Exercises: The sexagesimal number system

The sexagesimal number system is positional and has a base of $60$. State whether the following statements are true or false:

• a) In the sexagesimal system, $60$ units of one order form one unit of the next higher order.

• b) The sexagesimal system is currently used to measure temperature.

• c) One hour has $3600$ seconds.

An architect is designing a plan and needs to work with very precise angles. If they have an angle of $45^\circ$ and $20'$, express it entirely in minutes (single-unit form).

Lucía is practicing with her telescope and has turned the lens through an angle of $120^\circ$. If she then decides to reverse the turn by $35^\circ$ $15'$, what angle does her telescope show now?

In an athletics race, the first-place runner reached the finish line with a time of $1$ min $42$ s. The runner in second place arrived $15$ seconds later. What time did the second athlete record? (Express the result in minutes and seconds).

Alejandro has taken part in a $10\text{ km}$ fun run in Madrid. His sports watch recorded a time of $52\text{ min } 18\text{ s}$, while his friend Sofía completed the course in $48\text{ min } 45\text{ s}$. To find the exact difference between the two, Alejandro does the subtraction $52\text{ min } 18\text{ s} - 48\text{ min } 45\text{ s}$ in his notebook as follows:

«Since I can't subtract $45$ from $18$, I borrow $1$ from the minutes; I'm left with $51\text{ min}$ and I put a $1$ in front of the $18$, getting $118\text{ s}$. Now I calculate $118 - 45 = 73\text{ s}$ and $51 - 48 = 3\text{ min}$. The result is $3\text{ min } 73\text{ s}$, which, converting $60$ seconds into minutes, comes to $4\text{ min } 13\text{ s}$».

1. a) Identify the conceptual error Alejandro makes when «borrowing» a higher-order unit in the sexagesimal system.
2. b) Explain why his final reasoning also makes no logical sense if you look at the starting data of the race.
3. c) Correct the calculation by solving the subtraction step by step in the right way to determine Sofía's exact lead.

To prepare for a Social Sciences debate in 2nd year of ESO, you have to watch a documentary series of four episodes whose exact running times are:

• Episode 1: $48\text{ min } 35\text{ s}$

• Episode 2: $51\text{ min } 40\text{ s}$

• Episode 3: $49\text{ min } 15\text{ s}$

• Episode 4: $52\text{ min } 10\text{ s}$

You have an afternoon with a continuous free block of $3\text{ and a half hours}$ ($3\text{ h } 30\text{ min}$) to watch them in a row.

1. a) Estimate: Without doing detailed operations on paper, mentally round the duration of each episode to the nearest whole number of minutes and predict whether you will have enough time to watch the full series or if you will run out of time. Justify how you made the approximation.
2. b) Calculate: Perform the exact addition of the durations in the sexagesimal system, expressing the total time in hours, minutes, and seconds.
3. c) Check and evaluate: Compare the exact value with your initial estimate. Was your prediction accurate? How much exact time will you have left over or be short of within that $3\text{ h } 30\text{ min}$ block?

In an esports championship, the first game of a final lasted $45$ min and $38$ s. After a short break, the second game lasted $52$ min and $45$ s. Calculate the total duration of both games combined and express the result in an appropriate time format.

A professional diving team has an air supply that allows them to stay underwater for $3$ hours and $15$ minutes. If they must divide this time equally to explore $5$ different sectors of a reef, how much time should they spend observing each sector?

In a technical drawing class, we are asked to draw the complementary angle of a given angle. If the original angle measures $34^\circ 52' 18''$, carry out the necessary calculations to find the exact measure of the angle needed to complete the right angle ($90^\circ$).

In a technology and robotics class, a servomotor must rotate an angle of $13^\circ 25'\ 40''$ on each control pulse. To complete a work cycle, the motor must repeat that rotation exactly $6$ consecutive times. Two students propose different strategies to calculate the total accumulated rotation:

• Marta's strategy: Multiply each unit order separately by $6$ ($13^\circ \cdot 6$, $25' \cdot 6$, and $40'' \cdot 6$) and then carry out the appropriate conversions, converting groups of $60''$ to minutes and groups of $60'$ to degrees.

• Hugo's strategy: First convert the entire initial angle to arcseconds ($13 \cdot 3600 + 25 \cdot 60 + 40$), multiply the resulting value by $6$, and finally perform successive integer divisions by $60$ to rewrite the solution back into degrees, minutes, and seconds.

1. a) Apply both methods to obtain the final result of the accumulated rotation and check that they give the exact same angle.
2. b) What advantages and disadvantages does each strategy present if you had to perform the calculation by hand without a calculator? Justify which one minimizes the risk of making arithmetic errors.
3. c) If you now had to divide an angle of $53^\circ 14'\ 25''$ into $5$ equal parts using division, which of the two approaches do you consider more convenient and why?

At the Nuevos Ministerios commuter rail station (Madrid), two transport lines converge with periodic service during the morning rush hour:

• A line C-2 train passes every $6\text{ min } 40\text{ s}$.

• A line C-7 train passes every $10\text{ min } 00\text{ s}$.

At $07:20:00$, both trains stop at the platform simultaneously.

1. a) Model the situation using a single unit of measurement: convert the headway intervals of both lines to seconds and find the least common multiple (LCM) to determine every how many seconds they coincide again at the platform.
2. b) Express that coincidence interval in complex form (minutes and seconds).
3. c) Determine the exact time at which they will coincide again for the first time after $07:20:00$.
4. d) How many times in total will both trains coincide at that station between $07:20:00$ and $09:00:00$, including both times?

The ancient Babylonians used a positional base-$60$ number system. To represent their astronomical calculations, they used symbols for $10$ ($\langle$) and for $1$ ($\curlyvee$). If an astronomer recorded the observation time of a star as the sexagesimal number $(12, 45, 18)$, where the first value is hours, the second minutes and the third seconds:

1. a) Represent this number using Babylonian symbols (grouping the wedge marks for each position).

1. b) Calculate how many total seconds the observation lasted, expressing the result in standard form.

A surveillance drone carries out a security sweep by rotating on its axis. It starts at a position of $0^\circ$ and makes the following consecutive movements:

1. Turns to the right through an angle of $125^\circ \ 45' \ 52''$.

2. Turns to the left by $87^\circ \ 50' \ 15''$.

3. Makes a third turn to the right that is exactly three times the angle resulting from subtracting the previous two movements.

Calculate the final position of the drone relative to its starting point in degrees, minutes and seconds.

A video editor is working on a short film for social media. The project has a strict time limit of exactly $10$ minutes. They have four clips with the following durations:

• Clip 1: $2$ min $14$ s $45$ hundredths.

• Clip 2: $155$ s $80$ hundredths.

• Clip 3: $3$ min $12$ s.

• Clip 4: $1$ min $58$ s $95$ hundredths.

Calculate the exact time left over or needed to reach the limit of $10$ minutes. (Remember that $1$ second has $100$ hundredths).

In the design of a part for a high-precision engine, an angle of $145^\circ \ 25'$ must be divided into 6 equal sectors in order to place some bearings. Due to machine limitations, the measure of each sector must be a whole number of minutes (no seconds) and as large as possible. When carrying out the division, the leftover angle is shared equally between two lateral safety margins.

The drawing is not to scale: the margins are so small that they cannot be seen.

1. a) Calculate the measure of each of the 6 sectors.

1. b) Calculate the leftover angle and the measure of the angle of each safety margin.

A telecommunications satellite orbits the Earth. On their control panel, a technician observes that the satellite has completed $4,72$ orbits in the last $24$ hours.

1. a) Express the decimal part of the orbits ($0,72$) in compound unit form using degrees, minutes and seconds (knowing that a complete orbit is $360^\circ$).

1. b) If the satellite takes the same time for each orbit, what is the exact time it takes to complete a single turn? Express the result in hours, minutes and seconds.

A lighthouse emits three different light signals to guide ships:

• The red light flashes every $18$ seconds.

• The green light flashes every $45$ seconds.

• The white light flashes every $1$ minute and $15$ seconds.

If at $22:00:00$ at night all three lights flash together for the first time:

1. a) How much time will pass until they flash together again? Express the result in minutes and seconds.
2. b) At what exact time will they flash together for the second time?
3. c) How many times will all three lights flash together from $22:00:00$ to $00:00:00$, both times included?

In an irregular hexagon, the measurements of five of its interior angles are known:

• $A = 110^\circ \ 15' \ 20''$

• $B = 125^\circ \ 40'$

• $C = 98^\circ \ 25''$

• $D = 130^\circ \ 12' \ 15''$

• $E = 115^\circ \ 50' \ 30''$

Given that the sum of the interior angles of a $n$-sided polygon is $S = (n - 2) \cdot 180^\circ$, calculate the measure of the sixth angle $F$ in compound form.

A municipal surveying team measures the angles of a triangular plot. One of the interior angles, named $\hat{A}$, was determined by dividing a wider sightline angle into $4$ identical sections. When checking the theodolite's log, the original notation of the sightline was partially erased, but the division data remain intact: the exact quotient was $18^\circ 42'\ 35''$ and the final non-divisible remainder was $2''$.

1. a) Inverse problem: Use the fundamental theorem of Euclidean division ($\text{Dividend} = \text{divisor} \cdot \text{quotient} + \text{remainder}$) to reconstruct the original measurement in complex form ($^\circ, ', ''$) of the full sightline measured by the theodolite.
2. b) If angle $\hat{A}$ of the triangular plot measures exactly what resulted from each section ($18^\circ 42'\ 35''$) and the plot is known to be an isosceles triangle (two equal interior angles and one different), calculate the possible measurements of the other two interior angles of the triangle in complex form. Bear in mind that the sum of the interior angles of any triangle is $(3-2) \cdot 180^\circ = 180^\circ$ and consider both possible cases: that $\hat{A}$ is the unequal angle or that $\hat{A}$ is one of the two equal angles.

Earth's rotational motion traces a full circle of $360^\circ$ in a mean solar day ($24\text{ hours}$). Because of this, the sexagesimal system directly links terrestrial geographic longitude with solar time difference.

1. a) Show, by performing the corresponding sexagesimal divisions, that a longitude difference of $1^\circ$ corresponds to exactly $4\text{ minutes}$ of solar time difference, and that a difference of $1'$ of arc corresponds to $4\text{ seconds}$ of time.
2. b) The town of A Frouxeira (A Coruña) is located at a geographic longitude of $8^\circ 15'\ 00''\text{ W}$ (west), while Castelló de la Plana is located on the Greenwich meridian ($0^\circ 0'\ 0''$). When it is exactly solar noon in Castelló (12:00:00 solar time, the moment when the Sun reaches its highest point in the sky), what solar time does a sundial in A Frouxeira show? Perform the operations in complex form, explaining whether you must add or subtract the calculated difference and why.
3. c) In peninsular Spain, during daylight saving time, the official clock time is set $2\text{ hours}$ ahead of the Greenwich meridian (UTC+2 time zone). At what official time does a sundial show 12:00:00 solar time (solar noon) in A Frouxeira during the summer? Based on your result, justify why the sun sets much later in Galicia according to official clock time than on the mainland Mediterranean coast.