Sexagesimal System

The sexagesimal system is based on the number $60$. Carry out the following time conversions:

1. a) How many minutes are $3$ hours and $15$ minutes?
2. b) A TikTok video lasts $150$ seconds. Express it in minutes and seconds (compound form).
3. c) If a TV series lasts $0.5$ hours, how many minutes is that?

The coach of a swimming team notes down session times in different units. Express each time in the requested unit:

1. a) Monday's session lasted $1.5$ hours. How many minutes is that?

1. b) The team's best relay was done in $210$ seconds. How many minutes and seconds is that?

1. c) A swimmer held her breath for $2$ minutes and $40$ seconds underwater. Express that time in seconds only.

1. d) Friday's session lasted $2.25$ hours. How many minutes is that?

Look at the four angles drawn below. Without using a protractor, classify them as acute, right, obtuse or straight. Then, order them from smallest to largest angle.

Marta started studying at $17:45$ and finished at $19:10$.

1. a) How much time has she spent studying in total? (Express the result in hours and minutes).
2. b) If she took a $15$-minute break in the middle of the afternoon, how much actual study time did she have?

Express the following durations in seconds and then order them from longest to shortest:

• $t_1 = 2.5\text{ hours}$

• $t_2 = 145\text{ minutes and } 30\text{ seconds}$

• $t_3 = 9,200\text{ seconds}$

• $t_4 = 2\text{ h } 30\text{ min } 15\text{ s}$

In the school music festival, the final concert is divided into three parts: the performance by the first-year group lasts $25$ minutes, that of the second-year group $35$ minutes and that of the third-year group $45$ minutes.

1. a) How many minutes does the concert last in total?

1. b) Express that duration in hours and minutes.

An eSports tournament consists of three stages. The first stage lasted $2\text{ h } 45\text{ min } 15\text{ s}$, the second stage lasted $1\text{ h } 58\text{ min } 50\text{ s}$ and the grand final lasted $55\text{ min } 20\text{ s}$.

1. a) What was the total duration of the tournament in compound form?
2. b) If the tournament was recorded in full and the video file takes up $12\text{ MB}$ per minute of recording, approximately how many $GB$ does the entire video take up? (Given: $1\text{ GB} = 1,024\text{ MB}$).

A family takes a train journey with a layover. The first leg takes $1\text{ h } 45\text{ min } 30\text{ s}$, the wait at the interchange station is $50\text{ min } 40\text{ s}$ and the second leg takes $2\text{ h } 15\text{ min } 50\text{ s}$.

1. a) How long does the entire journey take, from getting on the first train to getting off the second?

1. b) If they left home at $8:30$ in the morning and took $10$ minutes to walk to the first station, at what time did they reach their final destination?

In a school cycling time trial, a rider's start time is recorded at $10\text{ h } 15\text{ min } 30\text{ s}$ and her finish time at $11\text{ h } 2\text{ min } 15\text{ s}$.

1. a) How long did she take to complete the course?

1. b) If the winner of the race took $42\text{ min } 10\text{ s}$, how much time separates our rider from first place?

A marathon runner maintains a constant pace of $4\text{ minutes and } 15\text{ seconds}$ per kilometer run.

1. a) How much total time will it take to complete a $15\text{ km}$ race? Express the result in hours, minutes, and seconds.
2. b) If the race started at $10:45\text{ h}$, at what time will they cross the finish line?

A skater has completed a roller marathon of $42$ km in exactly $2\text{ h } 20\text{ min}$.

1. a) What was her average pace per kilometer? Express it in minutes and seconds.

1. b) In the next race, she wants to improve her personal best by maintaining a pace of $3\text{ min } 10\text{ s}$ per kilometer. How long would it take her to complete the same distance? Express the result in hours and minutes.

Your friend Lucía has gone on an exchange to Tokyo. In summer, Tokyo is $7$ hours ahead of central Mexico (when it is $12:00$ here, it is $19:00$ there).

1. a) If it is $18:30$ in central Mexico, what time is it in Tokyo? Keep in mind that it might already be the next day.

1. b) Lucía can only video call before going to bed, at $22:15$ Japanese time. At what time in central Mexico do you have to log on?

1. c) A music channel broadcasts a live concert from Tokyo starting at $9:00$ Japanese time and lasting $1\text{ h } 45\text{ min}$. At what time in central Mexico does the broadcast end?

A wall clock gains $18\text{ seconds}$ every $4\text{ hours}$. If it is set to the exact time on $1$ March at $12:00$ midday, what time will this clock show when the actual time is exactly $12:00$ midday on $11$ March? Give the answer in hours, minutes and seconds.

A communications satellite orbits the Earth and completes one full orbit every $103\text{ minutes and } 24\text{ seconds}$. If the satellite started its first orbit on a Monday at $08:15:00$, at what exact time, expressed in hours, minutes and seconds, will it have completed its fifteenth orbit? Also indicate whether it will still be Monday or if the day will have changed.

Two angles are complementary if their sum is $90^\circ$. Work out the angle needed for each of the following to be complementary:

1. a) $\hat{A} = 35^\circ$
2. b) $\hat{B} = 52^\circ$ $30'$
3. c) $\hat{C} = 89^\circ$

Two angles are supplementary if their sum is $180^\circ$. Work out the supplementary angle for each of the following:

1. a) $65^\circ$

1. b) $110^\circ$

1. c) $90^\circ$

1. d) $179^\circ$

1. e) $45.5^\circ$

A graphic designer is drawing the logo for a new app. The logo uses a main angle of $125^\circ \ 40' \ 12''$.

1. a) The designer needs to divide that angle into three equal parts to create a fan effect. What will each of the three resulting angles measure?
2. b) What is the supplementary angle of the original angle?

Remember that two angles are complementary when they add up to $90^\circ$. In the Design and Technology workshop, a cutting guide forms an angle of $67^\circ \ 25' \ 40''$ with the edge of the workbench. For the cut to be perpendicular to the edge, a wedge whose angle is its complement must be placed.

1. a) Calculate the angle of the wedge, expressing the result in degrees, minutes and seconds.

1. b) If the guide is then adjusted to $45^\circ \ 30'$, what would the angle of the wedge be then?

In the triangular logo of a hiking club, two of the angles of the triangle measure $45^\circ \ 30'$ and $72^\circ \ 45' \ 30''$.

1. a) Find the measure of the third angle, expressing the result in degrees, minutes and seconds.

1. b) Classify the triangle according to its angles (acute-angled, right-angled or obtuse-angled) and justify your answer.

Look at the logo diagram, where the value of the third angle is missing:

Calculate the measure of the interior angle of a regular polygon with $9$ sides (nonagon). Express the result in compound form (degrees, minutes and seconds). Remember that the sum of the interior angles of a polygon with $n$ sides is calculated using the formula: $S = (n - 2) \times 180^\circ$.