A two digit number is such that the ten's digit exceeds twice the unit's digit by 2 and the number obtained by interchanging the digits is 5 more than three times the sum of the digits . Find the two digits number.

Answer

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

### Step 1: Define the variables
Let:
- \( x \) = the ten's digit of the number
- \( y \) = the unit's digit of the number

### Step 2: Set up the first equation
According to the problem, the ten's digit exceeds twice the unit's digit by 2. This can be expressed as:
\[ x = 2y + 2 \]
This is our first equation.

### Step 3: Set up the second equation
The problem also states that the number obtained by interchanging the digits is 5 more than three times the sum of the digits. The number with digits interchanged is \( 10y + x \). The sum of the digits is \( x + y \). Therefore, we can write the second equation as:
\[ 10y + x = 3(x + y) + 5 \]
Expanding this gives:
\[ 10y + x = 3x + 3y + 5 \]

### Step 4: Rearranging the second equation
Now, we will rearrange the second equation:
1. Move all terms involving \( x \) and \( y \) to one side:
\[ 10y + x - 3x - 3y = 5 \]
2. Simplifying this gives:
\[ 10y - 3y + x - 3x = 5 \]
\[ 7y - 2x = 5 \]
This is our second equation.

### Step 5: Substitute the first equation into the second equation
Now we will substitute the expression for \( x \) from the first equation into the second equation:
\[ 7y - 2(2y + 2) = 5 \]
Expanding this gives:
\[ 7y - 4y - 4 = 5 \]

### Step 6: Solve for \( y \)
Now, we simplify and solve for \( y \):
1. Combine like terms:
\[ 3y - 4 = 5 \]
2. Add 4 to both sides:
\[ 3y = 9 \]
3. Divide by 3:
\[ y = 3 \]

### Step 7: Find \( x \)
Now that we have \( y \), we can find \( x \) using the first equation:
\[ x = 2y + 2 \]
Substituting \( y = 3 \):
\[ x = 2(3) + 2 = 6 + 2 = 8 \]

### Step 8: Form the two-digit number
The two-digit number can now be formed using \( x \) and \( y \):
\[ \text{Number} = 10x + y = 10(8) + 3 = 80 + 3 = 83 \]

### Final Answer
The two-digit number is **83**.

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