What is the two digit number ?
(i) The difference between the two digits is 9.
(ii) The sum of the digits is equal to the difference between the two digits.

Answer

To solve the problem of finding the two-digit number based on the given conditions, let's break it down step by step.

### Step 1: Define the Digits
Let’s denote the two-digit number as \( 10x + y \), where \( x \) is the tens digit and \( y \) is the units digit. Since it is a two-digit number, \( x \) must be between 1 and 9 (inclusive), and \( y \) must be between 0 and 9 (inclusive).

### Step 2: Use the First Condition
The first condition states that the difference between the two digits is 9. This can be expressed mathematically as:
\[ x - y = 9 \]
This implies that \( x \) is greater than \( y \).

### Step 3: Use the Second Condition
The second condition states that the sum of the digits is equal to the difference between the two digits. This can be expressed as:
\[ x + y = x - y \]
Rearranging this gives:
\[ x + y = 9 \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( x - y = 9 \)
2. \( x + y = 9 \)

We can solve these equations simultaneously.

#### Adding the two equations:
\[ (x - y) + (x + y) = 9 + 9 \]
This simplifies to:
\[ 2x = 18 \]
Thus, we find:
\[ x = 9 \]

#### Substituting \( x \) back into one of the equations:
Using the first equation \( x - y = 9 \):
\[ 9 - y = 9 \]
This simplifies to:
\[ y = 0 \]

### Step 5: Form the Two-Digit Number
Now that we have \( x = 9 \) and \( y = 0 \), the two-digit number is:
\[ 10x + y = 10(9) + 0 = 90 \]

### Final Answer
The two-digit number is **90**.

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