Qations A transponder for a toll bridge costs \( \$ 12.50 \). With the transponder, the toll is \( \$ 5 \) each time you cross the bridge. The only other option is toll-by-plate, for which the toll is \( \$ 5.25 \) each time you cross the bridge with an additional administrative fee of \( \$ 2.25 \) for each crossing. How many times would you need to cross the bridge for the costs of the two toll options to be the same? You need to cross the bridge times.

To find when the costs of both toll options are the same, we first set up equations for each option. With the transponder, the total cost after crossing the bridge \( n \) times is: \[ C_{transponder} = 12.50 + 5n \] For toll-by-plate, the cost equation is: \[ C_{toll-by-plate} = 2.25 + 5.25n \] Now, we will set them equal to find the number of crossings \( n \): \[ 12.50 + 5n = 2.25 + 5.25n \] Subtract \( 5n \) from both sides: \[ 12.50 = 2.25 + 0.25n \] Subtract \( 2.25 \) from both sides: \[ 10.25 = 0.25n \] Now, divide by \( 0.25 \): \[ n = \frac{10.25}{0.25} = 41 \] You need to cross the bridge **41 times** for the costs of the two toll options to be the same!