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 any two elements, greater than 50, belong to Set A are transferred to Set C, then the average of Set C :

Options

A. A. remains constant
B. B. decreases
C. C. increases
D. D. can't say

Answer

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

### Step 1: Identify the elements of Set A, Set B, and Set C
- **Set A**: Contains all prime numbers up to 100.
- Prime numbers up to 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}.

- **Set B**: Contains all non-prime even natural numbers up to 100.
- Non-prime even numbers up to 100 are: {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**: Contains all non-prime odd natural numbers up to 100.
- Non-prime odd numbers up to 100 are: {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}.

### Step 2: Calculate the initial average of Set C
- **Elements in Set C**: {1, 9, 15, 21, 25, 27, 33, 95, 99}
- **Number of elements in Set C**: 9
- **Sum of elements in Set C**:
\[ 1 + 9 + 15 + 21 + 25 + 27 + 33 + 95 + 99 = 325 \]
- **Average of Set C**:
\[ \text{Average} = \frac{\text{Sum of elements}}{\text{Number of elements}} = \frac{325}{9} \approx 36.11 \]

### Step 3: Transfer two elements greater than 50 from Set A to Set C
- **Elements in Set A greater than 50**: {53, 59, 61, 67, 71, 73, 79, 83, 89, 97}
- We can choose any two elements, for example, let's choose 89 and 97.

### Step 4: Update Set C and recalculate the average
- **New Set C**: {1, 9, 15, 21, 25, 27, 33, 95, 99, 89, 97}
- **New number of elements in Set C**: 11
- **New sum of elements in Set C**:
\[ 325 + 89 + 97 = 511 \]
- **New average of Set C**:
\[ \text{New Average} = \frac{\text{New Sum of elements}}{\text{New Number of elements}} = \frac{511}{11} \approx 46.45 \]

### Conclusion
The average of Set C after transferring two elements from Set A is approximately 46.45.

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