Problem 30 - College Algebra Essentials

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 number of times the bridge must be crossed in a month for the cost to break even is 20 times.

Step 1: Express the total cost of using the bridge with and without the discount pass.

The total cost without the discount pass is given as \( \$5.00 \times \text{number of trips} \). The total cost with the discount pass is the cost of the pass plus the reduced cost per trip \( \$30.00 + \$3.50 \times \text{number of trips} \) .

Step 2: Set up an equation

The problem asks for the number of times 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, so set up the equation as follows: \( 5 \times X = 30 + 3.5 \times X \) where X represents the number of trips across the bridge in a month.

Step 3: Solve for the unknown

Now, solve for X by first subtracting \( 3.5X \) from both sides of the equation to get \( 1.5X = 30 \). Then divide both sides of the equation by 1.5 to solve for X, giving \( X = 30 ÷ 1.5 = 20 \).