C1.15; E1.15 - Time, clocks and timetables

Units of time

A coach leaves at $22\ 45$ and arrives at $00\ 20$ the next day. Is the journey really $22$ hours long? We will solve this by treating time as a continuous journey across midnight, rather than subtracting the printed digits.

First, units matter. There are $60$ seconds in a minute, $60$ minutes in an hour, $24$ hours in a day and $7$ days in a week. There are $12$ months in a year. Use $365$ days for a year in these syllabus calculations; if a problem supplies particular calendar dates, read that information carefully. Months have different lengths, so there is no single exact number of days in a month.

To change to a smaller unit, multiply: $2$ weeks is $2\times7=14$ days. To change to a larger unit, divide: $730$ days is $730\div365=2$ years. Similarly, $18$ months is $1$ year $6$ months. A remainder stays in the smaller unit: $135$ seconds is $2$ minutes $15$ seconds because $135=2\times60+15$.

Convert just the minutes to a fraction of an hour, then add the whole hours. This gives a number you can use in rate calculations later.

Change only the minutes. The conversion divisor stays $60$.

Read the requested unit before writing the answer.

Use the same conversion in this new case.

Twelve-hour and twenty-four-hour clocks

A clock face shows a time within a twelve-hour cycle. The long minute hand completes one turn in $60$ minutes; each numbered step is $5$ minutes. The short hour hand moves gradually, so at twenty minutes past three it has moved one third of the way from $3$ to $4$.

[DIAGRAM: asset_name: Reading two clock faces; asset_slug: reading_clock_faces; caption: The minute hand changes from twenty past to twenty to four. The hour hand also moves.; description: Two exact analogue clock faces labelled Clock A and Clock B. Clock A has its long blue minute hand at 4 and its short grey hour hand one third of the way from 3 to 4. Clock B has its minute hand at 8 and its hour hand two thirds of the way from 3 to 4. A key identifies the long blue hand as minutes and the shorter grey hand as hours.]

A twelve-hour time needs a.m. for midnight up to before noon, or p.m. for noon up to before midnight. A twenty-four-hour time gives hours since midnight and needs neither. Use two hour digits and two minute digits, as in $03\ 20$.

For afternoon/evening hours $1$ to $11$, add $12$ to the hour. For twenty-four-hour hours $13$ to $23$, subtract $12$ and use p.m. Noon and midnight are the exceptions shown above.

Only the minute-hand position changes in Clock B, but the hour hand must move with it.

The hour hand tells you which hour has begun, rather than whichever numeral it is nearest.

Now switch between the two written notations.

Durations and timetables

A duration is an elapsed length of time. A timetable gives clock readings at particular events. Read down one service's column to follow that service; do not take a departure from one service and an arrival from another.

For a duration, count forwards through a convenient whole hour or midnight. Hours and minutes do not behave like the digits of a decimal subtraction. A clock reading after midnight belongs to the following day if the service has crossed midnight.

The two services have the same departure time and arrival minutes. Their arrival hours make a large difference.

Now change only the arrival hour, to $00$ on the next day. The clock passes through midnight.

The same forward-counting method also finds an arrival: add the duration to the departure. For connections, compare the next departure with your arrival and allow any stated transfer time.

Try another journey that crosses midnight.

Time zones and local time

Two local clocks can show different times at the same instant. If City B is three hours ahead of City A, add three hours to A's clock to get B's clock. “Ahead” describes the clocks; it does not make a journey three hours longer.

Use the offset supplied in the problem. You do not need to know a city's real time zone or any seasonal clock changes. For a journey, put departure and arrival on the same city's clock before finding elapsed time.

Keep the route and flight time, but depart six hours later. This time the date changes.

To find an unknown time difference instead, calculate the arrival on the departure city's clock first. Compare that with the stated destination time, keeping the dates attached.

Apply the same idea to a different supplied offset.

Quick Recap

• Convert minutes to hours by dividing by $60$; decimal hours are not clock notation.
• Twenty-four-hour times distinguish morning and afternoon; $00\ 00$ is midnight and $12\ 00$ is noon.
• Follow one timetable service and count forwards to find durations, keeping track of the date.
• Use one local clock for a journey calculation, then apply any supplied time difference.

The coach from $22\ 45$ to $00\ 20$ next day travels for $1$ hour $35$ minutes: midnight resets the clock, not the elapsed time.