Of the three numbers, the first is twice the second and the second is 3 times the third. If their average is 100, the largest of the three numbers is :

Options

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

Answer

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

1. Let the first number be \( a \).
2. Let the second number be \( b \).
3. Let the third number be \( c \).

### Step 1: Set up the relationships between the numbers
According to the problem:
- The first number \( a \) is twice the second number \( b \):
\[ a = 2b \]
- The second number \( b \) is three times the third number \( c \):
\[ b = 3c \]

### Step 2: Express all numbers in terms of \( c \)
From the equation \( b = 3c \), we can substitute \( b \) into the equation for \( a \):
\[ a = 2b = 2(3c) = 6c \]

Now we have:
- \( a = 6c \)
- \( b = 3c \)
- \( c = c \)

### Step 3: Calculate the average of the three numbers
The average of the three numbers \( a, b, \) and \( c \) is given as 100. The formula for the average is:
\[ \text{Average} = \frac{a + b + c}{3} \]
Substituting the expressions for \( a \), \( b \), and \( c \):
\[ \frac{6c + 3c + c}{3} = 100 \]
This simplifies to:
\[ \frac{10c}{3} = 100 \]

### Step 4: Solve for \( c \)
To find \( c \), multiply both sides by 3:
\[ 10c = 300 \]
Now, divide by 10:
\[ c = 30 \]

### Step 5: Find the values of \( a \) and \( b \)
Now that we have \( c \), we can find \( b \) and \( a \):
- For \( b \):
\[ b = 3c = 3 \times 30 = 90 \]
- For \( a \):
\[ a = 6c = 6 \times 30 = 180 \]

### Step 6: Identify the largest number
Now we have the values:
- \( a = 180 \)
- \( b = 90 \)
- \( c = 30 \)

The largest of the three numbers is \( a \), which is \( 180 \).

### Conclusion
Thus, the largest of the three numbers is:
\[ \boxed{180} \]