If a two-digit number has a digit sum of 9 and adding 27 reverses its digits, determine the unit digit of the number.

Question

If a two-digit number has a digit sum of 9 and adding 27 reverses its digits, determine the unit digit of the number.

1. A. 3
2. B. 5
3. C. 4
4. D. 6 (Correct answer)

Correct Answer

Option D — 6

Detailed Solution & Explanation

The correct answer is 6.

Shortcut Trick

• Reversing a two-digit number changes it by $9 \times$ (difference of digits). Since adding 27 reverses it, the difference of digits is $\dfrac{27}{9} = 3$, with the units digit larger.
• With digit sum 9 and difference 3: the digits are $\dfrac{9+3}{2} = 6$ and $\dfrac{9-3}{2} = 3$.
• The number is 36, so the units digit is 6.

Alternate Method

• Let the number be $10a + b$ with $a + b = 9$.
• Reversal condition: $10a + b + 27 = 10b + a$
  • $9a + 27 = 9b \Rightarrow b = a + 3$
• Substituting: $a + (a+3) = 9 \Rightarrow 2a = 6 \Rightarrow a = 3$, $b = \textbf{6}$
• Check: $36 + 27 = 63$ ✓

Additional Information

• The key identity: $(10a+b) - (10b+a) = 9(a-b)$. Any difference between a two-digit number and its reversal is therefore always a multiple of 9 — worth knowing, as it instantly rules out impossible values.
• The sum $(10a+b) + (10b+a) = 11(a+b)$ is likewise always a multiple of 11.
• For three-digit numbers the same reversal difference is $99(a-c)$, depending only on the outer digits.