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Let's break down the problem and solve it step by step.

### First, summarize the given dataLet's break down the problem and solve it step by step.

### First, summarize the given 

- **Black "Congratulations GRAD" balloons**: 1 package = 48 balloons, $12.99 per package.
- **Red metallic latex balloons**: 1 package = 144 balloons, $14.99 per package.
- **White latex balloons**: 1 package = 200 balloons, $8.57 per package.

The committee has already made **8 arrangements**, each using:
- 6 black balloons per arrangement.
- 18 red balloons per arrangement.
- 25 white balloons per arrangement.

Now, we move to the second part of the problem:

### Question 1: What is the smallest number of balloon arrangements to use exactly one of each type of balloon?
Since they don’t want any leftover balloons, we need to find the **Greatest Common Divisor (GCD)** of the total number of balloons in a full package of each type.

- **Black balloons total**: 48 balloons per package.
- **Red balloons total**: 144 balloons per package.
- **White balloons total**: 200 balloons per package.

The smallest number of balloon arrangements will be based on the GCD of these values. The GCD of 48, 144, and 200 is **8**. Therefore, they can make **8 arrangements** where each arrangement uses exactly one black, one red, and one white balloon.

### Question 2: How many packages of each balloon type will they need to buy?
We divide the total number of balloons used for 8 arrangements by the number of balloons per package.

- **Black balloons**: $ \frac{8}{48} = \frac{1}{6} $ → So they need **1 package**.
- **Red balloons**: $ \frac{8}{144} = \frac{1}{18} $ → So they need **1 package**.
- **White balloons**: $ \frac{8}{200} = \frac{1}{25} $ → So they need **1 package**.

### Question 3: Total cost of balloons (excluding tax and shipping):
The cost for each package is given:
- **Black balloons**: 1 package = $12.99
- **Red balloons**: 1 package = $14.99
- **White balloons**: 1 package = $8.57

So the total cost will be:
\[
12.99 + 14.99 + 8.57 = 36.55
\]

Thus, the total cost is **$36.55**.

---

Would you like more detailed explanations for any of these steps, or have any other questions? Here are some follow-up questions to expand on the information:

1. What happens if they decide to buy more than one package of each balloon type?
2. How would you calculate the cost if tax and shipping were included?
3. What if the committee decides to increase the number of arrangements?
4. How can you check if there are leftover balloons in any scenario?
5. What would the total cost be if they needed to buy more packages of balloons?

**Tip**: Always check the GCD when distributing items evenly to avoid leftovers.

The student activities council ordered balloons from a party supplier for graduation decorations. They ordered 11 inch black "Congratulations GRAD" balloons (1 package of 48 balloons for $12.99 per package), 11 inch red metallic latex balloons (1 package of 144 balloons for $14.99 per package), and 5 inch bulk white latex balloons (1 package of 200 balloons for $8.57 per package). The committee wants to group the balloons in arrangements such that each type of balloon is equally spread across 8 arrangements, using all the balloons.

Learn how to calculate the greatest number of balloon arrangements using the GCD of different balloon types. The problem also includes calculating the cost of purchasing balloons in whole packages. Suitable for middle school students (Grades 6-8).

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