Problem 12 - Algebra and Trigonometry

Question

A discount pass for a bridge costs \(\$ 30\) per month. The toll for the bridge is normally \(\$ 5.00,\) but it is reduced to \(\$ 3.50\) for people who have purchased the discount pass. Determine the number of times in a month the bridge must be crossed so that the total monthly cost without the discount pass is the same as the total monthly cost with the discount pass.

Step-by-Step Solution

Answer

You must cross the bridge 20 times in a month to make the total monthly cost with a discount pass equal to the total monthly cost without a discount pass.

Step 1: Formulate The Two Cost Scenarios

For the scenario without the discount pass, every time the bridge is crossed costs \(\$ 5\). Therefore, if the bridge is crossed \(x\) times, the total cost will be \(5x\). On the other hand, having purchased the discount pass, each crossing costs \(\$ 3.50\) and the total cost will be \(\$ 30\) upfront for the pass and then \(\$ 3.50x\) for crossing the bridge \(x\) times, a total of \(30 + 3.50x\).

Step 2: Set Up the Cost Equations

The problem asks when the total cost with the discount pass equals the total cost without the pass, which leads to the equation: \(5x = 30 + 3.50x\).

Step 3: Solve for x

Subtract \(3.50x\) from both sides of the equation to isolate x terms on one side, resulting in \(1.50x = 30\). Then divide both sides by \(1.50\) to solve for \(x\), giving \(x = 30 / 1.50 = 20\).