The minor angle between the hour hand and minute hand of a clock was observed at **8:48 a.m.**. The minimum duration, in minutes, after 8:48 a.m., when this angle increases by **50%**, is:

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To find the minimum time after 8:48 a.m. when the angle increases by 50%, we calculate as follows:

1. **Calculate the initial angle at 8:48**:  
   - The hour hand completes one full rotation (360°) in 12 hours. At 8:48 (or 8.8 hours), the hour hand has moved:  
     $$
     	ext{Hour hand angle} = \frac{8.8}{12} 	imes 360 = 264^\circ.
     $$
   - The minute hand completes one full rotation (360°) in 60 minutes. At 48 minutes, the minute hand has moved:  
     $$
     	ext{Minute hand angle} = \frac{48}{60} 	imes 360 = 288^\circ.
     $$
   - The initial minor angle between the two hands is:  
     $$
     |288 - 264| = 24^\circ.
     $$

2. **Find the target angle**:  
   A 50% increase in the angle means the target angle is:  
   $$
   24 + (0.5 	imes 24) = 36^\circ.
   $$

3. **Relative speed of separation**:  
   The minute hand moves faster than the hour hand. The relative speed of separation between them is:  
   $$
   	ext{Relative speed} = \left(\frac{360}{60}ight) - \left(\frac{360}{12 	imes 60}ight) = 6 - 0.5 = 5.5^\circ/	ext{minute}.
   $$

4. **Time required to achieve the additional separation**:  
   To increase the angle from $$24^\circ$$ to $$36^\circ$$, an additional $$12^\circ$$ separation is required. The time taken for this is:  
   $$
   t = \frac{	ext{Additional separation}}{	ext{Relative speed}} = \frac{12}{5.5} = \frac{24}{11}~	ext{minutes}.
   $$

Thus, the minimum duration after 8:48 a.m., when the angle increases by 50%, is **$$ \frac{24}{11}~	ext{minutes} $$**.

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