We are given four numbers: 2, 6, 12, and 15, along with their frequencies as follows:

• Frequency of 2: $x + 6$
• Frequency of 6: $x + 2$
• Frequency of 12: $x - 3$
• Frequency of 15: $x$

The mean formula for a frequency distribution is:

$\text{Mean} = \frac{\sum (\text{value} \times \text{frequency})}{\sum (\text{frequency})}$

We are also given that the mean is 6.7. So we have:

$6.7 = \frac{(2(x + 6) + 6(x + 2) + 12(x - 3) + 15x)}{(x + 6) + (x + 2) + (x - 3) + x}$

Step 1: Simplify the numerator

Let's expand the terms in the numerator:

$2(x + 6) = 2x + 12$ $6(x + 2) = 6x + 12$ $12(x - 3) = 12x - 36$ $15x = 15x$

Now, sum them up:

$(2x + 12) + (6x + 12) + (12x - 36) + 15x = 2x + 6x + 12x + 15x + 12 + 12 - 36$ $= 35x - 12$

Step 2: Simplify the denominator

Sum the frequencies:

$(x + 6) + (x + 2) + (x - 3) + x = 4x + 5$

Step 3: Set up the equation

Now substitute into the mean formula:

$6.7 = \frac{35x - 12}{4x + 5}$

Step 4: Solve for $x$

Multiply both sides by $4x + 5$:

$6.7(4x + 5) = 35x - 12$

Expand both sides:

$26.8x + 33.5 = 35x - 12$

Move all terms involving $x$ to one side:

$35x - 26.8x = 33.5 + 12$

Simplify:

$8.2x = 45.5$

Now solve for $x$:

$x = \frac{45.5}{8.2} \approx 5.55$

The next highest integer is 6.

Thus, $x = 6$.

Would you like further details or have any questions?

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Here are five related questions for deeper understanding:

1. How would the calculation change if the given mean were different?
2. What happens if one of the frequencies is zero?
3. Can the frequencies be negative, and how would that affect the mean?
4. How can you verify the solution using another method?
5. What is the significance of each number in the frequency formula?

Tip: Always check the denominator after solving to ensure the terms match the number of data points in frequency-based problems.