### Theory:

Case I: If the given times are in $$24$$-hour format, you can directly subtract those times and find the duration.

Case II: If the given times are in $$12$$-hour format, first convert them into $$24$$-hour format, and then subtract those times and find the duration.
Example:
1. Calculate the duration between $$10\!:\!40$$ hours to $$16\!:\!15$$ hours.

Solution:

Starting time $$=$$ $$10\!:\!40$$ hours

Ending time $$=$$ $$16\!:\!15$$ hours

Duration $$=$$ Ending time $$-$$ Starting time

$$=$$ $$16\!:\!15$$ $$-$$ $$10\!:\!40$$

$\begin{array}{l}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}16:15\\ \underset{¯}{-10:40\phantom{\rule{0.294em}{0ex}}}\end{array}$

Here, $$40$$ minutes cannot be subtracted from $$15$$ minutes.

So, borrow $$1$$ hour from $$16$$ hours, add this to $$15$$ minutes.

$$1$$ hour $$+$$ $$15$$ minutes:

$$=$$ $$60$$ minutes $$+$$ $$15$$ minutes

$$=$$ $$75$$ minutes

Now, subtract $$15\!:\!75$$ hours from $$10\!:\!40$$ hours.

$\begin{array}{l}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}15\phantom{\rule{0.147em}{0ex}}\underset{\to }{}75\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}16:15\\ -10:40\\ \overline{\underset{¯}{\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}5:35\phantom{\rule{0.441em}{0ex}}\phantom{\rule{0.147em}{0ex}}}}\end{array}$

Therefore, duration between $$10\!:\!40$$ hours to $$16\!:\!15$$ hours is $$5$$ hours $$35$$ minutes.

2. Find the duration between $$11\!:\!20$$ a.m. to $$5\!:\!30$$ p.m.

Solution:

Let us first convert the given times to $$24$$-hour format.

Starting time $$=$$ $$11\!:\!20$$ a.m. $$=$$ $$11\!:\!20$$ hours

Ending time $$=$$ $$5\!:\!30$$ p.m. $$=$$ $$5\!:\!30$$ $$+$$ $$12\!:\!00$$ $$=$$ $$17\!:\!30$$ hours

Duration $$=$$ Ending time $$-$$ Starting time

$$=$$ $$17\!:\!30$$ hours $$-$$ $$11\!:\!20$$ hours

$\begin{array}{l}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}17:30\\ -11:20\\ \overline{\underset{¯}{\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}6:\phantom{\rule{0.147em}{0ex}}10\phantom{\rule{0.441em}{0ex}}}}\end{array}$

Therefore, the duration between $$11\!:\!20$$ a.m. to $$5\!:\!30$$ p.m. is $$6$$ hours $$10$$ minutes.