The sum of the digits of a two-digit number is 8. If the original number is subtracted from the number formed by reversing its digits, the result is 54. What is the original number?

Introduction / Context:
This question is about a two-digit number defined by conditions on its digit sum and the effect of reversing its digits. Such problems are standard in number system topics and involve representing the number using algebra and place value concepts. By forming equations from the given conditions, we can solve for the digits and hence determine the original number.

Given Data / Assumptions:

• Let the original two-digit number be 10a + b, where a is the tens digit and b is the units digit.
• The sum of the digits is 8: a + b = 8.
• The number formed by reversing its digits is 10b + a.
• When the original number is subtracted from the reversed number, the result is 54: (10b + a) − (10a + b) = 54.
• We must find the original number 10a + b.

Concept / Approach:
We use place value representation for the original and reversed numbers. The problem gives two equations: one for the digit sum and one for the difference between the reversed number and the original. Simplifying the second equation will give a relationship between a and b, which combined with a + b = 8 allows us to solve for both digits. Once the digits are known, the original number is easily constructed and checked against the options.

Step-by-Step Solution:
Step 1: Let the original number be 10a + b. Step 2: The sum of its digits is a + b = 8. Step 3: The reversed number is 10b + a. Step 4: According to the problem, reversed number minus original number equals 54: (10b + a) − (10a + b) = 54. Step 5: Simplify the left side: 10b + a − 10a − b = 9b − 9a. Step 6: So 9b − 9a = 54. Step 7: Factor out 9: 9(b − a) = 54, so b − a = 54 / 9 = 6. Step 8: Now we have a + b = 8 and b − a = 6. Step 9: Add the two equations: (a + b) + (b − a) = 8 + 6, giving 2b = 14. Step 10: Solve for b: b = 14 / 2 = 7. Step 11: Substitute b = 7 into a + b = 8 to get a + 7 = 8, so a = 1. Step 12: Therefore, the original number is 10a + b = 10 * 1 + 7 = 17.

Verification / Alternative check:
Check the digit sum: 1 + 7 = 8, which matches. The reversed number is 71. Compute the difference: 71 − 17 = 54, exactly as given. Hence, the number 17 satisfies both conditions, confirming it as the correct answer.

Why Other Options Are Wrong:
Option (a) 28: Digits sum to 2 + 8 = 10, not 8.
Option (b) 19: Digit sum is 10, and difference with 91 is 72, not 54.
Option (c) 37: Digit sum is 10, and reversed difference 73 − 37 = 36, not 54.

Common Pitfalls:
Some learners may reverse the subtraction order and consider original minus reversed, which leads to a negative value or wrong equation. Others may mis-handle the simplification of (10b + a) − (10a + b). Correctly setting up and simplifying to 9(b − a) is crucial. Also, care must be taken when solving the system of equations for a and b so that arithmetic errors do not occur.

Final Answer:
The original two-digit number is 17.