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Let's break down the problem and solve it step by step using the "M method," which likely refers to determining the **Greatest Common Factor (GCF)**. The problem asks for the greatest number of identical balloon decorations that can be made, where the store uses 20 orange, 30 black, and 15 purple balloons.

### Step 1: Identify the numbers involved
- Orange balloons: 20
- Black balloons: 30
- Purple balloons: 15

### Step 2: Find the Greatest Common Factor (GCF)
To find how many identical balloon decorations can be made, we need to determine the GCF of 20, 30, and 15. The GCF is the largest number that divides each of these numbers exactly.

#### Prime factorization:
- \( 20 = 2^2 \times 5 \)
- \( 30 = 2 \times 3 \times 5 \)
- \( 15 = 3 \times 5 \)

The common factor among all these numbers is **5**.

### Step 3: Solution for part 8A
Thus, the greatest number of balloon decorations the store can make is the GCF, which is **5**.

### Step 4: Determine the number of each balloon in each decoration (for part 8B)
Now, divide the total number of balloons by the GCF (5) to find out how many of each balloon type will be in each decoration:

- Orange balloons: \( \frac{20}{5} = 4 \)
- Black balloons: \( \frac{30}{5} = 6 \)
- Purple balloons: \( \frac{15}{5} = 3 \)

### Final Answer:
- 8A: GCF is **5** decorations.
- 8B:
  - Orange balloons per decoration: **4**
  - Black balloons per decoration: **6**
  - Purple balloons per decoration: **3**

Do you want more details, or do you have any other questions?

### Related Questions:
1. What is the Least Common Multiple (LCM) of 20, 30, and 15?
2. How do you calculate the prime factorization of a number?
3. How would the solution change if you added 25 blue balloons?
4. What happens if the store wants to use all the balloons in fewer decorations?
5. How can you check if the GCF is correct?

### Tip:
When solving for the GCF, always break the numbers into their prime factors and identify the common factors to ensure accuracy!

A party store is making balloon decorations for a party. The store will use 20 orange, 30 black, and 15 purple balloons altogether. What is the greatest number of balloon decorations the store will put together? How many of each balloon is in each decoration?

Solve a real-world problem by finding the greatest common factor (GCF) of 20, 30, and 15 to determine the maximum number of balloon decorations that can be made. Includes step-by-step prime factorization and balloon distribution calculations.

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