Bike-Rental Comparison with Rounding, Discount, and Tax
A youth club is comparing two bike-rental companies for a 9-person outing. Company A charges a fixed 18 lei booking fee plus 2.40 lei per bicycle per hour, billed in whole hours with any part hour rounded up. Company B charges a fixed 30 lei booking fee plus 2.00 lei per bicycle per hour, also rounded up. The outing lasts 5 hours 20 minutes. Company B offers a 12% discount on the total bill for groups of at least 8, but only if the booking is made online; company A offers no discount. The club must choose the cheaper total, including a 5% local tax added after discounts. Determine billed hours, compare, and find whether a 4-hour 40-minute outing would change the choice.
1. For the 5 h 20 min outing, determine billed hours, pre-tax costs, discount, and final totals. Which company is cheaper and by how much?
Read the worked answer
5 h 20 min rounds up to 6 billed hours. Company A: 18+2.40·9·6=18+129.60=147.60 before tax. With 5% tax, A costs 147.60·1.05=154.98.
Company B: 30+2.00·9·6=30+108.00=138.00 before discount. The 12% discount gives 138.00·0.88=121.44. With 5% tax, B costs 121.44·1.05=127.512, or 127.51 rounded to the nearest cent.
Company B is cheaper by 154.98−127.51=27.47 lei.
2. Repeat the comparison for a 4 h 40 min outing. Does the choice change?
Read the worked answer
4 h 40 min rounds up to 5 billed hours. Company A: 18+2.40·9·5=18+108.00=126.00 before tax; with tax, 126.00·1.05=132.30.
Company B: 30+2.00·9·5=30+90.00=120.00; after the 12% discount, 120.00·0.88=105.60; with tax, 105.60·1.05=110.88.
Company B is cheaper by 132.30−110.88=21.42 lei. The choice does not change.
3. For what whole numbers of billed hours is Company B cheaper after its discount and after tax? Include the one-hour case explicitly.
Read the worked answer
Let h be billed hours. The 5% tax multiplies both totals, so it cancels in the comparison. Company A costs 18+2.40·9h=18+21.6h. Company B costs after discount 0.88(30+2.00·9h)=0.88(30+18h).
B is cheaper when 0.88(30+18h)<18+21.6h. Expanding gives 26.4+15.84h<18+21.6h, so 8.4<5.76h and h>1.4583. Since h is a whole number of billed hours, B is cheaper for h≥2. For h=1, B costs 0.88(30+18)=42.24 before tax and A costs 18+21.6=39.60 before tax, so A is cheaper for exactly one billed hour.