By Interchanging the digits of a two digit number we get a number which is four times the original number minus 24. If the unit's digit of the original number exceeds its ten's digit by 7, then original number is

Options

A. 29
B. 36
C. 58
D. 18

Answer

To solve the problem, we need to find a two-digit number based on the given conditions. Let's break it down step by step.

### Step 1: Define the digits
Let the tens digit of the two-digit number be \( a \) and the unit digit be \( b \). Therefore, the original number can be represented as:
\[ 10a + b \]

### Step 2: Interchanging the digits
When we interchange the digits, the new number becomes:
\[ 10b + a \]

### Step 3: Set up the first equation
According to the problem, the number obtained by interchanging the digits is equal to four times the original number minus 24. This gives us the equation:
\[ 10b + a = 4(10a + b) - 24 \]

### Step 4: Simplify the equation
Expanding the right side:
\[ 10b + a = 40a + 4b - 24 \]
Now, rearranging the equation:
\[ 10b - 4b + a - 40a = -24 \]
This simplifies to:
\[ 6b - 39a = -24 \]
Dividing the entire equation by 3 gives:
\[ 2b - 13a = -8 \quad \text{(Equation 1)} \]

### Step 5: Set up the second condition
The problem states that the unit's digit exceeds the tens digit by 7:
\[ b = a + 7 \quad \text{(Equation 2)} \]

### Step 6: Substitute Equation 2 into Equation 1
Substituting \( b = a + 7 \) into Equation 1:
\[ 2(a + 7) - 13a = -8 \]
Expanding this gives:
\[ 2a + 14 - 13a = -8 \]
Combining like terms results in:
\[ -11a + 14 = -8 \]
Now, isolating \( a \):
\[ -11a = -8 - 14 \]
\[ -11a = -22 \]
Dividing both sides by -11:
\[ a = 2 \]

### Step 7: Find \( b \)
Using the value of \( a \) in Equation 2:
\[ b = a + 7 = 2 + 7 = 9 \]

### Step 8: Form the original number
Now we can form the original number using \( a \) and \( b \):
\[ \text{Original number} = 10a + b = 10(2) + 9 = 20 + 9 = 29 \]

### Conclusion
The original number is:
\[ \boxed{29} \]