2018 YJC P1 Q8

Eden is a new engineer for the Small Rail Transit (SRT) train. His current project is to devise a new schedule for the train to increase its efficiency. In the current schedule, the train will run for \(5\) days in one cycle. For the first day of the first cycle, the train runs for \(100\) km. On the \({{2}^{\mathrm{nd}}}\) and on the \({{3}^{\mathrm{rd}}}\) day of each cycle, the train will run \(30\) km more than the first day of that cycle. On the \({{4}^{\mathrm{th}}}\) and on the \({{5}^{\mathrm{th}}}\) day of each cycle, the train will run \(70\) km more compared to the first day of that cycle. On the first day of the \({{k}^{\mathrm{th}}}\) cycle, the train will run 30 km more compared to the last day of the \({{\left( k-1 \right)}^{\mathrm{th}}}\) cycle.

1. It was found out that the train broke down after travelling \(10\mathrm{ }000\) km. By forming an inequality, state the cycle and the day of the cycle that the train was in during the breakdown.[6]

Eden decides to modify the schedule such that the train will now operate every week from Monday to Saturday. On the first day of operations, a Monday, the train runs for \(100\) km. On every Tuesday, Thursday and Saturday, the distance travelled will be increased by \(20\%\) compared to the previous operating day. However, on every Monday, Wednesday and Friday, there will be a \(8\%\) decrease in the distance travelled compared to the previous operating day.

2. Find the total distance travelled by the train in \(2n\) days of operations.[3]
3. Find the minimum number of days that the train has run, such that Eden would have improved the previous schedule (i.e. the total distance run before breakdown is more than \(10\mathrm{ }000\) km).[2]