Three friends were traveling by express train to New York. After traveling for 10 minutes, Michelle fell asleep. She woke an hour later and said, “Where are we?”. Jo whispered her reply as Tony had fallen asleep 15 minutes earlier. Jo said, “In 20 minutes’ time we’ll have 100 miles to go.” Tony woke 30 minutes later and asked the others, “Where are we?” and they replied, “In 10 minutes’ time we will have exactly half as far to go as we’ve already gone.” How long (time and distance) do they have left in order to complete their journey, and how long (time and distance) was their entire train trip? Be sure to describe in detail how you determined the four solution

You want the total time and distance, and the time and distance remaining on a train trip such that after 90 minutes, 100 miles remain, and after 110 minutes, 1/3 of the trip remains.

Timeline

The scenario seems to describe this timeline:

Time from start

0:10 Michelle sleeps

0:55 Tony sleeps

1:10 Michelle wakes

1:30 Jo says 100 miles remain

1:40 Tony wakes (30 min after Michelle wakes)

1:50 Others say 1/3 of the trip remains

Time and distance

If the ratio of traveled distance to remaining distance is 2:1, then remaining distance is 1/(2+1) = 1/3 of the total. Traveled distance will be 1-1/3 = 2/3 of the total.

The total trip time can be computed from what the others told Tony. He wakes up after traveling 1:40, and is told that 2/3 of the trip will have passed in 10 minutes' time. That is 1:50 = 110 minutes is 2/3 of the trip. Then the entire trip must be 110(1 +1/2) = 165 minutes.

The trip is 165 minutes, or 2 hours 45 minutes long.

After 1:30 = 90 minutes, 100 miles remain. That means 100 miles will be covered in 165 -90 = 75 minutes. Since time is proportional to distance, we have ...

trip/(165 min) = (100 mi)/(75 min)

trip = 165(100 mi/75) = 220 mi

The trip is 220 miles long.

Remaining

When Michelle wakes after 70 minutes, the remaining trip is ...

time: 95 minutes, distance: 126 2/3 miles

When Tony wakes after 100 minutes, the remaining trip is ...

time: 65 minutes, distance: 86 2/3 miles

__

Additional comment

The phrasing "Tony woke 30 minutes later" is somewhat ambiguous. We have assumed that is 30 minutes after Michelle woke. It could be interpreted as 30 minutes after he fell asleep.

The question "how long do they have left" is also ambiguous, as there are 6 different time points mentioned in the scenario.

The problem mentions "four solution." We don't know if this is a typo intended to be "your solution," or if there are four solutions wanted.

The rate of speed is 80 mi/h, or 1 1/3 miles per minute. Knowing remaining minutes, the remaining miles can be found by multiplying by 4/3. The remaining time is calculated by subtracting the time traveled from the trip time of 165 minutes.