The digits of a two digit number are in the ratio of 2:3 and the number obtained by interchanging the digits is bigger than the original number by 27. What was the original number?

Options

A. 64
B. 46
C. 96
D. 69

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the digits
Let the digits of the two-digit number be represented as \(2x\) and \(3x\), where \(2x\) is the tens digit and \(3x\) is the units digit.

### Step 2: Formulate the original number
The original two-digit number can be expressed as:
\[ \text{Original Number} = 10 \times \text{(tens digit)} + \text{(units digit)} = 10(2x) + (3x) = 20x + 3x = 23x \]

### Step 3: Formulate the number after interchanging the digits
When the digits are interchanged, the new number becomes:
\[ \text{New Number} = 10 \times \text{(units digit)} + \text{(tens digit)} = 10(3x) + (2x) = 30x + 2x = 32x \]

### Step 4: Set up the equation based on the problem statement
According to the problem, the number obtained by interchanging the digits is bigger than the original number by 27. Therefore, we can write the equation:
\[ 32x = 23x + 27 \]

### Step 5: Solve the equation
Now, we will solve for \(x\):
\[ 32x - 23x = 27 \]
\[ 9x = 27 \]
\[ x = 3 \]

### Step 6: Find the digits
Now that we have \(x\), we can find the actual digits:
- Tens digit: \(2x = 2 \times 3 = 6\)
- Units digit: \(3x = 3 \times 3 = 9\)

### Step 7: Form the original number
Thus, the original number is:
\[ \text{Original Number} = 10 \times 6 + 9 = 60 + 9 = 69 \]

### Conclusion
The original number is **69**.

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