If the positions of the digits of a two -digit number are interchanged the number obtained is smaller than the original number by 27. If the digits of the number are in the ratio of ` 1: 2`, what is the original number?

Options

A. 36
B. 63
C. 48
D. Cannot be determined

Explanation

To solve the problem step by step, we will follow the logic presented in the video transcript:

### Step 1: Define the digits of the two-digit number
Let the unit digit be \( x \). According to the problem, the tens digit is in the ratio of \( 1:2 \), so the tens digit will be \( 2x \).

### Step 2: Write the original two-digit number
The original two-digit number can be expressed as:
\[ \text{Original Number} = 10 \times \text{(tens digit)} + \text{(unit digit)} = 10(2x) + x = 20x + x = 21x \]

### Step 3: Write the number after interchanging the digits
When the digits are interchanged, the new number becomes:
\[ \text{New Number} = 10 \times \text{(unit digit)} + \text{(tens digit)} = 10(x) + 2x = 10x + 2x = 12x \]

### Step 4: Set up the equation based on the problem statement
According to the problem, the new number is smaller than the original number by 27. Therefore, we can write the equation:
\[ 21x - 12x = 27 \]

### Step 5: Simplify the equation
Simplifying the left side gives:
\[ 9x = 27 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 9:
\[ x = \frac{27}{9} = 3 \]

### Step 7: Find the digits of the original number
Now that we have \( x \), we can find the digits:
- Unit digit \( = x = 3 \)
- Tens digit \( = 2x = 2 \times 3 = 6 \)

### Step 8: Write the original number
Thus, the original two-digit number is:
\[ \text{Original Number} = 10 \times \text{(tens digit)} + \text{(unit digit)} = 10(6) + 3 = 60 + 3 = 63 \]

### Final Answer
The original number is **63**.

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