In a two-digit number, the digit at the unit's place is 1 less than twice the digit at the ten's place. If the digits at unit's and ten's place are interchanged, the difference between the new and the original number is less then the original number by 20. the original number is

Options

A. 59
B. 23
C. 35
D. 47

Answer

To solve the problem step by step, let's define the variables and set up the equations based on the information given in the question.

### Step 1: Define the Variables
Let:
- \( y \) = digit at the ten's place
- \( x \) = digit at the unit's place

### Step 2: Write the Original Number
The original two-digit number can be expressed as:
\[ \text{Original Number} = 10y + x \]

### Step 3: Set Up the First Equation
According to the problem, the digit at the unit's place \( x \) is one less than twice the digit at the ten's place \( y \). This gives us our first equation:
\[ x = 2y - 1 \quad \text{(Equation 1)} \]

### Step 4: Write the New Number After Interchanging Digits
When the digits are interchanged, the new number becomes:
\[ \text{New Number} = 10x + y \]

### Step 5: Set Up the Second Equation
The problem states that the difference between the new number and the original number is less than the original number by 20. Therefore, we can write:
\[ (10x + y) - (10y + x) = (10y + x) - 20 \]

### Step 6: Simplify the Second Equation
Now, simplify the equation:
\[ 10x + y - 10y - x = 10y + x - 20 \]
This simplifies to:
\[ 9x - 9y = 10y + x - 20 \]
Rearranging gives:
\[ 9x - 10y - x + 9y = -20 \]
This simplifies to:
\[ 8x - 19y = -20 \quad \text{(Equation 2)} \]

### Step 7: Substitute Equation 1 into Equation 2
Now, substitute \( x \) from Equation 1 into Equation 2:
\[ 8(2y - 1) - 19y = -20 \]
Expanding this gives:
\[ 16y - 8 - 19y = -20 \]
Combining like terms results in:
\[ -3y - 8 = -20 \]
Adding 8 to both sides:
\[ -3y = -12 \]
Dividing by -3:
\[ y = 4 \]

### Step 8: Find \( x \) Using Equation 1
Now, substitute \( y \) back into Equation 1 to find \( x \):
\[ x = 2(4) - 1 = 8 - 1 = 7 \]

### Step 9: Find the Original Number
Now that we have both digits, we can find the original number:
\[ \text{Original Number} = 10y + x = 10(4) + 7 = 40 + 7 = 47 \]

### Final Answer
Thus, the original number is:
\[ \boxed{47} \]