26. DIG DEEPER An armusement park gives everyone who rides a new roller coaster at full price a \( 20 \% \) discount on all other rides. The prices of 4 rides at the park before the discount are shown. You ride the roller coaster once and then 3 other rides. You spend \( \$ 10 \). Identify several possible prices to ride the roller coaster. Which do you think is the most reasonable? Explain your reasoning. Rumpor Cars: \( \$ 1.75 \) \( \$ 1.25 \) more than Ferris Wheel Giant Slide: \( \$ 0.50 \) less than Bumper Cars Ferris Wheel:

To solve this problem, we need to identify the prices of the rides at the amusement park and then determine the possible price of the roller coaster based on the total amount spent. ### Step 1: Define the Prices of the Rides Let's denote the price of the Ferris Wheel as \( F \). According to the problem: 1. **Bumper Cars**: \( B = F + 1.25 \) 2. **Giant Slide**: \( G = B - 0.50 = (F + 1.25) - 0.50 = F + 0.75 \) Now we have the prices of the rides in terms of \( F \): - Ferris Wheel: \( F \) - Bumper Cars: \( F + 1.25 \) - Giant Slide: \( F + 0.75 \) ### Step 2: Calculate the Total Cost You ride the roller coaster once and then 3 other rides, spending a total of \( \$10 \). The discount on the other rides is \( 20\% \) because you rode the roller coaster at full price. Let \( R \) be the price of the roller coaster. The discounted price for the other rides will be \( 80\% \) of their total price. The total price of the three rides (Ferris Wheel, Bumper Cars, and Giant Slide) is: \[ \text{Total Price of Rides} = F + (F + 1.25) + (F + 0.75) = 3F + 2 \] The discounted price for these rides is: \[ \text{Discounted Price} = 0.8 \times (3F + 2) = 2.4F + 1.6 \] ### Step 3: Set Up the Equation The total amount spent is the price of the roller coaster plus the discounted price of the other rides: \[ R + (2.4F + 1.6) = 10 \] Rearranging gives us: \[ R = 10 - (2.4F + 1.6) = 8.4 - 2.4F \] ### Step 4: Identify Possible Prices for the Roller Coaster Now we need to find reasonable values for \( F \) such that \( R \) remains a positive price. 1. **Set \( R > 0 \)**: \[ 8.4 - 2.4F > 0 \] \[ 8.4 > 2.4F \] \[ F < \frac{8.4}{2.4} = 3.5 \] Thus, \( F \) must be less than \( 3.5 \). ### Step 5: Calculate Possible Prices Let's consider some reasonable values for \( F \) (the price of the Ferris Wheel): - If \( F = 1.50 \): - Bumper Cars: \( 1.50 + 1.25 = 2.75 \) - Giant Slide: \( 1.50 + 0.75 = 2.25 \) - Total Price of Rides: \( 1.50 + 2.75 + 2.25 = 6.50 \) - Discounted Price: \( 0.8 \times 6.50 = 5.20 \) - Roller Coaster Price: \( 10 - 5.20 = 4.80 \) - If \( F = 2.00 \): - Bumper Cars: \( 2.00 + 1.25 = 3.25 \) - Giant Slide: \( 2.00 + 0.75 = 2.75 \) - Total Price of Rides: \( 2.00 + 3.25 + 2.75 = 8.00 \) - Discounted Price: \( 0.8 \times 8.00 = 6.40 \) - Roller Coaster Price: \( 10 - 6.40 = 3.60 \) - If \( F = 2.50 \): - Bumper Cars: \( 2.50 + 1.25 = 3.75 \) - Giant Slide: \( 2.50 + 0.75 = 3.25 \) - Total Price of Rides: \( 2.50 + 3.75 + 3.25 = 9.50 \) - Discounted Price: \( 0.8 \times 9.50 = 7.60 \) - Roller Coaster Price: \( 10 - 7.60 = 2.40 \) ### Conclusion The possible prices for the roller coaster based on the values of \( F \) are: - If \( F = 1.50 \), then \( R = 4.80 \) - If \( F = 2.00 \), then \( R = 3.60 \) - If \( F = 2.50 \), then \( R = 2.40 \) **Most Reasonable Price**: Among these, \( R = 3.60 \) seems the most reasonable as it is a common price for amusement park rides, while \( R = 4.80 \) might be on the higher side, and \( R = 2.40 \) might be too low for a roller coaster.