Measuring Time
A bus leaves Santo Domingo at $08{:}45{:}00$ and should arrive in Santiago at $11{:}17{:}25$, but it arrives at $11{:}23{:}10$. How long was the planned trip, and how late was the bus? To solve this you need to handle clock times well. You will also see the units of time used when talking about history.
Reading the Time on a Clock with Roman Numerals
Some wall clocks, like those on churches and in parks, use Roman numerals. You only need a few letters: $I = 1$, $V = 5$ and $X = 10$.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| I | II | III | IV | V | VI | VII | VIII | IX | X | XI | XII |
A smaller letter to the left of a larger one is subtracted ($IV = 5 - 1$). To the right, it is added ($VI = 5 + 1$).
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.
Units of Time Longer 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
Time is not counted in tens but in sixties (the sexagesimal system): $1$ h $= 60$ min and $1$ min $= 60$ s. That is why $1$ h $= 3{,}600$ s.
A duration in hours, minutes and seconds is in complex form. A duration in a single unit (for example, only seconds) is in incomplex form.
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 and End Times
To find out how long something lasts that starts at one time and ends at another, you subtract the times in complex form, "borrowing" in sixties if necessary.
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 Santo Domingo at $08:45:00$ and is scheduled to arrive in Santiago at $11:17:25$. Due to a minor technical delay, the train arrived at…
The high-speed train leaves Santo Domingo at $08:45:00$ and is scheduled to arrive in Santiago 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?
Sharing, Multiplying and Speeds with Times
Sharing a total time among several turns, multiplying it, or calculating an average speed requires converting everything to the same unit first. Most of the time that is 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).
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?
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.
The Angle Between the Hands of a Clock
The hands of a clock also move in base $60$. The minute hand travels $360°$ in $60$ min. The hour hand travels $360°$ in $12$ h and moves a little with each minute that passes.
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.
Go back to the beginning: the bus was due to take $2$ h $32$ min $25$ s from Santo Domingo to Santiago and arrived $5$ min $45$ s late.