If the average of the numberd ,1,2,3 ….. 98,99 x is 100 x then the value of x is

Options

A. `(51 )/(100)`
B. `(50 )/(99)`
C. `(50 )/(101)`
D. `(51)/(99)`

Answer

To solve the problem step-by-step, we need to find the value of \( x \) given that the average of the numbers \( 1, 2, 3, \ldots, 99, x \) is equal to \( 100x \).

### Step 1: Understand the average formula
The average of a set of numbers is calculated by dividing the sum of the numbers by the total count of the numbers.

### Step 2: Calculate the sum of the first 99 natural numbers
The sum of the first \( n \) natural numbers is given by the formula:
\[ S_n = \frac{n(n + 1)}{2} \]
For \( n = 99 \):
\[ S_{99} = \frac{99 \times 100}{2} = 4950 \]

### Step 3: Set up the equation for the average
The total number of terms in the set \( 1, 2, 3, \ldots, 99, x \) is \( 100 \) (99 numbers + 1 number \( x \)). The average is given as:
\[ \text{Average} = \frac{S + x}{100} \]
where \( S \) is the sum of the first 99 numbers, which we calculated as \( 4950 \).

Thus, we can write:
\[ \frac{4950 + x}{100} = 100x \]

### Step 4: Solve the equation
Now, we will multiply both sides by \( 100 \) to eliminate the denominator:
\[ 4950 + x = 10000x \]

Rearranging gives:
\[ 4950 = 10000x - x \]
\[ 4950 = 9999x \]

### Step 5: Isolate \( x \)
Now, we can solve for \( x \):
\[ x = \frac{4950}{9999} \]

### Step 6: Simplify the fraction
To simplify \( \frac{4950}{9999} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 4950 and 9999 is 99:
\[ x = \frac{4950 \div 99}{9999 \div 99} = \frac{50}{101} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{50}{101}} \]