Logical Reasoning Pattern Solving

This question presents a set of equations that use the multiplication symbol (×) but do not follow standard arithmetic rules. This indicates that the symbol represents a hidden pattern or a specific calculation rule that we need to discover. Our goal is to identify this rule and then apply it to find the value of $5 \times 3$.

Analyzing the Given Equations and Patterns

We are given three examples:

• $9 \times 6 = 45$
• $7 \times 4 = 33$
• $6 \times 4 = 20$

Let's represent the numbers on the left side as A and B, so the format is $A \times B = \text{Result}$. We need to find a function or rule, say $f(A, B)$, that gives the specified Result for each pair (A, B).

Let's list the inputs and results and consider simple combinations like the sum (A+B) and the standard product (A×B):

Discovering the Secret Calculation Rule

Let's look for a relationship between the Result and the values of A, B, A+B, or A×B. Comparing the Result with A+B values:

• For $9 \times 6$: $A+B = 15$. The Result is 45. We see that $15 \times 3 = 45$.
• For $7 \times 4$: $A+B = 11$. The Result is 33. We see that $11 \times 3 = 33$.
• For $6 \times 4$: $A+B = 10$. The Result is 20. We see that $10 \times 2 = 20$.

This suggests a pattern where the Result is obtained by multiplying the sum (A+B) by a certain value, let's call it C. The value of C seems to be 3 for the first two examples and 2 for the third example.

Now, let's try to figure out how the multiplier C is determined. Let's look at the values of A in each case:

• When A=9, C=3.
• When A=7, C=3.
• When A=6, C=2.

A clear pattern emerges based on the value of A. If A is 7 or greater ($A \ge 7$), the multiplier C is 3. If A is less than 7 ($A < 7$), the multiplier C is 2.

So, the hidden rule is:

$A \times B = (A+B) \times C$, where

• $C = 3$ if $A \ge 7$
• $C = 2$ if $A < 7$

Verifying the Pattern with Examples

Let's double-check this rule with the given examples:

• For $9 \times 6$: A=9, B=6. Since $A=9$ and $9 \ge 7$, $C=3$. Result = $(9+6) \times 3 = 15 \times 3 = 45$. Correct.
• For $7 \times 4$: A=7, B=4. Since $A=7$ and $7 \ge 7$, $C=3$. Result = $(7+4) \times 3 = 11 \times 3 = 33$. Correct.
• For $6 \times 4$: A=6, B=4. Since $A=6$ and $6 < 7$, $C=2$. Result = $(6+4) \times 2 = 10 \times 2 = 20$. Correct.

The discovered pattern successfully explains all the given examples.

Applying the Pattern to Find $5 \times 3$

Now we apply the same rule to calculate the value of $5 \times 3$.

Here, $A=5$ and $B=3$.

First, determine the multiplier C based on the value of A. Since $A=5$ and $5 < 7$, the multiplier $C=2$.

Next, calculate the sum of A and B: $A+B = 5+3 = 8$.

Finally, multiply the sum by the multiplier C: $(A+B) \times C = 8 \times 2 = 16$.

Thus, the value of $5 \times 3$ according to this logical reasoning pattern is 16.

Final Answer

Based on the pattern identified, the value of $5 \times 3$ is 16.

Revision Table: Key Pattern Steps

Here is a summary of the steps to apply the identified pattern:

Additional Information: Logical Reasoning Patterns

Logical reasoning questions often test your ability to find underlying rules or patterns that are not immediately obvious. These patterns can involve various mathematical operations or conditions. Examples include patterns based on the sum, difference, product, or even the digits of the numbers involved. Sometimes, the rule for one part of the calculation (like the multiplier C in this case) depends on the value of one of the input numbers. To solve these puzzles, it's helpful to systematically examine how the inputs relate to the output in the given examples and test different possible rules until one fits all the cases. Once the rule is found and verified, applying it to the new problem provides the solution.