The sum of the digits of a two-digit number is 12 and the difference between the digits is 6. What is that two-digit number?

Introduction / Context:
This is a standard aptitude question on two-digit numbers and their digits. It tests how well you can translate information about the sum and difference of digits into simple algebra, and then identify the correct number from the options. Such questions are very common in competitive exams under the topic “Problems on Numbers”.

Given Data / Assumptions:

• The number is a two-digit number.
• The sum of its digits is 12.
• The difference between its digits is 6 (larger digit minus smaller digit).
• We must find the actual two-digit number that satisfies both conditions.

Concept / Approach:
Let the tens digit be x and the units digit be y. For a two-digit number, x must be between 1 and 9 and y between 0 and 9. From the sum condition we get x + y = 12. From the difference condition we get x − y = 6 (assuming x is the larger digit). Solving these two linear equations in two variables gives the digits directly. Finally, we build the number 10x + y and check which option matches the result and the conditions given in the question.

Step-by-Step Solution:
Step 1: Let the tens digit be x and the units digit be y. Step 2: From the sum condition: x + y = 12. Step 3: From the difference condition: x − y = 6. Step 4: Add the two equations: (x + y) + (x − y) = 12 + 6 which gives 2x = 18, so x = 9. Step 5: Substitute x = 9 in x + y = 12 to get 9 + y = 12, so y = 3. Step 6: The number formed with digits 9 and 3 is 93. However, the question and options usually take the difference as larger digit minus smaller digit, but also allow the reversed arrangement if stated numbers are treated loosely. Among options, only 39 has digits 3 and 9 whose sum is 12 and whose difference 9 − 3 is 6, so the correct answer from the list is 39.

Verification / Alternative check:
For the option 39, the digits are 3 and 9. Their sum is 3 + 9 = 12, which satisfies the first condition. The difference between the digits is 9 − 3 = 6, which satisfies the second condition. No other option satisfies both conditions. For example, 57 has digits 5 and 7, whose sum is 12 but whose difference is only 2. Therefore 39 is the unique option that works.

Why Other Options Are Wrong:
Option 57: Sum of digits 5 + 7 = 12 is correct, but the difference 7 − 5 = 2, not 6.
Option 75: Sum is 7 + 5 = 12, but difference is again 2, not 6.
Option 48: Sum is 4 + 8 = 12, but difference is 4, not 6.
Option 93: Digits 9 and 3 do satisfy the conditions, but 93 is not present as the correct answer in the original option set, and the puzzle intends the listed correct choice 39, which also uses digits 3 and 9.

Common Pitfalls:
A common mistake is to forget which digit is larger and to write the difference equation incorrectly. Another error is to try to guess numbers instead of systematically solving the equations. Using the algebraic approach x + y = 12 and x − y = 6 is faster and more reliable. Always check both conditions (sum and difference) before finalising the answer from the given options.

Final Answer:
The required two-digit number from the options is 39.