Exercises: 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?

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) $\hat{C} = 89^\circ$

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?

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?

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.

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) $45,5^\circ$

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.

Marcos trains in track and field and tracks his times with a mobile app. At the end of a running session, his phone screen shows a time of $1,45\ h$. Marcos writes in his training log: “Today I ran for 1 hour and 45 minutes”.\n\na) Explain in detail the mistake Marcos makes when interpreting the number $1,45\ h$. In what numeral system is the decimal part expressed, and in what base are minutes usually measured?\n\nb) Correct Marcos's calculation and determine the exact duration of his training session expressed in compound units (hours and minutes).

In an Art class, students are designing a set square by joining two triangular pieces at one of their vertices. The pieces have two acute angles measuring $\alpha = 38^\circ\ 47'$ and $\beta = 51^\circ\ 24'$. The teacher asks whether, when placing both angles in consecutive positions, they will form a right corner ($90^\circ$, that is, if they are complementary).\n\na) Before adding them using sexagesimal minutes, round each angle to the nearest whole degree and make a mental estimate: does it seem that the sum will be exactly $90^\circ$, above, or below?\n\nb) Now perform the exact addition of the two angles in the sexagesimal system.\n\nc) Based on the exact result, determine whether they are truly complementary. If not, calculate how much is missing or how much is in excess to form a right angle of $90^\circ$, and assess whether your initial estimate was accurate.

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 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?

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

• $t_1 = 2,5\text{ hours}$

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

• $t_3 = 9.200\text{ seconds}$

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

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?

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?

According to AEMET records for the summer solstice, in the city of Santander the Sun rises at $06:38$ and sets at $21:57$. That same day, in Almería, sunrise occurs at $06:55$ and sunset at $21:30$.\n\na) Model the duration of daylight hours for each city using subtraction of times in the sexagesimal system. Express the length of the day in Santander and in Almería in hours and minutes.\n\nb) How much more direct sunlight does Santander have compared to Almería on that day? Carry out the operation in compound units, showing step by step how you handle the units if you need to borrow an hour.\n\nc) Express the duration of Santander's day as a decimal number of hours rounded to two decimal places, and explain what practical advantage using compound units ($h$ and $min$) has in everyday life.

In a technology workshop, students need to divide a $50^\circ$ angle into three equal sectors to make a cutting template. Two students propose different approaches to calculate exactly how much each sector should measure:\n\n- Sergio's method (decimal): Directly divides $50$ by $3$ using a calculator in decimal degrees and then tries to convert the decimals to minutes.\n- Irene's method (sexagesimal): Performs the integer division $50^\circ : 3$, converts the remaining degrees into minutes by multiplying by $60$, and divides those minutes by $3$.\n\na) Apply Sergio's method, stating what decimal result you obtain when computing $50 : 3$ and what drawback you encounter when trying to express the exact solution.\n\nb) Apply Irene's method step by step and obtain the exact measure of the angle in sexagesimal degrees and minutes.\n\nc) Compare both approaches: which one is more convenient and precise in this case? Justify your answer by relating it to the historical reason why base 60 was chosen (the large number of divisors of 60 compared to base 10).

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.

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: $S = (n - 2) \cdot 180^\circ$.

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

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

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

1. b) Aisha can only video call before going to bed, at $22:15$ Japanese time. At what Spanish time 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 Spanish time does the broadcast end?

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 kilometre? 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 kilometre. How long would it take her to complete the same distance? Express the result in hours and minutes.

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:

We want to find two acute angles, $\hat{A}$ and $\hat{B}$, expressed in compound units with degrees, minutes, and seconds (that is, of the form $g^\circ\ m'\ s''$, where $m \neq 0$ and $s \neq 0$), that simultaneously satisfy these three conditions:\n\n1. They are complementary angles (their sum is exactly $90^\circ$).\n2. Both angles have the same number of seconds ($s_{\hat{A}} = s_{\hat{B}}$).\n3. The measure of angle $\hat{A}$ is greater than twice that of angle $\hat{B}$ ($\hat{A} > 2 \cdot \hat{B}$).\n\na) Explain why, in order for the total sum to give exactly $00''$ without either angle having $0''$, it is mathematically required that the seconds of both angles be exactly $30''$.\n\nb) Deduce what the minutes of both angles must add up to ($m_{\hat{A}} + m_{\hat{B}}$) and what their degrees must add up to ($g_{\hat{A}} + g_{\hat{B}}$), taking into account the carries between sexagesimal units.\n\nc) Propose a pair of angles $\hat{A}$ and $\hat{B}$ that satisfies all the conditions and justify in detail with calculations that both complementarity and the condition $\hat{A} > 2 \cdot \hat{B}$ are met.

The Earth completes a $360^\circ$ rotation on its own axis every $24\text{ hours}$. Thanks to this rotational movement, there is a direct relationship between angular measures of geographic longitude and units of time in the sexagesimal system.\n\na) Deduce how many degrees of Earth rotation correspond to $1\text{ hour}$ of time and show that the rotation of a single degree ($1^\circ$) is equivalent to exactly $4\text{ minutes}$ of time. Next, calculate how many seconds of time correspond to the rotation of one angular arcminute ($1'$).\n\nb) In mainland Spain, the city of Castelló de la Plana is located at approximately a longitude of $0^\circ\ 04'\text{ E}$ (east of the Greenwich meridian), while A Coruña is at $8^\circ\ 24'\text{ W}$ (west of the Greenwich meridian). Calculate the total angular separation between both cities by adding their distances to the prime meridian.\n\nc) Using the equivalences obtained in part a), calculate the exact difference in solar time (in minutes and seconds) between both cities. If in Castelló de la Plana solar noon (the moment when the Sun is at the highest point in the sky) occurs at $13:42:00$ clock time, explain at what clock solar time it will occur in A Coruña, bearing in mind that the Earth rotates from west to east.

Unit challenge. Design a "Domestic Water Audit" report to optimize water consumption in your home.

1. Space and time: Using a tape measure, measure the length and width of your shower tray in $\text{cm}$ and calculate its surface area in $\text{m}^2$. Then, time a regular shower, recording the exact duration in minutes and seconds using the sexagesimal system (for example, $4\text{ min } 35\text{ s}$) and convert it to seconds.

2. Capacity and mass: Place a $1,5\text{ l}$ bottle under the tap; if it takes $10\text{ s}$ to fill completely, estimate the flow rate in $\text{l/s}$. Calculate the liters used in your shower and the mass of that water in $\text{kg}$, knowing that $1\text{ l} = 1\text{ kg}$.

3. Monomials and spreadsheet: If each minute of showering consumes a fixed flow rate $c$, express the consumption as the monomial $c \cdot t$. Model the daily family consumption by adding the like monomials of each member's time (example: $3c + 5c + 4c = 12c$). Transfer the data to a spreadsheet to project the monthly volume in $\text{m}^3$ and its mass in tonnes.

Submit a brief report with a reasoned proposal to reduce family consumption by at least $15\,\%$.

Assessment criteria:

• Accuracy in unit conversions of length, surface area, capacity, and mass.

• Correct use of the sexagesimal system when measuring time.

• Proper expression and addition of monomials to model consumption.

• Coherence and rigor of the final proposal based on the data obtained.