🕰️ Study Notes: Chapter 1.14 Time (Number Section)
Welcome to the chapter on Time! This topic is incredibly practical—you use these skills every day, whether you are reading a train schedule, calculating how long a movie lasts, or planning a trip across time zones.
Mastering time calculations involves understanding units, converting between clock systems (12-hour and 24-hour), and dealing with time differences and time zones.
1. Understanding and Calculating with Time Units
When we calculate with time, we need to be fluent in converting between the common units. Unlike our decimal number system (base 10), time often uses base 60 (for minutes and seconds) or other factors (like 24 for hours in a day).
Key Relationships (The Conversion Factors):
- 1 minute = 60 seconds (s)
- 1 hour = 60 minutes (min)
- 1 day = 24 hours (h)
- 1 week = 7 days
- 1 year = 365 days (Remember this exact figure for the exam unless otherwise specified, ignoring leap years!)
Converting Minutes/Seconds to Decimals (and Vice Versa)
This is where students often struggle. You must remember that time works on base 60, not base 100.
💡 Step-by-Step Conversion:
- To change minutes into a decimal fraction of an hour, you must divide the minutes by 60.
- To change a decimal fraction of an hour back into minutes, you must multiply the decimal by 60.
Example: Convert 45 minutes into hours.
Calculation: \(45 \div 60 = 0.75\)
Result: 45 minutes is 0.75 hours.
Example: Convert 3.25 hours into hours and minutes.
The whole number is 3 hours.
The decimal part is 0.25 hours.
Calculation: \(0.25 \times 60 = 15\)
Result: 3.25 hours is 3 hours 15 minutes.
🛑 Common Mistake Alert!
Do NOT assume 30 minutes is 0.3 hours. 30 minutes is \(30 \div 60 = 0.5\) hours.
Quick Review: Time Units
The most important skill is confidently converting between minutes and decimal hours using the 60 factor.
2. The 12-Hour and 24-Hour Clocks
We use two clock systems. You must be able to convert between them accurately, especially when reading timetables.
The 12-Hour Clock (AM/PM)
- Uses the labels AM (Ante Meridiem, before midday) and PM (Post Meridiem, after midday).
- The hours run from 1 to 12.
- Example: 3.15 a.m. (3:15 in the morning).
The 24-Hour Clock
- Uses four digits in the format HH MM (Hour Hour Minute Minute).
- Hours run from 00 00 (midnight) to 23 59 (one minute before midnight).
- No AM or PM labels are used.
- Midnight is 00 00. Midday (noon) is 12 00.
- Example: 3.15 p.m. is denoted by 15 15.
Converting from 12-Hour to 24-Hour Clock (Step-by-Step)
1. Morning (AM) Times:
If the time is between 12:01 AM and 12:59 PM, use 00 for the hour.
If the time is between 1:00 AM and 11:59 AM, keep the hour the same, but ensure it has two digits (0X).
- 12:30 AM (after midnight) becomes 00 30.
- 9:20 AM becomes 09 20.
2. Afternoon/Evening (PM) Times:
If the time is between 1:00 PM and 11:59 PM, add 12 to the hour.
- 1:45 PM becomes \(1 + 12 = 13\), so 13 45.
- 8:00 PM becomes \(8 + 12 = 20\), so 20 00.
Exception: 12:00 PM (Midday) is simply 12 00.
Key Takeaway
The 24-hour clock simplifies calculating time differences because you don't have to worry about crossing the midday line (AM to PM).
3. Calculating Time Differences (Duration)
Finding the time elapsed (the duration) between a start time and an end time requires careful calculation, especially if the event spans multiple hours or days.
Method: The "Bridging" Technique
This is often the easiest method for non-calculator papers. Break the calculation into logical, easy-to-handle steps, usually bridging to the next whole hour.
Example: Find the duration from 07 50 to 11 15.
- Step 1: Bridge to the next full hour.
From 07 50 to 08 00: 10 minutes. - Step 2: Calculate the full hours.
From 08 00 to 11 00: 3 hours. - Step 3: Calculate the remaining minutes.
From 11 00 to 11 15: 15 minutes. - Step 4: Add them up.
Total time = 3 hours + 10 minutes + 15 minutes = 3 hours 25 minutes.
Did you know? This technique is known as the "subtraction by complements" method in some places, but you don't need the fancy name—just know the steps!
Dealing with Time Calculations in Decimal Form
Sometimes you need the final answer as a decimal hour for further rate calculations (like speed or pay rates). In this case, convert the total duration (3 hours 25 minutes) back to hours:
Hours: \(3\)
Minutes in decimal: \(25 \div 60 \approx 0.4167\)
Total time: \(3.4167\) hours.
You can then use this decimal value in calculations for average speed or hourly wages (C1.11).
4. Time Zones, Local Times, and Differences
Time zone problems combine duration calculation with addition/subtraction across zones.
Understanding Time Zones
Time zones are defined by their offset from UTC (Coordinated Universal Time, often related to GMT, Greenwich Mean Time).
If a city is UTC +3, it means the local time there is 3 hours later than UTC. If it is UTC -5, the time is 5 hours earlier than UTC.
✈️ Analogy: The Time Travel Line
Imagine a straight line representing UTC. If you travel East (e.g., London to Dubai), you move towards the future, so you add hours. If you travel West (e.g., London to New York), you move towards the past, so you subtract hours.
Calculating Local Time
The calculation depends on whether the required location is ahead of or behind the starting location.
Case 1: Calculating Time Ahead
London (UTC 0) is 10 00 on Tuesday. What time is it in Sydney (UTC +10)?
Sydney is 10 hours ahead.
\(10 \ 00 + 10 \text{ hours} = 20 \ 00\).
Local time in Sydney is 20 00 on Tuesday.
Case 2: Crossing the Day Boundary
London (UTC 0) is 23 00 on Tuesday. What time is it in Tokyo (UTC +9)?
Tokyo is 9 hours ahead.
\(23 \ 00 + 9 \text{ hours} = 32 \ 00\). (Impossible time! Subtract 24 hours.)
\(32 \ 00 - 24 \ 00 = 08 \ 00\).
Because we crossed 24 00, the day moves forward.
Local time in Tokyo is 08 00 on Wednesday.
Key Strategy for Time Zone Problems:
When solving complex flight or journey problems involving time zones, you must separate the journey time (duration) from the time zone shift (local time change).
- Find the time zone difference (add/subtract).
- Calculate the time you arrive locally (using the time zone difference).
- Subtract the duration of the flight to find the departure time in the destination's local time (or vice versa).
5. Reading Clocks and Timetables
The syllabus requires you to read and interpret clocks and timetables. This is a skill of careful reading and interpretation.
Reading Clocks
Ensure you can read both analogue (hands) and digital clocks accurately, and instantly convert any analogue time into 24-hour digital format.
Interpreting Timetables
Timetables (for trains, buses, or flights) provide columns of times for different stages of a journey. Often, the times given are local to the station/airport listed.
Tip for Tables:
- Always check the header row/column to see if the times are Departure (Dep) or Arrival (Arr).
- If the table uses 24-hour notation, 13 45 means 1:45 PM.
- Be aware of times that cross midnight (sometimes indicated with a small plus sign or by a time earlier than the departure time, meaning it arrives the next day).
Example Timetable Question:
A train leaves City A at 08 20 and arrives at City B at 10 10. The journey includes a 15-minute stop.
1. Total time elapsed (08 20 to 10 10):
(08 20 to 09 00 = 40 min; 09 00 to 10 00 = 1 h; 10 00 to 10 10 = 10 min).
Total elapsed time = 1 hour 50 minutes.
2. Actual travel time:
1 hour 50 minutes (total) - 15 minutes (stop) = 1 hour 35 minutes.
Final Key Takeaway: Time problems in maths test your ability to handle non-decimal conversions and logical sequencing of events (start time, duration, end time, and location change).