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The problem asks how many miles Ravi needs to drive for the two plans to cost the same. We are given:

- Plan 1: Initial fee of \$51.96 and \$0.12 per mile.
- Plan 2: Initial fee of \$63.96 and \$0.08 per mile.

Let $ x $ represent the number of miles driven. 

The total cost for each plan can be written as:
- Plan 1: $ 51.96 + 0.12x $
- Plan 2: $ 63.96 + 0.08x $

We need to find when the two costs are equal, so we set the equations equal to each other:

\[
51.96 + 0.12x = 63.96 + 0.08x
\]

Now, let's solve for $ x $.

\[
51.96 + 0.12x = 63.96 + 0.08x
\]
Subtract $ 0.08x $ from both sides:

\[
51.96 + 0.04x = 63.96
\]
Subtract 51.96 from both sides:

\[
0.04x = 12
\]
Now divide by 0.04:

\[
x = \frac{12}{0.04} = 300
\]

Thus, Ravi would need to drive **300 miles** for the two plans to cost the same.

Would you like more details on any part of this solution?

Here are five related questions to deepen your understanding:
1. What would the total cost of each plan be if Ravi drives 200 miles?
2. If the per-mile cost in Plan 1 increases to \$0.15, how would the equation change?
3. How many miles would Ravi need to drive for Plan 1 to be cheaper than Plan 2?
4. If Ravi drives 500 miles, how much more expensive would Plan 2 be than Plan 1?
5. How does the initial fee impact the total cost as the number of miles increases?

**Tip**: Always ensure the units of measurement (e.g., dollars per mile) match when solving word problems involving rates.

Ravi will rent a car for the weekend. He can choose one of two plans. The first plan has an initial fee of $51.96 and costs an additional $0.12 per mile driven. The second plan has an initial fee of $63.96 and costs an additional $0.08 per mile driven. How many miles would Ravi need to drive for the two plans to cost the same?

Discover how to solve a car rental problem where two different rental plans have different initial fees and mileage rates. This step-by-step guide will show how to find the number of miles where both plans cost the same. Suitable for middle school students.

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