The number obtained by interchanging the two digits of a two digit number is less than the original number by 18 the sum of the two digits of the number is 16 what is the original number

Options

A. 97
B. 87
C. 79
D. can not be determined

Answer

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

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

Thus, the original two-digit number can be represented as:
\[ \text{Original Number} = 10y + x \]

### Step 2: Set up the first equation
According to the problem, the sum of the two digits is 16. Therefore, we can write our first equation as:
\[ x + y = 16 \] (Equation 1)

### Step 3: Set up the second equation
The problem states that the number obtained by interchanging the two digits is less than the original number by 18. The number formed by interchanging the digits is:
\[ \text{Interchanged Number} = 10x + y \]

According to the problem:
\[ 10x + y = (10y + x) - 18 \]

### Step 4: Simplify the second equation
Now, we can simplify the equation:
\[ 10x + y = 10y + x - 18 \]
Subtract \( x \) and \( y \) from both sides:
\[ 10x - x + y - y = 10y - y - 18 \]
This simplifies to:
\[ 9x - 9y = -18 \]
Dividing the entire equation by 9 gives us:
\[ x - y = -2 \] (Equation 2)

### Step 5: Solve the equations simultaneously
Now we have a system of equations:
1. \( x + y = 16 \) (Equation 1)
2. \( x - y = -2 \) (Equation 2)

We can solve these equations by adding them together:
\[ (x + y) + (x - y) = 16 - 2 \]
This simplifies to:
\[ 2x = 14 \]
Dividing by 2 gives:
\[ x = 7 \]

### Step 6: Substitute to find \( y \)
Now substitute \( x = 7 \) back into Equation 1:
\[ 7 + y = 16 \]
Solving for \( y \):
\[ y = 16 - 7 = 9 \]

### Step 7: Find the original number
Now that we have both digits, we can find the original number:
\[ \text{Original Number} = 10y + x = 10(9) + 7 = 90 + 7 = 97 \]

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