Question

The sum of a two-digit number and the number obtained by reversing the digits is 110. If the tens digit of the number is 6 more than the units digit, then find the number.

Hint:

For the digit problem, always remember that $x$ and $y$ must be single-digit whole numbers ($0-9$). If you get a fraction, re-check your equations!

Answer 1

Step 1: Let the digits of the number be variables.
Let the units digit be \(x\).
According to the question, the tens digit is 6 more than the units digit. Therefore, the tens digit is \(x + 6\).

Step 2: Form the original number.
A two-digit number is written as:
\(10 \times (\text{tens digit}) + (\text{units digit})\).
So the original number becomes:
\(10(x+6) + x\)
\(= 10x + 60 + x\)
\(= 11x + 60\).

Step 3: Form the number obtained after reversing the digits.
When the digits are reversed, the tens digit becomes \(x\) and the units digit becomes \(x+6\).
Thus, the reversed number becomes:
\(10x + (x+6)\)
\(= 11x + 6\).

Step 4: Use the condition given in the question.
The sum of the number and the reversed number is 110.
So we form the equation:
\((11x + 60) + (11x + 6) = 110\)

Step 5: Solve the equation.
\(22x + 66 = 110\)
\(22x = 110 - 66\)
\(22x = 44\)
\(x = 2\).

Step 6: Find the digits and the number.
Units digit \(= x = 2\).
Tens digit \(= x + 6 = 8\).

Therefore, the two-digit number is:
\(10 \times 8 + 2 = 82\).

Verification:
Reversed number \(= 28\).
\(82 + 28 = 110\), which satisfies the given condition.

Final Answer:
The required two-digit number is 82.