Amongh three numbers first numbr is twice the 2nd number and 2nd number is thrice the 3rd number. If average of all three numbers is 100 then the largest number will be
तीन संख्याओं में पहली संख्या दूसरी की दुगुनी और दूसरी संख्या तीसरी की 3 गुनी है। यदि उनका औसत 100 हो तो उनमें सबसे बड़ी संख्या होगी:

Options

A. 120
B. 150
C. 180
D. 300

Explanation

To solve the problem step by step, let's denote the three numbers as follows:

Let:
- The second number be \( x \).
- The first number, which is twice the second number, be \( 2x \).
- The third number, which is one-third of the second number, be \( \frac{x}{3} \).

### Step 1: Express the three numbers in terms of \( x \)
- First number: \( 2x \)
- Second number: \( x \)
- Third number: \( \frac{x}{3} \)

### Step 2: Calculate the sum of the three numbers
The sum of the three numbers can be calculated as:
\[ \text{Sum} = 2x + x + \frac{x}{3} \]
To simplify this, we need a common denominator. The common denominator for 1 and 3 is 3. Thus, we can rewrite the sum as:
\[ \text{Sum} = \frac{6x}{3} + \frac{3x}{3} + \frac{x}{3} = \frac{6x + 3x + x}{3} = \frac{10x}{3} \]

### Step 3: Use the average to find the total sum
We know that the average of the three numbers is 100. The average is given by:
\[ \text{Average} = \frac{\text{Sum}}{3} \]
Thus, we can write:
\[ 100 = \frac{\frac{10x}{3}}{3} \]
Multiplying both sides by 3 gives:
\[ 300 = \frac{10x}{3} \]
Now, to eliminate the fraction, multiply both sides by 3:
\[ 900 = 10x \]

### Step 4: Solve for \( x \)
Dividing both sides by 10 gives:
\[ x = 90 \]

### Step 5: Find the three numbers
Now that we have \( x \), we can find the three numbers:
- Second number: \( x = 90 \)
- First number: \( 2x = 2 \times 90 = 180 \)
- Third number: \( \frac{x}{3} = \frac{90}{3} = 30 \)

### Step 6: Identify the largest number
Among the three numbers \( 180 \), \( 90 \), and \( 30 \), the largest number is:
\[ \text{Largest number} = 180 \]

### Final Answer
The largest number is **180**.