The sexagesimal system and its operations
Define two numbers $a$ and $b$ in sexagesimal system.
a) Compute the sum of both $a+b$
b) Compute the subtractions $a-b$ and $b-a$
c) Multiply $a\cdot7$
d) Divide $\dfrac{b}{6}$
$a=28^\circ \ 36' \ 54''$ and $b=75^\circ \ 43' \ 12''$
a) Step 1:
$$\begin{eqnarray} & & \ \ 28^\circ \ 36' \ 54'' \\\\ &+ & \underline{\ \ 75^\circ \ 43' \ 12''} \\\\ & & 103^\circ \ 79' \ 66'' \end{eqnarray}$$
Step 2:
$$\dfrac{66}{60}=1+\dfrac{6}{60}$$
We obtain,
$$103^\circ \ 80' \ 6''$$
Step 3:
Same procedure for the minutes,
$$\dfrac{80}{60}=1+\dfrac{20}{60}$$
And we obtain,
$$a+b=104^\circ \ 20' \ 6''$$
$$\\\\$$
b) $b-a$ is reduced first since $a$ is less than $b$,
$$\begin{eqnarray} & & 75^\circ \ 43' \ \fbox{12}'' \\\\ &- & \underline{28^\circ \ 36' \ \fbox{54}''} \end{eqnarray}$$
Step 1:
We convert a minute into $60$ seconds to obtain a positive number of seconds after having subtracted.
$$\begin{eqnarray} & & 75^\circ \ 42' \ \fbox{72}'' \\\\ &- & \underline{28^\circ \ 36' \ \fbox{54}''} \\\\ & & \ \ \ \ \ \ \ \ \ \ \ \ \ 18'' \end{eqnarray}$$
Step 2:
Minutes and hours are subtracted
$$\begin{eqnarray} & & 75^\circ \ 42' \ 72'' \\\\ &-& \underline{28^\circ \ 36' \ 54''} \\\\ & & 47^\circ \ \ 6' \ 18'' \end{eqnarray}$$
The subtraction $a-b$ will give a result of:
$$-47^\circ \ \ 6' \ 18'' $$
$$\\\\$$
c) Step 1:
$$\begin{eqnarray} & & 28^\circ \ \ \ 36' \ \ \ 54'' \\\\ & \times & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5 \\\\ & & \overline{140^\circ \ 180' \ 270''} \end{eqnarray}$$
Step 2:
More than $60$ seconds are obtained,
$$\dfrac{270}{60}=4+\dfrac{30}{60}$$
And the product is
$$140^\circ \ 184' \ 30''$$
Step 3:
The same procedure for the minutes
$$\dfrac{184}{60}=3+\dfrac{4}{60}$$
Finally,
$$a\times5=143^\circ \ 4' \ 30''$$
$$\\\\$$
d) Step 1:
We start by dividing the hours (or degrees):
$$\dfrac{75}{6}=12+\dfrac{3}{6}$$
$12$ will be the final hours, and $3\times60$ will be added to the minutes.
Step 2:
The same with the minutes $180' + 43' = 223'$
$$\dfrac{223}{6}=37+\dfrac{1}{6}$$
$37$ will be the final minutes and $1\times60$ will be added to the seconds.
Step 3:
The same with the seconds $60'' + 12'' =72''$
$$\dfrac{72}{6}=12''$$
And so,
$$\dfrac{b}{6}=12^\circ \ 37' \ 12''$$
$a=28^\circ \ 36' \ 54''$ and $b=75^\circ \ 43' \ 12''$
a) $a+b=104^\circ \ 20' \ 6''$
b) $b-a=47^\circ \ \ 6' \ 18''$, $a-b=-47^\circ \ \ 6' \ 18''$
c) $a\times5=143^\circ \ 4' \ 30''$
d) $\dfrac{b}{6}=12^\circ \ 37' \ 12''$