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Part 1
A truck can be rented from Company A for ​$80 a day plus ​$0.40 per mile. Company B charges ​$50 a day plus ​$0.70 per mile to rent the same truck. Find the number of miles in a day at which the rental costs for Company A and Company B are the same

Let's solve for the number of miles, $ x $, at which the daily rental costs for Company A and Company B are equal.

### Step 1: Set up the cost equations
Let $ x $ represent the number of miles driven in a day.

1. **Cost for Company A**:
   \[
   \text{Cost}_{A} = 80 + 0.40x
   \]

2. **Cost for Company B**:
   \[
   \text{Cost}_{B} = 50 + 0.70x
   \]

### Step 2: Set the equations equal to each other
To find when the costs are the same, set the two equations equal to each other:
\[
80 + 0.40x = 50 + 0.70x
\]

### Step 3: Solve for $ x $
1. Subtract $ 0.40x $ from both sides:
   \[
   80 = 50 + 0.30x
   \]

2. Subtract 50 from both sides:
   \[
   30 = 0.30x
   \]

3. Divide by 0.30 to isolate $ x $:
   \[
   x = \frac{30}{0.30} = 100
   \]

### Conclusion
The number of miles at which the rental costs for Company A and Company B are the same is **100 miles**.

Would you like further clarification on any part of this process?

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Here are some additional related questions for practice:
1. What would the cost be for each company if 100 miles were driven?
2. How would the cost comparison change if Company A increased its daily charge?
3. How many miles would need to be driven for Company B to be the cheaper option?
4. If Company B charged $0.50 per mile, at what point would the costs be equal?
5. What if Company A charged $100 per day and $0.30 per mile?

**Tip:** Setting up equations is crucial for solving comparison problems effectively.

A truck can be rented from Company A for ​$80 a day plus ​$0.40 per mile. Company B charges ​$50 a day plus ​$0.70 per mile to rent the same truck. Find the number of miles in a day at which the rental costs for Company A and Company B are the same.

Discover how to compare rental costs between Company A and Company B, where daily costs depend on fixed charges and variable mileage rates. Solve for the break-even mileage point, where costs become equal at 100 miles driven. Ideal for students in Grades 7-9 exploring linear equations and algebraic cost analysis.

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