The sum of all the two-digit odd numbers is:
(a)2475
(b)2530
(c)4905
(d)5049

Options

A. 2475
B. 2530
C. 4905
D. 5049

Answer

To find the sum of all the two-digit odd numbers, we can follow these steps:

### Step 1: Identify the first and last two-digit odd numbers
- The first two-digit odd number is **11**.
- The last two-digit odd number is **99**.

### Step 2: Determine the number of terms
- The formula to find the number of terms \( N \) in an arithmetic sequence is given by:
\[ N = \frac{L - A}{d} + 1 \]
where:
- \( L \) = last term = 99
- \( A \) = first term = 11
- \( d \) = common difference = 2 (since we are dealing with odd numbers)

Substituting the values:
\[ N = \frac{99 - 11}{2} + 1 = \frac{88}{2} + 1 = 44 + 1 = 45 \]

### Step 3: Calculate the sum of the series
- The sum \( S \) of an arithmetic series can be calculated using the formula:
\[ S = \frac{N}{2} \times (A + L) \]
Substituting the values we have:
\[ S = \frac{45}{2} \times (11 + 99) = \frac{45}{2} \times 110 \]

### Step 4: Simplify the sum
- Calculate \( \frac{45 \times 110}{2} \):
\[ S = \frac{4950}{2} = 2475 \]

### Conclusion
Thus, the sum of all the two-digit odd numbers is **2475**.

### Final Answer
The correct option is (a) 2475.

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