What is the average of all the one digit and two digit natural numbers?

Options

A. 25
B. 40
C. 50
D. 99

Answer

To find the average of all the one-digit and two-digit natural numbers, we can follow these steps:

### Step 1: Identify the range of numbers
The one-digit natural numbers are from 1 to 9, and the two-digit natural numbers are from 10 to 99.

### Step 2: Calculate the sum of one-digit natural numbers
The one-digit natural numbers are: 1, 2, 3, 4, 5, 6, 7, 8, 9.
To find the sum, we can use the formula for the sum of the first n natural numbers:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
Here, \( n = 9 \):
\[ \text{Sum of one-digit numbers} = \frac{9(9 + 1)}{2} = \frac{9 \times 10}{2} = 45 \]

### Step 3: Calculate the sum of two-digit natural numbers
The two-digit natural numbers are from 10 to 99. To find the sum of these numbers, we can also use the sum formula:
- The first term (a) = 10
- The last term (l) = 99
- The number of terms (n) = 99 - 10 + 1 = 90

The sum of an arithmetic series can be calculated as:
\[ \text{Sum} = \frac{n}{2} \times (a + l) \]
Substituting the values:
\[ \text{Sum of two-digit numbers} = \frac{90}{2} \times (10 + 99) = 45 \times 109 = 4905 \]

### Step 4: Calculate the total sum of one-digit and two-digit natural numbers
Now, we add the sum of one-digit and two-digit numbers:
\[ \text{Total Sum} = \text{Sum of one-digit numbers} + \text{Sum of two-digit numbers} = 45 + 4905 = 4950 \]

### Step 5: Calculate the total number of terms
The total number of one-digit numbers is 9, and the total number of two-digit numbers is 90.
\[ \text{Total number of terms} = 9 + 90 = 99 \]

### Step 6: Calculate the average
Now, we can find the average using the formula:
\[ \text{Average} = \frac{\text{Total Sum}}{\text{Total number of terms}} = \frac{4950}{99} \]
Calculating this gives:
\[ \text{Average} = 50 \]

Thus, the average of all the one-digit and two-digit natural numbers is **50**.

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