We are given the arithmetic mean of $A + 7$ and $A + 3$, and it equals $3A + 9$. To find the value of $A$, let's break it down step by step.

Step 1: Formula for Arithmetic Mean

The arithmetic mean of two numbers $x$ and $y$ is given by:

$\text{Mean} = \frac{x + y}{2}$

In this case, the two numbers are $A + 7$ and $A + 3$. So, the arithmetic mean is:

$\frac{(A + 7) + (A + 3)}{2}$

Step 2: Simplify the Expression

Simplifying the numerator:

$(A + 7) + (A + 3) = 2A + 10$

So, the arithmetic mean becomes:

$\frac{2A + 10}{2}$

Simplify the fraction:

$A + 5$

Step 3: Set Equal to the Given Expression

We are told that this arithmetic mean is equal to $3A + 9$. Therefore:

$A + 5 = 3A + 9$

Step 4: Solve for $A$

To solve for $A$, subtract $A$ from both sides:

$5 = 2A + 9$

Next, subtract 9 from both sides:

$-4 = 2A$

Finally, divide by 2:

$A = -2$

Conclusion:

The value of $A$ is $-2$.

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Would you like more details or have any questions?

Here are 5 related questions:

1. How is the arithmetic mean used in statistics?
2. What are other types of means besides arithmetic mean?
3. Can you solve a similar problem but with different numbers?
4. How do you verify the correctness of a mean calculation?
5. What is the relationship between mean and median in data sets?

Tip: In arithmetic mean problems, always ensure the terms are correctly simplified before proceeding with comparisons.