The difference between a two-digit number and the number obtained by interchanging its digits is 63. Which is the smaller of the two numbers?

Introduction / Context:
This is a digit-based number puzzle involving a two-digit number and the number formed by reversing its digits. You are told that the difference between these two numbers is 63 and asked to find the smaller of the two numbers. Such questions test your understanding of place value and your ability to work with algebraic representations of digit-based numbers.

Given Data / Assumptions:

• Let the original two-digit number have tens digit a and units digit b.
• The original number is 10a + b.
• The reversed number is 10b + a.
• The difference between the original number and the reversed number is 63.
• We must find the smaller of the two numbers.

Concept / Approach:
We translate the problem into an equation using the algebraic representations of the numbers. The difference between the original number and its reverse is (10a + b) − (10b + a). This simplifies to 9a − 9b, which equals 63 according to the question. Solving for a − b gives us the difference between the digits. Then we list all possible valid digit pairs (a, b) that satisfy this relation, ensuring that a and b are digits between 0 and 9 and that a is non-zero (because it is the tens digit). From these possibilities, we determine if the smaller number is uniquely determined or not.

Step-by-Step Solution:
Step 1: Let the original number be 10a + b, where a is the tens digit and b is the units digit. Step 2: The reversed number is 10b + a. Step 3: According to the question, (10a + b) − (10b + a) = 63. Step 4: Simplify the left-hand side: (10a + b) − (10b + a) = 10a + b − 10b − a = 9a − 9b = 9(a − b). Step 5: So 9(a − b) = 63, which implies a − b = 63 / 9 = 7. Step 6: Since a and b are digits, the pairs that satisfy a − b = 7 are (a, b) = (7, 0), (8, 1) and (9, 2). Step 7: These correspond to original numbers 70, 81 and 92. Their reversed numbers are 07 (that is 7), 18 and 29 respectively.

Verification / Alternative check:
Check the differences: 70 − 7 = 63, 81 − 18 = 63 and 92 − 29 = 63. All three pairs satisfy the condition in the question. The smaller numbers in these pairs are 7, 18 and 29 respectively. Since there are multiple possible smaller numbers that satisfy the condition, the problem does not have a unique answer for “the smaller of the two numbers”. Therefore, none of the specific numerical options like 12, 15 or 17 can be chosen as a unique correct answer.

Why Other Options Are Wrong:
Option 12, 15, 17 and 18: These are specific numbers, but each of them corresponds either to digit differences that do not give 63 when reversed or to only one of several valid solutions. The question expects a unique smaller number, which is not possible here, so no single one of these values can be the correct answer.
Option “none of these answers can be determined”: This correctly reflects the fact that there are several valid pairs of numbers giving a difference of 63, and thus the smaller number is not uniquely determined from the information provided.

Common Pitfalls:
A frequent mistake is to assume that there is only one possible pair of digits and to stop after finding the first valid solution, for example (8, 1). This leads to incorrectly choosing 18 as the smaller number without checking for other digit pairs. Another error is to forget that b can be zero, which gives additional valid pairs. Always list all digit pairs satisfying the algebraic condition and then check whether the problem has a unique answer or multiple possibilities.

Final Answer:
Because more than one pair of numbers satisfies the condition, the smaller number cannot be uniquely determined. Hence the correct choice is none of these answers can be determined.