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. Define the variable: let \(x\) represent the number of times the road is used.
2. Write the expression for the total cost without the electronic pass: since each toll costs \$5.00\(, the total cost is \)5x$.
3. Write the expression for the total cost with the electronic pass: the pass costs \$30\( upfront, and each toll is reduced by 30%, so each toll costs \)5 \times (1 - 0.30) = 5 \times 0.70 = 3.5\(. Therefore, the total cost is \)30 + 3.5x$.
4. Set the total costs equal to find the break-even point: \$5x = 30 + 3.5x$.
5. Solve the equation for \(x\) by isolating \(x\) on one side: subtract \$3.5x\( from both sides to get \)5x - 3.5x = 30\(, then simplify and solve for \)x$.

Key Concepts

Setting Up Equations for Cost Comparison

To solve the problem, you need to express the total costs with and without the electronic pass as algebraic expressions. This involves defining variables (e.g., number of uses) and writing equations that represent each total cost scenario for comparison.

Percentage Discount Calculation

Understanding how to calculate a percentage discount is essential. Here, a 30% reduction on the $5 toll means multiplying $5 by 0.30 to find the discount amount, then subtracting it from the original toll to find the reduced toll cost.

Solving Linear Equations

Once the cost expressions are set equal, solving for the variable requires knowledge of linear equations. This involves isolating the variable on one side through algebraic manipulation to find the number of uses where costs are equal.