The average of three number is 77 . The first number is twice the second and the second number is twice the third. The first number is :

Options

A. 33
B. 66
C. 77
D. 132

Answer

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

- Let the third number be \( x \).
- The second number is twice the third number, so the second number will be \( 2x \).
- The first number is twice the second number, so the first number will be \( 2(2x) = 4x \).

Now, we know that the average of these three numbers is 77. The average is calculated by adding the numbers together and dividing by the number of numbers (which is 3 in this case). Therefore, we can set up the following equation:

\[ \text{Average} = \frac{\text{First number} + \text{Second number} + \text{Third number}}{3} \]

Substituting the expressions we have for the numbers into the equation gives us:

\[ \frac{4x + 2x + x}{3} = 77 \]

Now, simplify the left side:

\[ \frac{7x}{3} = 77 \]

Next, we will multiply both sides of the equation by 3 to eliminate the fraction:

\[ 7x = 77 \times 3 \]

Calculating \( 77 \times 3 \):

\[ 7x = 231 \]

Now, divide both sides by 7 to solve for \( x \):

\[ x = \frac{231}{7} \]

Calculating \( \frac{231}{7} \):

\[ x = 33 \]

Now that we have the value of \( x \), we can find the first number:

\[ \text{First number} = 4x = 4 \times 33 = 132 \]

Thus, the first number is **132**.

### Summary of Steps:
1. Define the variables for the three numbers based on the relationships given.
2. Set up the equation for the average.
3. Simplify the equation and solve for \( x \).
4. Calculate the first number using the value of \( x \).