The sum of the digits of a two-digits number is 15. If the number formed by reversing the digits is less than the original number by 27, find the original number. Check your solution.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the Variables
Let the two-digit number be represented as \( 10a + b \), where:
- \( a \) is the digit in the tens place,
- \( b \) is the digit in the units place.

### Step 2: Set Up the Equations
From the problem, we have two key pieces of information:
1. The sum of the digits is 15:
\[ a + b = 15 \quad \text{(Equation 1)} \]
2. The number formed by reversing the digits is less than the original number by 27:
\[ (10b + a) = (10a + b) - 27 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
Rearranging Equation 2:
\[ 10b + a = 10a + b - 27 \]
Subtract \( b \) and \( a \) from both sides:
\[ 10b - b + a - a = 10a - a - 27 \]
This simplifies to:
\[ 9b - 9a = -27 \]
Dividing the entire equation by 9 gives:
\[ b - a = -3 \quad \text{(Equation 3)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( a + b = 15 \) (Equation 1)
2. \( b - a = -3 \) (Equation 3)

From Equation 3, we can express \( b \) in terms of \( a \):
\[ b = a - 3 \]

### Step 5: Substitute into Equation 1
Substituting \( b = a - 3 \) into Equation 1:
\[ a + (a - 3) = 15 \]
This simplifies to:
\[ 2a - 3 = 15 \]
Adding 3 to both sides:
\[ 2a = 18 \]
Dividing by 2:
\[ a = 9 \]

### Step 6: Find \( b \)
Now substitute \( a = 9 \) back into Equation 1 to find \( b \):
\[ 9 + b = 15 \]
So,
\[ b = 15 - 9 = 6 \]

### Step 7: Form the Original Number
The original number is:
\[ 10a + b = 10(9) + 6 = 90 + 6 = 96 \]

### Step 8: Check the Solution
1. The sum of the digits \( 9 + 6 = 15 \) (correct).
2. The number formed by reversing the digits is \( 69 \), and \( 96 - 69 = 27 \) (correct).

Thus, the original number is **96**.