The sum of two digits of a number is 10. If the digits are interchanged, then its value increases by 18. Find the numbers.
A. 46
B. 64
C. 19
D. 28

Options

A. D
B. A
C. C
D. B

Answer

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

### Step 1: Define the digits
Let the two digits of the number be \( x \) and \( y \). According to the problem, the sum of these two digits is given as:
\[ x + y = 10 \]

### Step 2: Express the number
The two-digit number can be expressed as:
\[ 10x + y \]
where \( x \) is the tens digit and \( y \) is the units digit.

### Step 3: Interchange the digits
When the digits are interchanged, the new number becomes:
\[ 10y + x \]

### Step 4: Set up the equation for the increase
According to the problem, when the digits are interchanged, the value of the number increases by 18. Therefore, we can set up the equation:
\[ 10y + x = (10x + y) + 18 \]

### Step 5: Simplify the equation
Now, let's simplify the equation:
\[ 10y + x - 10x - y = 18 \]
This simplifies to:
\[ 9y - 9x = 18 \]
Dividing the entire equation by 9 gives:
\[ y - x = 2 \]

### Step 6: Solve the system of equations
Now we have a system of two equations:
1. \( x + y = 10 \)
2. \( y - x = 2 \)

We can solve these equations simultaneously. From the second equation, we can express \( y \) in terms of \( x \):
\[ y = x + 2 \]

Substituting this expression for \( y \) into the first equation:
\[ x + (x + 2) = 10 \]
This simplifies to:
\[ 2x + 2 = 10 \]
Subtracting 2 from both sides:
\[ 2x = 8 \]
Dividing by 2 gives:
\[ x = 4 \]

### Step 7: Find the value of y
Now substituting \( x = 4 \) back into the equation for \( y \):
\[ y = 4 + 2 = 6 \]

### Step 8: Form the number
Thus, the two digits are \( x = 4 \) and \( y = 6 \), which means the original number is:
\[ 46 \]

### Conclusion
The number is **46**.