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.