If 164 = 2, 9-3 = 2, 81 – 3 = 4, then the value of 64 - 4 is:

Question

Accepted Answer

The correct option is 1:
The question presents a number puzzle where standard mathematical operators are replaced by a hidden pattern. Our task is to decipher this pattern from the three given equations and apply it to the final expression, $64 - 4$. This type of question tests the ability of students to identify complex logical rules embedded within numerical data.

Let us analyze the three established relationships:1.  $164 = 2$
2.  $9 - 3 = 2$
3.  $81 - 3 = 4$A common technique in number pattern problems involving squares or higher numbers is to use roots. We can observe that the first number in each expression (16, 9, 81) is a perfect square. Let $A$ be the first number and $B$ be the second number in the expression. We find that a consistent rule involves taking the square root of the first number, dividing it by the second number, and then adding 1 to the result.

**Rule Derivation:** $(\sqrt{A} / B) + 1 = \text{Result}$  * **Testing 1: $164 = 2$**
    $A=16, B=4$.
    $(\sqrt{16} / 4) + 1 = (4 / 4) + 1 = 1 + 1 = 2$. This result matches the given output.

* **Testing 2: $9 - 3 = 2$**
    $A=9, B=3$.
    $(\sqrt{9} / 3) + 1 = (3 / 3) + 1 = 1 + 1 = 2$. This result matches the given output.

* **Testing 3: $81 - 3 = 4$**
    $A=81, B=3$.
    $(\sqrt{81} / 3) + 1 = (9 / 3) + 1 = 3 + 1 = 4$. This result matches the given output.Since the rule $(\sqrt{A} / B) + 1$ holds true for all three examples, we can confidently apply it to find the value of $64 - 4$.

**Applying the Rule to $64 - 4$:**
$A=64, B=4$.
Value $= (\sqrt{64} / 4) + 1$
Value $= (8 / 4) + 1$
Value $= 2 + 1$
Value $= 3$

Therefore, the value of $64 - 4$ according to the established number pattern is 3, which corresponds to Option 1.

The incorrect option is 2 (Value: 2):
This option would only be correct if the hidden rule was simply $(\sqrt{A} / B)$. In this case, $(\sqrt{64} / 4) = 8 / 4 = 2$. However, this ignores the necessary '+1' operation required to satisfy the established relationships $164 = 2$, $9 - 3 = 2$, and $81 - 3 = 4$. Thus, 2 is incorrect because it fails to incorporate the final additive step of the pattern.

The incorrect option is 3 (Value: 4):
The value 4 would imply a result greater than the actual solution. It might arise from an incorrect application of the square root, perhaps if a student assumed the operation was related to addition, like $(\sqrt{64} - 4) + 1 = 8 - 4 + 1 = 5$, or some other flawed calculation. Specifically, for the correct expression, 4 is incorrect as $3 \neq 4$.

The incorrect option is 4 (Value: 6):
This option is significantly higher than the calculated result and would be derived from a fundamentally incorrect reading of the pattern. For instance, if a student simply multiplied the square root of 64 by some other constant, or applied a common algebraic substitution without verifying the initial examples. Since the established rule leads uniquely to 3, 6 must be incorrect.