Mathematics: Class 5 — Chapter 12: Racing Seconds

12.1 Introduction to Time and Elapsed Time

We begin by exploring daily activities and measuring the time taken to complete them. Understanding how to read clocks and calculate the duration of activities is essential.

Let Us Find Out: Yoga Practice

Raghav practices yoga in the morning. By observing the starting and finishing times on the clocks, we can determine the duration of his practice:

Start Time: Observe the first clock.
Finish Time: Observe the second clock.
Time Elapsed: Calculate the difference between the finish time and the start time.

Calculating Elapsed Time

To find the time elapsed between two given time periods, we calculate the difference:

From 01:15 p.m. to 01:42 p.m. is 27 minutes.
From 03:18 p.m. to 08:18 p.m. is 5 hours.
From 09:15 a.m. to 11:30 a.m. is 2 hours 15 minutes.

Comparing Durations

Consider the time taken by three children to paint a picture:

Raghav: 1 hour 20 minutes
Rani: 2 hours 10 minutes
Ritu: 1 hour 35 minutes

By comparing these durations, we can identify who took the longest time (Rani) and who took the least time (Raghav).

12.2 12-Hour and 24-Hour Time Formats

Time can be represented in two common formats: the 12-hour format (using a.m. and p.m.) and the 24-hour format (using hours from 00:00 to 24:00).

Conversion Rules

For times from 1:00 a.m. to 12:59 p.m., the hour remains the same (except 12 a.m. which becomes 00:00 hours).
For times from 1:00 p.m. to 11:59 p.m., add 12 to the hours to convert to the 24-hour format.

Examples:

$05:30 \text{ a.m.} = 05:30 \text{ hours}$
$08:35 \text{ a.m.} = 08:35 \text{ hours}$
$11:55 \text{ a.m.} = 11:55 \text{ hours}$
$02:30 \text{ p.m.} = 14:30 \text{ hours}$
$05:30 \text{ p.m.} = 17:30 \text{ hours}$
$09:35 \text{ p.m.} = 21:35 \text{ hours}$

12.3 Introducing Seconds

For very short durations, minutes are too large a unit. We introduce a smaller unit of time called seconds.

$$1 \text{ minute} = 60 \text{ seconds}$$

Estimating Seconds vs. Minutes

Different activities take different amounts of time. We can classify them based on whether they take seconds or minutes to complete:

Seconds: Blinking of eyes, switching on/off lights, snapping fingers, sitting down on the floor.
Minutes: Filling a glass from the tap, melting of an ice-cube, making a phone call, drinking a glass of water, washing hands.

Reading Seconds on a Clock

The thin, fast-moving hand on a clock is the seconds hand. It completes one full rotation (60 divisions) in 1 minute.

60 movements of the seconds hand lead to a shift of 1 minute on the minute hand.

12.4 Conversion of Hours, Minutes, and Seconds

We can convert between hours, minutes, and seconds using the following relationships:

To convert hours to minutes, multiply by 60: $$\text{Minutes} = \text{Hours} \times 60$$
To convert minutes to seconds, multiply by 60: $$\text{Seconds} = \text{Minutes} \times 60$$

Examples of Conversions

1 hour 10 minutes: $1 \times 60 + 10 = 70 \text{ minutes}$
2 hours 45 minutes: $2 \times 60 + 45 = 165 \text{ minutes}$
150 minutes: $150 \div 60 = 2 \text{ hours } 30 \text{ minutes}$
320 seconds: $320 \div 60 = 5 \text{ minutes } 20 \text{ seconds}$
4 minutes 8 seconds: $4 \times 60 + 8 = 248 \text{ seconds}$

12.5 Word Problems and Applications

Applying time calculations to real-world scenarios:

Practice Problems

Study Time: Raghav studies 4 subjects, taking 50 minutes for each.
Total time $= 4 \times 50 = 200 \text{ minutes}$.
Converting to hours: $200 \text{ minutes} = 3 \text{ hours } 20 \text{ minutes}$.
Travel Duration: Raghu left at 08:00 hours and arrived at 09:05 hours.
Time taken $= 1 \text{ hour } 5 \text{ minutes}$.
Play Time: Jyoti went to play at 06:15 PM and returned after 1 hour 45 minutes.
Arrival time $= 06:15 \text{ PM} + 1 \text{ hour } 45 \text{ minutes} = 08:00 \text{ PM}$.
Homework Start Time: Ragini finished her homework at 09:40 PM after working for 1 hour 10 minutes.
Start time $= 09:40 \text{ PM} - 1 \text{ hour } 10 \text{ minutes} = 08:30 \text{ PM}$.
Picnic Return: A group left at 08:30 AM and returned after 4 hours 10 minutes.
Return time $= 08:30 \text{ AM} + 4 \text{ hours } 10 \text{ minutes} = 12:40 \text{ PM}$.
Future Time Calculation: If it is 08:35 PM right now, what time will it be after 8 hours and 25 minutes?
Adding hours: $08:35 \text{ PM} + 8 \text{ hours} = 04:35 \text{ AM}$ (next day).
Adding minutes: $04:35 \text{ AM} + 25 \text{ minutes} = 05:00 \text{ AM}$.