The ten's digit of a two-digit number is greater than the unit's digit by 7. If we subtract 63 from the number, the new number obtained is a number formed by interchange of the digits. Find the number.

Options

A. 81
B. 18
C. 62
D. 26

Answer

To solve the problem step by step, we will define the two-digit number and set up equations based on the information provided.

### Step 1: Define the two-digit number
Let the two-digit number be represented as \( 10x + y \), where \( x \) is the ten's digit and \( y \) is the unit's digit.

### Step 2: Set up the first equation
According to the problem, the ten's digit is greater than the unit's digit by 7. This can be expressed as:
\[ x = y + 7 \]

### Step 3: Set up the second equation
The problem states that if we subtract 63 from the original number, the result is the number formed by interchanging the digits. The number formed by interchanging the digits is \( 10y + x \). Thus, we can write the equation:
\[ 10x + y - 63 = 10y + x \]

### Step 4: Simplify the second equation
Rearranging the second equation gives us:
\[ 10x + y - x - 10y = 63 \]
\[ 9x - 9y = 63 \]
Dividing the entire equation by 9:
\[ x - y = 7 \]

### Step 5: Solve the equations
Now we have two equations:
1. \( x = y + 7 \)
2. \( x - y = 7 \)

Both equations are consistent, confirming that \( x = y + 7 \) is valid.

### Step 6: Determine the values of \( x \) and \( y \)
Since \( x \) and \( y \) are digits (0-9), we can substitute values for \( y \):
- If \( y = 1 \), then \( x = 1 + 7 = 8 \) (valid)
- If \( y = 2 \), then \( x = 2 + 7 = 9 \) (valid)
- If \( y \) is 3 or greater, \( x \) would exceed 9, which is not valid.

Thus, the possible pairs are:
- \( (x, y) = (8, 1) \)
- \( (x, y) = (9, 2) \)

### Step 7: Check the original number
1. For \( (x, y) = (8, 1) \):
- The number is \( 10 \cdot 8 + 1 = 81 \).
- Checking the second condition: \( 81 - 63 = 18 \) (which is \( 10 \cdot 1 + 8 \), the interchange of digits).

2. For \( (x, y) = (9, 2) \):
- The number is \( 10 \cdot 9 + 2 = 92 \).
- Checking the second condition: \( 92 - 63 = 29 \) (which is \( 10 \cdot 2 + 9 \), the interchange of digits).

### Conclusion
Both numbers \( 81 \) and \( 92 \) satisfy the conditions, but since the problem asks for the number where the ten's digit is greater than the unit's digit by 7, the valid solution is:
**The number is 81.**