Problem 29 - College Algebra Essentials

Question

The bus fare in a city is \(\$ 1.25 .\) People who use the bus have the option of purchasing a monthly discount pass for \(\$ 15.00 .\) With the discount pass, the fare is reduced to \(\$ 0.75\) Determine the number of times in a month the bus must be used 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 bus must be used 30 times in a month for the total cost to be the same with or without the discount pass.

Step 1: Define the Variables

Let's define \(x\) as the number of times the bus is used in a month. The total cost per month without the pass is \(1.25x\) while the total cost per month with the pass is \(15 + 0.75x\).

Step 2: Set Up the Equation

Next, the two costs are equal when the break-even point is reached. This gives the equation: \(1.25x = 15 + 0.75x\).

Step 3: Solve the Equation

Subtract \(0.75x\) from both sides of the equation to isolate \(x\) on one side, giving \(0.5x = 15\). Then divide both sides of the equation by 0.5 to solve for \(x\), which gives \(x = 30\). So for the total cost of the two options to be equal, the bus must be used 30 times in a month.