A number has two digits whose sum is 9. if 27 is added to the number , its digits get interchanged. Find the number.

Answer

To solve the problem step by step, let's break it down systematically.

### Step 1: Define the digits
Let the unit digit of the two-digit number be \( x \). Since the sum of the digits is 9, the tens digit can be expressed as \( 9 - x \).

### Step 2: Write the number
The two-digit number can be represented as:
\[ 10 \times (\text{tens digit}) + (\text{unit digit}) = 10(9 - x) + x = 90 - 10x + x = 90 - 9x \]

### Step 3: Set up the equation based on the problem statement
According to the problem, if 27 is added to the number, the digits get interchanged. Therefore, we can write the equation:
\[ (90 - 9x) + 27 = 10x + (9 - x) \]
This simplifies to:
\[ 117 - 9x = 10x + 9 - x \]
\[ 117 - 9x = 9x + 9 \]

### Step 4: Combine like terms
Now, let's combine like terms:
\[ 117 - 9 = 9x + 9x \]
\[ 108 = 18x \]

### Step 5: Solve for \( x \)
Now, divide both sides by 18:
\[ x = \frac{108}{18} = 6 \]

### Step 6: Find the tens digit
Now that we have \( x \), we can find the tens digit:
\[ \text{Tens digit} = 9 - x = 9 - 6 = 3 \]

### Step 7: Write the final number
Thus, the two-digit number is:
\[ 10 \times (\text{tens digit}) + (\text{unit digit}) = 10 \times 3 + 6 = 30 + 6 = 36 \]

### Conclusion
The number is **36**.

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