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 $$\dfrac{b}{6}$$
Development:
$$a=28^\circ \ 36' \ 54''$$ 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,
$$\dfrac{b}{6}=12^\circ \ 37' \ 12''$$
Solution:
$$a=28^\circ \ 36' \ 54''$$ 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) $$\dfrac{b}{6}=12^\circ \ 37' \ 12''$$