The bus fare in a city is \( \$ 1.50 \). People who use the bus have the option of purchasing a monthly coupon book for \( \$ 25.00 \). With the coupon book, the fare is reduced to \( \$ 0.50 \). Determine the number of times in a month the bus must be used so that the total monthly cost without the coupon book is the same as the total monthly cost with the coupon book. The bus must be used \( \square \) times.

To find the number of times the bus must be used for the total costs to be the same, we can set up an equation. Let \( x \) be the number of times the bus is used in a month. Without the coupon book, the total cost is: \[ 1.50x \] With the coupon book, the total cost is: \[ 25.00 + 0.50x \] We want these two costs to be equal: \[ 1.50x = 25.00 + 0.50x \] Now, let's solve for \( x \): 1. Subtract \( 0.50x \) from both sides: \[ 1.50x - 0.50x = 25.00 \] \[ 1.00x = 25.00 \] 2. Divide both sides by \( 1.00 \): \[ x = 25.00 \] So, the bus must be used \( \boxed{25} \) times to equate the costs.