There are 3 sets of natural numbers 1 ot 100. Set A contains all the natural numbers which are prime, upto 100. Set B contains all the non-prime even natural numbers upto 100. Set C contains all the non-prime odd natural numbers upto 100 i.e.,
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}
If a least and a greatest element of Set C are transferred from Set C to Set A then the average of Set A :

Options

A. a. increases
B. b. decreases
C. c. remains constant
D. d. can't be determined

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the sets
- **Set A**: Contains all prime numbers up to 100.
- **Set B**: Contains all non-prime even natural numbers up to 100.
- **Set C**: Contains all non-prime odd natural numbers up to 100.

### Step 2: List the elements of Set A
The prime numbers up to 100 are:
\[ A = \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97\} \]
Count of elements in Set A = 25.

### Step 3: List the elements of Set C
The non-prime odd natural numbers up to 100 are:
\[ C = \{1, 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 77, 81, 85, 87, 91, 93, 95, 99\} \]
Count of elements in Set C = 25.

### Step 4: Identify the least and greatest elements of Set C
- Least element of Set C = 1
- Greatest element of Set C = 99

### Step 5: Transfer the least and greatest elements from Set C to Set A
After transferring:
- New Set A = \( A \cup \{1, 99\} \)
- New Set A = \( \{1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 99\} \)

### Step 6: Count the new number of elements in Set A
New count of elements in Set A = 27 (25 original + 2 transferred).

### Step 7: Calculate the sum of the original elements in Set A
The sum of the original prime numbers in Set A is:
\[ \text{Sum of Set A} = 1060 \]

### Step 8: Calculate the new sum of Set A after transferring elements
New sum of Set A = Original sum + Least element + Greatest element
\[ \text{New Sum} = 1060 + 1 + 99 = 1160 \]

### Step 9: Calculate the average of the new Set A
Average of Set A = New Sum / New Count of Elements
\[ \text{Average} = \frac{1160}{27} \approx 42.96 \]

### Conclusion
The average of Set A after transferring the least and greatest elements from Set C is approximately 42.96.

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