Question

The sum of the digits of a two-digit number is 9. If the digits of the number be interchanged then the new number is 27 more than the original number. Find the number.

Hint:

For digit-based number problems, always represent the number as $10x + y$ and apply given conditions systematically.

Answer 1

Step 1: Let the digits of the number be $x$ and $y$.
Then, the number can be written as $10x + y$.
Step 2: Write the given conditions.
Sum of digits = 9 \[ x + y = 9 \quad \text{...(1)} \] When digits are interchanged, the new number = $10y + x$, and it is 27 more than the original number. \[ 10y + x = 10x + y + 27 \] Step 3: Simplify the equation.
\[ 10y + x - 10x - y = 27 \Rightarrow 9y - 9x = 27 \Rightarrow y - x = 3 \quad \text{...(2)} \]
Step 4: Solve equations (1) and (2).
From (1): $x + y = 9$ From (2): $y - x = 3$ Add both equations: \[ 2y = 12 \Rightarrow y = 6 \] Substitute in (1): \[ x + 6 = 9 \Rightarrow x = 3 \]
Step 5: Find the number.
\[ \text{Number} = 10x + y = 10(3) + 6 = 36 \]
Step 6: Verification.
Interchanging digits gives 63, and $63 - 36 = 27$. Final Answer: The required number is $\boxed{36}$.