = A \cup \{4, 100\} \)

### Step 3: Calculate the new average of Set A
- The number of elements in Set A before the transfer is 25 (the number of prime numbers).
- After adding 4 and 100, the number of elements in Set A becomes 27.
- The sum of the original elements in Set A can be calculated as follows:
\[ \text{Sum of A} = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 = 1060 \]
- Adding the transferred elements:
\[ \text{New Sum of A} = 1060 + 4 + 100 = 1164 \]
- The average of Set A is:
\[ \text{Average of A} = \frac{1164}{27} \approx 43.11 \]

### Step 4: Calculate the average of Set B
- The number of elements in Set B is 49 (from 4 to 100).
- The sum of the elements in Set B can be calculated using the formula for the sum of an arithmetic series:
\[ \text{Sum of B} = \frac{n}{2} \times (\text{first term} + \text{last term}) = \frac{49}{2} \times (4 + 100) = 49 \times 52 = 2548 \]
- The average of Set B is:
\[ \text{Average of B} = \frac{2548}{49} = 52 \]

### Step 5: Calculate the average of Set C
- The number of elements in Set C is 17 (3 to 99, every 6th number).
- The sum of the elements in Set C can be calculated as follows:
\[ \text{Sum of C} = 3 + 9 + 15 + 21 + 27 + 33 + 39 + 45 + 51 + 57 + 63 + 69 + 75 + 81 + 87 + 93 + 99 = 816 \]
- The average of Set C is:
\[ \text{Average of C} = \frac{816}{17} = 48 \]

### Final Averages
- Average of Set A: \( \approx 43.11 \)
- Average of Set B: \( 52 \)
- Average of Set C: \( 48 \)