To determine the time that goes in the gap, we need to ensure that the journey times from Waterloo to Dartford are consistent across the different journeys recorded in the table. We'll first calculate the duration of the journeys that we do have times for, and then use that information to find the missing time.
### Step-by-Step Solution
1. Calculate the first journey time (from 16:15 to 17:20):
- Departure from Waterloo: 16:15
- Arrival at Dartford: 17:20
- Journey time: [tex]\( \text{17:20} - \text{16:15} = 1 \text{ hour and } 5 \text{ minutes} = 65 \text{ minutes} \)[/tex].
2. Calculate the second journey time (from 16:45 to 17:50):
- Departure from Waterloo: 16:45
- Arrival at Dartford: 17:50
- Journey time: [tex]\( \text{17:50} - \text{16:45} = 1 \text{ hour and } 5 \text{ minutes} = 65 \text{ minutes} \)[/tex].
3. Confirm both journey times are equal:
- Both journeys are 65 minutes long.
4. Use the consistent journey time to find the departure time for the third journey:
- Arrival at Dartford: 18:10
- Journey duration: 65 minutes
- Departure time from Waterloo = [tex]\( \text{18:10} - 65 \text{ minutes} \)[/tex].
To calculate the exact departure time:
[tex]\[ \text{18:10} = 18 \times 60 + 10 = 1090 \text{ minutes past midnight} \][/tex]
[tex]\[ \text{1090 minutes} - 65 \text{ minutes} = 1025 \text{ minutes past midnight} \][/tex]
[tex]\[ \text{Converting back to hours and minutes:} \][/tex]
[tex]\[ 1025 \div 60 = 17 \text{ hours and remainder } 5 \text{ minutes} \][/tex]
Therefore, the time in the gap is:
[tex]\[ \boxed{17:05} \][/tex]
