The sum of digits of a two - digit number is 9. When the digits are reversed, the number decreases by 45. Find the changed number.
A. 45
B. 72
C. 63
D. 27

Options

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

Answer

To solve the problem step by step, let's define the two-digit number and set up the equations based on the information provided.

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

### Step 2: Set up the first equation
According to the problem, the sum of the digits is 9. Therefore, we can write the first equation as:
\[ x + y = 9 \]

### Step 3: Set up the second equation
When the digits are reversed, the new number formed is \( 10x + y \) (where \( x \) is now in the ten's place and \( y \) is in the unit place). The problem states that this new number is 45 less than the original number \( 10y + x \). Thus, we can write the second equation as:
\[ 10x + y = (10y + x) - 45 \]

### Step 4: Simplify the second equation
Rearranging the second equation gives us:
\[ 10x + y + 45 = 10y + x \]
\[ 10x - x - 10y + y = -45 \]
\[ 9x - 9y = -45 \]
Dividing the entire equation by 9 gives:
\[ x - y = -5 \]
This can be rewritten as:
\[ y - x = 5 \] (Equation 2)

### Step 5: Solve the equations
Now we have two equations:
1. \( x + y = 9 \) (Equation 1)
2. \( y - x = 5 \) (Equation 2)

We can solve these equations simultaneously. Adding both equations:
\[ (x + y) + (y - x) = 9 + 5 \]
This simplifies to:
\[ 2y = 14 \]
Thus:
\[ y = 7 \]

### Step 6: Find the value of \( x \)
Substituting \( y = 7 \) back into Equation 1:
\[ x + 7 = 9 \]
\[ x = 9 - 7 \]
\[ x = 2 \]

### Step 7: Form the original number
Now we have:
- \( x = 2 \) (unit place)
- \( y = 7 \) (ten's place)

Thus, the original number is:
\[ 10y + x = 10(7) + 2 = 70 + 2 = 72 \]

### Step 8: Find the changed number
When the digits are reversed, the new number becomes:
\[ 10x + y = 10(2) + 7 = 20 + 7 = 27 \]

### Conclusion
The changed number is **27**.