8. Four friends throw a party for their school's basketball team. To have enough food for the whole team, each friend buys 1 medium pizza, 2 orders of breadsticks, and 1 salad. Let $$ p $$ represent the price of 1 medium pizza. Let $$ b $$ represent the price of 1 order of breadsticks. Let $$ s $$ represent the price of 1 salad. All prices are in dollars. a. Write two different expressions that represent the total amount of money the friends spend. Explain how each expression describes the situation in a different way. b. The party is running out of food, so another friend orders 1 medium pizza and 1 salad. Write an expression to represent the total cost of all the food after this order.

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Let's tackle each part of the problem step by step.

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### **Problem 8.a**

**Write two different expressions that represent the total amount of money the friends spend. Explain how each expression describes the situation in a different way.**

**Given:**
- Four friends are buying food.
- Each friend buys:
  - 1 medium pizza ($$ p $$)
  - 2 orders of breadsticks ($$ b $$)
  - 1 salad ($$ s $$)

**Expression 1: Aggregating Like Items**

$$
\text{Total Cost} = 4p + 8b + 4s
$$

**Explanation:**
- **4p** represents the total cost for 4 medium pizzas (1 pizza per friend).
- **8b** represents the total cost for 8 orders of breadsticks (2 orders per friend × 4 friends).
- **4s** represents the total cost for 4 salads (1 salad per friend × 4 friends).
  
This expression adds up the total cost of each type of item separately and then sums them all to find the total expenditure.

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**Expression 2: Factoring Per Friend**

$$
\text{Total Cost} = 4 \times (p + 2b + s)
$$

**Explanation:**
- **$$ p + 2b + s $$** represents the total cost for one friend’s order.
- Multiplying by **4** accounts for all four friends.

This expression groups the cost per friend first and then multiplies by the number of friends to find the total expenditure.

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### **Problem 8.b**

**The party is running out of food, so another friend orders 1 medium pizza and 1 salad. Write an expression to represent the total cost of all the food after this order.**

**Initial Total Cost (from part a):**
$$
4p + 8b + 4s
$$

**Additional Order:**
$$
1p + 1s
$$

**New Total Cost:**
$$
(4p + 8b + 4s) + (p + s) = 5p + 8b + 5s
$$

**Final Expression:**
$$
5p + 8b + 5s
$$

**Explanation:**
- **5p** accounts for the original 4 medium pizzas plus the additional 1 medium pizza.
- **8b** remains unchanged since no extra breadsticks were ordered.
- **5s** accounts for the original 4 salads plus the additional 1 salad.

This expression represents the updated total cost after the additional friend’s order.

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### **Summary**

- **Part a:**
  1. $$ 4p + 8b + 4s $$
  2. $$ 4(p + 2b + s) $$
  
- **Part b:**
  $$
  5p + 8b + 5s
  $$

These expressions effectively model the total expenditure in different formats, providing flexibility depending on how you want to approach the problem.
**Part a:** - **Expression 1:** $$ 4p + 8b + 4s $$ (Total cost for pizzas, breadsticks, and salads for all friends) - **Expression 2:** $$ 4(p + 2b + s) $$ (Total cost per friend multiplied by the number of friends) **Part b:** - **Total Cost After Additional Order:** $$ 5p + 8b + 5s $$

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