In a certain puzzle, the symbol # represents a special operation on two numbers. It is given that 9 # 3 = 6, 15 # 3 = 9 and 60 # 4 = 32. Using the same rule, what should be the value of 27 # 3 ?

Introduction / Context:
This question introduces an artificial operation denoted by the symbol #. Instead of indicating a standard arithmetic function, the symbol hides a pattern that we must uncover by studying the given numerical examples. Once we discover the relationship that connects each pair of numbers to its result, we apply the same rule to a new pair. Such problems check our ability to recognise simple numeric patterns and to avoid guessing without verification, which is essential for many reasoning tests.

Given Data / Assumptions:

• 9 # 3 is defined to be 6.
• 15 # 3 is defined to be 9.
• 60 # 4 is defined to be 32.
• The same operation # is used consistently for all pairs.
• We need to evaluate 27 # 3.
• Only basic arithmetic operations are assumed unless otherwise indicated.

Concept / Approach:
When dealing with such coded operations, a good starting point is to try simple combinations of addition, subtraction, multiplication and division involving the two numbers. Because the outputs 6, 9 and 32 are relatively close to the average of each pair, it is natural to check whether # might represent an average or mean. We can test this idea by computing the arithmetic mean of each given pair and seeing if it matches the specified result. If the same pattern works for all three examples, we can safely use it for the new pair 27 and 3.

Step-by-Step Solution:
Step 1: Consider the first pair 9 and 3. Compute the arithmetic mean (9 + 3) / 2 = 12 / 2 = 6. Step 2: This matches the given value 9 # 3 = 6, so the mean works for the first example. Step 3: Now check the second pair 15 and 3. The arithmetic mean is (15 + 3) / 2 = 18 / 2 = 9, which again matches 15 # 3 = 9. Step 4: Check the third pair 60 and 4. The mean is (60 + 4) / 2 = 64 / 2 = 32, exactly matching 60 # 4 = 32. Step 5: The same rule works for all three examples, so we define the operation as a # b = (a + b) / 2. Step 6: Apply the rule to the pair 27 and 3. Compute (27 + 3) / 2 = 30 / 2 = 15. Step 7: Therefore, 27 # 3 should be equal to 15.

Verification / Alternative check:
We can be confident about this conclusion because there is a perfect match between the arithmetic mean and the given results for all the original pairs. There is no need to seek a more complicated formula, since any alternative pattern would have to produce exactly the same outputs for 9, 3, 15, 60 and 4, which is unlikely if it is not equivalent to the mean. The arithmetic mean is also a natural and widely recognised operation, which makes it a very plausible intended rule for such a reasoning question.

Why Other Options Are Wrong:
Option 24 would correspond to (27 + 3) instead of dividing by 2, and does not fit the given examples. Option 13 and option 21 are random guesses that do not arise from any simple mean based pattern and are inconsistent with the earlier calculations. Option 33 is larger than the sum of the two numbers divided by 2 and cannot be generated by the same rule. Since we have a clear, tested pattern that leads to 15, all other options are logically ruled out.

Common Pitfalls:
Common mistakes include trying to use only multiplication or only subtraction, which will usually match at most one of the examples but not all. Some learners focus only on 9 # 3 = 6 and build a rule that fails for 60 # 4 or 15 # 3, leading to contradictions. Another error is to forget to divide by 2 after adding the numbers, which gives the sum rather than the mean. Always verify a candidate rule against every piece of provided data before using it for the unknown expression.

Final Answer:
Since the operation # represents the arithmetic mean of the two numbers, the value of 27 # 3 is 15.