Understanding Time Calculation Over Long Periods

This question asks us to determine the time after a significant number of hours have passed from a given starting time. The key to solving this type of problem is understanding that time on a clock repeats every 24 hours. This means that after 24 hours, the time will be the same as the starting time. The same applies after 48 hours (2 x 24), 72 hours (3 x 24), and so on.

Calculating Time Elapsed in Full Days

To find the time after 101 hours, we need to figure out how many full 24-hour cycles (full days) are contained within 101 hours and what the remaining hours are. The remaining hours are what will determine the final time relative to the starting time.

We can find this by dividing the total number of hours (101) by the number of hours in a day (24). The quotient will tell us how many full days have passed, and the remainder will tell us how many extra hours have passed beyond the full days.

Let's perform the division:

\( \frac{101}{24} \)

We can find the largest multiple of 24 that is less than or equal to 101.

• \( 24 \times 1 = 24 \)
• \( 24 \times 2 = 48 \)
• \( 24 \times 3 = 72 \)
• \( 24 \times 4 = 96 \)
• \( 24 \times 5 = 120 \)

The largest multiple of 24 less than 101 is 96, which is \( 24 \times 4 \). This means 4 full days have passed.

Determining the Remaining Hours

To find the remainder, we subtract the total hours in the full days (96) from the total elapsed hours (101):

\( 101 - 96 = 5 \)

The remainder is 5 hours. This means that after 4 full days (which brings the time back to the starting time of 4 o'clock), an additional 5 hours will pass.

Finding the Final Time

The current time is 4 o'clock. After 101 hours, which is equivalent to 4 full days and 5 additional hours, the time will be 5 hours after 4 o'clock.

Starting time: 4 o'clock

Hours to add: 5 hours

Final time calculation: \( 4 + 5 = 9 \)

The time after 101 hours will be 9 o'clock.

Summary of Time Calculation Steps

Here is a breakdown of the steps taken to solve this time problem:

1. Identify the current time. (4 o'clock)
2. Identify the total number of hours elapsed. (101 hours)
3. Divide the total elapsed hours by 24 (hours in a day) to find the remainder. \( 101 \div 24 = 4 \) with a remainder of \( 5 \).
4. The remainder represents the number of hours to add to the starting time after accounting for full 24-hour cycles. (5 hours)
5. Add the remainder hours to the current time to find the final time. \( 4 + 5 = 9 \).

Revision Table: Time Calculation Concepts

Additional Information: Applying Remainder in Time Problems

The technique of using the remainder is very useful for any problem involving time that repeats in cycles, like days of the week (7-day cycle) or months of the year (12-month cycle). For time on a clock, the cycle is 24 hours.

For example, if the time is 7 o'clock and you want to know the time after 50 hours:

• Divide 50 by 24: \( 50 \div 24 \). \( 24 \times 2 = 48 \).
• Remainder: \( 50 - 48 = 2 \).
• Add the remainder (2 hours) to the current time (7 o'clock): \( 7 + 2 = 9 \).
• The time after 50 hours would be 9 o'clock.

This remainder method simplifies calculating the time after any number of hours by effectively ignoring the full 24-hour periods, as they do not change the clock position.