Let the sets `A={2, 4, 6, 8….} and B={3,6,9,12…}` such that `n(A)=200 and n(B)=250`. If `n(AuuB)=k,` then `(k)/(100)` is equal to

Answer

To solve the problem step by step, we will first define the sets A and B and then calculate the required values.

### Step 1: Define the sets A and B
- Set A consists of even numbers: \( A = \{2, 4, 6, 8, \ldots\} \)
- Set B consists of multiples of 3: \( B = \{3, 6, 9, 12, \ldots\} \)

### Step 2: Determine the number of elements in sets A and B
- The number of elements in set A is given as \( n(A) = 200 \).
- The number of elements in set B is given as \( n(B) = 250 \).

### Step 3: Find the maximum elements in sets A and B
- The largest element in set A can be calculated as:
\[ \text{Max element in } A = 2 \times 200 = 400 \]
- The largest element in set B can be calculated as:
\[ \text{Max element in } B = 3 \times 250 = 750 \]

### Step 4: Calculate the intersection of sets A and B
- The intersection \( A \cap B \) consists of numbers that are multiples of both 2 and 3, which means they are multiples of 6.
- We need to find the number of multiples of 6 that are less than or equal to 400:
\[ n(A \cap B) = \left\lfloor \frac{400}{6} \right\rfloor \]
\[ n(A \cap B) = \left\lfloor 66.67 \right\rfloor = 66 \]

### Step 5: Use the formula for the union of two sets
- The formula for the number of elements in the union of two sets is:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
- Substituting the values we have:
\[ n(A \cup B) = 200 + 250 - 66 \]
\[ n(A \cup B) = 384 \]

### Step 6: Calculate \( \frac{k}{100} \)
- We have found \( k = n(A \cup B) = 384 \).
- Now we need to calculate \( \frac{k}{100} \):
\[ \frac{k}{100} = \frac{384}{100} = 3.84 \]

### Final Answer
Thus, the value of \( \frac{k}{100} \) is \( 3.84 \).

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