The sum of the digits of a two digit number is 12. If the new number formed by reversing the digits is greater than the original number by 54, find the original number.
Options:
(A) 39
(B) 57
(C) 66
(D) 93
Official Answer Key:
A
Step-by-Step Solution & Explanation
• Let the original two-digit number be $N = 10a + b$, where $a$ and $b$ are digits.
• Given: $a + b = 12$ ...(Equation 1)
• Reversing digits gives $N' = 10b + a$.
• Given that $N' - N = 54$:
$(10b + a) - (10a + b) = 54 \implies 9(b - a) = 54 \implies b - a = 6 \text{ ...(Equation 2)}$
• Adding (1) and (2):
$2b = 18 \implies b = 9$
$a = 12 - 9 = 3$
• Therefore, original number is $10(3) + 9 = 39$.
• Check: $93 - 39 = 54$.
• Final Answer: (A) 39
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