Time

Units of Time

Time is measured using a variety of units, starting from the smallest common units:

• Seconds (s) 6 the base unit of time.
• Minutes (min) 6 1 minute = 60 seconds.
• Hours (h) 6 1 hour = 60 minutes = 3600 seconds.

Larger units include:

• Days 6 1 day = 24 hours.
• Weeks 6 1 week = 7 days.
• Months 6 vary between 28, 29, 30, or 31 days depending on the month.
• Years 6 usually 365 days, but a leap year has 366 days.

A leap year occurs every 4 years to keep the calendar year aligned with the Earth's orbit around the Sun. In a leap year, February has 29 days instead of 28.

Converting Time Units

To convert between seconds, minutes, and hours, multiply or divide by 60:

• Seconds to minutes: divide by 60.
• Minutes to seconds: multiply by 60.
• Minutes to hours: divide by 60.
• Hours to minutes: multiply by 60.

Converting between days, weeks, and months:

• Days to weeks: divide by 7.
• Weeks to days: multiply by 7.
• Months vary in length, so conversions often depend on the specific month. Therefore, direct conversion between months and other units is usually avoided in calculations.

For clock time, there are two common formats:

• 12-hour clock uses numbers 1 to 12 with AM (midnight to noon) and PM (noon to midnight).
• 24-hour clock uses numbers 00 to 23 to represent hours from midnight to midnight.

To convert 12-hour time to 24-hour time:

• For AM times, if the hour is 12, convert to 00; otherwise, keep the hour the same.
• For PM times, add 12 to the hour unless it is 12 PM (which stays 12).

To convert 24-hour time to 12-hour time:

• If the hour is 0, it is 12 AM.
• If the hour is between 1 and 11, it is AM.
• If the hour is 12, it is 12 PM.
• If the hour is greater than 12, subtract 12 and add PM.

For instance, 3:45 PM in 12-hour clock is $15:45$ in 24-hour clock because $3 + 12 = 15$.

Time Calculations

Adding and subtracting time requires careful handling of units:

• Add or subtract seconds, then minutes, then hours, converting between units when totals exceed their limits (e.g., 60 seconds = 1 minute).
• When subtracting, if the smaller unit is larger, borrow 1 from the next higher unit.

Calculating durations involves finding the difference between two times, often converting both to the same unit (e.g., minutes or seconds) before subtracting.

Time calculations are commonly used in problem solving, such as finding elapsed time, arrival times, or time intervals.

For example, to add 2 hours 45 minutes and 1 hour 30 minutes:

$2\,\mathrm{h} 45\,\mathrm{min} + 1\,\mathrm{h} 30\,\mathrm{min} = (2 + 1)\,\mathrm{h} + (45 + 30)\,\mathrm{min} = 3\,\mathrm{h} + 75\,\mathrm{min}$

Since $75\,\mathrm{min} = 1\,\mathrm{h} 15\,\mathrm{min}$, the total is $3\,\mathrm{h} + 1\,\mathrm{h} 15\,\mathrm{min} = 4\,\mathrm{h} 15\,\mathrm{min}$.

Time and the Calendar

The number of days in each month varies:

Month	Number of Days
January	31
February	28 (29 in leap years)
March	31
April	30
May	31
June	30
July	31
August	31
September	30
October	31
November	30
December	31

There are 52 weeks in a year (since $52 \times 7 = 364$), with 1 or 2 extra days depending on whether it is a leap year.

AM stands for "ante meridiem" (before midday) and covers the time from midnight to noon.

PM stands for "post meridiem" (after midday) and covers the time from noon to midnight.

Examples

Example: Convert $2$ hours $35$ minutes into minutes.

Since 1 hour = 60 minutes,

$2 \times 60 = 120$ minutes

Total minutes = $120 + 35 = 155\,\mathrm{minutes}$.

For example, adding 30 minutes and 45 minutes gives 75 minutes, which is $1$ hour and $15$ minutes.