Time

1. Overview

Time is a fundamental concept in IGCSE Mathematics, essential for calculations involving durations, schedules, and rates. This revision note covers time units, conversions between them, and calculations using both 12-hour and 24-hour clock formats. Mastering these skills is crucial for success in the IGCSE exam and for real-world applications.

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Key Definitions

• 12-hour Clock: A time system that divides the 24 hours of a day into two periods: a.m. (ante meridiem - before noon) and p.m. (post meridiem - after noon).
• 24-hour Clock: A time system where the day runs from midnight (00:00) to 23:59.
• Time Interval/Duration: The amount of time that passes between a start time and an end time.
• Leap Year: A year occurring every four years that has 366 days (February has 29 days instead of 28).

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Core Content

A. Units of Time and Conversions

Time does not use a base-10 system. You must memorize these conversions:

• 1 minute (min) = 60 seconds (s)
• 1 hour (h) = 60 minutes = 3600 seconds ($60 \times 60$)
• 1 day = 24 hours
• 1 week = 7 days
• 1 year = 12 months = 365 days (366 in a leap year), which is about 52 weeks

Worked example 1 — Converting hours and minutes to seconds

Question: Convert 3 hours and 22 minutes into seconds.

Step	Working	Explanation
1	$3 \times 3600 = 10800 \text{ s}$	Convert the hours to seconds ($1\text{ h} = 3600\text{ s}$)
2	$22 \times 60 = 1320 \text{ s}$	Convert the minutes to seconds ($1\text{ min} = 60\text{ s}$)
3	$10800 + 1320 = 12120 \text{ s}$	Add the results together
4		Answer: 12120 seconds

B. The 12-hour and 24-hour Clock

• To convert p.m. to 24-hour time: Add 12 to the hours (e.g., 3:00 p.m. $\rightarrow 3 + 12 = 15:00$).
• To convert 24-hour time (from 13:00 onwards) to p.m.: Subtract 12 from the hours.
• Note: 12:00 a.m. is 00:00; 12:00 p.m. is 12:00.

C. Calculating Time Intervals

When calculating how long a journey takes, it is often safer to "bridge" to the next hour rather than using column subtraction.

Worked example 2 — Calculating duration across days

Question: A bus departs at 22:35 and arrives at 03:20 the following day. Calculate the duration of the journey.

Step	Working	Explanation
1	$22:35 \rightarrow 23:00 = 25 \text{ mins}$	Calculate minutes to the next full hour
2	$23:00 \rightarrow 00:00 = 1 \text{ hour}$	Calculate hours to midnight
3	$00:00 \rightarrow 03:20 = 3 \text{ hours } 20 \text{ mins}$	Calculate time from midnight to arrival
4	$25\text{m} + 1\text{h} + 3\text{h } 20\text{m} = 4\text{h } 45\text{m}$	Sum the intervals
5		Answer: 4 hours 45 minutes

D. Reading Timetables

Timetables are read vertically (for one journey) and horizontally (to compare different transport times).

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Extended Content (Extended Only)

In the Extended curriculum, you are often required to convert between fractional/decimal time and hours/minutes to use in Speed, Distance, and Time formulas ($Speed = \frac{Distance}{Time}$). You also need to be comfortable with more complex time interval calculations, including those involving multiple days or time zones.

Worked example 3 — Decimal Time to Hours and Minutes

Question: A program runs for 2.75 hours. Express this duration in hours and minutes.

Step	Working	Explanation
1	$2 \text{ hours}$	The whole number represents the hours
2	$0.75 \times 60 = 45 \text{ minutes}$	Multiply the decimal part by 60 to find minutes
3		Answer: 2 hours 45 minutes

Worked example 4 — Minutes to Decimal Time

Question: Convert 5 hours 12 minutes into hours (decimal).

Step	Working	Explanation
1	$12 \div 60 = 0.2$	Divide the minutes by 60
2	$5 + 0.2 = 5.2 \text{ hours}$	Add to the whole hours
3		Answer: 5.2 hours

Worked example 5 — Calculating duration across multiple days

Question: A ship departs port at 14:40 on Monday and arrives at its destination at 09:15 on Wednesday. What is the total duration of the voyage?

Step	Working	Explanation
1	$14:40 \text{ Monday} \rightarrow 14:40 \text{ Tuesday} = 24 \text{ hours}$	Calculate the time to the same time the next day
2	$14:40 \text{ Tuesday} \rightarrow 14:40 \text{ Wednesday} = 24 \text{ hours}$	Calculate the time to the same time the next day
3	$14:40 \text{ Wednesday} \rightarrow 09:15 \text{ Wednesday} = -5 \text{ hours } 25 \text{ mins}$	Calculate the difference between 14:40 and 09:15. Note that since we are going backwards in time, this is negative.
4	$24 \text{ hours} + 24 \text{ hours} - 5 \text{ hours } 25 \text{ mins} = 42 \text{ hours } 35 \text{ mins}$	Sum the intervals
5		Answer: 42 hours 35 minutes

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Key Equations

• Time Conversion (Seconds): $\bf{T_{(s)} = T_{(h)} \times 3600}$
• Time Conversion (Minutes): $\bf{T_{(min)} = T_{(h)} \times 60}$
• Time Duration: $\bf{\text{Duration} = \text{End Time} - \text{Start Time}}$
• Decimal Time: $\bf{\text{Hours} + \frac{\text{Minutes}}{60}}$

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Common Mistakes to Avoid

• ❌ Wrong: Calculating 1 hour 45 minutes as 1.45 hours.
  • ✓ Right: 1 hour 45 minutes is $1 + \frac{45}{60} = 1.75$ hours.
• ❌ Wrong: Converting 150 seconds to hours by dividing by 60.
  • ✓ Right: Divide by 3600: $150 \div 3600 = 0.0417$ hours (approximately).
• ❌ Wrong: Subtracting times like $16:15 - 15:50 = 1: -35$.
  • ✓ Right: Borrow an hour (60 minutes) to make the subtraction work: $16:15 - 15:50 = 0:25$ (25 minutes). Or use the bridging method.
• ❌ Wrong: Subtracting the times for a journey that departs at 23:00 and arrives at 01:00 and getting 22 hours.
  • ✓ Right: Recognize the journey crosses midnight and add the segments: 1 hour (23:00 to 00:00) + 1 hour (00:00 to 01:00) = 2 hours.

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Exam Tips

• Command Words: "Calculate" or "Find the duration" are common. Always show the "bridging" steps to secure method marks even if your final addition is wrong.
• Calculator Tip: Most scientific calculators have a "DMS" button (marked as $^\circ \ ' \ ''$). You can use this to input hours and minutes directly. For example, pressing 2 [,] 30 [,] represents 2 hours 30 minutes.
• Real-world Context: Be prepared for questions involving "Time Zones." If a plane leaves London (GMT) and flies to a country at GMT+4, the local arrival time is 4 hours later than the London time on arrival; the length of the flight does not change.
• Values to Memorize:
  • 30 days: September, April, June, November.
  • 31 days: January, March, May, July, August, October, December.
  • 28/29 days: February.
• Final Check: Does your answer make sense? If a flight from London to New York says "25 hours," re-read the question for time zone differences or errors in your subtraction. Also, double-check that you've provided the answer in the units requested by the question (e.g., hours, minutes, or seconds).