Question

The sum of the digits of a two-digit number is 9. If the digits of the number are interchanged, then the new number will exceed the original number by 27. Find the number.

Hint:

When dealing with two-digit numbers and their digits, use the system of equations to represent the relationships between the digits and the number.

Answer 1

Let the two-digit number be \( 10a + b \), where \( a \) is the tens digit and \( b \) is the units digit. We are given the following conditions: 1. The sum of the digits is 9: \[ a + b = 9 \quad \text{(Equation 1)}. \] 2. When the digits are interchanged, the new number exceeds the original number by 27: \[ 10b + a = (10a + b) + 27. \] Step 1: Simplify the second equation. \[ 10b + a = 10a + b + 27, \] \[ 10b - b = 10a - a + 27, \] \[ 9b = 9a + 27, \] \[ b = a + 3 \quad \text{(Equation 2)}. \] Step 2: Solve the system of equations. Substitute \( b = a + 3 \) from Equation 2 into Equation 1: \[ a + (a + 3) = 9, \] \[ 2a + 3 = 9, \] \[ 2a = 6, \] \[ a = 3. \] Step 3: Find \( b \). Substitute \( a = 3 \) into Equation 2: \[ b = 3 + 3 = 6. \] Thus, the original number is \( 10a + b = 10(3) + 6 = 36 \).
Conclusion:
Therefore, the number is \( 36 \).