Let me start with Question 1, which is the classic LMT-to-UTC conversion — Page 414, Lesson 367

I want to walk you through the answers to these time and longitude problems, because this is where all the arc-to-time conversion work we've been doing comes together into actual flight planning. Let me start with Question 1, which is the classic LMT-to-UTC conversion. We have a position at 103°15'E, and the local mean time there is 17 hours, 10 minutes, 45 seconds. The first step is to convert that longitude into time. Since the position is east, we subtract the arc-to-time value. For 103°15'E, the arc-to-time conversion gives us 6 hours 53 minutes. So we take 17:10:45 LMT, subtract 6:53, and we get 17:03:52 UTC. That's the universal time at that moment. Now, the second part of Question 1 asks for the LMT at a different longitude, 007°15'E. We already have UTC as 17:03:52. Since this position is also east, we add the arc-to-time value. For 007°15'E, that's 29 minutes. So 17:03:52 UTC plus 29 minutes gives us 17:04:21 LMT at 007°15'E. Notice the pattern: east of Greenwich, LMT is ahead of UTC, so we add; west, we subtract. Now Question 2, part (a). We start with 2300 LMT on 9 May at a position we need to find. The difference in longitude is given as 100°00'. Converting that to time, 100 degrees is 6 hours 40 minutes. Since we're going from the unknown position to 108°30'W, and the LMT is later there, we subtract. So 2300 LMT minus 6:40 gives us 1620 LMT on 9 May at 108°30'W. Part (b) of Question 2 is a two-step conversion. We start with 2300 LMT on 9 May, and the difference in longitude is 117°00'. Converting that to time gives us 7 hours 48 minutes. Wait, let me check the arithmetic here. The answer shows we add 34 minutes first to get to UTC, then add 7 hours 14 minutes to get to the new LMT. So the first step is converting from LMT to UTC by subtracting the arc-to-time for the original longitude, which gives us 2334 UTC on 9 May. Then we add the arc-to-time for the new longitude, 7 hours 14 minutes, to get 0648 LMT on 10 May. So we've crossed midnight, and the date advances to 10 May. Question 3 gives us three separate conversions from UTC to LMT. Part (a): 1300 UTC, and we add 29 minutes for 07°15'E, giving 1329 LMT on 1 April. Part (b): 1300 UTC, and we subtract 7 hours 11 minutes for 107°43'W, giving 0549 LMT on 1 April. Part (c): 1300 UTC, and we add 11 hours 14 minutes for 168°35'E, giving 0014 LMT on 2 April. Notice in part (c), adding that much time pushes us past midnight, so the date becomes 2 April. Question 4 deals with converting LMT to UTC. Part (a): 0300 LMT, and we subtract 3 hours 6 minutes to get 2354 UTC on the previous day. So the date goes back to the day before. Part (b): we have an arc-to-time of 1 hour 5 minutes, and we subtract that from 2354 UTC to get 2249 LMT on the previous day. Now Question 5 is about Standard Time, not LMT. We're given 1400 UTC on 6th November, and we need to find the standard time in various places. Iraq is UTC+3, so 1400 plus 3 hours gives 1700 ST on 6 Nov. Libya is UTC+1, giving 1500 ST. Tonga is UTC+13, giving 0300 ST on 7 Nov — note the date change. Labrador is UTC-4, giving 1000 ST. New York is UTC-5, giving 0900 ST. And Ghana keeps UTC, so it's simply 1400 ST. Question 6 is a full flight planning problem. We have 1430 LMT on 10 February at some position, and we convert to UTC by adding 4 hours 16 minutes, giving 1846 UTC on 10 February. Then we apply the Standard Time correction for Hong Kong, which is +8 hours. So 1846 UTC plus 8 hours gives 0246 ST on 11 February in Hong Kong. That's the arrival time. Question 7 is the most involved. We're flying from Wellington to Samoa. We arrive at Samoa at 2350 LMT on 8 September, and we convert that to UTC by adding 11 hours 27 minutes, giving 1117 UTC on 9 September. Then we subtract the flight time of 6 hours 15 minutes to find the departure time from Wellington in UTC: 0502 UTC on 9 September. Finally, we apply Wellington's Standard Time correction of +12 hours, giving 1702 ST on 9 September as the departure time. And the note says, "You have won yourself a day!" — because by flying east across the International Date Line, you arrive on a date that's effectively a day earlier than where you started. Now, the last set of answers, Answers 3. Question 1: 0730 UTC plus 11 hours 39 minutes of arc-to-time gives 1909 LMT at Wellington. Question 2: 1200 LMT at Boston plus 4 hours 45 minutes of arc-to-time gives 1645 UTC. Question 3, part (a): 1335 UTC minus 10 hours 31 minutes gives 0304 LMT at Hawaii. Part (b): 1335 UTC plus 10 hours gives the standard time correction. Now, the opening question at the top of the page asks about a flight from New York to Madrid. New York is at 073°45'W, Madrid is at 003°33'W. The flight leaves at 2000 Standard Time on 16th February, and the flight time is 7 hours 45 minutes. To find the arrival time in Madrid, we first convert New York's departure time to UTC. New York is UTC-5, so 2000 ST becomes 0100 UTC on 17th February. Then we add the flight time of 7 hours 45 minutes, giving 0845 UTC on 17th February. Finally, we convert to Madrid's standard time. Madrid is UTC+1, so 0845 UTC becomes 0945 ST on 17th February. That's the arrival time and date. So the key takeaway from all these problems is the systematic approach: convert everything to UTC first, do your arithmetic there, then convert to the local time at your destination. That way, you never get confused about which direction to add or subtract.