The sum of the digits of a two digit number is 16. If the number formed by reversing the digits is less than the original number by 18. Find the original number.

Answer

To solve the problem step by step, we will use algebraic equations based on the information provided in the question.

### Step 1: Define the variables
Let:
- \( x \) = the digit at the unit place
- \( y \) = the digit at the ten's place

### Step 2: Write the equations based on the problem
From the problem statement, we have two key pieces of information:
1. The sum of the digits is 16:
\[ x + y = 16 \quad \text{(Equation 1)} \]

2. The number formed by reversing the digits is less than the original number by 18:
- The original number can be expressed as \( 10y + x \).
- The reversed number can be expressed as \( 10x + y \).
- Therefore, we can write the equation:
\[ 10x + y = (10y + x) - 18 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
Rearranging Equation 2 gives:
\[ 10x + y = 10y + x - 18 \]
Subtract \( x \) and \( y \) from both sides:
\[ 10x - x + y - 10y = -18 \]
This simplifies to:
\[ 9x - 9y = -18 \]
Dividing the entire equation by 9:
\[ x - y = -2 \quad \text{(Equation 3)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( x + y = 16 \) (Equation 1)
2. \( x - y = -2 \) (Equation 3)

We can solve these equations simultaneously.

Adding Equation 1 and Equation 3:
\[ (x + y) + (x - y) = 16 - 2 \]
This simplifies to:
\[ 2x = 14 \]
Dividing by 2:
\[ x = 7 \]

Now, substitute \( x = 7 \) back into Equation 1:
\[ 7 + y = 16 \]
Subtracting 7 from both sides:
\[ y = 9 \]

### Step 5: Find the original number
Now that we have both digits:
- \( x = 7 \)
- \( y = 9 \)

The original number can be calculated as:
\[ \text{Original Number} = 10y + x = 10 \times 9 + 7 = 90 + 7 = 97 \]

### Final Answer
The original number is **97**.

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