In a coded arithmetic operation, 6 * 9 – 4 is written as 58 and 3 * 9 – 7 is written as 34. Using the same hidden rule, in the expression A * 4 – 9 = 91, what value of A will correctly satisfy the pattern?

Introduction / Context:
This question is about identifying a hidden numerical pattern in a coded arithmetic expression. The normal arithmetic symbols appear to be used in an unusual way. By decoding how 6 * 9 – 4 becomes 58 and how 3 * 9 – 7 becomes 34, we can extend the same rule to find the unknown value A in A * 4 – 9 = 91.

Given Data / Assumptions:

• 6 * 9 – 4 is coded as 58.
• 3 * 9 – 7 is coded as 34.
• A * 4 – 9 is coded as 91.
• All three expressions follow the same hidden rule.

Concept / Approach:
The key idea is that the symbols may not mean their usual arithmetic operations together, or an extra step is being applied after the normal calculation. We test simple patterns like replacing the minus sign by plus, or adding the last number instead of subtracting it. Once a rule fits the first two examples exactly, we apply it to the third expression to solve for A.

Step-by-Step Solution:
Start with 6 * 9 – 4 and assume normal multiplication first. Compute 6 * 9 = 54. To reach 58 from 54, we must add 4 instead of subtracting 4. So effectively, 6 * 9 – 4 (coded) means 6 * 9 + 4 = 54 + 4 = 58. Check with the second example: 3 * 9 – 7. Compute 3 * 9 = 27. Now 27 + 7 = 34, which matches the given coded value. Thus the rule is: first multiply, then add the last number (instead of subtracting it). Apply the same rule to A * 4 – 9 = 91. This becomes A * 4 + 9 = 91. So A * 4 = 91 - 9 = 82. Therefore A = 82 / 4 = 20.5.

Verification / Alternative check:
If A = 20.5, then A * 4 – 9 in coded form is actually 20.5 * 4 + 9. That gives 82 + 9 = 91, which exactly matches the given coded result. Since the same rule works perfectly for both first examples and the target expression, the pattern is consistent and reliable.

Why Other Options Are Wrong:
6.5, 17.5 and 24.5 do not satisfy the relation A * 4 + 9 = 91. For example, 17.5 * 4 + 9 = 79, not 91. Similarly, 6.5 * 4 + 9 = 35 and 24.5 * 4 + 9 = 107. Therefore, these options do not match the coded value of 91.

Common Pitfalls:
A common mistake is to treat the minus sign as an actual subtraction sign and try to adjust with complicated formulas. Another error is to apply different rules to different examples. In coded arithmetic questions, once a simple and consistent rule fits all given cases, that is almost always the intended pattern.

Final Answer:
The value of A that fits the same coded pattern is 20.5.