Mathematics (0580) | Time

Welcome to the Time Chapter!

Hi there! Time is one of those essential mathematical topics that you use every single day, whether you are checking a bus schedule, planning your homework, or figuring out a flight arrival. In IGCSE Maths, we focus on making sure you are brilliant at converting between different units and confidently using the 12-hour and 24-hour clock systems.

Don't worry if calculations involving hours and minutes sometimes feel tricky—they are different from standard base-10 maths because there are 60 minutes in an hour, not 100! We will look at simple methods to handle these "base-60" conversions smoothly.

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Section 1: Time Units and Essential Conversions (C1.14/E1.14)

To master time calculations, you must be completely fluent with the relationships between the units. These are the foundations of the chapter.

Key Relationships Between Time Units

Memorise these relationships:

• \(1 \text{ minute} = 60 \text{ seconds}\)
• \(1 \text{ hour} = 60 \text{ minutes}\)
• \(1 \text{ day} = 24 \text{ hours}\)
• \(1 \text{ week} = 7 \text{ days}\)
• \(1 \text{ year} = 365 \text{ days}\) (This is the specific value required by the syllabus unless specified otherwise, like in a leap year problem, though 365 is standard for IGCSE).

How to Convert Units

When converting, remember this simple rule:

• To go from a larger unit to a smaller unit, you multiply.
• To go from a smaller unit to a larger unit, you divide.

Example Conversion: Hours to Minutes

Question: Convert 3 hours and 20 minutes into total minutes.

Step 1: Convert the hours to minutes.

\(3 \text{ hours} \times 60 = 180 \text{ minutes}\)

Step 2: Add the remaining minutes.

\(180 \text{ minutes} + 20 \text{ minutes} = 200 \text{ minutes}\)

Did you know? Many students make mistakes converting decimals of an hour. For instance, 1.5 hours is NOT 1 hour 50 minutes. It's 1 hour and \(0.5 \times 60 = 30\) minutes, so 1 hour 30 minutes.

Common Mistake to Avoid!

When calculating time duration, do not treat minutes like decimal numbers unless you convert them first. For example, if you subtract 30 minutes from 1 hour, the result is 30 minutes, not 70 minutes (like \(100 - 30\)).

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Section 2: The 12-Hour and 24-Hour Clock (C1.14/E1.14)

In mathematics and travel, the 24-hour clock is the standard. It uses four digits (HH MM) and removes the ambiguity of "a.m." and "p.m."

Understanding the Notation

The 24-hour clock runs from 00 00 (midnight) to 23 59 (one minute before midnight).

• Time is written as HH MM (Hours and Minutes).
• The syllabus specifies notation without a colon, e.g., 03 15.

Converting from 12-Hour to 24-Hour

This conversion relies on whether the time is before noon (a.m.) or after noon (p.m.).

1. A.M. Times (Morning)

For times between midnight (12:00 a.m.) and noon (12:00 p.m.):

• The hour stays the same, but always use two digits for the hour (add a leading zero if needed).
• Example 1: 3:15 a.m. is 03 15.
• Example 2: 10:45 a.m. is 10 45.

Special Case: Midnight (12 a.m.)

• 12:00 a.m. (Midnight) is 00 00.
• 12:35 a.m. is 00 35.

2. P.M. Times (Afternoon/Evening)

For times from 1:00 p.m. onwards, you add 12 to the hour:

$$\text{24-Hour Time} = \text{12-Hour PM Hour} + 12$$

• Example 1: 4:00 p.m. is \(4 + 12 = 16\). So, 16 00.
• Example 2: 9:55 p.m. is \(9 + 12 = 21\). So, 21 55.

Special Case: Noon (12 p.m.)

• 12:00 p.m. (Noon) is 12 00.

Converting from 24-Hour to 12-Hour

If the hour is 13 or greater, subtract 12 to find the p.m. time.

• Example 1: 17 30. \(17 - 12 = 5\). This is 5:30 p.m.
• Example 2: 06 10. This is 6:10 a.m. (since 06 is less than 12).

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Section 3: Calculating Time Differences (Durations)

Calculating the time elapsed between two points is a frequent exam question. The easiest method is usually the "Bridging Method", especially when you are not allowed a calculator (Paper 1).

The Bridging Method (Using the Next Full Hour)

We break the calculation into simple steps, using the next full hour as a milestone.

Example Duration Calculation

Question: How long is the duration from 07 48 to 11 15?

Step 1: Bridge to the next full hour (08 00).

From 07 48 to 08 00: \(60 - 48 = 12\) minutes.

Step 2: Calculate the full hours between the milestones.

From 08 00 to 11 00: \(11 - 8 = 3\) hours.

Step 3: Calculate the remaining minutes.

From 11 00 to 11 15: 15 minutes.

Step 4: Add up the durations.

Total duration = 3 hours + 12 minutes + 15 minutes = 3 hours 27 minutes.

Working Backwards: Finding the Start or End Time

If you need to find a time that is a certain duration before or after a given time, use the same bridging method, but be careful when 'borrowing' hours.

Example: Finding the Start Time

Question: A journey took 4 hours 40 minutes and ended at 18 25. What was the start time?

We need to subtract 4 hours 40 minutes from 18 25.

Step 1: Subtract the full hours first.

\(18 25 - 4 \text{ hours} = 14 25\).

Step 2: Subtract the remaining minutes (40 minutes) using the bridge.

• Take 25 minutes back to the hour mark: \(14 25 - 25 \text{ minutes} = 14 00\). (Remaining minutes to subtract: \(40 - 25 = 15 \text{ minutes}\)).
• Subtract the final 15 minutes from 14 00: \(14 00 - 15 \text{ minutes} = 13 45\).

The start time was 13 45.

Struggling Student Tip: Vertical Subtraction (The 'Borrow 60' Rule)

If you prefer vertical subtraction, remember that when you 'borrow' from the hours column, you add 60 (not 100) to the minutes column.

We need to calculate \(18 \text{ hours } 25 \text{ minutes} - 4 \text{ hours } 40 \text{ minutes}\).

$$ \begin{array}{rcl} \text{Hours} & & \text{Minutes} \\ \end{array} $$

Since 25 is less than 40, borrow 1 hour (60 minutes) from the 18 hours:

$$ \begin{array}{rcl} \text{Hours} & & \text{Minutes} \\ 17 & & (25 + 60 = 85) \\ -4 & & 40 \\ \hline 13 & & 45 \\ \end{array} $$

Result: 13 hours 45 minutes (13 45).

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Section 4: Real-World Time Problems (Timetables and Time Zones)

Reading Timetables

Timetables require you to read information accurately and then calculate duration (as shown in Section 3). Pay close attention to whether the times given are 12-hour (a.m./p.m.) or 24-hour.

Example: Reading a Timetable

A train leaves Station A at 19 25 and arrives at Station B at 23 05.

Question: Calculate the total journey time.

• 19 25 to 20 00: \(60 - 25 = 35\) minutes.
• 20 00 to 23 00: \(23 - 20 = 3\) hours.
• 23 00 to 23 05: 5 minutes.
• Total: 3 hours + 35 minutes + 5 minutes = 3 hours 40 minutes.

Time Zones and Local Times

When solving time zone problems, you deal with time differences. The syllabus specifically includes these types of problems.

The difference is given as a positive or negative offset (e.g., +3 hours means 3 hours ahead of your starting point).

The East/West Rule (Memory Aid)

• If you travel East (e.g., London to Dubai), the time gets later. You Add (+) the time difference.
• If you travel West (e.g., London to New York), the time gets earlier. You Subtract (-) the time difference.

Step-by-Step: Time Zone Calculation

Question: London local time is 11 00 on Monday. Tokyo is 9 hours ahead (\(+9\)). What is the local time and day in Tokyo?

Step 1: Calculate the time difference.

\(11 00 + 9 \text{ hours} = 20 00\).

The time in Tokyo is 20 00 Monday.

Example: Crossing Midnight (Changing Days)

Question: A flight leaves New York (local time 08 30) and flies to London. London is 5 hours ahead (\(+5\)). The flight takes 7 hours. What is the arrival time in London?

Step 1: Find the departure time in London local time.

New York departure: 08 30.

London is +5 hours: \(08 30 + 5 \text{ hours} = 13 30\). (London departure time is 13 30).

Step 2: Add the flight duration (7 hours).

\(13 30 + 7 \text{ hours} = 20 30\).

The arrival time in London is 20 30 on the same day.

Did you know? If the journey was longer, say 15 hours, you would need to cross the 24 00 boundary:

• \(13 30 + 15 \text{ hours} = 28 30\).
• To convert 28 30 back into 24-hour time, subtract 24 hours: \(28 30 - 24 00 = 04 30\).
• Since you crossed 24 00, the day advances. The arrival time would be 04 30 the next day.