Of the three numbers, the first is twice the second, and the second is twice the third. The average of the reciprocal of the numbers is `7/12`. The numbers are :

Options

A. 20,10,5
B. 4,2,1
C. 36,18,9
D. 16,8,4

Answer

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

1. Let the third number be \( x \).
2. The second number, being twice the third, will be \( 2x \).
3. The first number, being twice the second, will be \( 4x \).

Now we have the three numbers:
- First number: \( 4x \)
- Second number: \( 2x \)
- Third number: \( x \)

Next, we need to find the average of the reciprocals of these numbers. The reciprocals of the numbers are:
- Reciprocal of the first number: \( \frac{1}{4x} \)
- Reciprocal of the second number: \( \frac{1}{2x} \)
- Reciprocal of the third number: \( \frac{1}{x} \)

Now, we can calculate the average of these reciprocals:
\[ \text{Average} = \frac{\frac{1}{4x} + \frac{1}{2x} + \frac{1}{x}}{3} \]

To simplify the expression inside the average, we need a common denominator. The common denominator for \( 4x, 2x, \) and \( x \) is \( 4x \). Thus, we rewrite each term:
\[ \frac{1}{4x} + \frac{1}{2x} + \frac{1}{x} = \frac{1}{4x} + \frac{2}{4x} + \frac{4}{4x} = \frac{1 + 2 + 4}{4x} = \frac{7}{4x} \]

Now substituting this back into the average formula:
\[ \text{Average} = \frac{\frac{7}{4x}}{3} = \frac{7}{12x} \]

According to the problem, this average is equal to \( \frac{7}{12} \):
\[ \frac{7}{12x} = \frac{7}{12} \]

To solve for \( x \), we can cross-multiply:
\[ 7 \cdot 12 = 7 \cdot 12x \]
This simplifies to:
\[ 12 = 12x \]
Dividing both sides by 12 gives:
\[ x = 1 \]

Now that we have \( x \), we can find the three numbers:
- The third number \( x = 1 \)
- The second number \( 2x = 2 \)
- The first number \( 4x = 4 \)

Thus, the three numbers are \( 4, 2, \) and \( 1 \).

### Final Answer:
The numbers are \( 4, 2, \) and \( 1 \).