If 13 * 45 = 29, 24 * 58 = 41 and 74 * 32 = 53 under a certain rule, what should 97 * 47 be equal to?

Introduction / Context:
This is a number pattern question where the symbol * does not mean ordinary multiplication. Instead, it represents a special operation that combines two numbers into a new result. You are given three examples and then asked to apply the same rule to a fourth pair. This tests your ability to recognize patterns involving sums and averages of numbers.

Given Data / Assumptions:

- 13 * 45 = 29.
- 24 * 58 = 41.
- 74 * 32 = 53.
- We must find 97 * 47 using the same rule.
- All given results are between the two original numbers, suggesting an average or midpoint pattern.

Concept / Approach:
Notice that 13 and 45 have the result 29, which is exactly the average of 13 and 45. Similarly, 24 and 58 have 41 between them, and 74 and 32 have 53 between them. This suggests a simple rule:
a * b = (a + b) ÷ 2. If this holds for all the given examples, we can confidently apply it to 97 and 47.

Step-by-Step Solution:
Check the pattern for 13 * 45. Compute the sum: 13 + 45 = 58. Compute the average: 58 ÷ 2 = 29. This matches the given result. Check the pattern for 24 * 58. 24 + 58 = 82. 82 ÷ 2 = 41, which matches the second result. Check the pattern for 74 * 32. 74 + 32 = 106. 106 ÷ 2 = 53, which matches the third result. Apply the pattern to 97 * 47. 97 + 47 = 144. 144 ÷ 2 = 72.

Verification / Alternative check:
Since the rule a * b = (a + b) ÷ 2 is valid for all three examples, it is very likely the intended pattern. The computed result 72 appears as one of the options, reinforcing our conclusion. A quick mental check confirms that 72 indeed lies midway between 97 and 47, which is consistent with the idea of an average.

Why Other Options Are Wrong:

- 73, 63, and 64 are not equal to (97 + 47) ÷ 2.
- If we tried to fit those results into the same rule for the earlier examples, the pattern would break.

Thus, they do not represent a consistent continuation of the given relationship.

Common Pitfalls:
A common error is to attempt actual multiplication or do arbitrary operations on digits rather than looking for something as simple as the average. Others may miscalculate the sum or divide incorrectly. Carefully computing the sum first and then halving it helps avoid these mistakes.

Final Answer:
Using the rule a * b = (a + b) ÷ 2, the value of 97 * 47 is 72.