what is (@) doing to the numbers is 4@3=5  ; 6@5=9

The symbol "@" in the context of the equations \(4@3=5\) and \(6@5=9\) represents an operation or function that is applied to the two numbers on either side of the "@" symbol. To determine what the "@" operation does, we can analyze the given examples and try to identify a pattern.

### Step-by-Step Analysis:
1. **Example 1**: \(4@3=5\)
   - Here, applying the "@" operation to 4 and 3 gives 5.

2. **Example 2**: \(6@5=9\)
   - Here, applying the "@" operation to 6 and 5 gives 9.

To identify the operation, let's check for common arithmetic operations that might relate the pairs of numbers:

- **Addition**: \(4 + 3 = 7\) and \(6 + 5 = 11\) (doesn't match the results, so it's not addition).
- **Subtraction**: \(4 - 3 = 1\) and \(6 - 5 = 1\) (doesn't match, so not subtraction).
- **Multiplication**: \(4 \times 3 = 12\) and \(6 \times 5 = 30\) (doesn't match, so not multiplication).
- **Division**: \(4 \div 3 \approx 1.33\) and \(6 \div 5 = 1.2\) (doesn't match, so not division).

Now, let's explore another potential operation:
- **Average**: 
   - For \(4@3\), the average of 4 and 3 is \(\frac{4 + 3}{2} = 3.5\), but rounding up gives us 4 (not 5, so this doesn't work).
   - For \(6@5\), the average of 6 and 5 is \(\frac{6 + 5}{2} = 5.5\), but rounding up gives us 6 (not 9, so this doesn't work either).

Finally, consider the possibility that the operation involves adding a fixed number or using a custom formula:
- **Custom Operation Hypothesis**:
   - For \(4@3\), the result is \(4 + 1\), meaning the operation could be \(a@b = a + 1\) for this case.
   - For \(6@5\), the result is \(6 + 3\), meaning the operation could be \(a@b = a + 3\) for this case.

However, this would mean the operation isn't consistent and might depend on the relationship between \(a\) and \(b\). A likely rule might be adding the difference between the numbers:

- **Final Operation**: \( a@b = a + (b - a + 1) \)
  - For \(4@3\), the difference is \(3 - 4 = -1\), so \(4 + (-1 + 1) = 4 + 0 = 4\) (which doesn't match).
  - For \(6@5\), the difference is \(5 - 6 = -1\), so \(6 + (-1 + 3) = 6 + 2 = 8\) (doesn't match exactly).

There seems to be a slight modification needed, possibly involving the idea that the operation could involve addition and a fixed adjustment. One plausible pattern might be:

- **Alternative Hypothesis**: The operation \(a@b = \frac{a + b + 1}{2}\), rounding to the nearest integer.
   - For \(4@3\), \( \frac{4 + 3 + 1}{2} = \frac{8}{2} = 4 \) (adjust by adding +1).
   - For \(6@5\), \( \frac{6 + 5 + 1}{2} = \frac{12}{2} = 6 \) (adjust by adding +3).

However, we might need more examples to fully confirm this pattern.

**Summary**: Based on the examples provided, the "@" operation seems to involve adding the two numbers and adjusting by a constant. For the specific cases given:
- \(4@3\) results in 5, possibly by adding 1 after summing.
- \(6@5\) results in 9, possibly by adding 3 after summing.

To confirm this pattern or identify the exact operation, more examples or rules would be needed.

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Would you like more details, or do you have any questions?

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Here are 8 related questions you might find interesting:

1. How would the "@" operation behave for other pairs like \(5@2\) or \(7@4\)?
2. What are some common operations that can be defined between two numbers?
3. How can we identify the function or rule governing a custom operation from examples?
4. How does modular arithmetic apply in custom operations?
5. What are some examples of non-standard operations used in mathematical puzzles?
6. Can you create your own custom operation with specific rules?
7. How would you program a custom operation into a calculator or software?
8. How does rounding influence the result of arithmetic operations?

**Tip**: When trying to decode an unknown operation, look for consistent patterns, such as addition, multiplication, or a combination of arithmetic and logical operations.

Explore the '@' operation in mathematical equations such as 4@3=5 and 6@5=9. Learn the step-by-step analysis to identify the custom operation involved. Suitable for Grades 7-9 students studying arithmetic operations and custom mathematical rules.

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