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 an element less than 50 belongs to Set A is transferred to Set B,then the average of Set B :

Options

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

Answer

To find the new average of Set B after transferring an element from Set A that is less than 50, we will follow these steps:

### Step 1: Identify the Elements of Set A and Set B
- Set A consists of prime numbers less than 100: 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}.
- The elements of Set A that are less than 50 are: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}.

- 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}.

### Step 2: Calculate the Current Average of Set B
- The number of elements in Set B is 49 (from 4 to 100).
- The sum of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (a + l) \]
where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term.

- Here, \(n = 49\), \(a = 4\), and \(l = 100\):
\[ S_{49} = \frac{49}{2} \times (4 + 100) = \frac{49}{2} \times 104 = 49 \times 52 = 2548 \]

- The average of Set B is:
\[ \text{Average} = \frac{S_{49}}{49} = \frac{2548}{49} = 52 \]

### Step 3: Transfer an Element from Set A to Set B
- Let's denote the element transferred from Set A to Set B as \(x\), where \(x < 50\). The possible values for \(x\) are {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}.

### Step 4: Calculate the New Average of Set B
- After transferring \(x\), the number of elements in Set B becomes 50.
- The new sum of Set B becomes \(2548 + x\).
- The new average of Set B is:
\[ \text{New Average} = \frac{2548 + x}{50} \]

### Step 5: Analyze the Effect on the Average
- Since \(x\) is less than 52 (the original average), adding \(x\) will decrease the average of Set B.
- Therefore, the new average will be less than 52.

### Conclusion
The average of Set B after transferring an element less than 50 from Set A will be less than 52.

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