Calculating Marbles in Bags: Determining the 9 Count from an Average of 9.64

To solve this problem, we'll follow these steps:

• Step 1: Identify the number of bags and compute the total number of marbles.
• Step 2: Apply the formula for the average and solve for the unknown.
• Step 3: Solve the equation to determine the number of bags containing 9 marbles.

Let's work through these steps:

Step 1: The problem details that the average number of marbles per bag is 9.64. We need to find the total number of bags. Let $n$ be the number of bags that contain 9 marbles. Calculating the total number of bags is necessary as a starting point.

Given bags and their marbles:
- 1 bag containing 18 marbles
- 2 bags containing 12 marbles each: Total $= 2 \times 12 = 24$ marbles
- 3 bags containing 7 marbles each: Total $= 3 \times 7 = 21$ marbles

Total marbles in known bags = $18 + 24 + 21 = 63$ marbles

Step 2: Use the total average calculation to find the number of bags:

$\text{Number of unknown bags} = n$

Total marbles = $63 + 9n$

The number of total bags is $1 + 2 + 3 + n = 6 + n$.

Thus, by the average formula:

$\frac{63 + 9n}{6 + n} = 9.64$

Step 3: Solve for $n$.

$63 + 9n = 9.64(6 + n)$ $63 + 9n = 57.84 + 9.64n$

Subtract $57.84$ from both sides:

$63 - 57.84 = 9.64n - 9n$ $5.16 = 0.64n$

Divide by 0.64 to solve for $n$:

$n = \frac{5.16}{0.64} = 8.0625$

This result implies rounding is needed, thus $n = 8$ bags.

Therefore, the solution to the problem is $8$ bags.