problem solving time intervals

Time Interval Word Problems

Concept Development Look back at your work on Application Problem. We know that Lilly finished after Patrick. Let’s use a number line to figure out how many more minutes than Patrick Lilly took to finish.

Label the first tick mark 0 and the last tick mark 60. Label the hours and 5-minute intervals. T: Plot the times 5:31 p.m. and 5:43 p.m. Find the difference between Patrick and Lilly’s times.

How many more minutes than Patrick did it take Lilly to finish her chores? 12 minutes more.

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Time Word Problems Worksheets

Time riddles (harder).

Welcome to our Time Word Problems Worksheets page. Here you find our selection of more challenging Time Riddles to help your child learn to read, record and solve problems involving time and clocks.

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Looking for some 24 hour clock problem sheets? Do you need some challenging time problems for older children? Then hopefully you have found the right place!

The printable time sheets in this section involve being able to tell the time to the nearest minute, as well as converting times between the 12 and 24 clock.

These worksheets are great to use when your child is confident telling the time and needs to extend their knowledge by solving time problems.

The sheets are also very good at developing an understanding of mathematical language associated with time.

Each sheet has 2 different Time Riddles with 8 possible solutions.

The aim of each puzzle is to use the clues to work out the correct solution.

Using these sheets will help your child to:

• read times to the nearest minute;
• convert analogue to digital times;
• use 'past' and 'to' language correctly to tell the time;
• convert times between the 12 and 24 hour clock;
• solve problems involving time;

Our time word problems worksheets will help you practice applying your time skills and knowledge to solve problems.

Time Word Problems Worksheets : to the nearest minute

• Time Riddles 4a
• PDF version
• Time Riddles 4b
• Time Riddles 4c

Time Word Problems Worksheets: 24 hour clock

The printable time worksheets in this section involve converting times between the 12 and 24 hour clock.

• 24 Hour Time Riddles Sheet 1
• 24 Hour Time Riddles Sheet 2
• 24 Hour Time Riddles Sheet 3

Extension Activity Ideas

If you are looking for a way to extend learning with these Time Riddles, why not...

• Get children to work in partners with one child choosing one of the 8 possible times and the other child asking 'yes/no' questions. E.g. Is it earlier than...? Is the minute hand on number 6...? Is it later than 4pm? ... etc
• Children could write their own set of clues down to identify one of the clocks.

Looking for something easier?

Here you will find our selection of easier Time Riddles.

The sheets in this section are similar to those on this page, but involve telling the times: o'clock and half-past, quarter past and quarter to, and to 5 minute intervals.

• Time Riddles (easier)

Need help telling the time?

Here you will find our selection of telling time clock worksheets to help your child to learn their o'clock, half-past, quarter past and to, and 5 minute intervals.

• read o'clock, half-past times;
• read quarter past and quarter to;
• read time going up in 5 minute intervals;
• convert analogue times to digital;

The sheets in this section are a great way of starting your child off with learning to tell the time.

• Telling Time Worksheets o'clock and half past
• Clock Worksheets - Quarter Past and Quarter To
• Telling Time 5 minute intervals
• Telling Time Worksheets Grade 4 (1 minute intervals)

Time Interval Worksheets

These sheets will help you learn to add and subtract hours and minutes from times as well as working out a range of time intervals.

There are also sheets to help you practice adding and subtracting time intervals.

• Add and Subtract Time Worksheets
• Elapsed Time Worksheets
• Online Age Calculator

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Course: 4th grade   >   Unit 14

• Converting units of time
• Convert to smaller units (sec, min, & hr)
• Time word problem: travel time

Time word problem: Susan's break

• Time conversion word problems
• Converting units of time review (seconds, minutes, & hours)

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Video transcript

Time and Clock Worksheets

Welcome to the time worksheets page at Math-Drills.com where taking your time is encouraged! On this page you will find Time math worksheets including elapsed time, telling time on analog clocks, calendars and converting time worksheets.

The calendars come in two different formats: yearly (all on one page) and monthly when you need extra space or a larger layout. The calendars are very useful in conjunction with the elapsed time worksheets with days, weeks, months, and years. Students who have difficulty visualizing a calendar may need the actual calendars to use as a reference. Telling time on analog clocks is still an important skill despite the number of digital clocks around; many people still choose the analog clock design for aesthetic purposes.

Most Popular Time and Clock Worksheets this Week

The calendars on this page are meant for anyone to use for purposes including personal schedules, classroom planning, holiday calendars, business meetings, event calendars, or anything else. They can also be used in math activities such as elapsed date activities. A simple activity with the monthly calendars is to ask students to place an item or mark on specific spots on the calendar (e.g. "Place a bean on a Tuesday in March."). After students become familiar with how a calendar is laid out and works, you can create more challenging activities like finding elapsed dates, discovering the number of days in each month, scheduling activities, etc.

Yearly calendars are a great way to see an entire year on one page. Although not easy to write on all your appointments and anniversaries, they are a quick reference and can be very useful when completing math activities to familiarize students with calendars or more advanced activities with calendars.

Specific yearly calendars for the years 2000 to 2050 are available in this section. For any other year or if you want a custom title, you can use the general yearly calendars with fillable titles. There are seven general yearly calendars and seven general leap year calendars in two different formats (Sunday to Saturday and Monday to Sunday) which will cover any year from 1583 on.

• Yearly Calendars (Sunday to Saturday format) Yearly Calendars for Specific Years from 2000 to 2050 (Fillable Title) General Yearly Calendars ✎ (Fillable Title) General Leap Year Calendars ✎
• Yearly Calendars (Monday to Sunday Format) Yearly Calendars for Specific Years from 2000 to 2050 (Monday to Sunday Format) (Fillable Title) General Yearly Calendars (Monday to Sunday Format) ✎ (Fillable Title) General Leap Year Calendars (Monday to Sunday Format) ✎

A great number of math activities can be accomplished with monthly calendars. Familiarization activities include finding specific dates, determining which day of the week it is, marking important events on the calendar, and determining the number of days in each month, week or year. Further activities mainly include elapsed date activities where students find the number of days, weeks and/or months between two dates or find a date a certain number of months, weeks and/or days in the future or the past. Of course, these calendars can also be used as normal reference calendars by anyone.

Originally, Math-Drills calendars always started on Sundays, but there are many people in the world who use calendars starting on Mondays. A good argument can be made by thinking of the word, "weekend." The end of the week or week end is Saturday and Sunday, so why would you put Sunday at the beginning of the week? Luckily, both options exist, so pick the one that suits you the best.

Fillable means that you can type whatever you like into each date. It is possible to add up to seven short lines of text. This is useful if you want to write important dates onto the calendar or create activities for students (e.g. what date is 78 days from today?).

• Fillable Monthly Calendars (Sunday to Saturday Format) Fillable Monthly Calendars for Specific Years from 2023 to 2050 ✎ Fillable General Monthly Calendars ✎ Fillable General Leap Year Monthly Calendars ✎
• Fillable Monthly Calendars (Monday to Sunday Format) Fillable Monthly Calendars from 2023 to 2050 (Monday to Sunday Format) ✎ Fillable General Monthly Calendars (Monday to Sunday Format) ✎ Fillable General Leap Year Monthly Calendars (Monday to Sunday Format) ✎
• Retro (old versions) Monthly Calendars Retro Monthly Calendars for Specific Years from 2000 to 2050 Retro General Monthly Calendars Retro General Leap Year Monthly Calendars Retro Monthly Calendars for Specific Years from 2000 to 2050 (Monday to Sunday Format)

Reading and Sketching Time on Analog Clocks

Even though the time is displayed digitally in so many places these days—on cell phones, on computers, on microwaves—there are still quite a few analog clocks around. Besides being able to tell time on an analog clock, this is probably one of the first places that students encounter a number system other than base ten. Thanks to the Babylonians et. al. we have 60 seconds in a minute and 60 minutes in an hour. Once your students master the intricacies of the time system, they can start learning about other useful number systems like hexadecimal and binary, both of which are heavily used in computer programming.

Suns and moons are included with each clock to indicate the time of day. Think of the moon as midnight and the sun as noon. If the clock has a moon (midnight) on the left and a sun (noon) on the right, then the time is between midnight and noon (AM in North America). The reverse means that the time is between noon and midnight (PM in North America).

These clock worksheets include hour and minute hands, so students who are starting to learn reading time on analog clocks only have to worry about two arms. There are a variety of intervals available depending on the level of the student. The goal is to get students to be able to tell time to the minute. There are versions with twelve clocks and versions with four large clocks.

• Reading 12 Hour Time from Clocks with Minute Hands (12 Clocks per Page) Reading 12 Hour Time in One Hour Intervals (12 Clocks) Reading 12 Hour Time in 30 Minute Intervals (12 Clocks) Reading 12 Hour Time in 15 Minute Intervals (12 Clocks) Reading 12 Hour Time in 5 Minute Intervals (12 Clocks) Reading 12 Hour Time in 1 Minute Intervals (12 Clocks)
• Reading 12 Hour Time from Clocks with Minute Hands (4 Clocks per Page) Reading 12 Hour Time in One Hour Intervals ( 4 Large Clocks ) Reading 12 Hour Time in 30 Minute Intervals ( 4 Large Clocks ) Reading 12 Hour Time in 15 Minute Intervals ( 4 Large Clocks ) Reading 12 Hour Time in 5 Minute Intervals ( 4 Large Clocks ) Reading 12 Hour Time in 1 Minute Intervals ( 4 Large Clocks )

These worksheets also include second hands in various intervals. These are best to use after students have mastered reading time to the minute. The addition of a third hand challenges them more and helps them understand how many seconds are in a minute.

• Reading 12 Hour Time from Clocks with Second Hands (12 Clocks per Page) Reading 12 Hour Time in 30 Second Intervals (12 Clocks) Reading 12 Hour Time in 15 Second Intervals (12 Clocks) Reading 12 Hour Time in 5 Second Intervals (12 Clocks) Reading 12 Hour Time in 1 Second Intervals (12 Clocks)
• Reading 12 Hour Time from Clocks with Second Hands (4 Clocks per Page) Reading 12 Hour Time in 30 Second Intervals ( 4 Large Clocks ) Reading 12 Hour Time in 15 Second Intervals ( 4 Large Clocks ) Reading 12 Hour Time in 5 Second Intervals ( 4 Large Clocks ) Reading 12 Hour Time in 1 Second Intervals ( 4 Large Clocks )

Students encounter 24 hour time in various places such as on plane tickets, in computer programming and in literature. These worksheets are similar to reading 12 hour clocks, but include a second set of numbers on the inside of the minute ticks.

• Reading 24 Hour Time from Clocks with Minute Hands (12 Clocks per Page) Reading 24 Hour Time in One Hour Intervals (12 Clocks) Reading 24 Hour Time in 30 Minute Intervals (12 Clocks) Reading 24 Hour Time in 15 Minute Intervals (12 Clocks) Reading 24 Hour Time in 5 Minute Intervals (12 Clocks) Reading 24 Hour Time in 1 Minute Intervals (12 Clocks)
• Reading 24 Hour Time from Clocks with Minute Hands (4 Clocks per Page) Reading 24 Hour Time in One Hour Intervals ( 4 Large Clocks ) Reading 24 Hour Time in 30 Minute Intervals ( 4 Large Clocks ) Reading 24 Hour Time in 15 Minute Intervals ( 4 Large Clocks ) Reading 24 Hour Time in 5 Minute Intervals ( 4 Large Clocks ) Reading 24 Hour Time in 1 Minute Intervals ( 4 Large Clocks )

These 24 hour clocks also include second hands, so students can read time represented on clocks to the nearest second.

• Reading 24 Hour Time from Clocks with Second Hands (12 Clocks per Page) Reading 24 Hour Time in 30 Second Intervals (12 Clocks) Reading 24 Hour Time in 15 Second Intervals (12 Clocks) Reading 24 Hour Time in 5 Second Intervals (12 Clocks) Reading 24 Hour Time in 1 Second Intervals (12 Clocks)
• Reading 24 Hour Time from Clocks with Second Hands (4 Clocks per Page) Reading 24 Hour Time in 30 Second Intervals ( 4 Large Clocks ) Reading 24 Hour Time in 15 Second Intervals ( 4 Large Clocks ) Reading 24 Hour Time in 5 Second Intervals ( 4 Large Clocks ) Reading 24 Hour Time in 1 Second Intervals ( 4 Large Clocks )

Once students are able to read time off of analog clocks, they can be challenged to sketch time. This might seem easy, but analog clocks tend to have a small quirk in that the hands don't always point to the exact number. For example, if it is 6:30, the hour hand will be half way between the 6 and the 7. Taking time to point this out to students will ensure they succeed on these worksheets.

• Sketching Times to Minutes on 12 Hour Analog Clocks (12 Clocks per Page) Sketching 12 Hour Time in One Hour Intervals (12 Clocks) Sketching 12 Hour Time in 30 Minute Intervals (12 Clocks) Sketching 12 Hour Time in 15 Minute Intervals (12 Clocks) Sketching 12 Hour Time in 5 Minute Intervals (12 Clocks) Sketching 12 Hour Time in 1 Minute Intervals (12 Clocks)
• Sketching Times to Minutes on 12 Hour Analog Clocks (4 Clocks per Page) Sketching 12 Hour Time in One Hour Intervals ( 4 Large Clocks ) Sketching 12 Hour Time in 30 Minute Intervals ( 4 Large Clocks ) Sketching 12 Hour Time in 15 Minute Intervals ( 4 Large Clocks ) Sketching 12 Hour Time in 5 Minute Intervals ( 4 Large Clocks ) Sketching 12 Hour Time in 1 Minute Intervals ( 4 Large Clocks )

Once students have mastered sketching in the hour and minute hands, it is time to add the second hands. As with the hour hand, the minute hand doesn't always point exactly to the minute. For example, if it is 6:30:45, the minute hand will be about 3/4 of the way to the :31 mark. If they have learned this about the hour hand already, it shouldn't be too much of a leap to get them to understand this about the minute hand.

• Sketching Times to Seconds on 12 Hour Analog Clocks (12 Clocks per Page) Sketching 12 Hour Time in 30 Second Intervals (12 Clocks) Sketching 12 Hour Time in 15 Second Intervals (12 Clocks) Sketching 12 Hour Time in 5 Second Intervals (12 Clocks) Sketching 12 Hour Time in 1 Second Intervals (12 Clocks)
• Sketching Times to Seconds on 12 Hour Analog Clocks (4 Clocks per Page) Sketching 12 Hour Time in 30 Second Intervals ( 4 Large Clocks ) Sketching 12 Hour Time in 15 Second Intervals ( 4 Large Clocks ) Sketching 12 Hour Time in 5 Second Intervals ( 4 Large Clocks ) Sketching 12 Hour Time in 1 Second Intervals ( 4 Large Clocks )

Very much the same as sketching on 12 hour clocks, these worksheets use 24 hour time.

• Sketching Times to Minutes on 24 Hour Analog Clocks (12 Clocks per Page) Sketching 24 Hour Time in One Hour Intervals (12 Clocks) Sketching 24 Hour Time in 30 Minute Intervals (12 Clocks) Sketching 24 Hour Time in 15 Minute Intervals (12 Clocks) Sketching 24 Hour Time in 5 Minute Intervals (12 Clocks) Sketching 24 Hour Time in 1 Minute Intervals (12 Clocks)
• Sketching Times to Minutes on 24 Hour Analog Clocks (4 Clocks per Page) Sketching 24 Hour Time in One Hour Intervals ( 4 Large Clocks ) Sketching 24 Hour Time in 30 Minute Intervals ( 4 Large Clocks ) Sketching 24 Hour Time in 15 Minute Intervals ( 4 Large Clocks ) Sketching 24 Hour Time in 5 Minute Intervals ( 4 Large Clocks ) Sketching 24 Hour Time in 1 Minute Intervals ( 4 Large Clocks )
• Sketching Times to Seconds on 24 Hour Analog Clocks (12 Clocks per Page) Sketching 24 Hour Time in 30 Second Intervals (12 Clocks) Sketching 24 Hour Time in 15 Second Intervals (12 Clocks) Sketching 24 Hour Time in 5 Second Intervals (12 Clocks) Sketching 24 Hour Time in 1 Second Intervals (12 Clocks)
• Sketching Times to Seconds on 24 Hour Analog Clocks (4 Clocks per Page) Sketching 24 Hour Time in 30 Second Intervals ( 4 Large Clocks ) Sketching 24 Hour Time in 15 Second Intervals ( 4 Large Clocks ) Sketching 24 Hour Time in 5 Second Intervals ( 4 Large Clocks ) Sketching 24 Hour Time in 1 Second Intervals ( 4 Large Clocks )

Converting Time

It is a very useful skill to be able to convert between 12 and 24 hour time in a global world. Even in places, like the U.S., where 12 hour time is used a lot, students still see time formatted in 24 hour time in a wide variety of situations.

• Converting Between 12- and 24-Hour Times Converting 12- to 24-Hour time Converting 24- to 12-Hour time Converting Between 12- and 24-Hour time

Converting between time units worksheets are useful to test students understanding of time measurement and to give them more practice.

• Converting Between Seconds, Minutes, Hours, Days and Weeks (one step up or down) Converting between seconds, minutes and hours (one step up or down) Converting between seconds, minutes, hours and days (one step up or down) Converting between seconds, minutes, hours, days and weeks (one step up or down)
• Converting Between Seconds, Minutes, Hours, Days and Weeks (one or two steps up or down) Converting between seconds, minutes and hours (one- or two-steps up or down) Converting between seconds, minutes, hours and days (one- or two-steps up or down) Converting between seconds, minutes, hours, days and weeks (one- or two-steps up or down)

Elapsed Time Worksheets

Elapsed time refers to the amount of time that has passed between a start time and a finish time. This can be challenging to students if they are not completely familiar with the base 60 system used for second, minutes and hours, and the fact that there are 24 hours a day sometimes split into morning and afternoon times.

• Elapsed Time to Minutes Elapsed Time with 1 Minute Intervals; Max 5 Hours Elapsed Time with 5 Minute Intervals; Max 5 Hours Elapsed Time with 15 Minute Intervals; Max 5 Hours Elapsed Time with 1 Minute Intervals; Max 24 Hours Elapsed Time with 5 Minute Intervals; Max 24 Hours Elapsed Time with 15 Minute Intervals; Max 24 Hours
• Elapsed Time to Seconds Elapsed Time with 1 Minute/Second Intervals; Max 5 Hours Elapsed Time with 5 Minute/Second Intervals; Max 5 Hours Elapsed Time with 15 Minute/Second Intervals; Max 5 Hours Elapsed Time with 1 Minute/Second Intervals; Max 24 Hours Elapsed Time with 5 Minute/Second Intervals; Max 24 Hours Elapsed Time with 15 Minute/Second Intervals; Max 24 Hours

The elapsed date worksheets in this section are based on a three column table with ten rows. Each different version of the elapsed date table will challenge students in different ways. The worksheets that ask students to find the end date are given a start date and an elapsed time. These worksheets work well for starting at a specific point and counting up. The worksheets that ask students to find the start date, elapsed time or the end date will have random blanks in the table, so students may have to figure out the elapsed time forwards or backwards. Below you will find various challenges including elapsed time with days only, then we progressively add weeks, months, and years to the worksheets.

Please note that when finding future dates, it is important to start with the largest unit first and progress to the smaller units. For example, if the start date is February 8, 2020 and the elapsed time is 3 years, 2 months, 3 weeks and 6 days, you would add the three years first to get February 8, 2023. Next, add the months to get April 8, 2023. Finally, add the weeks and days to get May 5, 2023. You can add the weeks and days together as they are both exact lengths whereas years and months vary in size.

• Calculate End Dates from Start Dates and Elapsed Times in Days, Weeks, Months and Years Calculate End Date for Various Elapsed Days Calculate End Date for Various Elapsed Days + Weeks Calculate End Date for Various Elapsed Days + Weeks + Months Calculate End Date for Various Elapsed Days + Weeks + Months + Years
• Calculate Elapsed Time Between Two Dates in Days, Weeks, Months and Years Calculate Elapsed Time Between Two Dates in Days Calculate Elapsed Time Between Two Dates in Days + Weeks Calculate Elapsed Time Between Two Dates in Days + Weeks + Months Calculate Elapsed Time Between Two Dates in Days + Weeks + Months + Years
• Calculate Start Dates from End Dates and Elapsed Time in Days, Weeks, Months and Years Calculate Start Date from End Date and Elapsed Time in Days Calculate Start Date from End Date and Elapsed Time in Days + Weeks Calculate Start Date from End Date and Elapsed Time in Days + Weeks + Months Calculate Start Date from End Date and Elapsed Time in Days + Weeks + Months + Years
• Calculate Various Start Dates, Elapsed Time or End Dates with Days, Weeks, Months and Years Calculate Start Date, Elapsed Time or End Date (Days) Calculate Start Date, Elapsed Time or End Date (Days + Weeks) Calculate Start Date, Elapsed Time or End Date (Days + Weeks + Months) Calculate Start Date, Elapsed Time or End Date (Days + Weeks + Months + Years)

Adding and Subtracting Time

Adding and subtracting time is similar to adding any numbers, but the regrouping amounts are different. If you think of the decimal system, numbers are divided into places named: ones, tens, hundreds, etc. In time values, the places have different values based on an ancient Babylonian numbering system with a base of 60. In the "seconds place," there are 60 unique seconds from 0 to 59. The same is true for the "minutes place." The "hours place" can vary from 24 if one is interested in counting days as well, or can be greater than 24 if the largest place value is in the hours place. In these worksheets, students are challenged to regroup seconds and minutes as they add or subtract two time amounts.

• Adding and Subtracting Hours and Minutes Adding Hours and Minutes (Compact Format) Adding Hours and Minutes (Long Format) Subtracting Hours and Minutes (Compact Format) Subtracting Hours and Minutes (Long Format) Adding and Subtracting Hours and Minutes (Compact Format) Adding and Subtracting Hours and Minutes (Long Format)
• Adding and Subtracting Hours, Minutes and Seconds Adding Hours, Minutes and Seconds (Compact Format) Adding Hours, Minutes and Seconds (Long Format) Subtracting Hours, Minutes and Seconds (Compact Format) Subtracting Hours, Minutes and Seconds (Long Format) Adding and Subtracting Hours, Minutes and Seconds (Compact Format) Adding and Subtracting Hours, Minutes and Seconds (Long Format)

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Time Interval – Definition with Examples

What is time interval, calculating time interval, subtracting times , solved example, practice problems, frequently asked questions.

The amount of time between two given times is known as time interval. In other words, it is the amount of time that has passed between the beginning and end of the event. It is also known as elapsed time .

Let’s understand this concept with an example of time interval.

Example: Sam had soccer practice after school. Training started at 4:00 p.m. and ended at 5:00 p.m. For how long did he play?

The interval between 4:00 p.m. and 5:00 p.m. is one hour. So, Sam played soccer for one hour. 

Units for Measuring Time

Interval of time is measured in different units. Each unit describes a different amount of time. Some units are better suited to specific intervals of time. For example, if you were baking a cake in the oven, you would choose to measure the time in minutes or maybe in hours. If you were calculating the time for your birthday from a specific date, you would prefer to measure the time in days, weeks, or months (depending on how far away it was).

The smallest unit of time used every day is a second .

Seconds, minutes and hours can be calculated on a clock or watch. If we want to measure bigger time units, we prefer a calendar.

Also, 

• 10 years $= 1$ decade
• 100 years $= 1$ century
• 1000 years $= 1$ millennium

Related Worksheets

By definition, time interval is the amount of time between two given times. We can calculate the time interval in math by determining the difference between the start and end times. We can show differences in time using a number line or by simply subtracting two different times.

Time Interval on the Number Line

We can calculate the elapsed time using number lines . To calculate, we divide the number line into equal intervals of time . Suppose you have to calculate the amount of time between 2:05 p.m and 2:35 p.m. 

The first step would be to plot both the times on a number line:

We can count the number of tick marks between the two times and multiply that by the value of each tick mark (5 minutes), as shown in the figure above. So, the answer is $6 \times 5$ minutes or 30 minutes. 

We can calculate the amount of time between two events by finding the difference between the start and end times.

Time Interval $=$ Final Time – Starting Time

Here are the steps to subtract two times when both the times are either in a.m. or in p.m.

Step I: If the minutes of the first (end) time is less than the minutes of the second (start) time, then in the first (end) time,

• Take one hour away 
• Add 60 to the minutes of the first time.

Step II: Subtract the minutes. 

Step III: Subtract the hours. 

Hence, the resultant from Step II and Step III together is the answer.

For example, calculate the time covered between 4:15 p.m. and 4:45 p.m.

4: 45 pm $-$ 4:15 pm

Step I: Here, the minutes of the first (end) time is not less than the minutes of the second (start) time. So, we move on to the next step.

Step II: Subtract the minutes. $45 – 15 = 30$

Step III: Subtract the hours. $4 – 4 = 0$

So, the final answer is 30 minutes.

Here’s another example: Calculate the time between 7:10 p.m. and 9:00 p.m.

9:00 – 7:10. Here, we will use regrouping. 

Step I: Here, the minutes of the first (end) time is less than the minutes of the second (start) time. So, we borrow 1 hour and add 60 minutes.

Step II: Subtract the minutes. $60 – 10 = 50$

Step III: Subtract the hours. $8 – 7 = 1$.

So, the solution is 1 hour and 50 minutes.

Let’s look at another example: Calculate the time interval between 8:45 a.m. and 2:30 p.m.

If one of the times is in a.m. and the other is in p.m., we will follow the below steps:

Step I: Find the time elapsed between 8:45 a.m. and 12 noon. 

Between 8:45 a.m. and 12:00 noon, we have 3 hours and 15 minutes.

Step II: Find the amount of time between 12 noon and 2:30 p.m.

Between 12 noon and 2:30 p.m., the interval is 2 hours and 30 minutes.

Step III: Add the time of step I and step II to find the final answer.

3 hours and 15 minutes $+ 2$ hours and 30 minutes or 5 hours and 45 minutes

To conclude, as explained above, the amount of time between two given times. If you are looking for some more worksheets on the time, check out SplashLearn today! Visit the website to make learning a fun experience.

Example 1: Find the time interval between 17:20 and 18:50 on a number line.

Solution : 

The number line is divided into equal intervals of 5 minutes.

Between the two points marked, there are 18 intervals.

Therefore, the amount of time between 17:20 and 18:50 is 185 or 90 minutes $= 1$ hour and 30 minutes

By looking at the number line, we get that between 17:20 and 18:50, we have 10 minutes $+ 1$ hour $+ 20$ minutes $= 1$ hour 30 minutes

Example 2: How many hours and minutes are there between 10:10 a.m. and 2:45 p.m.?

Amount of time between 10:10 a.m. and 12 noon $= 1$ hour 50 minutes

Interval between 12 noon and 2:45 p.m. $= 2$ hours 45 minutes

So, the interval between 10:10 a.m. and 2:45 p.m. 

$= 1$ hour 50 minutes $+ 2$ hours 45 minutes

$= 4$ hours 35 minutes

Example 3: Anna plans to watch her favorite movie, which starts at 2:30 p.m. The time interval is 2 hours 34 minutes. At what time will the movie end? 

The time the movie ends $= 2:30$ p.m. $+ 2$ hours 34 minutes.

Adding 2 hours to 2:30 p.m., we get 4:30 p.m.

Adding 34 minutes to 4:30 p.m., we get 5:04 p.m.

Time Interval - Definition with Examples

Attend this quiz & Test your knowledge.

What is the time interval between 1:30 p.m. and 9:10 p.m.?

On an analog clock, what is the time covered when the hour hand moves from the number 4 to 5, sam started his journey and reached his destination in 7 hours. if he reached at 12:00 p.m., then at what time did he start for his destination.

How can we calculate the time interval on the number line?

To calculate the interval of time on a number line, we divide the number line into equal intervals and then jump from the start time to the end time.

What are some examples of time intervals in our daily lives?

The following are some examples of interval of time:

• The time taken to finish a race.
• The time taken to bake a cake.
• The duration of the journey is covered by a bus and so on.

What do you mean by estimated time interval?

The estimated time interval is the time that will be required for an event (that has not occurred yet) that is yet to be completed. It is generally used for flight duration.

RELATED POSTS

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• Subtracting Time

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Once Upon a Time

In the problem Once Upon a Time , students use measurement, number properties, and circular geometry to solve problems involving time and angles. The mathematical topics that underlie this problem are time measurement, including conversion between years, months, weeks, days, hours, minutes, and seconds; modular arithmetic that involves divisibility and remainders; pattern recognition; as well as circles and angular measurement. In each level, students must make sense of the problem and persevere in solving it ( MP.1 ). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.

PRE-K In this task, students will examine the parts of a clock. They are then presented with the task of determining how many minutes will pass before the large hand catches up to the small hand on a clock. Their task involves counting up and understanding how a clock measures time.

LEVEL A In this level, students are presented with the task of determining how many minutes will pass before the large hand catches up to the small hand on a clock. Their task involves counting up and understanding how a clock measures time.

This level supports Common Core standard 3.MD.A.1 by having students measure time intervals in minutes and solve problems involving addition and subtraction of time intervals.

LEVEL B In this level, the students are asked to convert their age from years into seasons, months, and weeks. Students are also asked to determine what day number the current date is in the year.

This level supports Common Core standard 4.OA.A.3 by having students solve problems using the four operations. Students will also need to be able to express units of time (years) in terms of a smaller unit (days, weeks, months).

LEVEL C In this level, students are given a problem that requires an understanding of divisibility and may be determined by using knowledge of relatively prime factors.

This level extends the work of Common Core standards 6.NS.B.2 and 6.NS.B.4 . Students apply understanding of divisibility, common factors, and multiples to solve the problem. 

LEVEL D In this level, students are presented with a problem that involves three different-sized alarm clocks that ring at varied intervals. The task is to determine if or when the three clocks chime simultaneously. 

This level supports Common Core standard 8.EE.C.8b as students can write equations and solve systems of linear equations in two variables algebraically to determine if or when three different-sized alarm clocks chime simultaneously.

LEVEL E In this level, students are asked to determine the times in a day that the minute and hour hands of a clock form an angle of 48 degrees.

This level supports Common Core standard A-CED.A.1 as students can write equations to describe the motion of the hands of a clock. They use these equations and their knowledge of the central angle of a circle between two radii to find when the hands are 48 degrees apart. Common Core standard G-C.A.2 is supported as students make sense of the clock and the motion of its hands through angles in a circle. The problem is complicated by the fact that time is measured in intervals of 60. As they persist in analyzing the problem, they will notice patterns and regularity ( MP.8 ) in the times for each hour of the day.  

PROBLEM OF THE MONTH  Download the complete packet of Once Upon a Time  Levels A-E here . 

You can learn more about how to implement these problems in a school-wide Problem of the Month initiative in “Jumpstarting a Schoolwide Culture of Mathematical Thinking: Problems of the Month,” a practitioner’s guide.  Download the guide as iBook with embedded videos or Download as PDF without embedded videos .

SOLUTIONS To request the Inside Problem Solving Solutions Guide, please get in touch with us via the  feedback form .

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5.4: Time Dilation

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Learning Objectives

By the end of this section, you will be able to:

• Explain how time intervals can be measured differently in different reference frames.
• Describe how to distinguish a proper time interval from a dilated time interval.
• Describe the significance of the muon experiment.
• Explain why the twin paradox is not a contradiction.
• Calculate time dilation given the speed of an object in a given frame.

The analysis of simultaneity shows that Einstein’s postulates imply an important effect: Time intervals have different values when measured in different inertial frames. Suppose, for example, an astronaut measures the time it takes for a pulse of light to travel a distance perpendicular to the direction of his ship’s motion (relative to an earthbound observer), bounce off a mirror, and return (Figure \(\PageIndex{1a}\)). How does the elapsed time that the astronaut measures in the spacecraft compare with the elapsed time that an earthbound observer measures by observing what is happening in the spacecraft?

Examining this question leads to a profound result. The elapsed time for a process depends on which observer is measuring it. In this case, the time measured by the astronaut (within the spaceship where the astronaut is at rest) is smaller than the time measured by the earthbound observer (to whom the astronaut is moving). The time elapsed for the same process is different for the observers, because the distance the light pulse travels in the astronaut’s frame is smaller than in the earthbound frame, as seen in Figure \(\PageIndex{1b}\). Light travels at the same speed in each frame, so it takes more time to travel the greater distance in the earthbound frame.

Definition: Time Dilation

Time dilation is the lengthening of the time interval between two events for an observer in an inertial frame that is moving with respect to the rest frame of the events (in which the events occur at the same location).

To quantitatively compare the time measurements in the two inertial frames, we can relate the distances in Figure \(\PageIndex{1b}\) to each other, then express each distance in terms of the time of travel (respectively either \(\Delta t\) or \(\Delta \tau\)) of the pulse in the corresponding reference frame. The resulting equation can then be solved for \(\Delta t\) in terms of \(\Delta \tau\).

The lengths \(D\) and \(L\) in Figure \(\PageIndex{1c}\) are the sides of a right triangle with hypotenuse \(s\). From the Pythagorean theorem ,

\[s^2 = D^2 + L^2. \nonumber \]

The lengths \(2s\) and \(2L\) are, respectively, the distances that the pulse of light and the spacecraft travel in time \(\Delta t\) in the earthbound observer’s frame. The length \(D\) is the distance that the light pulse travels in time \(\Delta \tau\) in the astronaut’s frame. This gives us three equations:

\[\begin{align*} 2s &= c\Delta t \\[4pt] 2L &= v\Delta t; \\[4pt] 2D &= c\Delta \tau. \end{align*} \nonumber \]

Note that we used Einstein’s second postulate by taking the speed of light to be c in both inertial frames. We substitute these results into the previous expression from the Pythagorean theorem:

\[ \begin{align*} s^2 &= D^2 + L^2 \\[4pt] \left(c\dfrac{\Delta t}{2}\right)^2 &= \left(c\dfrac{\Delta \tau}{2}\right)^2 + \left(v\dfrac{\Delta t}{2}\right)^2 \end{align*} \nonumber \]

Then we rearrange to obtain

\[(c\Delta t)^2 - (v\Delta t)^2 = (c\Delta \tau)^2. \nonumber \]

Finally, solving for \(\Delta t\) in terms of \(\Delta \tau\) gives us

\[\Delta t = \dfrac{\Delta \tau}{\sqrt{1 - (v/c)^2}}. \nonumber \]

This is equivalent to

\[\Delta t = \gamma \Delta \tau, \label{timedilation} \]

where \(\gamma\) is the relativistic factor (often called the Lorentz factor ) given by

\[\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}} \nonumber \]

and \(v\) and \(c\) are the speeds of the moving observer and light, respectively.

Note the asymmetry between the two measurements. Only one of them is a measurement of the time interval between two events—the emission and arrival of the light pulse—at the same position. It is a measurement of the time interval in the rest frame of a single clock. The measurement in the earthbound frame involves comparing the time interval between two events that occur at different locations. The time interval between events that occur at a single location has a separate name to distinguish it from the time measured by the earthbound observer, and we use the separate symbol \(\Delta \tau\) to refer to it throughout this chapter.

Definition: Proper Time

The proper time interval \(\Delta \tau\) between two events is the time interval measured by an observer for whom both events occur at the same location.

The equation relating \(\delta t\) and \(\Delta \tau\) is truly remarkable. First, as stated earlier, elapsed time is not the same for different observers moving relative to one another, even though both are in inertial frames. A proper time interval \(\Delta \tau\) for an observer who, like the astronaut, is moving with the apparatus, is smaller than the time interval for other observers. It is the smallest possible measured time between two events. The earthbound observer sees time intervals within the moving system as dilated (i.e., lengthened) relative to how the observer moving relative to Earth sees them within the moving system. Alternatively, according to the earthbound observer, less time passes between events within the moving frame. Note that the shortest elapsed time between events is in the inertial frame in which the observer sees the events (e.g., the emission and arrival of the light signal) occur at the same point.

This time effect is real and is not caused by inaccurate clocks or improper measurements. Time-interval measurements of the same event differ for observers in relative motion. The dilation of time is an intrinsic property of time itself. All clocks moving relative to an observer, including biological clocks, such as a person’s heartbeat, or aging, are observed to run more slowly compared with a clock that is stationary relative to the observer.

Note that if the relative velocity is much less than the speed of light (v << c), then \(v^2/c^2\) is extremely small, and the elapsed times \(\Delta t\) and \(\Delta \tau\) are nearly equal. At low velocities, physics based on modern relativity approaches classical physics—everyday experiences involve very small relativistic effects. However, for speeds near the speed of light, \(v^2/c^2\) is close to one, so \(\sqrt{1 - v^2/c^2}\) is very small and \(\Delta t\) becomes significantly larger than \(\Delta \tau\).

Half-Life of a Muon

There is considerable experimental evidence that the equation \(\Delta t = \gamma \Delta \tau\) is correct. One example is found in cosmic ray particles that continuously rain down on Earth from deep space. Some collisions of these particles with nuclei in the upper atmosphere result in short-lived particles called muons . The half-life (amount of time for half of a material to decay) of a muon is 1.52 μs when it is at rest relative to the observer who measures the half-life. This is the proper time interval \(\Delta \tau\). This short time allows very few muons to reach Earth’s surface and be detected if Newtonian assumptions about time and space were correct. However, muons produced by cosmic ray particles have a range of velocities, with some moving near the speed of light. It has been found that the muon’s half-life as measured by an earthbound observer (\(\Delta t\)) varies with velocity exactly as predicted by the equation \(\Delta t = \gamma \Delta \tau\). The faster the muon moves, the longer it lives. We on Earth see the muon last much longer than its half-life predicts within its own rest frame. As viewed from our frame, the muon decays more slowly than it does when at rest relative to us. A far larger fraction of muons reach the ground as a result.

Before we present the first example of solving a problem in relativity, we state a strategy you can use as a guideline for these calculations.

PROBLEM-SOLVING STRATEGY: RELATIVITY

• Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Look in particular for information on relative velocity v .
• Identify exactly what needs to be determined in the problem (identify the unknowns).
• Make certain you understand the conceptual aspects of the problem before making any calculations (express the answer as an equation). Decide, for example, which observer sees time dilated or length contracted before working with the equations or using them to carry out the calculation. If you have thought about who sees what, who is moving with the event being observed, who sees proper time, and so on, you will find it much easier to determine if your calculation is reasonable.
• Determine the primary type of calculation to be done to find the unknowns identified above (do the calculation). You will find the section summary helpful in determining whether a length contraction, relativistic kinetic energy, or some other concept is involved.

Note that you should not round off during the calculation . As noted in the text, you must often perform your calculations to many digits to see the desired effect. You may round off at the very end of the problem solution, but do not use a rounded number in a subsequent calculation. Also, check the answer to see if it is reasonable: Does it make sense? This may be more difficult for relativity, which has few everyday examples to provide experience with what is reasonable. But you can look for velocities greater than c or relativistic effects that are in the wrong direction (such as a time contraction where a dilation was expected).

Example \(\PageIndex{1A}\): Time Dilation in a High-Speed Vehicle

The Hypersonic Technology Vehicle 2 (HTV-2) is an experimental rocket vehicle capable of traveling at 21,000 km/h (5830 m/s). If an electronic clock in the HTV-2 measures a time interval of exactly 1-s duration, what would observers on Earth measure the time interval to be?

Apply the time dilation formula to relate the proper time interval of the signal in HTV-2 to the time interval measured on the ground.

• Identify the knowns: \(\Delta \tau = 1 \, s\); \(v = 5830m/s.\)
• Identify the unknown: \(\Delta t\).

\[\Delta t = \gamma \Delta \tau = \dfrac{\Delta \tau}{\sqrt{1 - \dfrac{v^2}{c^2}}}. \nonumber \]

\[\begin{align*} \Delta t &= \dfrac{1 \, s}{\sqrt{1 - \left(\dfrac{5830 \, m/s}{3.00 \times 10^8 m/s}\right)^2}} \\[4pt] &= 1.000000000189 \, s \\[4pt] &= 1 \, s + 1.89 \times 10^{-10}s. \end{align*} \nonumber \]

Significance

The very high speed of the HTV-2 is still only 10 -5 times the speed of light. Relativistic effects for the HTV-2 are negligible for almost all purposes, but are not zero.

What Speeds are Relativistic?

How fast must a vehicle travel for 1 second of time measured on a passenger’s watch in the vehicle to differ by 1% for an observer measuring it from the ground outside?

Use the time dilation formula to find v/c for the given ratio of times.

\[\dfrac{\Delta \tau}{\Delta t} = \dfrac{1}{1.01}. \nonumber \]

• Identify the unknown: v/c .

\[ \begin{align*} \Delta t &= \gamma \Delta \tau \\[4pt] &= \dfrac{1}{\sqrt{1 - v^2/c^2}}\Delta \tau \\[4pt] \dfrac{\Delta \tau}{\Delta t} &= \sqrt{1 - v^2/c^2} \\[4pt] \left(\dfrac{\Delta \tau}{\Delta t}\right)^2 &= 1 - \dfrac{v^2}{c^2} \\[4pt] \dfrac{v}{c} &= \sqrt{1 - (\Delta \tau/\Delta t)^2}. \end{align*} \nonumber \]

\[\dfrac{v}{c} = \sqrt{1 - (1/1.01)^2} = 0.14. \nonumber \]

The result shows that an object must travel at very roughly 10% of the speed of light for its motion to produce significant relativistic time dilation effects.

Calculating \(\Delta t\) for a Relativistic Event

Suppose a cosmic ray colliding with a nucleus in Earth’s upper atmosphere produces a muon that has a velocity \(v = 0.950c\). The muon then travels at constant velocity and lives 2.20 μs as measured in the muon’s frame of reference. (You can imagine this as the muon’s internal clock.) How long does the muon live as measured by an earthbound observer (Figure \(\PageIndex{2}\))?

As we will discuss later, in the muon’s reference frame, it travels a shorter distance than measured in Earth’s reference frame.

A clock moving with the muon measures the proper time of its decay process, so the time we are given is \(\Delta \tau = 2.20 \mu s\). The earthbound observer measures \(\Delta t\) as given by the equation \(\Delta t = \gamma \Delta \tau\). Because the velocity is given, we can calculate the time in Earth’s frame of reference.

• Identify the knowns: \(v = 0.950c\); \(\delta \tau = 2.20 \mu s\).

\[\Delta t = \gamma \Delta \tau. \nonumber \]

\[\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}}. \nonumber \]

\[\begin{align*} \Delta t &= \gamma \Delta \tau. \\[4pt] &=\dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}}\delta \tau \\[4pt] &=\dfrac{2.20 \mu s}{\sqrt{1 - (0.950)^2}} \\[4pt] &= 7.05 \, \mu s.\end{align*} \nonumber \]

One implication of this example is that because \(\gamma = 3.20\) at 95.0% of the speed of light (\(v = 0.950c\)), the relativistic effects are significant. The two time intervals differ by a factor of 3.20, when classically they would be the same. Something moving at 0.950 c is said to be highly relativistic.

Example \(\PageIndex{1B}\): Relativistic Television

A non-flat screen, older-style television display (Figure \(\PageIndex{3}\)) works by accelerating electrons over a short distance to relativistic speed, and then using electromagnetic fields to control where the electron beam strikes a fluorescent layer at the front of the tube. Suppose the electrons travel at \(6.00 \times 10^7 m/s\) through a distance of 0.200m0.200m from the start of the beam to the screen.

• What is the time of travel of an electron in the rest frame of the television set?
• What is the electron’s time of travel in its own rest frame?

Strategy for (a)

(a) Calculate the time from \(vt = d\). Even though the speed is relativistic, the calculation is entirely in one frame of reference, and relativity is therefore not involved.

\[v = 6.00 \times 10^7 m/s \, d = 0.200 \, m. \nonumber \]

• Identify the unknown: the time of travel \(\Delta t\).

\[\Delta t = \dfrac{d}{v}. \nonumber \]

\[ \begin{align*} t &= \dfrac{0.200 \, m}{6.00 \times 10^7 \, m/s} \\[4pt] &= 3.33 \times 10^{-9} \, s. \end{align*} \nonumber \]

The time of travel is extremely short, as expected. Because the calculation is entirely within a single frame of reference, relativity is not involved, even though the electron speed is close to c .

Strategy for (b)

(b) In the frame of reference of the electron, the vacuum tube is moving and the electron is stationary. The electron-emitting cathode leaves the electron and the front of the vacuum tube strikes the electron with the electron at the same location. Therefore we use the time dilation formula to relate the proper time in the electron rest frame to the time in the television frame.

\[\Delta t = 3.33 \times 10^{-9} \, s; \, v = 6.00 \times 10^7 \, m/s; \, d = 0.200 \, m. \nonumber \]

• Identify the unknown: \(\tau\).

\[\Delta t = \gamma \Delta \tau = \dfrac{\Delta \tau}{\sqrt{1 - v^2/c^2}}. \nonumber \]

\[\begin{align*} \Delta \tau &= (3.33 \times 10^{-9}s)\sqrt{1 - \left(\dfrac{6.00 \times 10^7 m/s}{3.00 \times 10^8 m/s}\right)^2} \\[4pt] &= 3.26 \times 10^{-9}s. \end{align*} \nonumber \]

The time of travel is shorter in the electron frame of reference. Because the problem requires finding the time interval measured in different reference frames for the same process, relativity is involved. If we had tried to calculate the time in the electron rest frame by simply dividing the 0.200 m by the speed, the result would be slightly incorrect because of the relativistic speed of the electron.

Exercise \(\PageIndex{1}\)

What is \(\gamma\) if \(v = 0.650c\)?

\[\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}} = \dfrac{1}{\sqrt{1 - \dfrac{(0.650c)}{c^2}}} = 1.32 \nonumber \]

The Twin Paradox

An intriguing consequence of time dilation is that a space traveler moving at a high velocity relative to Earth would age less than the astronaut’s earthbound twin. This is often known as the twin paradox . Imagine the astronaut moving at such a velocity that \(\gamma = 30.0\), as in Figure \(\PageIndex{4}\). A trip that takes 2.00 years in her frame would take 60.0 years in the earthbound twin’s frame. Suppose the astronaut travels 1.00 year to another star system, briefly explores the area, and then travels 1.00 year back. An astronaut who was 40 years old at the start of the trip would be would be 42 when the spaceship returns. Everything on Earth, however, would have aged 60.0 years. The earthbound twin, if still alive, would be 100 years old.

The situation would seem different to the astronaut in Figure \(\PageIndex{4}\). Because motion is relative, the spaceship would seem to be stationary and Earth would appear to move. (This is the sensation you have when flying in a jet.) Looking out the window of the spaceship, the astronaut would see time slow down on Earth by a factor of \(\gamma = 30.0\). Seen from the spaceship, the earthbound sibling will have aged only 2/30, or 0.07, of a year, whereas the astronaut would have aged 2.00 years.

The paradox here is that the two twins cannot both be correct. As with all paradoxes, conflicting conclusions come from a false premise. In fact, the astronaut’s motion is significantly different from that of the earthbound twin. The astronaut accelerates to a high velocity and then decelerates to view the star system. To return to Earth, she again accelerates and decelerates. The spacecraft is not in a single inertial frame to which the time dilation formula can be directly applied. That is, the astronaut twin changes inertial references. The earthbound twin does not experience these accelerations and remains in the same inertial frame. Thus, the situation is not symmetric, and it is incorrect to claim that the astronaut observes the same effects as her twin. The lack of symmetry between the twins will be still more evident when we analyze the journey later in this chapter in terms of the path the astronaut follows through four-dimensional space-time.

In 1971, American physicists Joseph Hafele and Richard Keating verified time dilation at low relative velocities by flying extremely accurate atomic clocks around the world on commercial aircraft. They measured elapsed time to an accuracy of a few nanoseconds and compared it with the time measured by clocks left behind. Hafele and Keating’s results were within experimental uncertainties of the predictions of relativity. Both special and general relativity had to be taken into account, because gravity and accelerations were involved as well as relative motion.

Exercise \(\PageIndex{2A}\)

a. A particle travels at \(1.90 \times 10^8 \, m/s\) and lives \(2.1 \times 10^8 \, s\) when at rest relative to an observer. How long does the particle live as viewed in the laboratory?

\[\Delta t = \dfrac{\Delta \tau}{\sqrt{1 - \dfrac{v^2}{c^2}}} = \dfrac{2.10 \times 10^{-8}s}{\sqrt{1 - \dfrac{(1.90 \times 10^8 \, m/s)^2}{(3.00 \times 10^8 \, m/s)^2}}} = 2.71 \times 10^{-8} \, s. \nonumber \]

Exercise \(\PageIndex{2B}\)

Spacecraft A and B pass in opposite directions at a relative speed of \(4.00 \times 10^7 \, m/s\). An internal clock in spacecraft A causes it to emit a radio signal for 1.00 s. The computer in spacecraft B corrects for the beginning and end of the signal having traveled different distances, to calculate the time interval during which ship A was emitting the signal. What is the time interval that the computer in spacecraft B calculates?

Only the relative speed of the two spacecraft matters because there is no absolute motion through space. The signal is emitted from a fixed location in the frame of reference of A , so the proper time interval of its emission is \(\tau = 1.00 \, s\). The duration of the signal measured from frame of reference B is then

\[\Delta t = \dfrac{\Delta \tau}{\sqrt{1 - \dfrac{v^2}{c^2}}} = \dfrac{1.00 \, s}{\sqrt{1 - \dfrac{(4.00 \times 10^7 \, m/s)^2}{(3.00 \times 10^8 \, m/s)^2}}} = 1.01 \, s. \nonumber \]

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25 Time Word Problems for Year 2 to Year 6 With Tips On Supporting Pupils’ Progress

Emma Johnson

Time word problems are an important element of teaching children how to tell the time. Children are introduced to the concept of time in Year 1. At this early stage, they learn the basics of analogue time; reading to the hour and half past and learn how to draw hands on the clocks to show these times.

As they move through primary school, pupils progress onto reading the time in analogue, digital and 24 hour clocks and being able to compare the duration of events. By the time children reach upper Key Stage 2, they should be confident in reading the time in all formats and solving problems involving converting between units of time.

Time in Year 1

Time in year 2, time in year 3, time in year 4, time in year 5 & 6.

• Why are word problems important for children’s understanding of time

How to teach time word problem solving in primary school

Time word problems for year 2, time word problems for year 3, time word problems for year 4, time word problems for year 5, time word problems for year 6, more time and word problems resources.

Let's Practice Telling The Time

Download this free printable worksheet to let your students practice telling the time.

When students are first introduced to time and time word problems , it is important for them to have physical clocks, to hold and manipulate the hands. Pictures on worksheets are helpful, but physical clocks enable them to work out what is happening with the hands and to solve word problems involving addition word problems and subtraction word problems .

Time word problems are important for helping children to understand how time is used in the real-world. We have put together a collection of 25 time word problems, which can be used with pupils from Year 2 to Year 6.

Time word problems in the National Curriculum  

In Year 1, students are introduced to the basics of time. They learn to recognise the hour and minute hand and use this to help read the time to the hour and half past the hour. They also draw hands on clock faces to represent these times. 

By the end of Year 2, pupils should be able to tell the time to five minutes, including quarter past/to the hour and draw the hands on a clock face to show these times. They should also know the number of minutes in an hour and the number of hours in a day.

In Year 3, children read the time in analogue (including using Roman Numerals). By this stage they are also learning to read digital time in 12 and 24 hour clock, using the AM and PM suffixes. Pupils record and compare time in terms of seconds, minutes and hours; know the number of seconds in a minute, days in a month and year and compare durations of events.

By Year 4, pupils should be confident telling the time in analogue to the nearest minute, digital and 24 hour clock. They also need to be able to read, write and convert time between analogue and digital 12 and 24 hour clocks and solve problems involving converting from hours to minutes; minutes to seconds; years to months and weeks to days. 

By Year 5 and 6, there is only limited mention of time in the curriculum. Pupils continue to build on the knowledge they have picked up so far and should be confident telling the time and solving a range of problems, including: converting units of time; elapsed time word problems, working with timetables and tackling multi-step word problems .

Time word problems have been known to appear on Year 6 SATs tests. Third Space Learning’s online one-to-one SATs revision programme incorporates a wide range of word problems to develop students’ problem solving skills and prepare them the SATs tests. Available for all primary year groups as well as Year 7 and GCSE, our online tuition programmes are personalised to suit the needs of each individual student, fill learning gaps and build confidence in maths.

Why are word problems important for children’s understanding of time

Word problems are important for helping children to develop their understanding of time and the different ways time is used on an every-day basis. Confidence in telling the time and solving a range of time problems is a key life skill. Time word problems provide children with the opportunity to build on the skills they have picked up and apply them to real-world situations.

It’s important children learn the skills needed to solve word problems. Key things they need to remember are: to make sure they read the question carefully; to think whether they have fully understood what is being asked and then identify what they will need to do to solve the problem and whether there are any concrete resources or pictorial representations which will help them.

Here is an example:

Mr Arrowsmith drives to Birmingham. He sets off at 3:15pm. He stops for a break of 15 minutes at 4:50 and arrives in Birmingham at 6:15pm.

How long did Mr Arrowsmith spend driving?

How to solve:

What do you already know?

• We know that he set off at 3:15pm and stopped for a break at 4:50. We can calculate how long the first part of his journey was, by counting on from 3:15 to 4:50.
• He had a break at 4:50pm for 15 minutes, so we won’t include that in our driving time calculation.
• He then must have set off again at 5:05pm, before arriving at 6:15pm. We can use this information to work out the length of the second part of his journey.
• We can then add the 2 journey times together, to calculate the total amount of time spent driving. 

How can this be represented pictorially?

•  We can use a number line to calculate the length of time each journey takes.
• If we start by adding on an hour, we can then calculate how many more minutes for each section of the journey.
• Once we have calculated the journey time for each part of the journey, we can add these together to calculate the total journey time.

Time word problems in Year 2 require students to read the time to o’clock and half past the hour and compare and sequence time intervals. 

Oliver went for a bike ride with his friend. 

He left home at 2 o’clock and came home at 4 o’clock.

How long was he out on his bike for?

Answer: 2 hours

Count on from 2 o’ clock to 4 o’clock or subtract 2 from 4.

Mum went shopping at 3 o’clock and got home an hour later.

Draw the time she got home on the clock below.

Tom baked a cake.

The cake was in the oven for one hour. 

If he took the cake out at half past 11, what time did he put the cake in?

Answer: Half past 10

Use an hour from half past 11.

Arlo starts school at 9 o’clock and has his first break at half past 10. 

How long does he have to wait for his first break?

Answer: One and a half hours.

(Use a number line to count on from 9 to half past 10)

The Smith family are going to the beach.

They plan to leave home at 10 o’clock and the journey take two hours.

What time will they arrive at the beach?

Answer: 12 o’clock

(Use a number line to count on 2 hours from 10 o’clock)

With time word problems for year 3 , students build on their understanding of analogue time from Year 2 and also begin to  read the time in  digital (12 and 24 hour clock). Children also need to be able to compare time and durations of events.

Chloe is walking to football training.

She sets off at 8:40am and takes 17 minutes to get there.

What time does she arrive?

Answer: 8:57

(Count on 17 minutes from 8:40 – use a number line if needed)

(Picture of analogue clock with 2:30 showing here)

Maisie says that in 1 hour and 48 minutes it will be 4:28.

Do you agree? Explain how you worked out your answer.

Answer: Maisie is wrong. It will be 4:18. 

This can be worked out by counting on an hour from 2:30 to 3:30 and then another 48 minutes to 4:18.

The Baker family are driving to their campsite. 

They set off at 8:30 am, drive for 2 hours and 15 minutes, then had a 30 minute break. 

If they drive for another 1 hour and 45 minutes, what time do they arrive at the campsite?

Answer: 12:45pm

Use a number line to show what time they arrive at the break. From 8:30, count on 2 hours and 15 minutes to get to 10:45. Add on the 30 minute break. It is now 11:15. They count on another hour and a half  to 12:45

Ahmed looks at his watch and says ‘it is half past 4 in the afternoon’

Jude says that it is 17:30 in a 24 hour clock.

Is Jude correct? Explain your answer.

Answer: Jude is not correct. Half past 4 in the afternoon is 16:30 not 17:30

How many minutes are there in 2 hours and 30 minutes?

Answer: 150 minutes

60 + 60 + 30 = 150

When solving time word problems for year 4 , pupils need to be confident telling time in analogue, and digital, as well as converting between analogue, 12 hour and 24 hour clock. They also begin to solve more challenging problems involving duration of time and converting time.

If there are 60 seconds in 1 minute. How many seconds are there in 8 minutes?

Answer: 480 seconds

60 x 8 = 480 seconds (calculate 6 x 8, then multiply by 10)

Mason played on his VR from 3:35 to 5:25.

How long did he play on his VR?

Answer: 1 hour and 50 minutes.

Count on from 3:35 (using a numberline if needed)

Jamie started his homework at 3:45pm. He finished 43 minutes later.

What time did Jamie finish? Give your answer in 24 hour clock.

Answer: 16:28

Count on 43 minutes from 3:45 (use a number line, if needed) = 4:28. Convert to 24 hour clock.

Chloe and Freya went to the cinema to watch a film. The film started at 2:05pm and lasted for 1 hour and 43 minutes. 

What time did the film end?

Answer: 3:48pm

Count on one hour from 2:05 pm to 3:05pm, then add another 43 minutes – 3:48pm

A family is driving on their holiday.

They drive for 2 hours and 28 minutes, stop for 28 minutes and then drive a further 1 hour and 52 minutes.

If they left at 8:30am, what time did they arrive?

Answer: 1:18pm

2 hours and 28 minutes from 8:30am = 10:58am

10:58am with a 28 minute break = 11:26am

1 hour 52 minute drive from 11:26 am = 1:18pm

With word problems for year 5 , pupils should be confident telling the time in analogue and digital and solving a wider range of time problems including: converting units of time; interpreting and answering questions on timetables and elapsed time.

The sun set at 19:31 and rose again at 6:28. 

How many hours passed between the sun setting and rising again?

Answer: 10 hours and 57 minutes

Count on from 19:31 to 5:31 (10 hours)

Then count on from 5:31 to 6:28 (57 minutes)

A play started at 14:45 and finished at 16:58.

How long was the play?

Answer: 2 hours and 13 minutes

Count on 2 hours from 14:45 to 16:45, then add another 13 minutes to get to 16:58

How many seconds are there in 23 minutes?

Answer: 1380 seconds

Show as column method: 60 x 23 = 1380

Max ran a race in 2 minutes 13 seconds, Oscar ran it in 125 seconds. 

What was the difference in time between Max and Oscar?

Answer: Oscar was 8 seconds faster.

Max – 2 minutes 13 seconds, Oscar – 2 minutes 5 seconds (difference of 8 seconds)

4 children take part in a freestyle swimming relay.

There times were: 

Maisie: 42.8 seconds

Amber 36.3 seconds

Megan 48.7 seconds

Zymal 45.6 seconds

What was the final time for the relay in minutes and seconds?

Answer: 2:53.4

(Show as column method) 42.8 + 36.3 + 48.7 + 45.6 = 173.4 seconds

173.4 seconds = 2:53.4

No new time concepts are taught to pupils in word problems for year 6 . By this stage they are continuing to build confidence and develop skills within the concepts already taught.

Chess: 25 minutes

Basketball: 40 minutes.

Trampolining: 30 minutes

Gymnastics: 50 minutes

Tennis 40 minutes

Tri golf – 45 minutes

Hamza is choosing activities to take part in at his holiday club.

The activities can’t add up to more than 2 hours.

Which 3 activities could he do, which add up to exactly 2 hours?

Answer: Trampolining, gymnastics and tennis: Trampolining: 30 minutes, gymnastics: 50 minutes, tennis: 40 minutes.

5 children took part in a sponsored swim. The children swam for the following lengths of time:

Sam: 27 minutes 37 seconds

Jemma: 33 minutes 29 seconds.

Ben: 23 minutes 18 seconds

Lucy: 41 minutes 57 seconds

Oliver: 39 minutes 21 seconds

Answer: 18 minutes 30 seconds

Longest: Lucy: 41 minutes 57 seconds

Shortest: Ben: 23 minutes 18 seconds.

Difference – count up from 23 minutes 18 seconds to 41 minutes 57 seconds = 18 minutes 39 seconds

What is 6 minutes 47 seconds in seconds?

Answer: 407 minutes

60 x 6 = 360

360 + 47 = 407 minutes

Bethany’s goal is to run round her school running track in under 8 minutes.

She runs it in 440 seconds. Does she achieve her goal? How far above or below the target is she?

Answer: Bethany beats her target by 40 seconds

8 minutes = 8 x 60 = 480 minutes

Lucy’s favourite programme is on TV twice a week for 35 minutes. 

In 6 weeks, how many hours does Lucy spend watching her favourite programme?

Answer: 7 hours

420 minutes = 7 hours

(Show as column method) 35 x 12 = 420 minutes

420 ÷ 60 = 7 

For more time resources, take a look at our collection of printable time worksheets. Third Space Learning also offers a wide collection of word problems covering a range of topics such as place value, decimals and fractions word problems , percentages word problems , division word problems , ratio word problems , addition and subtraction word problems , multiplication word problems , money word problems and other word problem challenge cards.

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During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

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On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

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Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.

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Can you put these mixed-up times in order? You could arrange them in a circle.

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On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

These clocks have only one hand, but can you work out what time they are showing from the information?

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Elapsed time word problems

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