The sum of the digits of a two digit number is 10. If 18 is subtracted from the number, digits are reversed. Find the numbers.

Answer

To solve the problem, we need to find a two-digit number based on the given conditions. Let's break it down step by step.

### Step 1: Define the Variables
Let the two-digit number be represented as:
- \( X \) = the digit in the unit's place
- \( Y \) = the digit in the ten's place

Thus, the two-digit number can be expressed as:
\[ \text{Number} = 10Y + X \]

### Step 2: Set Up the First Equation
According to the problem, the sum of the digits is 10. Therefore, we can write:
\[ X + Y = 10 \]
This is our first equation.

### Step 3: Set Up the Second Equation
The problem states that if 18 is subtracted from the number, the digits are reversed. This means:
\[ 10Y + X - 18 = 10X + Y \]
Now, rearranging this equation gives:
\[ 10Y + X - Y - 10X = 18 \]
\[ 10Y - Y + X - 10X = 18 \]
\[ 9Y - 9X = 18 \]
Dividing the entire equation by 9, we get:
\[ Y - X = 2 \]
This is our second equation.

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( X + Y = 10 \) (Equation 1)
2. \( Y - X = 2 \) (Equation 2)

We can solve these equations simultaneously. From Equation 2, we can express \( Y \) in terms of \( X \):
\[ Y = X + 2 \]

### Step 5: Substitute and Solve for X
Now, substitute \( Y \) in Equation 1:
\[ X + (X + 2) = 10 \]
\[ 2X + 2 = 10 \]
Subtract 2 from both sides:
\[ 2X = 8 \]
Now divide by 2:
\[ X = 4 \]

### Step 6: Find Y
Now that we have \( X \), we can find \( Y \) using Equation 1:
\[ Y + 4 = 10 \]
\[ Y = 10 - 4 \]
\[ Y = 6 \]

### Step 7: Form the Two-Digit Number
Now we have both digits:
- \( X = 4 \)
- \( Y = 6 \)

Thus, the two-digit number is:
\[ 10Y + X = 10 \times 6 + 4 = 64 \]

### Final Answer
The two-digit number is **64**.

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