A truck can be rented from Company A for ​$100 a day plus ​$0.30 per mile. Company B charges ​$20 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 the number of miles driven in a day be represented by $ x $.

For Company A:
- Daily cost: \$100
- Per mile cost: \$0.30

Thus, the total cost for Company A is:
\[
C_A = 100 + 0.30x
\]

For Company B:
- Daily cost: \$20
- Per mile cost: \$0.70

Thus, the total cost for Company B is:
\[
C_B = 20 + 0.70x
\]

To find the number of miles at which the costs are the same, we set the two equations equal to each other:
\[
100 + 0.30x = 20 + 0.70x
\]

Now, solve for $ x $:

1. Subtract $ 20 $ from both sides:
   \[
   100 - 20 + 0.30x = 0.70x
   \]
   \[
   80 + 0.30x = 0.70x
   \]

2. Subtract $ 0.30x $ from both sides:
   \[
   80 = 0.40x
   \]

3. Divide both sides by $ 0.40 $:
   \[
   x = \frac{80}{0.40} = 200
   \]

Thus, the number of miles at which the rental costs are the same is $ \boxed{200} $ miles.

Would you like further explanation on this, or do you have any questions?

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Here are 5 related questions you might find helpful:

1. How would the costs compare if you drive fewer than 200 miles?
2. What if Company B also added a surcharge after a certain number of miles?
3. How would you model the costs if you rented the truck for multiple days?
4. What if Company A offered a discount for miles over 300 in a day?
5. Can you find the breakeven point if both companies charge per hour instead of per mile?

**Tip:** When solving problems involving comparisons between two linear equations, always look for where the equations intersect to find the breakeven point!

Find out how to calculate the breakeven point for truck rental costs between two companies using linear equations. This problem involves solving for the number of miles driven in a day where the costs of Company A and Company B are equal. Suitable for students in Grades 6-8.

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