Problem 30 - College Algebra

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

The bridge must be crossed 20 times in a month.

Step 1: Define variables

Let \(x\) be the number of times in a month the bridge is crossed.

Step 2: Set up the equation

Then, the total cost per month without the discount pass is \(5x\) and the cost with the discount pass is \(30 + 3.5x\). You should set these two expressions equal to each other to form the equation in order to find out \(x\). So, the equation becomes \(5x = 30 + 3.5x \).

Step 3: Solve the equation

To solve for \(x\), subtract \(3.5x\) from both sides of the equation: \(5x - 3.5x = 30\). This simplifies to \(1.5x = 30\). Finally, divide both side of the equation by \(1.5\) to obtain \(x = 30/1.5\).

Step 4: Determine the number of times

Carry out the division to find the value of \(x\). So, \(x = 20\). This is the number of times the bridge should be crossed in a month to make the total monthly toll cost the same whether one has the discount pass or not.