Question

On the Bologna-Rome route of a high-speed train, 3/4 of the reservations are with departure from Bologna, while the remainder are with departure from Florence, which is the only intermediate stop. In Bologna all the passengers who have booked get on, while in Florence 20 people get on, corresponding to 2/5 of bookings from Florence, and 36 passengers from Bologna get off the train. How many passengers had booked the train? And when the train arrives in Rome, how many passengers are left on board? the answer is 200 and 134 but I would like the procedure to get there.​

Answer 1

Step-by-step explanation:

First, we consider the calculation towards finding the total number of passengers who had booked the train in Florence. The main question to ask in answering this problem is

      $\text{``If 20 passengers that board the train from Florence correlates to only} \ \displaystyle\frac{2}{5} \\ \text{of the bookings from Florence, then what is the total number of passengers} \\ \text{that booked regardless if the passengers boarded the train or not?"}$

This statement can be phrase mathematically using the method of ratio and proportion as shown below

                                                    $\displaystyle\frac{2}{5} : 1 \ = \ 20:x$,

in which the variable $x$ is the total number of passengers who booked the train from Florence.

Now, rewriting the above ratio into the form of fractions, yields

                                                         $\displaystyle\frac{\displaystyle\frac{2}{5}}{1} \ = \ \displaystyle\frac{20}{x} \\ \\ \displaystyle\frac{2}{5} \ = \ \displaystyle\frac{20}{x} \\ \\ x \ = \ \displaystyle\frac{20}{\frac{2}{5}} \\ \\ x \ = \ 20 \ \times \ \displaystyle\frac{5}{2} \\ \\ x \ = \ 50$

Given that a quarter out of all reservations is from Florence, then using the same method as before to evaluate the number of passengers that had booked the train (from both Bologna and Florence).

                                                         $\displaystyle\frac{\displaystyle\frac{1}{4}}{1} \ = \ \displaystyle\frac{50}{y} \\ \\ y \ = \ \dsiplaystyle\frac{50}{\frac{1}{4}} \\ \\ y \ = \ 50 \ \times \ 4 \\ \\ y \ = \ 200$,

where the variable $y$ denotes the total number of bookings.

Therefore, the total number of passengers that had booked the train is 200.

Now, it is stated that all passengers who booked had boarded the train from Bologna which is 150 passengers (three-quarters of total bookings). Then, 36 passengers from earlier got off in Florence, while 20 passengers boarded.

Number of passengers left when arriving Rome = 150 - 36 + 20

                                                                                = 134 passengers