The numbers are given in which the second is triple the first and is also double the third . If the average of the three numbers is 66 , find the first number .

Options

A. 36
B. 54
C. 108
D. 72

Answer

To solve the problem step by step, let's define the variables and use the information given in the question.

### Step 1: Define the variables
Let the first number be \( x \).

### Step 2: Express the second and third numbers
According to the problem:
- The second number is triple the first number, so the second number will be \( 3x \).
- The second number is also double the third number. Therefore, if the second number is \( 3x \), the third number will be \( \frac{3x}{2} \).

### Step 3: Set up the equation for the average
The average of the three numbers is given as 66. The formula for the average is:
\[ \text{Average} = \frac{\text{Sum of all numbers}}{\text{Number of observations}} \]
In this case, the sum of the three numbers is:
\[ x + 3x + \frac{3x}{2} \]
The number of observations is 3. Therefore, we can set up the equation:
\[ \frac{x + 3x + \frac{3x}{2}}{3} = 66 \]

### Step 4: Simplify the equation
Combine the terms in the numerator:
\[ x + 3x = 4x \]
So, the sum becomes:
\[ 4x + \frac{3x}{2} \]
To combine these, we need a common denominator. The common denominator between 1 and 2 is 2:
\[ 4x = \frac{8x}{2} \]
Thus, the sum becomes:
\[ \frac{8x}{2} + \frac{3x}{2} = \frac{11x}{2} \]
Now, substitute this back into the average equation:
\[ \frac{\frac{11x}{2}}{3} = 66 \]

### Step 5: Solve for \( x \)
Multiply both sides by 3:
\[ \frac{11x}{2} = 198 \]
Now, multiply both sides by 2 to eliminate the fraction:
\[ 11x = 396 \]
Finally, divide by 11:
\[ x = \frac{396}{11} = 36 \]

### Conclusion
The first number is \( 36 \).