An electronic pass for a toll road costs $30. The toll is normally $5.00 but is reduced by 30% for people who have purchased the electronic pass. Determine the number of times the road must be used so that the total cost without the pass is the same as the total cost with the pass.

Verified step by step guidance

1. Let the number of times the road is used be represented by the variable \( x \).
2. Calculate the cost per use with the electronic pass: The toll is reduced by 30%, so the cost per use with the pass is \( 5.00 - 0.30 \times 5.00 \).
3. Set up the equation for the total cost with the pass: \( 30 + (5.00 - 0.30 \times 5.00) \times x \).
4. Set up the equation for the total cost without the pass: \( 5.00 \times x \).
5. Set the two total cost equations equal to each other and solve for \( x \): \( 30 + (5.00 - 0.30 \times 5.00) \times x = 5.00 \times x \).

Key Concepts

Cost Analysis

Cost analysis involves comparing different expenses to determine the most economical option. In this scenario, we need to analyze the total costs associated with using the toll road with and without the electronic pass. This includes calculating the upfront cost of the pass and the reduced toll rate for pass holders.

Percentage Reduction

Understanding percentage reduction is crucial for calculating the new toll rate for electronic pass holders. In this case, the toll is reduced by 30%, meaning we need to find 30% of the original toll ($5.00) and subtract that from the original amount to determine the discounted toll rate.

Equation Setup

Setting up an equation is essential for solving the problem mathematically. We need to create an equation that equates the total cost of using the toll road with the electronic pass to the total cost without the pass, allowing us to solve for the number of uses required to break even.