The digit of a three digit numder are in G.P.When the digit of this number are reversed and this resultenc number is subtracted from the original number the difference comes out to be 792. The actual number is:

Options

A. A)842
B. B)961
C. C)421
D. D)931

Answer

To solve the problem step by step, we will follow the reasoning outlined in the transcript while adding clarity and detail to each step.

### Step 1: Understand the problem
We need to find a three-digit number where the digits are in a geometric progression (G.P.). When the digits are reversed and subtracted from the original number, the result is 792.

### Step 2: Define the digits
Let the three-digit number be represented as \(abc\), where:
- \(a\) is the hundreds place,
- \(b\) is the tens place,
- \(c\) is the units place.

Since the digits are in G.P., we can express this relationship mathematically:
\[ b^2 = ac \]

### Step 3: Express the original and reversed numbers
The original number can be expressed as:
\[ 100a + 10b + c \]
The reversed number will be:
\[ 100c + 10b + a \]

### Step 4: Set up the equation based on the difference
According to the problem, the difference between the original number and the reversed number is 792:
\[ (100a + 10b + c) - (100c + 10b + a) = 792 \]
This simplifies to:
\[ 99a - 99c = 792 \]
Dividing the entire equation by 99 gives:
\[ a - c = 8 \]

### Step 5: Relate the digits
From the equation \(a - c = 8\), we can express \(a\) in terms of \(c\):
\[ a = c + 8 \]

### Step 6: Consider the digit constraints
Since \(a\), \(b\), and \(c\) are digits (0 to 9), and \(a\) must be a single-digit number, the maximum value for \(c\) can be calculated:
- The maximum value of \(c\) is 1 (since \(a\) must be less than or equal to 9).
- Therefore, the possible values are:
- If \(c = 1\), then \(a = 9\).

### Step 7: Find \(b\) using the G.P. condition
Now we substitute \(a\) and \(c\) into the G.P. condition \(b^2 = ac\):
\[ b^2 = 9 \times 1 = 9 \]
Taking the square root gives:
\[ b = 3 \]

### Step 8: Form the original number
Now we have:
- \(a = 9\)
- \(b = 3\)
- \(c = 1\)

Thus, the original number is:
\[ 931 \]

### Step 9: Check the reversed number
The reversed number is:
\[ 139 \]

### Step 10: Verify the difference
Now, we check the difference:
\[ 931 - 139 = 792 \]
This confirms our solution.

### Final Answer
The actual number is **931**.