Measuring Time
A bus leaves Puebla at $06{:}50{:}00$ and is supposed to arrive in Mexico City at $09{:}12{:}20$, but it arrives at $09{:}19{:}05$. How long was the trip supposed to last, and how late was it? To solve this you need to work with hours, minutes, and seconds, and you also need to know the units of time we use when we talk about history.
Units of time larger than a year
In the study of history and time, we use units larger than a year. Complete the following equivalences: a) $3$ centuries = $\dots$ years. b) $5$ decades = $\dots$ years.…
In the study of history and time, we use units larger than a year. Complete the following equivalences:
- a) $3$ centuries = $\dots$ years.
- b) $5$ decades = $\dots$ years.
- c) $8$ lustrums = $\dots$ years.
- d) $200$ years = $\dots$ centuries.
Complex form: hours, minutes, and seconds
An hour has $60$ minutes and a minute has $60$ seconds. That is why time is called sexagesimal (base $60$). Writing a time as "$2$ h $12$ min $15$ s" is writing it in complex form.
To add in complex form, add each column and, if the seconds or the minutes reach $60$ or more, "carry" one to the next unit.
A short video on social media lasts exactly $145$ seconds. Express the duration of this video in compound form, that is, stating the minutes and seconds it lasts ($145$…
A short video on social media lasts exactly $145$ seconds. Express the duration of this video in compound form, that is, stating the minutes and seconds it lasts ($145$ s = $\dots$ min $\dots$ s).
This afternoon you spent two periods of time studying for the math exam. The first lasted $1$ h $40$ min and the second $55$ min. Calculate the total study time,…
This afternoon you spent two periods of time studying for the math exam. The first lasted $1$ h $40$ min and the second $55$ min. Calculate the total study time, expressing the result in hours and minutes.
An elite athlete completed a marathon in an official time of $2$ hours, $12$ minutes and $15$ seconds. a) Express the total duration of the race in seconds only. b) If…
An elite athlete completed a marathon in an official time of $2$ hours, $12$ minutes and $15$ seconds.
- a) Express the total duration of the race in seconds only.
- b) If their coach tells them to reduce their time by $450$ seconds for the next competition, what would their new target time be, expressed in compound form ($h$, $min$, $s$)?
Calculating a duration from the start time and the end time
To find out how long something lasts, subtract the start time from the end time. If a column doesn't have enough, you "borrow" $1$ hour ($60$ min) or $1$ minute ($60$ s).
An animated film starts at $17:15$ and ends at $18:50$. What is the exact duration of the film in hours and minutes?
An animated film starts at $17:15$ and ends at $18:50$. What is the exact duration of the film in hours and minutes?
The high-speed train leaves Mexico City at $08:45:00$ and is scheduled to arrive in Guadalajara at $11:17:25$. Due to a minor technical delay, the train arrived at…
The high-speed train leaves Mexico City at $08:45:00$ and is scheduled to arrive in Guadalajara at $11:17:25$. Due to a minor technical delay, the train arrived at $11:23:10$.
- a) What was the scheduled duration of the trip?
- b) How much delay did the train experience compared to the scheduled time?
Splitting, multiplying, and speeds
To split a time, multiply it, or calculate a speed, the safest approach is to convert everything to seconds.
A youth association organises a park clean-up day that lasts a total of $5$ hours, $42$ minutes and $36$ seconds. If $4$ work teams have been formed and they want to…
A youth association organises a park clean-up day that lasts a total of $5$ hours, $42$ minutes and $36$ seconds. If $4$ work teams have been formed and they want to split the total time equally so that each team covers a different time slot without overlapping:
How long should each team's shift last? (Carry out the division in the sexagesimal system).
The planet Mars has a rotation period (a "sol") of approximately $24 \text{ h } 37 \text{ min } 22 \text{ s}$. An astronaut arrives on Mars and starts an Earth stopwatch…
The planet Mars has a rotation period (a "sol") of approximately $24 \text{ h } 37 \text{ min } 22 \text{ s}$. An astronaut arrives on Mars and starts an Earth stopwatch and a stopwatch set to Martian time right at sunrise. What Earth time difference will there be between both clocks when exactly one Martian week (7 sols) has elapsed? Express the result in compound form (hours, minutes and seconds).
An internet server processes user requests. It is known to take an average of $1.25$ hundredths of a second to handle each request. If during a cyberattack the server…
An internet server processes user requests. It is known to take an average of $1.25$ hundredths of a second to handle each request. If during a cyberattack the server receives a total of $2^{18}$ requests:
- a) Calculate the total number of requests in the decimal system.
- b) Calculate the total time it will take for the server to process them all. Express the final result in hours, minutes and seconds.
An ultra-trail runner has completed a $105 \text{ km}$ race in a time of $14 \text{ h } 18 \text{ min } 36 \text{ s}$. a) Calculate their average speed in meters per…
An ultra-trail runner has completed a $105 \text{ km}$ race in a time of $14 \text{ h } 18 \text{ min } 36 \text{ s}$.
- a) Calculate their average speed in meters per second ($m/s$), rounding to hundredths.
- b) If the course record is $12 \text{ h } 45 \text{ min } 50 \text{ s}$, by how many seconds per kilometer would they have needed to reduce their time to match the record?
Reading the clock and the angle between the hands
Some clocks use Roman numerals: $\text{I}=1$, $\text{II}=2$, $\text{III}=3$, $\text{IV}=4$, $\text{V}=5$, $\text{VI}=6$, $\text{VII}=7$, $\text{VIII}=8$, $\text{IX}=9$, $\text{X}=10$, $\text{XI}=11$, $\text{XII}=12$. The short hand shows the hours. The long hand shows the minutes: if it points to the number $n$, that is $5 \times n$ minutes. For example, pointing at $\text{VII}$ means $35$ minutes.
The hands also follow the sexagesimal system: the minute hand travels $360°$ in $60$ min, and the hour hand travels $360°$ in $12$ h, moving a little bit with each minute.
Look at the following image of a wall clock that uses the Roman numeral system. State the time shown on the clock, expressing the hours and minutes in the decimal system.
Look at the following image of a wall clock that uses the Roman numeral system. State the time shown on the clock, expressing the hours and minutes in the decimal system.
An analogue clock shows exactly $10:15:30$ (10 hours, 15 minutes and 30 seconds). Calculate the exact angle formed by the hour hand and the minute hand at that precise…
An analogue clock shows exactly $10:15:30$ (10 hours, 15 minutes and 30 seconds). Calculate the exact angle formed by the hour hand and the minute hand at that precise moment. Note: Take into account that the hour hand moves continuously as the minutes and seconds pass.
Back to the beginning: the Puebla bus had a planned trip of $2$ h $22$ min $20$ s and arrived $6$ min $45$ s late.