The tens digit of a two digit number exceeds its units digit by 6. The number itself is 10 times the sum of is digits. Find the number.

Options

A. 82
B. 90
C. 93
D. 60

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information given in the question.

### Step 1: Define the Variables
Let the unit digit of the two-digit number be \( x \).
Then, the tens digit, which exceeds the unit digit by 6, can be expressed as \( x + 6 \).

**Hint:** Identify the digits of the number and express them in terms of a variable.

### Step 2: Write the Number
The two-digit number can be represented as:
\[ 10 \times \text{(tens digit)} + \text{(units digit)} = 10(x + 6) + x = 10x + 60 + x = 11x + 60 \]

**Hint:** Remember that the two-digit number is formed by multiplying the tens digit by 10 and adding the units digit.

### Step 3: Set Up the Equation
According to the problem, the number itself is 10 times the sum of its digits. The sum of the digits is:
\[ \text{(tens digit)} + \text{(units digit)} = (x + 6) + x = 2x + 6 \]
Thus, we can set up the equation:
\[ 11x + 60 = 10 \times (2x + 6) \]

**Hint:** Use the relationship given in the problem to form an equation.

### Step 4: Simplify the Equation
Expanding the right side of the equation gives:
\[ 11x + 60 = 20x + 60 \]

**Hint:** Make sure to distribute correctly when multiplying.

### Step 5: Solve for \( x \)
Now, we can simplify the equation:
\[ 11x + 60 - 60 = 20x \]
This simplifies to:
\[ 11x = 20x \]
Rearranging gives:
\[ 20x - 11x = 0 \implies 9x = 0 \]
Thus,
\[ x = 0 \]

**Hint:** Isolate the variable to find its value.

### Step 6: Find the Tens Digit
Now that we have \( x = 0 \), we can find the tens digit:
\[ \text{Tens digit} = x + 6 = 0 + 6 = 6 \]

**Hint:** Substitute back to find the other digit.

### Step 7: Write the Final Number
The two-digit number is:
\[ 10 \times \text{(tens digit)} + \text{(units digit)} = 10 \times 6 + 0 = 60 \]

**Hint:** Combine the digits to form the final answer.

### Conclusion
The two-digit number is **60**.

**Final Answer:** 60