The 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?

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:

• $t_1 = 2,5\text{ hours}$

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

• $t_3 = 9.200\text{ seconds}$

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

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?

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) $45,5^\circ$

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?