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1.14 Time

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Before you start

  • Multiplying and dividing by 60, by 24 and by 7, with and without a calculator.
  • Changing a fraction to a decimal and back, e.g. 34=0.75 and 0.2=15.
  • The speed, distance and time relationship from 1.11: distance = speed × time.
  • Subtracting a negative number, e.g. 9−(−5)=14.
Syllabus objectives covered (3)
  • Calculate with time: seconds (s), minutes (min), hours (h), days, weeks, months, years, including the relationship between units.
  • Calculate times in terms of the 24-hour and 12-hour clock.
  • Read clocks and timetables.

Part 1 of 6

The units of time and how they link

  1. The units of time do not go up in tens. Learn these links by heart, because every time question uses them.
    60 s=1 min60 min=1 h24 h=1 day7 days=1 week
  2. For a year, the syllabus uses 1 year =365 days. A year also has 12 months. Use 365 days unless a question tells you something different.
    1 year=365 days=12 months
  3. To change to a smaller unit, multiply. To change to a bigger unit, divide. A smaller unit means more of them, so the number must get bigger. 3 hours is 3×60=180 minutes, and 420 seconds is 420÷60=7 minutes.
  4. For a jump of more than one unit, chain the links. One day in seconds: multiply by 24 to get hours, by 60 to get minutes, and by 60 again to get seconds.
    1 day=24×60×60=86400 s
  5. A year is not exactly 52 weeks. 52×7=364, so 365 days is 52 weeks and 1 day. If you need the number of weeks in a year exactly, it is 365÷7=5217.
  6. Months are the awkward unit, because they are not all the same length. April, June, September and November have 30 days. February has 28 days, or 29 in a leap year. Every other month has 31 days.
  7. So a month is not 4 weeks, and it is not always 30 days. To count days across months, add up the real length of each month.
  8. When you count days between two dates, decide whether both end days are included. 10 May is 7 days after 3 May. But a holiday from 3 May to 10 May, with both days included, lasts 8 days.

Part 2 of 6

Hours and minutes are not a decimal

  1. A calculator works in decimals, and time does not. When you divide a distance by a speed, the time comes out in hours as a decimal, such as 2.4 hours.
  2. The part after the point is a fraction of an hour, not a number of minutes. 0.4 of an hour is 0.4×60=24 minutes. So 2.4 hours is 2 h 24 min, not 2 h 40 min.
    2.4 h=2 h+(0.4×60) min=2 h 24 min
  3. To go the other way, divide the minutes by 60. 2 h 45 min is 2+4560=2.75 hours. Do this whenever a formula needs the time in hours, such as distance = speed × time.
  4. Some numbers of minutes give recurring decimals. 20 minutes is 2060=13 of an hour, which is 0.333… hours. 50 minutes is 56 of an hour, which is 0.8333… hours.
  5. Keep the fraction, or the full calculator value, in the middle of a question. Do not round it to 0.83. At 18 km/h for 2 h 50 min, the exact distance is 18×256=51 km. Using 2.83 hours gives 50.94 km, which is wrong.
  6. The same rule works one level down. 0.35 of a minute is 0.35×60=21 seconds, so 4.35 minutes is 4 min 21 s.

Part 3 of 6

Adding and subtracting times

  1. To find how long something lasts, count on from the start time in easy chunks: up to the next whole hour, then whole hours, then the minutes left over. From 09 45 to 14 20:
    09 45→ 15 min 10 00→ 4 h 14 00→ 20 min 14 20
  2. Add the chunks: 15 min + 4 h + 20 min = 4 h 35 min.
  3. Do not subtract the times as ordinary numbers. 1420−945=475, which looks like 4 h 75 min. It is wrong because an hour has 60 minutes, not 100. Each time the subtraction borrows across an hour, the answer comes out 40 minutes too long.
  4. If you prefer columns, borrow 1 hour as 60 minutes. 14 h 20 min becomes 13 h 80 min.
    13 h 80 min−9 h 45 min=4 h 35 min
  5. To add a time, add the hours and the minutes separately, then change each 60 minutes into 1 hour. 08 50 plus 2 h 35 min gives 10 h 85 min. 85 min is 1 h 25 min, so the answer is 11 25.
  6. If the hours go past 24, you have crossed midnight. Subtract 24 and write 'the next day'. 21 30 + 5 h 15 min = 26 45, which is 02 45 the next day.
  7. For a duration across midnight, count on to midnight first. From 21 30 to 06 15 the next day: 2 h 30 min takes you to midnight, then 6 h 15 min more. The total is 8 h 45 min.

Part 4 of 6

The 24-hour and the 12-hour clock

  1. The 12-hour clock counts from 12 round to 12 twice a day. a.m. runs from midnight up to noon (midday), and p.m. runs from noon up to midnight. A 12-hour time without a.m. or p.m. is ambiguous: 7.30 could be morning or evening.
  2. The 24-hour clock counts the hours straight through, from 00 00 at midnight to 23 59. It needs no a.m. or p.m. It always has four digits. The syllabus writes 3.15 a.m. as 03 15 and 3.15 p.m. as 15 15. A digital clock shows the same time with a colon, 15:15.
  3. 12-hour to 24-hour. For an a.m. time, keep the digits and put a zero in front if the hour has one digit: 7.05 a.m. is 07 05. For a p.m. time, add 12 to the hour: 7.05 p.m. is 19 05.
  4. The exceptions are the 12 o'clock hours. 12.40 p.m. is 40 minutes after noon, so it is 12 40, with nothing added. 12.40 a.m. is 40 minutes after midnight, so it is 00 40.
    12-hour clock24-hour clock12.40 a.m.00 407.05 a.m.07 0512.40 p.m.12 407.05 p.m.19 0511.59 p.m.23 59
  5. 24-hour to 12-hour. If the hour is 13 or more, subtract 12 and write p.m.: 16 25 is 4.25 p.m. If the hour is 00, write 12 and a.m.: 00 10 is 12.10 a.m. Hours 01 to 11 are a.m., and hour 12 is p.m.
  6. A quick check: if you changed a p.m. time from 1 p.m. onwards and got a 24-hour hour below 13, you forgot to add 12.

Part 5 of 6

Time zones and local time

  1. Different parts of the world set their clocks to different times. The time on the clocks in a place is its local time.
  2. A time difference says how far apart two local times are. 'In January, Tokyo is 9 hours ahead of London' means that Tokyo's clocks show a time 9 hours later. When it is 10 00 in London, it is 19 00 in Tokyo.
  3. To change to the local time of a place that is ahead, add the difference. For a place that is behind, subtract it. Check with the words: ahead means later.
  4. The date can change. 20 00 in London is 20+9=29, so 29 00, in Tokyo. Subtract 24: it is 05 00 the next day in Tokyo.
  5. Use only the time difference the question gives you. It may be in words, or in a table of differences from one place, such as +9 for Tokyo and −5 for New York. To compare two places in such a table, subtract: 9−(−5)=14, so Tokyo is 14 hours ahead of New York.
  6. A journey takes one real stretch of time, whatever the clocks say. So do the journey on one clock only. For an arrival time, add the journey time on the departure clock, then change to the local time of the arrival place at the end.
  7. For a journey time, change both times to the same clock first, then count on. Never subtract a local time in one place from a local time in another place.
  8. Expect odd-looking answers. Flying to a place far ahead, the local arrival time can look much too late. Flying to a place far behind, the plane can land at a clock time earlier than the one it left at. Both are correct if every step was done on one clock.

Part 6 of 6

Reading clocks and timetables

  1. On an analogue clock, the long hand shows the minutes and the short hand shows the hours. The minute hand moves 5 minutes for each number: at the 3 it is 15 minutes past, and at the 9 it is 45 minutes past. Each small mark between two numbers is 1 minute.
  2. The hour hand does not jump from number to number. It moves steadily during the hour. At 4.30 it is halfway between the 4 and the 5. At 4.50 it is almost on the 5.
    graph diagram — not drawn yet

    An analogue clock face with the numbers 1 to 12 and 60 small minute marks. The long minute hand points exactly at the 10. The short hour hand points five-sixths of the way from the 4 to the 5, so it is very close to the 5 but has not reached it.

    4.50: the hour hand is nearly on the 5, but the hour is still 4.
  3. So read the hour as the number the hour hand has most recently passed. If the minute hand is on the 10 (50 minutes) and the hour hand is nearly on the 5, the time is 4.50, not 5.50.
  4. A clock face cannot show a.m. or p.m. Decide from the words in the question, such as 'in the evening', then write the time in the form asked. 4.50 in the evening is 16 50.
  5. A timetable is a grid of times. In the bus timetable below, the stops are down the side and each column is one bus. Read down a column to follow one bus. Some timetables are turned the other way, with one row for each journey, so read the headings before you read any times.
    Bus 1Bus 2Bus 3Bus 4Bus station08 1508 5009 2510 05Park Road08 2709 0209 3710 17Market Square08 41—09 5110 31Hospital08 5809 2610 0810 48
  6. A dash (—) means the bus does not stop there. Bus 2 goes straight past Market Square, so you cannot get on or off it there.
  7. For a journey time between two stops, take both times from the same column and count on. Bus 1 leaves Park Road at 08 27 and reaches the Hospital at 08 58, so the journey takes 31 minutes.
  8. 'The next bus after 09 10' means: go along the row for your stop and find the first time at or after 09 10. 'The latest bus that arrives by 10 00' means: go along the row for where you are going, find the last time that is not after 10 00, then read up that column to your stop.

Worked examples

Example 1. (a) Work out the number of seconds in one week. (b) A cyclist rides at a steady 18 km/h for 2 hours 50 minutes. Calculate the distance she rides. (c) A walker covers 13 km at a steady 5 km/h. Work out the time this takes, in hours and minutes.

Show the working
  1. (a) Chain the links from days down to seconds. Multiply at each step, because each new unit is smaller.
    7×24×60×60=604800 s
  2. (b) The speed is per hour, so the time must be in hours. 50 minutes is 5060=56 of an hour.
    2 h 50 min=256 h=176 h
  3. Multiply the speed by the time. Keeping the fraction gives an exact answer.
    18×176=51 km
  4. (c) Time = distance ÷ speed.
    13÷5=2.6 h
  5. The 0.6 is a fraction of an hour, not 60 minutes. Multiply it by 60 to get minutes.
    0.6×60=36 min
  6. Check: in 2 hours she walks 2×5=10 km, and in 36 minutes (0.6 h) she walks 0.6×5=3 km. The total is 13 km, as it should be.

Answer: (a) 604800 seconds (b) 51 km (c) 2 h 36 min

Example 2. A night bus leaves at 22 47. The journey takes 5 hours 38 minutes. (a) Work out the time the bus arrives, using the 24-hour clock. (b) Write this time using the 12-hour clock.

Show the working
  1. (a) Add the hours and the minutes separately.
    22 h 47 min+5 h 38 min=27 h 85 min
  2. Every 60 minutes is one more hour. 85 min is 1 h 25 min, so this is 28 h 25 min.
  3. A day has only 24 hours. Subtract 24 and note that the day has changed: 28−24=4, so the bus arrives at 04 25 the next day.
  4. Check by counting on. From 22 47 to midnight is 1 h 13 min. That leaves 5 h 38 min − 1 h 13 min = 4 h 25 min after midnight, which is 04 25.
  5. (b) The hour is 04, which is between midnight and noon, so it is a.m. Drop the zero at the front.

Answer: (a) 04 25 (the next day) (b) 4.25 a.m.

Example 3. In January, the local time in Tokyo is 9 hours ahead of the local time in London. (a) A plane leaves London at 13 20 London time. It lands in Tokyo at 10 50 Tokyo time the next day. Work out how long the flight takes. (b) The plane flies back. It leaves Tokyo at 11 25 Tokyo time and the flight takes 14 hours 10 minutes. Find the local time in London when it lands.

Show the working
  1. (a) The two times are on different clocks, so put them on the same clock first. Tokyo is ahead, so subtract 9 hours to get London time.
    10 50 (Tokyo)−9 h=01 50 (London)
  2. That is 01 50 the next day on the London clock. Now count on from 13 20 on the London clock: 10 h 40 min to midnight, then 1 h 50 min more.
    10 h 40 min+1 h 50 min=12 h 30 min
  3. (b) Do the journey on the Tokyo clock: 11 25 + 14 h 10 min = 25 h 35 min. Subtract 24: it lands at 01 35 the next day, Tokyo time.
  4. Now change to London time. London is behind, so subtract 9 hours. 01 35 − 9 h goes back past midnight: 01 35 is the same moment as 25 35 on the day before, and 25 35 − 9 h = 16 35.
  5. So it lands at 16 35 London time, on the same date it left Tokyo. Check on the London clock: it left at 11 25 − 9 h = 02 25 London time, and 02 25 + 14 h 10 min = 16 35. The answers agree.

Answer: (a) 12 h 30 min (b) 16 35 London time, on the same date as it left Tokyo

Example 4. Use this bus timetable. Bus 1Bus 2Bus 3Bus 4Bus station08 1508 5009 2510 05Park Road08 2709 0209 3710 17Market Square08 41—09 5110 31Hospital08 5809 2610 0810 48 (a) Rania reaches the Park Road stop at 08 55. She must be at the Hospital by 10 00. Which bus does she catch, how long does she wait, and how long is her bus journey? (b) Tomas gets on at Market Square. He must also be at the Hospital by 10 00. Find the latest time he can get on a bus.

Show the working
  1. (a) Go along the Park Road row and find the first time at or after 08 55. Bus 1 left at 08 27, so the next is Bus 2 at 09 02.
  2. Read down the Bus 2 column. It reaches the Hospital at 09 26, which is before 10 00, so Bus 2 works. (Bus 3 would arrive at 10 08, which is too late.)
  3. Her wait is from 08 55 to 09 02, which is 7 minutes. Her journey is from 09 02 to 09 26, which is 24 minutes.
  4. (b) Start from where he is going. Along the Hospital row, the last time that is not after 10 00 is 09 26, on Bus 2.
  5. Read up the Bus 2 column to Market Square. It is a dash: Bus 2 does not stop there, so it is no use to Tomas.
  6. Try the bus before it. Bus 1 leaves Market Square at 08 41 and reaches the Hospital at 08 58. That is the latest bus he can take.

Answer: (a) Bus 2, from Park Road at 09 02. She waits 7 minutes and the journey takes 24 minutes. (b) 08 41 (Bus 1)

Where people go wrong

Mistake: Reading a decimal time as hours and minutes, e.g. writing 2.4 hours as 2 h 40 min.

Instead: Multiply the decimal part by 60: 0.4×60=24, so 2.4 hours is 2 h 24 min.

The part after the point is a fraction of an hour. Decimals count in tenths and hundredths, but an hour is split into 60 minutes.

Mistake: Subtracting times as ordinary numbers, e.g. 14 20 − 09 45 = 475, read as 4 h 75 min.

Instead: Count on: 09 45 to 10 00 is 15 min, to 14 00 is 4 h more, and to 14 20 is 20 min more. The total is 4 h 35 min.

Column subtraction borrows 100, but an hour holds only 60 minutes. Each borrow across an hour makes the answer 40 minutes too long.

Mistake: Getting the 12 o'clock hours wrong: writing 12.30 a.m. as 12 30, or 12.30 p.m. as 24 30.

Instead: 12.30 a.m. is half an hour after midnight: 00 30. 12.30 p.m. is half an hour after noon: 12 30.

The 'add 12 for p.m.' rule works only for 1 p.m. to 11.59 p.m. The 12 o'clock hour is the exception at both ends of the day.

Mistake: Adding a time difference when the other place is behind, or subtracting it when the other place is ahead.

Instead: Ahead means its clocks show a later time, so add. Behind means an earlier time, so subtract. Say it in words before you calculate: 'New York is behind, so its clock shows less.'

Going the wrong way puts the answer twice the time difference away from the correct time.

Mistake: Reading an analogue clock that shows 4.50 as 5.50, because the hour hand is almost on the 5.

Instead: The hour is the number the hour hand has most recently passed. When the minute hand shows 50 minutes, the hour hand should be close to the next number, but it has not reached it.

The hour hand moves steadily through the whole hour, so near the end of the hour it sits just before the next number.

Check yourself

  1. Write 4.6 hours in hours and minutes.
    Answer0.6×60=36, so 4 h 36 min.
  2. How many hours are there in one year of 365 days?
    Answer365×24=8760 hours.
  3. A course starts on 25 March and ends on 12 April. Both days are included. How many days does the course last?
    AnswerMarch has 31 days, so 25 to 31 March is 7 days. 1 to 12 April is 12 days. The total is 19 days.
  4. (a) Write 7.40 p.m. using the 24-hour clock. (b) Write 00 25 using the 12-hour clock.
    Answer(a) 19 40 (b) 12.25 a.m.
  5. A film starts at 20 50 and ends at 23 15. How long is the film?
    Answer20 50 to 21 00 is 10 min, 21 00 to 23 00 is 2 h, and 23 00 to 23 15 is 15 min. The film lasts 2 h 25 min.
  6. In January, Dubai is 4 hours ahead of London and New York is 5 hours behind London. When it is 15 00 in Dubai, what time is it in New York?
    AnswerDubai is 4+5=9 hours ahead of New York. 15−9=6, so it is 06 00 in New York, on the same day.
  7. On an analogue clock, the minute hand points at the 3 and the hour hand is a quarter of the way from the 11 to the 12. It is the evening. Write the time using the 24-hour clock.
    Answer15 minutes past 11 in the evening is 11.15 p.m., which is 23 15.
  8. Use the bus timetable in 'Reading clocks and timetables'. You reach the Bus station at 09 30. Which bus do you catch, and at what time does it reach Market Square?
    AnswerBus 3 left the Bus station at 09 25, so you catch Bus 4 at 10 05. It reaches Market Square at 10 31.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. A ferry leaves Port Allan at 21 35. It arrives at Bray Island at 01 25 the next morning. The crossing is 69 km long.

(a) Write 21 35 using the 12-hour clock. 1 mark No calculator

Mark scheme
  • B1 9.35 p.m. oe, e.g. 9:35 pm; p.m. must be shown

(b) Work out how long the crossing takes. Give your answer in hours and minutes. 2 marks No calculator

Mark scheme
  • M1 a correct counting-on method, e.g. 2 h 25 min to midnight and 1 h 25 min after it, or 25 h 25 min − 21 h 35 min with 60 minutes borrowed, or 230 minutes seen
  • A1 3 h 50 min cao (230 minutes alone is M1 A0)

(c) Calculate the average speed of the ferry. Give your answer in km/h. 2 marks

Mark scheme
  • M1 69 ÷ (3 + 50/60) oe, e.g. 69 ÷ 3.83… or 69 ÷ 23/6; FT their (b) correctly changed to hours. 69 ÷ 3.5 or 69 ÷ 3.50 is M0
  • A1 18 cao

Question 2. In January, the local time in Mumbai is 5 hours 30 minutes ahead of the local time in London.

(a) A plane leaves London at 20 50 London time and flies to Mumbai. The flight takes 9 hours 25 minutes. Find the local time in Mumbai when the plane lands. 3 marks No calculator

Mark scheme
  • M1 9 h 25 min added to a time, e.g. 20 50 + 9 h 25 min giving 30 15 or 06 15 (London time) seen
  • M1 5 h 30 min added to a London time, e.g. 20 50 + 5 h 30 min = 02 20, or their 06 15 + 5 h 30 min
  • A1 11 45 cao; accept 11:45 or 11.45 a.m.

(b) Another plane leaves Mumbai at 02 10 Mumbai time. It lands in London at 07 05 London time on the same date. Work out how long the flight takes. 2 marks No calculator

Mark scheme
  • M1 one time changed to the other clock: 02 10 Mumbai time = 20 40 London time the day before, or 07 05 London time = 12 35 Mumbai time, or the clock-time gap 4 h 55 min with 5 h 30 min added
  • A1 10 h 25 min cao

Question 3. Here is part of a train timetable. All the trains run in the morning. Train ATrain BTrain CNorthgate06 5207 3408 16Elm Park07 09—08 33Riverside07 3108 0508 55Central07 4808 2209 12

(a) Ama arrives at Northgate station. On the station clock, the minute hand points at the 5 and the hour hand is a little less than halfway from the 7 to the 8. Write down the time on the station clock, using the 24-hour clock. 1 mark No calculator

Mark scheme
  • B1 07 25 cao; accept 0725 or 07:25

(b) Ama catches the next train from Northgate to Central. Work out the number of minutes from the time on the station clock until she arrives at Central. 2 marks No calculator

Mark scheme
  • M1 Train B identified, or 07 34 and 08 22 used, or a duration from their (a) to 08 22 attempted; FT their (a)
  • A1 57 cao, or FT the correct number of minutes from their (a) to the Central arrival of the next train after their (a)

(c) Compare the time Train A takes to travel from Northgate to Central with the time Train B takes. 2 marks No calculator

Mark scheme
  • B1 56 (minutes) for Train A and 48 (minutes) for Train B, both seen
  • B1 Train B is 8 minutes quicker oe; FT their two times

Question 4.

(a) Show that there are 10080 minutes in one week. 2 marks No calculator

Mark scheme
  • M1 7 × 24 (= 168 hours in a week) or 24 × 60 (= 1440 minutes in a day) seen
  • A1 a complete calculation such as 168 × 60 = 10 080 or 7 × 1440 = 10 080, with no errors

(b) A phone contract lasts for 100 days, counting the first day. The first day is 18 May. May, July and August have 31 days, and June has 30 days. Find the date of the last day of the contract. 2 marks No calculator

Mark scheme
  • M1 14 days in May (18 to 31 May, both included) used, or a running total such as 14 + 30 + 31 = 75 seen
  • A1 25 August cao

How this is examined

  • Finding a local arrival time, or a flight time, when the two places are in different time zones: A 3-mark Find or Work out, with a departure time, a flight time and a time difference given in the stem in words or in a table Every step on one clock: the flight time added to the departure time on the departure clock (e.g. 20 50 + 9 h 25 min = 06 15 the next day), then the time difference applied in the right direction (+5 h 30 min because the destination is ahead). Each of those two steps is a method mark. The final answer is a real clock time such as 11 45, not 35 45.
  • Reading a timetable or a clock face to choose the right journey and time it: A 1-mark Write down of a time from a clock or a timetable, then a 2-mark Work out of a waiting time or a journey time The right column: the first departure at or after the given time, with a dash read as 'does not stop here'. Both times for a journey taken from the same column, and the answer in the unit asked for, e.g. '57 minutes', not '0 57'.
  • Working out a duration that crosses the hour or midnight: A 2-mark Work out from a start time and a finish time, with the answer asked for in hours and minutes A counting-on method with each chunk shown, e.g. 2 h 25 min to midnight + 1 h 25 min after it, is the method mark. The answer is 3 h 50 min. Plain subtraction such as 125−2135 or 2135−125 is not a valid method.
  • Changing between hours-and-minutes and decimal hours inside a speed calculation: A 2-mark Calculate of an average speed straight after a duration has been found The time written in hours before dividing: 35060 or 3.833… or 236. That is the method mark. Using 3.5 or 3.50 for 3 h 50 min is not a correct method. Then the speed in the units asked, e.g. 18 km/h.
  • Changing a time between the 12-hour and 24-hour clock: A 1-mark Write that gives a time in one form and asks for the other Four digits in the 24-hour form (07 05, not 7 05), and a.m. or p.m. in the 12-hour form (9.35 p.m., not 9.35). The 12 o'clock hours: 00 40 is 12.40 a.m. and 12 40 is 12.40 p.m.

Words to know

Duration
How long something lasts, from its start time to its finish time.
12-hour clock
Times counted from 12 round to 12 twice a day, with a.m. or p.m. to show which half: 3.15 p.m.
24-hour clock
Times counted straight through from 00 00 at midnight to 23 59, always with four digits: 15 15.
a.m.
The time from midnight up to noon. Short for the Latin ante meridiem, 'before midday'.
p.m.
The time from noon up to midnight. Short for the Latin post meridiem, 'after midday'.
Midnight
The start of a day: 00 00 in the 24-hour clock and 12.00 a.m. in the 12-hour clock.
Noon
Midday: 12 00 in the 24-hour clock and 12.00 p.m. in the 12-hour clock.
Local time
The time shown on the clocks in a particular place.
Time difference
How far the local time in one place is ahead of, or behind, the local time in another place.
Timetable
A table of the times a bus, train or other service reaches each stop. Each journey has its own column, or in some timetables its own row.
Decimal hours
A time written as a decimal number of hours, e.g. 2.75 hours = 2 h 45 min.
Leap year
A year in which February has 29 days, so the year has 366 days. The syllabus uses 1 year =365 days.
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