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, we will follow these steps:

### Step 1: Identify the elements in sets A and B
Set A consists of all even numbers, which can be represented as:
\[ A = \{2, 4, 6, 8, \ldots\} \]
Set B consists of multiples of 3, which can be represented as:
\[ B = \{3, 6, 9, 12, \ldots\} \]

### Step 2: Determine the number of elements in sets A and B
We are given:
- \( n(A) = 200 \)
- \( n(B) = 250 \)

### Step 3: Find the last element in each set
The last element in set A (the 200th even number) is:
\[ 2 \times 200 = 400 \]

The last element in set B (the 250th multiple of 3) is:
\[ 3 \times 250 = 750 \]

### Step 4: Find the intersection of sets A and B
The intersection \( A \cap B \) will consist of the common elements, which are the multiples of 6 (the least common multiple of 2 and 3).

### Step 5: Calculate the number of multiples of 6 in set A
To find the number of multiples of 6 up to 400:
\[ \text{Number of multiples of 6} = \left\lfloor \frac{400}{6} \right\rfloor \]

Calculating:
\[ 400 \div 6 = 66.67 \]
Thus, the number of multiples of 6 is:
\[ \left\lfloor 66.67 \right\rfloor = 66 \]

### Step 6: Use the formula for the union of two sets
Using the formula:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

Substituting the values we have:
\[ n(A \cup B) = 200 + 250 - 66 \]
\[ n(A \cup B) = 450 - 66 = 384 \]

### Step 7: Find the value of \( k \) and calculate \( \frac{k}{100} \)
Here, \( k = n(A \cup B) = 384 \).
Now, we need to calculate:
\[ \frac{k}{100} = \frac{384}{100} = 3.84 \]

### Final Answer
Thus, the value of \( \frac{k}{100} \) is:
\[ \boxed{3.84} \]

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