I need a refresher on degrees/minutes/seconds
When we have
$a:= \frac{7^\circ}{48}$=7(1' 15'')
This is because we have $a:= \frac{7^\circ}{48} = 7\frac{1}{48} = 7 \frac{75}{3600} = 7 \frac{60}{3600} + \frac{15}{3600} = 7 \frac{1}{60} + \frac{15}{3600} = 7(1' 15'')$
But how dow we get
7(1' 15'') = 8' 45''?
On
Start by converting everything to seconds. $$1' 15'' = 1 \times 60 + 15 = 75 \text { seconds}$$ Then do the multiplication. $$7 \times 75 = 525 \text{ seconds}$$ Then divide by $60$ and keep the remainder in order to convert seconds to minutes and seconds. $$\frac{525}{60} = 8 \text{ with remainder } 45$$ so $$7 \; (1' 15'') = 8' 45''$$ The only tricky part may be doing the division with remainder. Some calculators and programming languages have special functions for this task. But if you are using a simple four-function calculator, you can first divide $525$ by $60$ with result $8.75$. You then know the integer part of the division is $8$. To find the remainder, one way is to compute $525 - 8 \times 60 = 45$. Alternatively, $0.75 \times 60 = 45$ will also work to find the remainder.
That's a computation in basis $60$: $$7\cdot(1'\, 15'')=7(1'+15'')=7'+7\cdot 15''=7'+105''$$ Now $105 >60$ so we subtract $60$ from $105$ and carry $1$ to the minutes, i.e. $$7'+105''=7'+(1'+45')=(7'+1')+45''=8' \,45''.$$
Answer for the real question (which I misread): \begin{align}7°\div 48&=(7\cdot 60)'\div 48=420'\div 48= 8'+(36'\div 48)= 8'+(3'\div 4)\\&=8'+(3\cdot 60)''\div 4= 8'\;45''. \end{align}