In a two-digit number, the units digit is twice the ten's digit and the difference between the number and the number formed by reversing the digits is 18. Find the original number

Answer

To solve the problem step by step, we will define the digits of the two-digit number and set up equations based on the given conditions.

### Step 1: Define the digits
Let the tens digit be \( y \) and the units digit be \( x \).

### Step 2: Set up the equations
According to the problem, we have two conditions:
1. The units digit is twice the tens digit:
\[ x = 2y \quad \text{(Equation 1)} \]
2. The difference between the number and the number formed by reversing the digits is 18. The original number can be expressed as \( 10y + x \), and the number formed by reversing the digits is \( 10x + y \). Therefore, we can write:
\[ (10y + x) - (10x + y) = 18 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
Let's simplify Equation 2:
\[ 10y + x - 10x - y = 18 \]
Combine like terms:
\[ 9y - 9x = 18 \]
Dividing the entire equation by 9 gives:
\[ y - x = 2 \quad \text{(Equation 3)} \]

### Step 4: Substitute Equation 1 into Equation 3
Now we can substitute Equation 1 into Equation 3. From Equation 1, we know \( x = 2y \). Substituting this into Equation 3:
\[ y - 2y = 2 \]
This simplifies to:
\[ -y = 2 \]
Thus, we find:
\[ y = -2 \]
This is incorrect since \( y \) must be a positive digit. Let's correct our substitution.

Instead, we should substitute \( y = \frac{x}{2} \) from Equation 1 into Equation 3:
\[ \frac{x}{2} - x = 2 \]
This simplifies to:
\[ -\frac{x}{2} = 2 \]
Multiplying both sides by -2 gives:
\[ x = -4 \]
Again, this is incorrect. Let's go back to our equations.

### Step 5: Solve the equations correctly
From Equation 1, we have:
\[ x = 2y \]
Substituting \( x \) into Equation 3:
\[ y - 2y = 2 \]
This leads to:
\[ -y = 2 \implies y = -2 \]
This is not valid. Let's try substituting \( y = 2 \) into Equation 1:
If \( y = 2 \), then:
\[ x = 2 \times 2 = 4 \]
Now we check if this satisfies Equation 2:
The original number is \( 10y + x = 10(2) + 4 = 24 \).
The reversed number is \( 10x + y = 10(4) + 2 = 42 \).
The difference is:
\[ 42 - 24 = 18 \]
This is correct.

### Step 6: Find the original number
Thus, the original number is:
\[ \text{Original number} = 10y + x = 10(2) + 4 = 24 \]

### Conclusion
The original two-digit number is **24**.